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1202.3490
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# Meson Scattering in a Pion Superfluid
Shijun Mao and Pengfei Zhuang Physics Department, Tsinghua University,
Beijing 100084, China
###### Abstract
Instead of the fermion-fermion scattering which identifies the BCS-BEC
crossover in cold atom systems, boson-boson scattering is measurable and
characterizes the BCS-BEC crossover at quark level. We study $\pi$-$\pi$
scattering in a pion superfluid described by the Nambu–Jona-Lasinio model. We
found that the scattering amplitude drops down monotonically with decreasing
isospin density and finally vanishes at the boundary of the phase transition.
This indicates a BCS-BEC crossover in the pion superfluid.
###### pacs:
21.65.Qr, 74.90.+n, 12.39.-x
There are two kinds of condensed states in usual fermion gas, the
Bardeen–Cooper–Shrieffer condensation (BCS) of fermions where the pair size is
large and the pairs overlap each other, and the Bose–Einstein condensation
(BEC) of molecules where the pair size is small and the pairs are
distinguishable. The BCS wave function can be generalized to arbitrary
attraction which leads to a smooth crossover from BCS to BEC eagles ; leggett
. In cold atom systems, the experimental observable to identify the BCS-BEC
crossover is the $s$-wave scattering between two fermions courteille ; greiner
; zwieriein ; bourdel .
Recently the study on quantum chromodynamics (QCD) phase structure is extended
to finite isospin density. For a QCD system at finite temperature and baryon
and isospin density, the phase transitions include not only color
deconfinement hwa , chiral symmetry restoration hwa and color superconductor
alford ; rapp , but also pion superfluid son ; he1 . The increasing isospin
density induces a phase transition from normal nuclear matter to pion
superfluid, due to the spontaneous isospin symmetry breaking. By analogy with
the usual superfluid, the BCS-BEC crossover in pion superfluid can be
theoretically described matsuo ; margueron ; mao ; sun ; mu1 by the quark
chemical potential which is positive in BCS and negative in BEC, the size of
the Cooper pair which is large in BCS and small in BEC, and the scaled pion
condensate which is small in BCS and large in BEC. However, unlike the
fermion-fermion scattering in cold atom systems, quarks are unobservable
degrees of freedom, and thus the quark-quark scattering can not be measured or
used to experimentally identify the BCS-BEC crossover.
In pion superfluid, the pairs themselves, namely the pion mesons, are
observable objects. One can measure the $\pi-\pi$ scattering to probe the
properties of the pion condensate and in turn the BCS-BEC crossover. Since
pions are Goldstone modes corresponding to the chiral symmetry spontaneous
breaking, the $\pi-\pi$ scattering provides a direct way to link chiral
theories and experimental data and has been widely studied in many chiral
models gasser ; bijnens ; schulze ; quack ; huang . Note that pions are also
the Goldstone modes of the isospin symmetry spontaneous breaking, the
$\pi-\pi$ scattering should be a sensitive signature of the pion superfluid
phase transition.
While the perturbative QCD can well describe the properties of the new phases
at extremely high temperature and density, the study on the phase transitions
at moderate temperature and density depends on lattice QCD calculations kogut
and effective models with QCD symmetries. One of the widely used effective
models is the Nambu–Jona-Lasinio (NJL) model nambu , which is originally
inspired by the BCS theory and its version at quark level vogl ; klevansky ;
volkov ; hatsuda ; buballa gives simple and direct description of the dynamic
mechanisms of spontaneous chiral symmetry breaking, color symmetry breaking
and isospin symmetry breaking. The $s$-wave $\pi-\pi$ scattering calculated
schulze ; quack ; huang in the model is consistent with the Weinberg limit
weinberg and the experimental data pocanic in vacuum. In this work, we
extend the calculation to finite isospin chemical potential and focus on its
relation to the BCS-BEC crossover in the pion superfluid.
The Lagrangian density of the two flavor NJL model at quark level is defined
as vogl ; klevansky ; volkov ; hatsuda ; buballa
${\cal
L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m_{0}+\gamma_{0}\mu\right)\psi+G\Big{[}\left(\bar{\psi}\psi\right)^{2}+\left(\bar{\psi}i\gamma_{5}\tau_{i}\psi\right)^{2}\Big{]}$
(1)
with scalar and pseudoscalar interactions corresponding to $\sigma$ and $\pi$
excitations, where $m_{0}$ is the current quark mass, $G$ is the four-quark
coupling constant with dimension GeV-2, $\tau_{i}\ (i=1,2,3)$ are the Pauli
matrices in flavor space, and
$\mu=diag\left(\mu_{u},\mu_{d}\right)=diag\left(\mu_{B}/3+\mu_{I}/2,\mu_{B}/3-\mu_{I}/2\right)$
is the quark chemical potential matrix with $\mu_{u}$ and $\mu_{d}$ being the
$u$\- and $d$-quark chemical potentials and $\mu_{B}$ and $\mu_{I}$ the baryon
and isospin chemical potentials. At $\mu_{I}=0$, the Lagrangian density has
the symmetry of $U_{B}(1)\bigotimes SU_{I}(2)\bigotimes SU_{A}(2)$,
corresponding to baryon, isospin and chiral symmetry. At $\mu_{I}\neq 0$, the
symmetries $SU_{I}(2)$ and $SU_{A}(2)$ are firstly explicitly broken down to
$U_{I}(1)$ and $U_{A}(1)$, and then the nonzero pion condensate leads to a
spontaneous breaking of $U_{I}(1)$, with pions as the corresponding Goldstone
modes. At $\mu_{B}=0$, the Fermi surface of $u(d)$ and anti-$d(u)$ quarks
coincide and hence the condensate of $u$ and anti-$d$ is favored at
$\mu_{I}>0$ and the condensate of $d$ and anti-$u$ quarks is favored at
$\mu_{I}<0$. Finite $\mu_{B}$ provides a mismatch between the two Fermi
surfaces and will reduce the pion condensation.
Introducing the chiral and pion condensates
$\sigma=\langle\bar{\psi}\psi\rangle$ and
$\pi=\langle\bar{\psi}i\gamma_{5}\tau_{1}\psi\rangle$ and taking them to be
real, the quark propagator ${\cal S}$ in mean field approximation can be
expressed as a matrix in the flavor space
${\cal
S}^{-1}(p)=\left(\begin{array}[]{cc}\gamma^{\mu}p_{\mu}+\mu_{u}\gamma_{0}-m_{q}&2iG\pi\gamma_{5}\\\
2iG\pi\gamma_{5}&\gamma^{\mu}p_{\mu}+\mu_{d}\gamma_{0}-m_{q}\end{array}\right)$
(2)
with the dynamical quark mass $m_{q}=m_{0}-2G\sigma$ generated by the chiral
symmetry breaking. By diagonalizing the propagator, the thermodynamic
potential can be simply expressed as a condensation part plus a summation of
four quasiparticle contributions he1 . The gap equations to determine the
condensates $\sigma$ (or quark mass $m_{q}$) and $\pi$ can be obtained by the
minimum of the thermodynamic potential.
In the NJL model, the meson modes are regarded as quantum fluctuations above
the mean field. The two quark scattering via a meson exchange can be
effectively expressed at quark level in terms of quark bubble summation in the
random phase approximation (RPA) vogl ; klevansky ; volkov ; hatsuda ; buballa
. The quark bubbles are defined as
$\Pi_{mn}(k)=i\int{d^{4}p\over(2\pi)^{4}}Tr\left(\Gamma_{m}^{*}{\cal
S}(p+k)\Gamma_{n}{\cal S}(p)\right)$ (3)
with indexes $m,n=\sigma,\pi_{+},\pi_{-},\pi_{0}$, where the trace
$Tr=Tr_{C}Tr_{F}Tr_{D}$ is taken in color, flavor and Dirac spaces, the four
momentum integration is defined as $\int d^{4}p/(2\pi)^{4}=iT\sum_{j}\int
d^{3}{\bf p}/(2\pi)^{3}$ with fermion frequency $p_{0}=i\omega_{j}=i(2j+1)\pi
T\ (j=0,\pm 1,\pm 2,\cdots)$ at finite temperature $T$, and the meson vertices
are from the Lagrangian density (1),
$\Gamma_{m}=\left\\{\begin{array}[]{ll}1&m=\sigma\\\
i\gamma_{5}\tau_{+}&m=\pi_{+}\\\ i\gamma_{5}\tau_{-}&m=\pi_{-}\\\
i\gamma_{5}\tau_{3}&m=\pi_{0}\ ,\end{array}\right.\ \
\Gamma_{m}^{*}=\left\\{\begin{array}[]{ll}1&m=\sigma\\\
i\gamma_{5}\tau_{-}&m=\pi_{+}\\\ i\gamma_{5}\tau_{+}&m=\pi_{-}\\\
i\gamma_{5}\tau_{3}&m=\pi_{0}\ .\\\ \end{array}\right.$ (4)
Since the quark propagator ${\cal S}$ contains off-diagonal elements, we must
consider all possible channels in the bubble summation in RPA. Using matrix
notation for the meson polarization function $\Pi(k)$ in the $4\times 4$ meson
space, the meson propagator can be expressed as
${\cal D}(k)={2G\over 1-2G\Pi(k)}.$ (5)
Since the isospin symmetry is spontaneously broken in the pion superfluid, the
original meson modes $\sigma,\pi_{+},\pi_{-},\pi_{0}$ with definite isospin
quantum number are no longer the eigen modes of the Hamiltonian of the system,
the new eigen modes
$\overline{\sigma},\overline{\pi}_{+},\overline{\pi}_{-},\overline{\pi}_{0}$
are linear combinations of the old ones, their masses
$M_{i}(i=\overline{\sigma},\overline{\pi}_{+},\overline{\pi}_{-},\overline{\pi}_{0})$
are determined by the poles of the meson propagator at $k_{0}=M_{i}$ and ${\bf
k=0}$, $\text{det}\left[1-2G\Pi(M_{i},{\bf 0})\right]=0$ he1 , and their
coupling constants $g_{iq\overline{q}}$ are defined as the residues of the
propagator at the poles hao .
The condition for a meson to decay into a $q$ and a $\overline{q}$ is that its
mass lies above the $q-\overline{q}$ threshold. From the pole equation, the
heaviest mode in the pion superfluid is $\overline{\sigma}$ and its mass is
beyond the threshold value. Therefore, there will be no $\bar{\sigma}$ mesons
at $\mu_{I}>\mu^{c}_{I}$ hao .
Figure 1: The lowest order diagrams for $\pi-\pi$ scattering in the pion
superfluid. The solid and dashed lines are respectively quarks and pions, and
the dots denote meson-quark vertices.
We now study $\pi-\pi$ scattering at finite isospin chemical potential. To the
lowest order in $1/N_{c}$ expansion, where $N_{c}$ is the number of colors,
the invariant amplitude ${\cal T}$ is calculated from the diagrams shown in
Fig.1 for the $s$ channel. Different from the calculation in normal state
schulze ; quack ; huang where both the box and $\sigma$-exchange diagrams
contribute, the $\sigma$-exchange diagrams vanish in the pion superfluid due
to the disappearance of the $\overline{\sigma}$ meson. This greatly simplifies
the calculation in the pion superfluid.
For the calculation in normal matter at $\mu_{I}=0$, people are interested in
the $\pi$-$\pi$ scattering amplitude with definite isospin, ${\cal
T}_{I=0,1,2}$, which can be measured in experiments due to isospin symmetry.
However, the nonzero isospin chemical potential breaks down the isospin
symmetry and makes the scattering amplitude ${\cal T}_{I=0,1,2}$ not well
defined. In fact, the new meson modes in the pion superfluid do not carry
definite isospin quantum numbers. Unlike the chiral dynamics in normal matter,
where the three degenerated pions are all the Goldstone modes corresponding to
the chiral symmetry spontaneous breaking, the pion mass splitting at finite
$\mu_{I}$ results in only one Goldstone mode $\overline{\pi}_{+}$ in the pion
superfluid.
The scattering amplitude for any channel of the box diagrams can be expressed
as
$i{\cal
T}_{s,t,u}(k)=-2g_{\overline{\pi}q\overline{q}}^{4}\int{d^{4}p\over(2\pi)^{4}}Tr\prod_{l=1}^{4}\left[\gamma_{5}\tau{\cal
S}_{l}\right]$ (6)
with the quark propagators ${\cal S}_{1}={\cal S}_{3}={\cal S}(p)$, ${\cal
S}_{2}={\cal S}(p+k)$, and ${\cal S}_{4}={\cal S}(p-k)$ for the $s$ and $t$
channels and ${\cal S}_{1}={\cal S}_{3}={\cal S}(p+k)$ and ${\cal S}_{2}={\cal
S}_{4}={\cal S}(p)$ for the $u$ channel. To simplify the numerical
calculation, we consider in the following the limit of the scattering at
threshold $\sqrt{s}=2M_{\overline{\pi}}$ and $t=u=0$, where $s,t$ and $u$ are
the Mandelstam variables. In this limit, the amplitude approaches to the
scattering length. Note that the threshold condition can be fulfilled by a
simple choice of the pion momenta, $k_{a}=k_{b}=k_{c}=k_{d}=k$ and
$k^{2}=M_{\overline{\pi}}^{2}=s/4$, which facilitates a straightforward
computation of the diagrams. Doing the fermion frequency summation over the
internal quark lines, the scattering amplitude for the process of
$\overline{\pi}_{+}\ +\ \overline{\pi}_{+}\rightarrow\overline{\pi}_{+}\ +\
\overline{\pi}_{+}$ in the pion superfluid is simplified as
$\displaystyle{\cal
T}_{+}=18g_{\overline{\pi}_{+}q\overline{q}}^{4}\int{d^{3}{\bf
p}\over(2\pi)^{3}}$ $\displaystyle\Bigg{\\{}$ $\displaystyle{1\over
E_{+}^{3}}\left[\left(f(E_{+}^{-})-f(-E_{+}^{+})\right)-E_{+}\left(f^{\prime}(E_{+}^{-})+f^{\prime}(-E_{+}^{+})\right)\right]$
(7) $\displaystyle+$ $\displaystyle{1\over
E_{-}^{3}}\left[\left(f(E_{-}^{-})-f(-E_{-}^{+})\right)-E_{-}\left(f^{\prime}(E_{-}^{-})+f^{\prime}(-E_{-}^{+})\right)\right]\Bigg{\\}},$
where $E_{\pm}^{\mp}=E_{\pm}\mp\mu_{B}/3$ are the energies of the four
quasiparticles with
$E_{\pm}=\sqrt{\left(E\pm\mu_{I}/2\right)^{2}+4G^{2}\pi^{2}}$ and
$E=\sqrt{{\bf p}^{2}+m_{q}^{2}}$, $f(x)$ is the Fermi-Dirac distribution
function $f(x)=\left(e^{x/T}+1\right)^{-1}$, and $f^{\prime}(x)=df/dx$ is the
first order derivative of $f$. For the scattering amplitude outside the pion
superfluid, one should consider both the box and $\sigma$-exchange diagrams.
The calculation is straightforward.
Since the NJL model is non-renormalizable, we can employ a hard three momentum
cutoff $\Lambda$ to regularize the gap equations for quarks and pole equations
for mesons. In the following numerical calculations, we take the current quark
mass $m_{0}=5$ MeV, the coupling constant $G=4.93$ GeV-2 and the cutoff
$\Lambda=653$ MeVzhuang . This group of parameters correspond to the pion mass
$m_{\pi}=134$ MeV, the pion decay constant $f_{\pi}=93$ MeV and the effective
quark mass $M_{q}=310$ MeV in the vacuum.
Figure 2: (Color online) The scaled scattering amplitude ${\cal T}_{+}$ as a
function of isospin chemical potential $\mu_{I}$ at two values of temperature
$T$.
In Fig.2, we plot the scattering amplitude $|{\cal T}_{+}|$ as a function of
isospin chemical potential $\mu_{I}$ at two temperatures $T=0$ and $T=100$
MeV, keeping baryon chemical potential $\mu_{B}=0$. The normal matter with
$\mu_{I}<\mu_{I}^{c}$ is dominated by the explicit isospin symmetry breaking
and spontaneous chiral symmetry breaking, and the pion superfluid with
$\mu_{I}>\mu_{I}^{c}$ and the corresponding BEC-BCS crossover is controlled by
the spontaneous isospin symmetry breaking and chiral symmetry restoration.
From (6), the scattering amplitude is governed by the meson coupling constant,
${\cal T}_{+}\sim g_{\overline{\pi}_{+}q\overline{q}}^{4}$. In the pion
superfluid, the meson mode $\overline{\pi}_{+}$ is always a bound state, its
coupling to quarks drops down with decreasing $\mu_{I}$ hao , and therefore
the scattering amplitude $\left|{\cal T}_{+}\right|$ decreases when the system
approaches to the phase transition and reaches zero at the critical value
$\mu^{c}_{I}$ due to $g_{\overline{\pi}_{+}q\overline{q}}=0$ at this point,
where the critical isospin chemical potential $\mu^{c}_{I}=m_{\pi}=134$ MeV at
$T=0$ and $142$ MeV at $T=100$ MeV. After crossing the border of the phase
transition, the coupling constant changes its moving trend and start to go up
with decreasing isospin chemical potential in the normal matter hao , and the
scattering amplitude smoothly increases and finally approaches its vacuum
value for $\mu_{I}\to 0$.
The above $\mu_{I}$-dependence of the meson-meson scattering amplitude in the
pion superfluid with $\mu_{I}>\mu_{I}^{c}$ can be understood well from the
point of view of BCS-BEC crossover. We recall that the BCS and BEC states are
defined in the sense of the degree of overlapping among the pair wave
functions. The large pairs in BCS state overlap each other, and the small
pairs in BEC state are individual objects. Therefore, the cross section
between two pairs should be large in the BCS state and approach zero in the
limit of BEC. From our calculation shown in Fig.2, the $\pi-\pi$ scattering
amplitude is a characteristic quantity for the BCS-BEC crossover in pion
superfluid. The overlapped quark-antiquark pairs in the BCS state at higher
isospin density have large scattering amplitude, while in the BEC state at
lower isospin density with separable pairs, the scattering amplitude becomes
small. This provides a sensitive observable for the BCS-BEC crossover at quark
level, analogous to the fermion scattering in cold atom systems.
Figure 3: (Color online) The scattering amplitude ${\cal T}_{+}$ as a function
of temperature $T$ at two values of isospin chemical potential $\mu_{I}$ in
the pion superfluid.
The minimum of the scattering amplitude at the critical point can generally be
understood in terms of the interaction between the two quarks. A strong
interaction means a tightly bound state with small meson size and small meson-
meson cross section, and a weak interaction means a loosely bound state with
large meson size and large meson-meson cross section. Therefore, the minimum
of the meson scattering amplitude at the critical point indicates the most
strong quark interaction at the phase transition. This result is consistent
with theoretical calculations for the ratio $\eta/s$ kovtun ; csernai of
shear viscosity to entropy and for the quark potential mu2 ; jiang , which
show a strongly interacting quark matter around the phase transition.
With increasing temperature, the pairs will gradually melt and the coupling
constant $g_{\overline{\pi}q\overline{q}}$ drops down in the hot medium,
leading to a smaller scattering amplitude at $T=100$ MeV in the pion
superfluid, in comparison with the case at $T=0$, as shown in Fig.2. To see
the continuous temperature effect on the scattering amplitude in the BCS and
BEC states, we plot in Fig.3 $\left|{\cal T}_{+}\right|$ as a function of $T$
at $\mu_{I}=160$ and $\mu_{I}=400$ MeV, still keeping $\mu_{B}=0$. While the
temperature dependence is similar in both cases, the involved physics is
different. In the BCS state at $\mu_{I}=400$ MeV, $\left|{\cal T}_{+}\right|$
is large and drops down with increasing temperature and finally vanishes at
the critical temperature $T_{c}=188$ MeV. Above $T_{c}$ the system becomes a
fermion gas with weak coupling and without any pair. In the BEC state at
$\mu_{I}=160$ MeV, the scattering amplitude becomes much smaller (multiplied
by a factor of $10$ in Fig.3). At a lower critical temperature $T_{c}=136$
MeV, the condensate melts but the still strong coupling between quarks makes
the system be a gas of free pairs.
In summary, we proposed the meson-meson scattering as a sensitive probe of the
BCS-BEC crossover at quark level. Different from the fermion-fermion
scattering which is often used to experimentally identify the BCS-BEC
crossover in cold atom systems, quark scattering can not be measured and its
function to characterize the BCS-BEC crossover at quark level is replaced by
the meson scattering. In the BCS quark superfluid, the large and overlapped
pairs lead to a large pair-pair cross section, but the small and individual
pairs in the BEC superfluid interact weakly with small cross section. In the
frame of a two flavor NJL model at finite temperature and isospin density, we
calculated the $\pi-\pi$ scattering amplitude in the pion superfluid. It is
large at high isospin chemical potential and drops down monotonically with
decreasing isospin chemical potential and finally approaches zero at the
border of the pion superfluid, indicating a BCS-BEC crossover.
The meson scattering amplitude $\left|{\cal T}_{+}\right|$ shown in Figs.2 and
3 are obtained in a particular model, the NJL model, which has proven to be
rather reliable in the study on chiral, color and isospin condensates at low
temperature. Since there is no confinement in the model, one may ask the
question to what degree the conclusions obtained here can be trusted. From the
general picture for BCS and BEC states, the feature that the meson scattering
amplitude approaches to zero in the process of BCS-BEC crossover can be
geometrically understood in terms of the degree of overlapping between the two
pairs. Therefore, the qualitative conclusion of taking meson scattering as a
probe of BCS-BEC crossover at quark level may survive any model dependence.
Our result that the molecular scattering amplitude approaches to zero in the
BEC limit is consistent with the recent work for a general fermion gas he2 .
Different from a system with finite baryon density where the fermion sign
problem muroya makes it difficult to simulate QCD on lattice, there is in
principle no problem to do lattice QCD calculations at finite isospin density
kogut . From the recent lattice QCD results detmold at nonzero isospin
chemical potential in a canonical approach, the scattering length in the pion
superfluid increases with increasing isospin density, which qualitatively
supports our conclusion here.
Acknowledgement: The work is supported by the NSFC (Grant Nos. 10975084 and
11079024), RFDP (Grant No.20100002110080 ) and MOST (Grant No. 2013CB922000).
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|
arxiv-papers
| 2012-02-16T01:42:27 |
2024-09-04T02:49:27.451110
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shijun Mao and Pengfei Zhuang",
"submitter": "Shijun Mao",
"url": "https://arxiv.org/abs/1202.3490"
}
|
1202.3523
|
# Mean Field Effect on $J/\psi$ Production in Heavy Ion Collisions
Baoyi Chen, Kai Zhou, and Pengfei Zhuang Physics Department, Tsinghua
University, Beijing 100084, China
###### Abstract
The mass shift effect at finite temperature on $J/\psi$ production in
relativistic heavy ion collisions is calculated in a detailed transport
approach, including both mean field and collision terms. While the momentum-
integrated nuclear modification factor $R_{AA}$ is not sensitive to the mass
shift, the reduced threshold for $J/\psi$ regeneration in the quark-gluon
plasma leads to a remarkable enhancement for the differential $R_{AA}$ at low
transverse momentum.
###### pacs:
25.75.-q, 12.38.Mh, 24.85.+p
$J/\psi$ is a tightly bound state of charm quarks $c$ and $\overline{c}$, its
dissociation temperature $T_{d}$ in hot medium is much higher than the
critical temperature $T_{c}$ of the deconfinement phase transition. Therefore,
the measured $J/\psi$s in nuclear collisions at Relativistic Heavy Ion
Collider (RHIC) and Large Hadron Collider (LHC) carry the information of the
hot medium and has long been considered as a signature matsui of the new
state of matter, the so-called quark-gluon plasma (QGP) qgp .
The $J/\psi$ properties are significantly affected by the deconfinement phase
transition. From the calculations with quantum chromodynamics (QCD) sum rules
megias ; morita and QCD second-order Stark effect lee , both the $J/\psi$
width and mass are largely changed in a static hot medium. For a dynamically
evolutive QGP phase created in the early stage of heavy ion collisions, the
width is induced by $J/\psi$ decay like the gluon dissociation $g+J/\psi\to
c+\overline{c}$ na50 ; blaizot ; capella ; hufner ; polleri ; bratkovskaya ;
zhuang ; zhu ; wong , and the mass shift comes from the mean field effect of
the medium. Since the mass shift for a heavy quark system is expected to be
weak at low temperature, it is neglected in almost all the model calculations.
However, in the region above and close to the critical temperature, there is a
sudden change in the mass of $J/\psi$. For instance, at temperature
$T/T_{c}=1.1$ the mass shift $\delta m_{J/\psi}=m_{J/\psi}(T)-m_{J/\psi}(0)$
can reach $-(100-200)$ MeV megias ; morita ; lee , which is already comparable
with the mass change for light hadrons leupold and should have remarkable
consequence in $J/\psi$ production. In this paper, we study the hot nuclear
matter effect on $J/\psi$ production in heavy ion collisions at RHIC and LHC
energies, including not only the gluon dissociation but also the mean field.
Let’s first qualitatively estimate the mean field effect on the $J/\psi$
distribution. The decreased mass reduces the threshold value for the $J/\psi$
production in QGP, and should result in an enhancement for the $J/\psi$ yield.
Secondly, the attractive force between the inhomogeneous medium and $J/\psi$,
${\bf F}({\bf x},{\bf p})=-{m_{J/\psi}\over
E_{J/\psi}}{\bf\nabla}m_{J/\psi}=-{m_{J/\psi}\over E_{J/\psi}}{\partial
m_{J/\psi}\over\partial T}{\bf\nabla}T$ (1)
with $J/\psi$ energy $E_{J/\psi}=\sqrt{m_{J/\psi}^{2}+{\bf p}^{2}}$, pulls
$J/\psi$ to the center of the fireball and kicks $J/\psi$ to a lower momentum
region. This will lead to an enhancement at low momentum and a reduction at
high momentum.
Since charmonia are so heavy and difficult to be thermalized in hot medium, we
use a detailed transport approach yan to describe their distribution
functions $f_{\Psi}({\bf p},{\bf x},\tau|{\bf b})$ in the phase space at fixed
impact parameter ${\bf b}$ of a nucleus-nucleus collision,
${\partial f_{\Psi}\over\partial\tau}+{{\bf p}\over
E_{\Psi}}\cdot{\bf\nabla}_{x}f_{\Psi}+{\bf
F}\cdot{\bf\nabla}_{p}f_{\Psi}=-\alpha_{\Psi}f_{\Psi}+\beta_{\Psi},$ (2)
where ${\bf\nabla}_{x}$ and ${\bf\nabla}_{p}$ indicate three dimensional
derivatives in coordinate and momentum spaces. Considering the fact that in
proton-proton collisions the contribution from the decay of the excited
charmonium states to the $J/\psi$ yield is about $40\%$ herab , we must take
transport equations not only for the ground state $\Psi=J/\psi$ but also the
excited states $\Psi=\psi^{\prime}$ and $\chi_{c}$. The collision terms on the
right hand side are from the charmonium inelastic interaction with the medium.
The lose term $\alpha$ zhu ; yan arising from the gluon dissociation process
controls the $J/\psi$ suppression, and the gain term $\beta$ yan related to
$\alpha$ by detailed balance governs the $J/\psi$ regeneration pbm ;
gorenstein ; thews1 ; rapp ; thews2 ; zhao in the QGP phase. The gluons and
charm quarks in $\alpha$ and $\beta$ are assumed to be thermalized at RHIC and
LHC energies. The cross section for the gluon dissociation $\sigma_{\Psi}(0)$
in vacuum has been calculated through the method of operator production
expansion peskin ; bhanot , and its value at finite temperature can be
obtained from its classical relation to the charmonium size $r_{\Psi}$,
$\sigma_{\Psi}(T)=\langle r_{\Psi}^{2}(T)\rangle/\langle
r_{\Psi}^{2}(0)\rangle\sigma_{\Psi}(0)$, where the averaged $r$-square can be
derived from the Schrödinger equation satz with lattice simulated potential
karsch ; shuryak . In Fig.1, we show the $J/\psi$ decay rate
$\Gamma\equiv\alpha_{J/\psi}$ as a function of transverse momentum at fixed
temperature $T/T_{c}=1.1$ and $1.5$. In the calculation here we have chosen a
typical medium velocity $v_{QGP}=0.5$ and assumed that $\vec{v}_{QGP}$ and
$J/\psi$ momentum $\vec{p}$ have the same direction. In the vicinity of the
phase transition, the width is not sensitive to the momentum, and the
amplitude is in qualitative agreement with the result calculated from QCD sum
rules morita . When the temperature increases from $1.1T_{c}$ to $1.5T_{c}$,
the width increases by a factor of about 2.
Figure 1: The $J/\psi$ decay width as a function of transverse momentum
$p_{t}$ at temperature $T/T_{c}=1.1$ and $1.5$. The medium velocity is fixed
as $v_{QGP}=0.5$ and its direction is chosen as the same as the $J/\psi$
momentum.
Different from the previous study, we consider here not only the inelastic
processes, but also a mean field term ${\bf F}\cdot{\bf\nabla}_{p}f_{\Psi}$.
The local temperature $T({\bf x},\tau)$ appeared in the elastic and inelastic
terms are obtained from the hydrodynamic equations. At RHIC and LHC energies,
one can take the $2+1$ dimensional version heinz1 ; hirano1 ; zhu of the
relativistic hydrodynamic equations
$\partial_{\mu}T^{\mu\nu}=0,\ \ \partial_{\mu}N^{\mu}=0$ (3)
to describe the space-time evolution of the QGP, where $T_{\mu\nu}$ and
$N_{\mu}$ are the energy-momentum tensor and baryon number current of the
system. In our numerical calculation, the QGP formation time is chosen to be
$\tau_{0}=0.6$ fm/c and the initial thermodynamics is determined by the
colliding energy and the nuclear geometry zhu ; hirano2 .
The charm quark mass $m_{c}$ is determined by the charmonium mass $m_{\Psi}$
and its binding energy $\epsilon_{\Psi}$ satz ,
$m_{c}=\left(m_{\Psi}-\epsilon_{\Psi}\right)/2$. The cold nuclear matter
effects on $J/\psi$ production, such as nuclear absorption gerschel , Cronin
effect gavin ; hufner2 and shadowing effect vogt can be reflected in the
initial charmonium distributions $f_{\Psi}({\bf p},{\bf x},\tau_{0}|{\bf b})$.
In the following we neglect the nuclear absorption at RHIC and LHC energies
due to the small QGP formation time, and consider the Cronin effect with a
Gaussian smearing treatment zhao ; liu .
The transport equation with the mean field force can not be solved
analytically. Supposing that the very small elastic cross section for
charmonia at hadron level is still true at parton level, the mean field term
can be considered as a perturbation, and the transport equation can be solved
perturbatively. Fully neglecting the mass shift, the zeroth-order transport
equation can be strictly solved with the solution yan
$f_{\Psi}^{(0)}({\bf p},{\bf x},\tau|{\bf b})=f_{\Psi}({\bf p},{\bf
x}_{0},\tau_{0}|{\bf b})e^{-\int_{\tau_{0}}^{\tau}d\tau_{1}\alpha_{\Psi}({\bf
p},{\bf x}_{1},\tau_{1}|{\bf
b})}+\int_{\tau_{0}}^{\tau}d\tau_{1}\beta_{\Psi}({\bf p},{\bf
x}_{1},\tau_{1}|{\bf b})e^{-\int_{\tau_{1}}^{\tau}d\tau_{2}\alpha_{\Psi}({\bf
p},{\bf x}_{2},\tau_{2}|{\bf b})}$ (4)
with the coordinate shift ${\bf x}_{n}={\bf x}-{\bf
p}/E_{\Psi}(\tau-\tau_{n})$ which reflects the leakage effect matsui ; hufner3
during the time period $\tau-\tau_{n}$. The first term on the right hand side
is the contribution from the initial production, it suffers from the gluon
dissociation during the whole evolution of QGP. The second term arises from
the regeneration and suffers also the suppression from the production time
$\tau_{1}$ to the end of QGP.
Taking into account the fact that the dissociation temperatures for the
excited states are around $T_{c}$ satz , we do not consider their mass shifts
in QGP. Therefore, for the transport equations for $\psi^{\prime}$ and
$\chi_{c}$, their masses are temperature independent and the mean field forces
disappear, the zeroth-order solutions become the full distributions,
$f_{\psi^{\prime}}=f_{\psi^{\prime}}^{(0)}$ and
$f_{\chi_{c}}=f_{\chi_{c}}^{(0)}$.
We now consider the mean field effect on the $J/\psi$ distribution
$f_{J/\psi}$. We extract the mass shift from the QCD second-order Stark effect
lee ,
$\displaystyle\delta m_{J/\psi}(T)$ $\displaystyle=$ $\displaystyle
m_{J/\psi}(T)-m_{J/\psi}(0)$ $\displaystyle=$ $\displaystyle-{7\pi^{2}\over
18}{a^{2}\over e}\left[{2\over 11}M_{0}(T)+{3\over
4}{\alpha_{s}(T)\over\pi}M_{2}(T)\right],$
which is used as input for the spectral function analysis with QCD sum rules
morita , where the constants are taken as $a=0.271$ fm and $e=640$ MeV, the
coupling constant $\alpha_{s}$ is temperature dependent lee ; morita , and the
functions $M_{0}=\epsilon-3P$ and $M_{2}=\epsilon+P$ determined by the energy
density $\epsilon$ and pressure $P$ of the medium are extracted from the
lattice QCD simulations boyd . Leaving out first the mean field force induced
by the inhomogeneous property of the fireball but keeping the temperature
dependence of the mass, the solution of the transport equation has the same
form with the zeroth-order distribution, the only difference is the
replacement of $m_{J/\psi}(0)\rightarrow m_{J/\psi}(T)$,
$f_{J/\psi}^{(1)}=f_{J/\psi}^{(0)}\big{|}_{m_{J/\psi}(0)\rightarrow
m_{J/\psi}(T)}.$ (6)
If the effect of the mean field force is not very strong, we may infer that it
is not such a bad approximation to replace $f_{J/\psi}$ in the mean field term
by the known $f_{J/\psi}^{(1)}$, the approximate transport equation
${\partial f_{J/\psi}\over\partial\tau}+{{\bf p}\over
E_{J/\psi}}\cdot{\bf\nabla}_{x}f_{J/\psi}=-\alpha_{J/\psi}f_{J/\psi}+\beta_{J/\psi}-{\bf
F}\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}$ (7)
is then similar to the equation for $f_{J/\psi}^{(1)}$ but with a replacement
for the regeneration
$\beta_{J/\psi}\to\beta_{J/\psi}-{\bf F}\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}.$
(8)
This means that the mean field force can be considered as an effective
regeneration, which is not from the coalescence of heavy quarks but due to the
$J/\psi$ mass shift. The approximate solution so obtained is known as the
second-order $J/\psi$ distribution,
$f_{J/\psi}^{(2)}({\bf p},{\bf x},\tau|{\bf b})=f_{J/\psi}^{(1)}({\bf p},{\bf
x},\tau|{\bf b})-\int_{\tau_{0}}^{\tau}d\tau_{1}\left[{\bf F}({\bf
x}_{1})\cdot{\bf\nabla}_{p}f_{J/\psi}^{(1)}({\bf p},{\bf x}_{1},\tau_{1}|{\bf
b})\right]e^{-\int_{\tau_{1}}^{\tau}d\tau_{2}\alpha({\bf p},{\bf
x}_{2},\tau_{2}|{\bf b})}.$ (9)
Substituting the obtained $f_{J/\psi}^{(2)}$ into the mean field term, and
solving the transport equation again, we can derive the third-order
distribution function $f_{J/\psi}^{(3)}$. With the similar way, the
distribution to the $n$-th order can be generally expressed as a series of the
mean field force,
$\displaystyle f_{\Psi}^{(n)}({\bf p},{\bf x},\tau|{\bf b})$ $\displaystyle=$
$\displaystyle f_{\Psi}^{(1)}({\bf p},{\bf x},\tau|{\bf
b})-\sum_{m=1}^{n-1}(-1)^{m-1}\int_{\tau_{0}}^{\tau}d\tau_{1}\int_{\tau_{0}}^{\tau_{1}}d\tau_{2}\cdots\int_{\tau_{0}}^{\tau_{m-1}}d\tau_{m}$
(10) $\displaystyle\times$ $\displaystyle\left[\left(\prod_{i=1}^{m}{\bf
F}({\bf x}_{i})\cdot{\bf\nabla}_{p}\right)f_{\Psi}^{(1)}({\bf p},{\bf
x}_{m},\tau_{m}|{\bf
b})\right]e^{-\int_{\tau_{m}}^{\tau}d\tau^{\prime}\alpha({\bf p},{\bf
x}^{\prime},\tau^{\prime}|{\bf b})}.$
Note again that the mass shift affects the $J/\psi$ production in two aspects.
The reduced threshold, which is considered already in the first-order
distribution $f_{J/\psi}^{(1)}$, makes the production more easily, and the
attractive force between $J/\psi$ and the matter, which is included only in
the higher-order distributions $f_{J/\psi}^{(n)}\ (n=2,3,\cdots)$, makes a
momentum shift and leads to a low $p_{t}$ enhancement and a corresponding high
$p_{t}$ suppression.
By integrating the charmonium distribution function $f_{\Psi}({\bf p},{\bf
x},\tau|{\bf b})$ over the hyper surface of hadronization determined by
$T({\bf x},\tau)=T_{c}$, and considering the decay of the excited states to
the ground state, we can calculate the number $N_{AA}$ of survived $J/\psi$s
in a heavy ion collision. Let’s first examine the differential nuclear
modification factor
$R_{AA}(p_{t})=N_{AA}(p_{t})/\left(N_{c}N_{pp}(p_{t})\right)$ as a function of
transverse momentum $p_{t}$ in mid rapidity region, where $N_{pp}$ is the
number of $J/\psi$s in a nucleon-nucleon collision and $N_{c}$ the number of
nucleon-nucleon collisions. To see the largest mean field effect, we calculate
first $R_{AA}(p_{t})$ in central collisions with ${\bf b}=0$ where the formed
fireball is hot and large and the surviving time is long. Fig.2 shows our
numerical results for Au+Au collisions at top RHIC energy $\sqrt{s_{NN}}=200$
GeV. The thick and thin lines indicate our results with and without
considering the $J/\psi$ mass shift, by taking $f_{J/\psi}$ and
$f_{J/\psi}^{(0)}$, respectively. The dotted, dashed and solid lines are the
calculations with initial production only, regeneration only and the total.
From our numerical results, the series $f_{J/\psi}^{(n)}\ (n=0,1,2,\cdots)$
converges rapidly, and the deviation
$\left|R_{AA}^{(2)}-R_{AA}^{(1)}\right|/\left|R_{AA}^{(1)}\right|<3\%$ is
valid in any case. This confirms our qualitative estimation that the mean
field effect is mainly reflected in the change in the threshold, the
attractive force is a second order effect and the higher order corrections
with $n>2$ can be safely neglected. For the following numerical calculations
we will take $f_{J/\psi}=f_{J/\psi}^{(2)}$. Since the initial production has
ceased before the formation of the hot medium, the change in the threshold
does not affect it, and the correction is only from the small mean field
force. That is the reason why the initial contribution shown in Fig.2 is
almost independent of the mass shift. However, the regeneration happens in the
hot medium, it is affected by both the reduced threshold and the mean field
force, the overall correction should be much larger in comparison with the
correction to the initial production. Considering the fact that the
regenerated $J/\psi$s in the early stage of the QGP will be eaten up by the
hot medium and only the $J/\psi$s regenerated in the later stage of the QGP
can survive, the enhanced regeneration is mainly in the low $p_{t}$ region, as
shown in Fig.2. At $p_{t}=0$, the total $R_{AA}$ goes up from $0.38$ to
$0.48$, the enhancement is $26\%$. Since some $J/\psi$s are kicked to low
$p_{t}$ region by the mean field force, there is a little reduction of
$R_{AA}$ in the mid $p_{t}(2-3$ GeV) region. At high enough $p_{t}$, there is
almost no effect of the mass shift, and the $J/\psi$ distribution is dominated
by the perturbative QCD in the initial production. In comparison with our
previous calculations liu2 ; zhou , the Gaussian smearing treatment zhao ; liu
for the Cronin effect used here reduces the $R_{AA}$ at high $p_{t}$.
Figure 2: (color online) The differential nuclear modification factor
$R_{AA}(p_{t})$ as a function of transverse momentum $p_{t}$ in Au+Au
collisions at impact parameter $b=0$ and top RHIC energy $\sqrt{s_{NN}}=200$
GeV. The initial production, the regeneration, and the total are shown by
dotted, dashed and solid lines. The thick and thin lines are the calculations
with and without considering the mean field effect. Figure 3: (color online)
The differential nuclear modification factor $R_{AA}(p_{t})$ as a function of
transverse momentum $p_{t}$ in Au+Au collisions at impact parameter $b=4.2$ fm
and top RHIC energy $\sqrt{s_{NN}}=200$ GeV. The data are from the PHENIX
phenix and STAR star collaborations, and the thick and thin lines are the
full calculations with and without considering the mean field effect.
To compare with the experimental data, we show in Fig.3 the RHIC data for
Au+Au collisions at centrality $0\%-20\%$ phenix ; star and our calculation
at $b=4.2$ fm. With decreasing centrality, the fireball temperature drops down
and the mean field effect should be gradually reduced. However, from $b=0$ to
$4.2$ fm, the change in the mean field is small, and the model calculations
are almost the same. In our calculation we used the EKS98 parton distribution
function eks98 for the shadowing evolution and incorporated it with our
transport model through the initial distribution. We show in Fig.3 only the
total calculation, the results with and without the mean field effect can both
describe the data reasonably well. Since the current data are still with large
uncertainty even in low $p_{t}$ region, we need more precise data to
distinguish the mean field effect in the distribution.
Note that in the above calculations the shadowing effect reduces the charm
quark number and in turn the $J/\psi$ regeneration in the medium. Since the
parton momentum fraction $x_{F}$ at RHIC energy does not lie in the remarkable
shadowing region, the shadowing effect on $J/\psi$ production is not
remarkable. However, $x_{F}$ at LHC energy becomes very small and lies in the
strong shadowing region. This would give rise to a considerable reduction of
the $J/\psi$ regeneration. We calculated the $J/\psi$ $R_{AA}(p_{t})$ in Pb+Pb
collisions at LHC energy $\sqrt{s_{NN}}=2.76$ TeV and compared it with the
data from the ALICE collaboration alice , as shown in Fig.4. Besides the
shadowing effect, another important effect one should include in the
comparison with the ALICE data is the B meson decay, which leads to the non-
prompt part in the inclusive $J/\psi$ yield. We use the experimentally
measured p+p data from the CDF cdf , CMS cms and ATLAS atlas collaborations
to estimate the decay contribution and take the B meson quench in the medium.
The contribution from the B meson decay is important at high $p_{t}$ but the
influence is small at low $p_{t}$. The charm quark production cross section
$\sigma_{c\bar{c}}=0.38$ mb comes from the combination of the FONLL
calculation fonll and the p+p data pp . From the numerical calculation, the
shadowing effect at LHC results in a $25\%$ reduction of the $J/\psi$
regeneration, and this makes a better agreement between the calculation
including shadowing effect and the data at low $p_{t}$, see Fig.4.
Figure 4: (color online) The differential nuclear modification factor
$R_{AA}(p_{t})$ as a function of transverse momentum $p_{t}$ in Pb+Pb
collisions at LHC energy $\sqrt{s_{NN}}=2.76$ TeV. The data are from the ALICE
collaboration alice at centrality $0\%-90\%$ and at forward rapidity
$2.5<y<4$, and the model calculation is for the impact parameter $b=7.2$ fm.
The initial production, the regeneration, and the total are shown by dotted,
dashed and solid lines, and the thick and thin lines are the calculations with
and without considering the mean field effect.
While the mean field effect changes significantly the differential nuclear
modification factor at low transverse momentum, it does not affect the global
yield remarkably. From Figs.2-4, the most important mean field effect is at
$p_{t}=0$ and the effective region is around $p_{t}=0.5$ GeV which is much
less than the averaged $J/\psi$ transverse momentum $\langle
p_{t}\rangle\simeq 2-3$ GeV at RHIC and LHC energies. Therefore, the
$p_{t}$-integrated nuclear modification factor $R_{AA}(N_{p})$ as a function
of the number $N_{p}$ of participant nucleons becomes not sensitive to the
mean field effect. From our numerical calculations, the mass shift induced
enhancement of $R_{AA}(N_{p})$ is very small in peripheral and semi-central
collisions and less than $10\%$ even in most central collisions. Different
from $R_{AA}$ which is a summation of the initial production and regeneration,
the averaged transverse momentum square $\langle p_{t}^{2}\rangle$ is governed
by the fraction of the regeneration liu2 ; zhou . We found that the enhanced
$J/\psi$ yield at low $p_{t}$ leads to a less than $10\%$ suppression of
$\langle p_{t}^{2}\rangle$ in most central collisions.
In summery, we developed a perturbative expansion to study the mean field
effect on $J/\psi$ production in heavy ion collisions. Taking the mean field
term as a perturbation, we analytically solved the $J/\psi$ transport equation
with elastic and inelastic terms to any order and found a rapid convergence of
the perturbative expansion. The initial $J/\psi$ production, which happens
before the formation of QGP, is not sensitive to the mean field force, but the
continuous regeneration, which happens in the evolution of QGP, is
significantly affected by the mean field. As a result, the differential
nuclear modification factor of $J/\psi$ is enhanced at low $p_{t}$ in heavy
ion collisions at RHIC and LHC.
Acknowledgement: We thank useful discussions with Yunpeng Liu, Zhen Qu and Nu
Xu. The work is supported by the NSFC (Grant Nos. 10975084 and 11079024) and
RFDP (Grant No.20100002110080 ).
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|
arxiv-papers
| 2012-02-16T07:23:31 |
2024-09-04T02:49:27.458963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Baoyi Chen, Kai Zhou, Pengfei Zhuang",
"submitter": "Baoyi Chen",
"url": "https://arxiv.org/abs/1202.3523"
}
|
1202.3564
|
# Regular Reduction of Controlled Hamiltonian System with Symplectic Structure
and Symmetry
Jerrold E. Marsden
Control and Dynamical Systems,
California Institute of Technology, Pasadena, CA 91125 USA
Hong Wang, Zhenxing Zhang
School of Mathematical Sciences and LPMC,
Nankai University, Tianjin 300071, P.R.China
February 16, 2012 Corresponding Author: Hong Wang, E-mail:
hongwang@nankai.edu.cn
Abstract: In this paper, our goal is to study the regular reduction theory of
regular controlled Hamiltonian (RCH) systems with symplectic structure and
symmetry, and this reduction is an extension of regular symplectic reduction
theory of Hamiltonian systems under regular controlled Hamiltonian equivalence
conditions. Thus, in order to describe uniformly RCH systems defined on a
cotangent bundle and on the regular reduced spaces, we first define a kind of
RCH systems on a symplectic fiber bundle. Then introduce regular point and
regular orbit reducible RCH systems with symmetry by using momentum map and
the associated reduced symplectic forms. Moreover, we give regular point and
regular orbit reduction theorems for RCH systems to explain the relationships
between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with
symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as
an application we regard rigid body and heavy top as well as them with
internal rotors as the regular point reducible RCH systems on the rotation
group $\textmd{SO}(3)$ and on the Euclidean group $\textmd{SE}(3)$,
respectively, and discuss their RCH-equivalence. We also describe the RCH
system and RCH-equivalence from the viewpoint of port Hamiltonian system with
a symplectic structure.
Keywords: regular controlled Hamiltonian system, symplectic structure,
momentum map, regular Hamiltonian reduction, RCH-equivalence.
AMS Classification: 70H33, 53D20, 70Q05
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Momentum map
2. 2.2 Symplectic fiber bundles
3. 2.3 Lie group lifted action on (co-)tangent bundles and reduction
3. 3 Regular Controlled Hamiltonian Systems
4. 4 Regular Point Reduction of RCH Systems
5. 5 Regular Orbit Reduction of RCH Systems
6. 6 Applications
1. 6.1 Regular Point Reducible RCH System on a Lie Group
2. 6.2 Rigid Body and Heavy Top
3. 6.3 Port Hamiltonian System with a Symplectic Structure
## 1 Introduction
Symmetry is a general phenomenon in the natural world, but it is widely used
in the study of mathematics and mechanics. The reduction theory for mechanical
system with symmetry has its origin in the classical work of Euler, Lagrange,
Hamilton, Jacobi, Routh, Liouville and Poincaré and its modern geometric
formulation in the general context of symplectic manifolds and equivariant
momentum maps is developed by Meyer, Marsden and Weinstein; see Abraham and
Marsden [1] or Marsden and Weinstein [23] and Meyer [24]. The main goal of
reduction theory in mechanics is to use conservation laws and the associated
symmetries to reduce the number of dimensions required to describe a
mechanical system. So, such reduction theory is regarded as a useful tool for
simplifying and studying concrete mechanical systems. Reduction is a very
general procedure that is applied to arbitrary symmetric dynamical systems.
However, it is particularly powerful for conservative systems when the
symmetries induce a momentum map; see Abraham and Marsden [1], Arnold [3],
Marsden [20], Marsden et al [21], Marsden and Ratiu [22] and Ortega and Ratiu
[26].
It is well-known that Hamiltonian reduction theory is one of the most active
subjects in the study of modern analytical mechanics and applied mathematics,
in which a lot of deep and beautiful results have been obtained, see the
studies by Abraham and Marsden [1], Arnold [3], Leonard and Marsden [19],
Marsden et al [20, 21, 22, 23], Ortega and Ratiu [26] etc. on regular point
reduction and regular orbit reduction, singular point reduction and singular
orbit reduction, optimal reduction and reduction by stages for Hamiltonian
systems and so on; there is still much to be done in this subject.
On the other hand, just as we have known that the theory of mechanical control
systems presents a challenging and promising research area between the study
of classical mechanics and modern nonlinear geometric control theory and there
have been a lot of interesting results. Such as, Bloch et al. in [5, 6, 7, 8],
referred to the use of feedback control to realize a modification to the
structure of a given mechanical system; Blankenstein et al. in [4], Crouch and
Van der Schaft in [12], Nijmeijer and Van der Schaft in [25], van der Schaft
in [27, 28, 29, 30, 31], referred to the reduction and control of implicit
(port) Hamiltonian systems, and the use of feedback control to stabilize
mechanical systems and so on.
Nevertheless, we also note that Chang et al. in [9], defined a controlled
Hamiltonian (CH) system by using the almost Poisson tensor, and studied the
reduction of CH systems with symmetry in [11]. Unfortunately, there is a
serious wrong of rigor in their above work, that is, all of CH systems and
reduced CH systems given in [9, 11], have not the spaces on which these
systems are defined, see Definition 3.1 in [9] and Definition 3.1, 3.3 in
[11]. Thus, it is impossible to give the actions of a Lie group on the phase
of systems and their momentum maps, also impossible to determine the reduced
spaces of CH systems, and it is not that all of CH systems in [11] have same
space $T^{*}Q$, same action of Lie group $G$, and same reduced space
$T^{*}Q/G$. For example, we consider the cotangent bundle $T^{*}Q$ of a smooth
manifold $Q$ with a free and proper action of Lie group $G$, and the Poisson
tensor $B$ on $T^{*}Q$ is determined by canonical symplectic form $\omega_{0}$
on $T^{*}Q$. Then there is an $\operatorname{Ad}^{\ast}$-equivariant momentum
map $\mathbf{J}:T^{*}Q\rightarrow\mathfrak{g}^{\ast}$ for the symplectic, free
and proper cotangent lifted $G$-action, where $\mathfrak{g}^{\ast}$ is the
dual of Lie algebra $\mathfrak{g}$ of $G$. For $\mu\in\mathfrak{g}^{\ast}$, a
regular value of $\mathbf{J}:M\rightarrow\mathfrak{g}^{\ast}$, from Abraham
and Marsden [1], we know that the regular point reduced space
$\mathbf{J}^{-1}(\mu)/G_{\mu}$ and regular orbit reduced space
$\mathbf{J}^{-1}(\mathcal{O}_{\mu})/G$ at $\mu$ are not $T^{*}Q/G$, and the
two reduced spaces are determined by the momentum map $\mathbf{J}$, where
$G_{\mu}$ is the isotropy subgroup of the coadjoint $G$-action at the point
$\mu$, and $\mathcal{O}_{\mu}$ is the orbit of the coadjoint $G$-action
through the point $\mu$. Thus, in the two cases, it is impossible to determine
the reduced CH systems by using the method given in Chang et al [11].
In order to deal with the above problems and determine the reduced CH systems,
our idea in this paper is that we first define a CH system on $T^{*}Q$ by
using the symplectic form, and such system is called a RCH system, and then
regard a Hamiltonian system on $T^{*}Q$ as a spacial case of a RCH system
without external force and control. Thus, the set of Hamiltonian systems on
$T^{*}Q$ is a subset of the set of RCH systems on $T^{*}Q$. We hope to study
regular reduction theory of RCH systems with symplectic structure and
symmetry, as an extension of regular symplectic reduction theory of
Hamiltonian systems under regular controlled Hamiltonian equivalence
conditions. The main contributions in this paper is given as follows. (1) In
order to describe uniformly RCH systems defined on a cotangent bundle and on
the regular reduced spaces, we define a kind of RCH systems on a symplectic
fiber bundle by using its symplectic form; (2) We give regular point and
regular orbit reducible RCH systems by using momentum map and the associated
reduced symplectic forms, and prove regular point and regular orbit reduction
theorems (Theorem 4.3 and 5.3) for RCH systems to explain the relationships
between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with
symmetry and RCH-equivalence for associated reduced RCH systems; (3) We prove
that rigid body with external force torque, rigid body with internal rotors
and heavy top with internal rotors are all RCH systems, and as a pair of
regular point reduced RCH systems, rigid body with internal rotors (or
external force torque) and heavy top with internal rotors are RCH-equivalent;
(4) We describe the RCH system from the viewpoint of port Hamiltonian system
with a symplectic structure, and state the relationship between RCH-
equivalence of RCH system and equivalence of port Hamiltonian system.
A brief of outline of this paper is as follows. In the second section, we
review some relevant definitions and basic facts about momentum map,
symplectic fiber bundle, Lie group lifted actions on (co-)tangent bundles and
reduction, which will be used in subsequent sections. The RCH systems are
defined by using the symplectic forms on a symplectic fiber bundle and on the
cotangent bundle of a configuration manifold, respectively, and RCH-
equivalence is introduced in the third section. From the fourth section we
begin to discuss the RCH systems with symmetry by combining with regular
symplectic reduction theory. The regular point and regular orbit reducible RCH
systems are considered respectively in the fourth section and the fifth
section, and give the regular point and regular orbit reduction theorems for
RCH systems to explain the relationships between the RpCH-equivalence, RoCH-
equivalence for reducible RCH systems with symmetry and the RCH-equivalence
for associated reduced RCH systems. As the applications of the theoretical
results, in sixth section, we first give the regular point reduced RCH system
on a Lie group $G$, which is a RCH system on a coadjoint orbit of $G$. Then we
regard the rigid body and heavy top as well as them with internal rotors as
the regular point reducible RCH systems on the rotation group $\textmd{SO}(3)$
and on the Euclidean group $\textmd{SE}(3)$, respectively, and give their
regular point reduced RCH systems and discuss their RCH-equivalence. In order
to understand well the abstract definition of RCH system, we also describe the
RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system
with a symplectic structure. These research work develop the theory of
Hamiltonian reduction for the regular controlled Hamiltonian systems with
symmetry and make us have much deeper understanding and recognition for the
structure of controlled Hamiltonian systems.
## 2 Preliminaries
In order to study the regular reduction theory of RCH systems, we first give
some relevant definitions and basic facts about momentum maps, symplectic
fiber bundle, Lie group lifted actions on (co-)tangent bundles and reduction,
which will be used in subsequent sections, we shall follow the notations and
conventions introduced in Abraham et al [1, 2], Marsden [20], Marsden et al
[21], Marsden and Ratiu [22], Ortega and Ratiu [26], Kobayashi and Nomizu
[16]. In this paper, we assume that all manifolds are real, smooth and finite
dimensional and all actions are smooth left actions.
### 2.1 Momentum map
Let $(M,\omega)$ be a symplectic manifold, $G$ a Lie group with Lie algebra
$\mathfrak{g}$. We say that $G$ acts on $M$ and the action of any $g\in G$ on
$z\in M$ will be denoted by $\Phi:G\times M\rightarrow M:\Phi(g,z)=g\cdot z$.
For any $g\in G$, the map $\Phi_{g}:=\Phi(g,\cdot):M\rightarrow M$ is a
diffeomorphism of $M$ and if the map $\Phi_{g}$ satisfies
$\Phi_{g}^{\ast}\omega=\omega,\;\forall g\in G,$ we say that $G$ acts
symplectically on a symplectic manifold $(M,\omega)$. The isotropy subgroup of
a point $z\in M$ is $G_{z}=\\{g\in G|\;g\cdot z=z\\}.$ An action is free if
all the isotropy subgroups $G_{z}$ are trivial; and is proper if the map
$(g,z)\rightarrow(g,g\cdot z)$ is proper (i.e., the pre-image of every compact
set is compact). For a proper action, all isotropy subgroups are compact. The
$G$-orbit of $z\in M$ is denoted $\mathcal{O}_{z}=G\cdot
z=\\{\Phi_{g}(z)|\;g\in G\\},$ and the orbit space
$M/G=\\{\mathcal{O}_{z}|\;z\in M\\}.$ If $G$ acts freely and properly on $M$,
then $M/G$ has a unique smooth structure such that $\pi_{G}:M\rightarrow M/G$
is a surjective submersion. If $G$ acts only properly on $M$, does not act
freely, then $M/G$ is not necessarily smooth manifold, but just a quotient
topological space.
For each $\xi\in\mathfrak{g}$, the infinitesimal generator of $\xi$ is the
vector field $\xi_{M}$ defined by
$\xi_{M}(z)=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\exp(t\xi)\cdot
z,\forall z\in M$. We will also write $\xi_{M}(z)$ as $\xi\cdot z$, and refer
to the map $(\xi,z)\mapsto\xi\cdot z$ as the infinitesimal action of
$\mathfrak{g}$ on $M$. A momentum map
$\mathbf{J}:M\rightarrow\mathfrak{g}^{\ast}$ is defined by
$<\mathbf{J}(z),\xi>=J_{\xi}(z)$, for every $\xi\in\mathfrak{g}$, where the
function $J_{\xi}:M\rightarrow\mathbb{R}$ satisfies $X_{J_{\xi}}=\xi_{M}$, and
$\mathfrak{g}^{\ast}$ is the dual of Lie algebra $\mathfrak{g}$, and
$<,>:\mathfrak{g}^{\ast}\times\mathfrak{g}\rightarrow\mathbb{R}$ is the
duality pairing between the dual $\mathfrak{g}^{\ast}$ and $\mathfrak{g}$. If
the adjoint action of $G$ on $\mathfrak{g}$ is denoted by $\operatorname{Ad}$,
and the infinitesimal adjoint action by $\operatorname{ad}$, then the
coadjoint action of $G$ on $g^{\ast}$ is the inverse dual to the adjoint
action, given by
$g\cdot\nu=\operatorname{Ad}_{g^{-1}}^{\ast}\nu=(\operatorname{Ad}_{g^{-1}})^{\ast}\nu,\forall\;\nu\in\mathfrak{g}^{\ast}$.
The infinitesimal coadjoint action is given by
$\xi\cdot\nu=-\operatorname{ad}_{\xi}^{\ast}\nu,\forall\;\nu\in\mathfrak{g}^{\ast}$.
For $\mu\in\mathfrak{g}^{\ast}$, a value of
$\mathbf{J}:M\rightarrow\mathfrak{g}^{\ast}$, $G_{\mu}$ denotes the isotropy
subgroup of $G$ with respect to the coadjoint $G$-action
$\operatorname{Ad}_{g^{-1}}^{\ast}$ at the point $\mu$, and
$\mathcal{O}_{\mu}$ denotes the $G$-orbit of through the point $\mu$ in
$\mathfrak{g}^{\ast}$. The momentum map $\mathbf{J}$ is
$\operatorname{Ad}^{\ast}$-equivariant if
$\mathbf{J}(\Phi_{g}(z))=\operatorname{Ad}_{g^{-1}}^{\ast}\mathbf{J}(z)$, for
any $z\in M$.
The following proposition is very important for the regular reduction and
singular reduction of Hamiltonian systems with symmetry; see Marsden [20] and
Ortega and Ratiu [26].
###### Proposition 2.1
(Bifurcation Lemma) Let $(M,\omega)$ be a symplectic manifold and $G$ a Lie
group acting symplectically on $M$ (not necessarily freely). Suppose that the
action has an associated momentum map $\mathbf{J}:M\to\mathfrak{g}^{\ast}$.
Then for any $z\in M$, $(\mathfrak{g}_{z})^{0}=\mbox{range}(T_{z}\mathbf{J})$,
where $\mathfrak{g}_{z}=\\{\xi\in\mathfrak{g}|\;\xi_{M}(z)=0\\}$ is the Lie
algebra of the isotropy subgroup $G_{z}=\\{g\in G|\;g\cdot z=z\\}$ and
$(\mathfrak{g}_{z})^{0}=\\{\mu\in\mathfrak{g}^{\ast}|\;\mu|_{\mathfrak{g}_{z}}=0\\}$
denotes the annihilator of $\mathfrak{g}_{z}$ in $\mathfrak{g}^{\ast}$.
An immediate consequence of this proposition is the fact that when the action
of $G$ is free, each value $\mu\in\mathfrak{g}^{\ast}$ of the momentum map
$\mathbf{J}$ is a regular value of $\mathbf{J}$. Thus, if $\mu$ is a singular
value of $\mathbf{J}$, then the $G$-action is not free. Moreover, if $\mu$ is
a regular value of $\mathbf{J}$ and $\mathcal{O}_{\mu}$ is an embedded
submanifold of $\mathfrak{g}^{\ast}$, the $\mathbf{J}$ is transverse to
$\mathcal{O}_{\mu}$ and hence $\mathbf{J}^{-1}(\mathcal{O}_{\mu})$ is
automatically an embedded submanifold of $M$. In this paper, we consider only
that the $G$-action is free, and the Hamiltonian reductions are regular.
### 2.2 Symplectic fiber bundles
Let $E$ and $M$ be two smooth manifolds, Lie group $G$ acts freely on $E$ from
the left side. Denote by $(E,M,\pi,G)$ a (left) principal fiber bundle over
$M$ with group $G$, where $E$ is the bundle space, $M$ is the base space, $G$
is the structure group and the projection $\pi:E\rightarrow M$ is a surjective
submersion. For each $x\in M$, $\pi^{-1}(x)$ is a closed submanifold of $E$,
which is called the fiber over $x$. Each fiber of the principal bundle
$(E,M,\pi,G)$ is diffeomorphic to $G$. In the following we shall give a
construction of the associated bundle of $G$-principal bundle. Assume that $F$
is another smooth manifold and Lie group $G$ acts on $F$ from the left side.
We can define a fiber bundle associated to principal bundle $(E,M,\pi,G)$ with
fiber $F$ as follows. Consider the left action of $G$ on the product manifold
$E\times F$, $\Phi:G\times(E\times F)\rightarrow E\times F$ given by
$\Phi(g,(z,y))=(gz,g^{-1}y),\;\forall\;g\in G,\;z\in E,\;y\in F.$ Denote by
$E\times_{G}F$ is the orbit space $(E\times F)/G$, and the map
$\rho:E\times_{G}F\rightarrow M$ is uniquely determined by the condition
$\rho\cdot\pi_{/G}=\pi\cdot\pi_{E}$, that is, the following commutative
Diagram-1,
$\begin{CD}E\times F@>{\pi_{/G}}>{}>E\times_{G}F\\\
@V{\pi_{E}}V{}V@V{}V{\rho}V\\\ E@>{\pi}>{}>M\end{CD}$ Diagram-1
where $\pi_{/G}:E\times F\rightarrow E\times_{G}F$ is the canonical projection
and $\pi_{E}:E\times F\rightarrow E$ is the projection onto the first factor.
Then $(E\times_{G}F,M,F,\rho,G)$, simply written as $(E,M,F,\pi,G)$, is a
fiber bundle with fiber $F$ and structure group $G$ associated to principal
bundle $(E,M,\pi,G)$. In particular, if $F=V$ is a vector space, then
$(E,M,V,\pi,G)$ is a vector bundle associated to principal bundle
$(E,M,\pi,G)$.
A bundle of symplectic manifolds is such a fiber bundle $(E,M,F,\pi,G)$, all
of whose fibers are symplectic and whose structure group $G$ preserves the
symplectic structure on $F$. From Gotay et al. [14] we know that there exists
a presymplectic form $\omega_{E}$ on $E$ under some topological conditions,
whose pull-back to each fiber is the given fiber symplectic form. We assume
that if a symplectic form $\omega_{E}$ is given on $E$, then $(E,\omega_{E})$
is called a symplectic fiber bundle. In particular, if $E$ is a vector bundle,
then $(E,\omega_{E})$ is called a symplectic vector bundle; see Libermann and
Marle [18].
### 2.3 Lie group lifted action on (co-)tangent bundles and reduction
For a smooth manifold $Q$, its cotangent bundle $T^{\ast}Q$ has a canonical
symplectic form $\omega_{0}$, which is given in natural cotangent bundle
coordinates $(q^{i},p_{i})$ by
$\omega_{0}=\mathbf{d}q^{i}\wedge\mathbf{d}p_{i}$, so $T^{\ast}Q$ is a
symplectic vector bundle. Let $\Phi:G\times Q\rightarrow Q$ be a left smooth
action of a Lie group $G$ on the manifold $Q$. The tangent lift of this action
$\Phi:G\times Q\rightarrow Q$ is the action of $G$ on $TQ$, $\Phi^{T}:G\times
TQ\rightarrow TQ$ given by $g\cdot v_{q}=T\Phi_{g}(v_{q}),\;\forall\;v_{q}\in
T_{q}Q,q\in Q$. The cotangent lift is the action of $G$ on $T^{\ast}Q$,
$\Phi^{T^{\ast}}:G\times T^{\ast}Q\rightarrow T^{\ast}Q$ given by
$g\cdot\alpha_{q}=(T\Phi_{g^{-1}})^{\ast}\cdot\alpha_{q},\;\forall\;\alpha_{q}\in
T^{\ast}_{q}Q,\;q\in Q$. The tangent or cotangent lift of any proper (resp.
free) $G$-action is proper(resp. free). Each cotangent lift action is
symplectic with respect to the canonical symplectic form $\omega_{0}$, and has
an $\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}:T^{\ast}Q\to\mathfrak{g}^{\ast}$ given by
$<\mathbf{J}(\alpha_{q}),\xi>=\alpha_{q}(\xi_{Q}(q)),$ where
$\xi\in\mathfrak{g}$, $\xi_{Q}(q)$ is the value of the infinitesimal generator
$\xi_{Q}$ of the $G$-action at $q\in Q$,
$<,>:\mathfrak{g}^{\ast}\times\mathfrak{g}\rightarrow\mathbb{R}$ is the
duality pairing between the dual $\mathfrak{g}^{\ast}$ and $\mathfrak{g}$.
The reduction theory of cotangent bundle is a very important special case of
general reduction theory. Let $\mu\in\mathfrak{g}^{\ast}$ is a regular value
of the momentum map $\mathbf{J}$, the simplest case of symplectic reduction of
cotangent bundle $T^{\ast}Q$ is regular point reduction at zero, in this case
the symplectic reduced space formed at $\mu=0$ is given by
$((T^{\ast}Q)_{\mu},\omega_{\mu})=(T^{\ast}(Q/G),\omega_{0})$, where
$\omega_{0}$ is the canonical symplectic form of cotangent bundle
$T^{\ast}(Q/G)$. Thus, the reduced space $((T^{\ast}Q)_{\mu},\omega_{\mu})$ at
$\mu=0$ is a symplectic vector bundle. If $\mu\neq 0$, from Marsden et al [21]
we know that, when $G_{\mu}=G$, the regular point reduced space
$((T^{*}Q)_{\mu},\omega_{\mu})$ is symplectic diffeomorphic to symplectic
vector bundle $(T^{\ast}(Q/G),\omega_{0}-B_{\mu})$, where $B_{\mu}$ is a
magnetic term; If $G$ is not abelian and $G_{\mu}\neq G$, the regular point
reduced space $((T^{*}Q)_{\mu},\omega_{\mu})$ is symplectic diffeomorphic to a
symplectic fiber bundle over $T^{\ast}(Q/G_{\mu})$ with fiber to be the
coadjoint orbit $\mathcal{O}_{\mu}$. In the case of regular orbit reduction,
from Ortega and Ratiu [26] and the regular reduction diagram, we know that the
regular orbit reduced space
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$ is symplectic
diffeomorphic to the regular point reduced space
$((T^{*}Q)_{\mu},\omega_{\mu})$, and hence is symplectic diffeomorphic to a
symplectic fiber bundle. Thus, if we may define a RCH systems on a symplectic
fiber bundle, then it is possible to describe uniformly the RCH systems on
$T^{*}Q$ and their regular reduced RCH systems on the associated reduced
spaces.
## 3 Regular Controlled Hamiltonian Systems
In this paper, our goal is to study regular reduction theory of RCH systems
with symplectic structure and symmetry, as an extension of regular symplectic
reduction theory of Hamiltonian systems under regular controlled Hamiltonian
equivalence conditions. Thus, in order to describe uniformly RCH systems
defined on a cotangent bundle and on regular reduced spaces, in this section
we first define a RCH system on a symplectic fiber bundle. In particular, we
obtain the RCH system by using the symplectic structure on the cotangent
bundle of a configuration manifold as a special case, and discuss RCH-
equivalence. In consequence, we can study the RCH systems with symmetry by
combining with regular symplectic reduction theory of Hamiltonian systems.
Let $(E,M,N,\pi,G)$ be a fiber bundle and $(E,\omega_{E})$ be a symplectic
fiber bundle. If for any function $H:E\rightarrow\mathbb{R}$, we have a
Hamiltonian vector field $X_{H}$ by $i_{X_{H}}\omega_{E}=\mathbf{d}H$, then
$(E,\omega_{E},H)$ is a Hamiltonian system. Moreover, if considering the
external force and control, we can define a kind of regular controlled
Hamiltonian (RCH) system on the symplectic fiber bundle $E$ as follows.
###### Definition 3.1
(RCH System) A RCH system on $E$ is a 5-tuple $(E,\omega_{E},H,F,W)$, where
$(E,\omega_{E},H)$ is a Hamiltonian system, and the function
$H:E\rightarrow\mathbb{R}$ is called the Hamiltonian, a fiber-preserving map
$F:E\rightarrow E$ is called the (external) force map, and a fiber submanifold
$W$ of $E$ is called the control subset.
Sometimes, $W$ also denotes the set of fiber-preserving maps from $E$ to $W$.
When a feedback control law $u:E\rightarrow W$ is chosen, the 5-tuple
$(E,\omega_{E},H,F,u)$ denotes a closed-loop dynamic system. In particular,
when $Q$ is a smooth manifold, and $T^{\ast}Q$ its cotangent bundle with a
symplectic form $\omega$ (not necessarily canonical symplectic form), then
$(T^{\ast}Q,\omega)$ is a symplectic vector bundle. If we take that
$E=T^{*}Q$, from above definition we can obtain a RCH system on the cotangent
bundle $T^{\ast}Q$, that is, 5-tuple $(T^{\ast}Q,\omega,H,F,W)$. Where the
fiber-preserving map $F:T^{*}Q\rightarrow T^{*}Q$ is the (external) force map,
that is the reason that the fiber-preserving map $F:E\rightarrow E$ is called
an (external) force map in above definition.
In order to describe the dynamics of the RCH system $(E,\omega_{E},H,F,W)$
with a control law $u$, we need to introduce a notations for vertical lifts
along fiber. For the bundle $\pi:E\rightarrow M$ and any a point $x\in M$,
$E_{x}=\pi^{-1}(x)$ is the fiber over $x$, the vertical lift operator
$\mbox{vlift}:E\times E\rightarrow TE$ is defined as follows:
$T_{u_{x}}E\ni\mbox{vlift}(v_{x},u_{x})=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{s=0}(u_{x}+sv_{x}),\;\forall\;u_{x},v_{x}\in
E_{x}.$
The vertical lift of a fiber-preserving map $F:E\rightarrow E$ is a section,
$\mbox{vlift}(F):E\rightarrow TE$, defined by
$\mbox{vlift}(F)(v_{x})=\mbox{vlift}(F(v_{x}),v_{x}),\;\;\forall\;v_{x}\in
E_{x},$
and $\mbox{vlift}(u)$ is defined in the similar manner. The vertical lift of a
fiber submanifold $W$ of $E$ is the subset of $TE$ defined by
$\mbox{vlift}(W)=\bigcup_{x\in M}\\{\mbox{vlift}(v_{x},u_{x})|\;u_{x}\in
E_{x},v_{x}\in W_{x}\\}.$
For the RCH System $(T^{\ast}Q,\omega,H,F,W)$, when a feedback control law
$u:T^{\ast}Q\rightarrow W$ is chosen, by using the notations for vertical
lifts along fiber, we can give a expression of vector field
$X_{(T^{\ast}Q,\omega,H,F,u)}$ of the RCH system $(T^{\ast}Q,\omega,H,F,W)$
with a control law $u$ as follows
$X_{(T^{\ast}Q,\omega,H,F,u)}=(\mathbf{d}H)^{\sharp}+\textnormal{vlift}(F)+\textnormal{vlift}(u)$
(1)
where $\sharp:T^{\ast}T^{\ast}Q\rightarrow
TT^{\ast}Q;\mathbf{d}H\mapsto(\mathbf{d}H)^{\sharp}$ such that
$i_{(\mathbf{d}H)^{\sharp}}\omega=\mathbf{d}H$, that is,
$(\mathbf{d}H)^{\sharp}=X_{H}$.
Next, we note that when a RCH system is given, the force map $F$ is
determined, but the feedback control law $u:T^{\ast}Q\rightarrow W$ could be
chosen. In order to describe the feedback control law to modify the structure
of RCH system, the Hamiltonian matching conditions and RCH-equivalence are
induced as follows.
###### Definition 3.2
(RCH-equivalence) Suppose that we have two RCH systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2,$ we say them to be RCH-
equivalent, or simply,
$(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RCH}}{{\sim}}(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},W_{2})$, if there exists a
diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$, such that the following
Hamiltonian matching conditions hold:
RHM-1: The cotangent lift map of $\varphi$, that is,
$\varphi^{\ast}=T^{\ast}\varphi:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is
symplectic, and $W_{1}=\varphi^{\ast}(W_{2}).$
RHM-2:
$Im[(\mathbf{d}H_{1})^{\sharp}+\textnormal{vlift}(F_{1})-((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}-\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})]\subset\textnormal{vlift}(W_{1})$,
where the map $\varphi_{\ast}=(\varphi^{-1})^{\ast}:T^{\ast}Q_{1}\rightarrow
T^{\ast}Q_{2}$, and
$(\varphi^{\ast})_{\ast}=(\varphi_{\ast})^{\ast}=T^{\ast}\varphi_{\ast}:T^{\ast}T^{\ast}Q_{2}\rightarrow
T^{\ast}T^{\ast}Q_{1}$, and $Im$ means the pointwise image of the map in
brackets.
It is worthy of note that our RCH system is defined by using the symplectic
structure on the cotangent bundle of a configuration manifold, we must keep
with the symplectic structure when we define the RCH-equivalence, that is, the
induced equivalent map $\varphi^{*}$ is symplectic on the cotangent bundle. In
the same way, for the RCH systems on the symplectic fiber bundles, we can also
define the RCH-equivalence by replacing $T^{\ast}Q_{i}$ and
$\varphi:Q_{1}\rightarrow Q_{2}$ by $E_{i}$ and
$\varphi^{\ast}:E_{2}\rightarrow E_{1}$, respectively. Moreover, the following
Theorem 3.3 explains the significance of the above RCH-equivalence relation.
###### Theorem 3.3
Suppose that two RCH systems $(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i})$,
$i=1,2,$ are RCH-equivalent, then there exist two control laws
$u_{i}:T^{\ast}Q_{i}\rightarrow W_{i},\;i=1,2,$ such that the two closed-loop
systems produce the same equations of motion, that is,
$X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}\cdot\varphi^{\ast}=T(\varphi^{\ast})X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}$,
where the map $T(\varphi^{\ast}):TT^{\ast}Q_{2}\rightarrow TT^{\ast}Q_{1}$ is
the tangent map of $\varphi^{\ast}$. Moreover, the explicit relation between
the two control laws $u_{i},i=1,2$ is given by
$\textnormal{vlift}(u_{1})-\textnormal{vlift}(\varphi^{\ast}u_{2}\varphi_{\ast})=-(\mathbf{d}H_{1})^{\sharp}-\textnormal{vlift}(F_{1})+((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}+\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})$
(2)
Proof: From (1), we have that
$X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}=(\mathbf{d}H_{1})^{\sharp}+\textnormal{vlift}(F_{1})+\textnormal{vlift}(u_{1})$
and
$\displaystyle
T(\varphi^{\ast})X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}$
$\displaystyle=T(\varphi^{\ast})[(\mathbf{d}H_{2})^{\sharp}+\textnormal{vlift}(F_{2})+\textnormal{vlift}(u_{2})]$
$\displaystyle=T(\varphi^{\ast})(\mathbf{d}H_{2})^{\sharp}+T(\varphi^{\ast})\textnormal{vlift}(F_{2})+T(\varphi^{\ast})\textnormal{vlift}(u_{2})$
$\begin{CD}T^{\ast}Q_{2}@>{\textnormal{vlift}}>{}>TT^{\ast}Q_{2}@<{\sharp}<{}<T^{\ast}T^{\ast}Q_{2}\\\
@V{\varphi^{\ast}}V{}V@V{T\varphi^{\ast}}V{}V@V{(\varphi_{\ast})^{\ast}}V{}V\\\
T^{\ast}Q_{1}@>{}>{\textnormal{vlift}}>TT^{\ast}Q_{1}@<{}<{\sharp}<T^{\ast}T^{\ast}Q_{1}\end{CD}$
Diagram-2
From the commutative Diagram-2 and the definition of the vertical lift
operator vlift, we have that for $\alpha\in T^{\ast}Q_{2}$,
$\displaystyle
T(\varphi^{\ast})\textnormal{vlift}(F_{2})(\alpha)=T(\varphi^{\ast})\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=0}(\alpha+sF_{2}(\alpha))$
$\displaystyle=\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=0}(\varphi^{\ast}\alpha+s\varphi^{\ast}F_{2}\varphi_{\ast}(\varphi^{\ast}\alpha))=\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})(\varphi^{\ast}\alpha).$
In the same way, we have that
$T(\varphi^{\ast})\textnormal{vlift}(u_{2})=\textnormal{vlift}(\varphi^{\ast}u_{2}\varphi_{\ast})\cdot\varphi^{\ast}$.
Since $\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is symplectic,
and $i_{(\mathbf{d}H_{i})^{\sharp}}\omega_{i}=\mathbf{d}H_{i}$, we have that
$T(\varphi^{\ast})(\mathbf{d}H_{2})^{\sharp}=((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}\cdot\varphi^{\ast}$.
Thus,
$T(\varphi^{\ast})X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}=((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}\cdot\varphi^{\ast}+\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})\cdot\varphi^{\ast}+\textnormal{vlift}(\varphi^{\ast}u_{2}\varphi_{\ast})\cdot\varphi^{\ast}.$
From
$X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}\cdot\varphi^{\ast}=T(\varphi^{\ast})X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}$,
we have that (2) holds. $\blacksquare$
In the following we shall introduce the regular point and regular orbit
reducible RCH with symplectic form and symmetry, and show a variety of
relationships of their regular reduced RCH-equivalences.
## 4 Regular Point Reduction of RCH Systems
Let $Q$ be a smooth manifold and $T^{\ast}Q$ its cotangent bundle with the
symplectic form $\omega$. Let $\Phi:G\times Q\rightarrow Q$ be a smooth left
action of the Lie group $G$ on $Q$, which is free and proper. Then the
cotangent lifted left action $\Phi^{T^{\ast}}:G\times T^{\ast}Q\rightarrow
T^{\ast}Q$ is symplectic, free and proper, and admits an
$\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}:T^{\ast}Q\rightarrow\mathfrak{g}^{\ast}$, where $\mathfrak{g}$ is
a Lie algebra of $G$ and $\mathfrak{g}^{\ast}$ is the dual of $\mathfrak{g}$.
Let $\mu\in\mathfrak{g}^{\ast}$ be a regular value of $\mathbf{J}$ and denote
by $G_{\mu}$ the isotropy subgroup of the coadjoint $G$-action at the point
$\mu\in\mathfrak{g}^{\ast}$, which is defined by $G_{\mu}=\\{g\in
G|\operatorname{Ad}_{g}^{\ast}\mu=\mu\\}$. Since $G_{\mu}(\subset G)$ acts
freely and properly on $Q$ and on $T^{\ast}Q$, then $Q_{\mu}=Q/G_{\mu}$ is a
smooth manifold and that the canonical projection $\rho_{\mu}:Q\rightarrow
Q_{\mu}$ is a surjective submersion. It follows that $G_{\mu}$ acts also
freely and properly on $\mathbf{J}^{-1}(\mu)$, so that the space
$(T^{\ast}Q)_{\mu}=\mathbf{J}^{-1}(\mu)/G_{\mu}$ is a symplectic manifold with
symplectic form $\omega_{\mu}$ uniquely characterized by the relation
$\pi_{\mu}^{\ast}\omega_{\mu}=i_{\mu}^{\ast}\omega.$ (3)
The map $i_{\mu}:\mathbf{J}^{-1}(\mu)\rightarrow T^{\ast}Q$ is the inclusion
and $\pi_{\mu}:\mathbf{J}^{-1}(\mu)\rightarrow(T^{\ast}Q)_{\mu}$ is the
projection. The pair $((T^{\ast}Q)_{\mu},\omega_{\mu})$ is called the
symplectic point reduced space of $(T^{\ast}Q,\omega)$ at $\mu$.
###### Remark 1
If $T^{\ast}Q$ is a connected symplectic manifold, and
$\mathbf{J}:T^{\ast}Q\rightarrow\mathfrak{g}^{\ast}$ is a non-equivariant
momentum map with a non-equivariance group one-cocycle
$\sigma:G\rightarrow\mathfrak{g}^{\ast}$, which is defined by
$\sigma(g):=\mathbf{J}(g\cdot
z)-\operatorname{Ad}^{\ast}_{g^{-1}}\mathbf{J}(z)$, where $g\in G$ and $z\in
T^{\ast}Q$. Then we know that $\sigma$ produces a new affine action
$\Theta:G\times\mathfrak{g}^{\ast}\rightarrow\mathfrak{g}^{\ast}$ defined by
$\Theta(g,\mu):=\operatorname{Ad}^{\ast}_{g^{-1}}\mu+\sigma(g)$, where
$\mu\in\mathfrak{g}^{\ast}$, with respect to which the given momentum map
$\mathbf{J}$ is equivariant. Assume that $G$ acts freely and properly on
$T^{\ast}Q$, and $G_{\mu}$ denotes the isotropy subgroup of
$\mu\in\mathfrak{g}^{\ast}$ relative to this affine action $\Theta$ and $\mu$
is a regular value of $\mathbf{J}$. Then the quotient space
$(T^{\ast}Q)_{\mu}=\mathbf{J}^{-1}(\mu)/G_{\mu}$ is also a symplectic manifold
with symplectic form $\omega_{\mu}$ uniquely characterized by (3).
If $H:T^{\ast}Q\rightarrow\mathbb{R}$ is a $G$-invariant Hamiltonian, the flow
$F_{t}$ of the Hamiltonian vector field $X_{H}$ leaves the connected
components of $\mathbf{J}^{-1}(\mu)$ invariant and commutes with the
$G$-action, then it induces a flow $f_{t}^{\mu}$ on $(T^{\ast}Q)_{\mu}$,
defined by $f_{t}^{\mu}\cdot\pi_{\mu}=\pi_{\mu}\cdot F_{t}\cdot i_{\mu}$, and
the vector field $X_{h_{\mu}}$ generated by the flow $f_{t}^{\mu}$ on
$((T^{\ast}Q)_{\mu},\omega_{\mu})$ is Hamiltonian with the associated regular
point reduced Hamiltonian function
$h_{\mu}:(T^{\ast}Q)_{\mu}\rightarrow\mathbb{R}$ defined by
$h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$, and the Hamiltonian vector fields
$X_{H}$ and $X_{h_{\mu}}$ are $\pi_{\mu}$-related. See Ortega and Ratiu [26].
On the other hand, from section 2, we know that the regular point reduced
space $((T^{*}Q)_{\mu},\omega_{\mu})$ is symplectic diffeomorphic to a
symplectic fiber bundle. Thus, we can introduce a regular point reducible RCH
systems as follows.
###### Definition 4.1
(Regular Point Reducible RCH System) A 6-tuple $(T^{\ast}Q,G,\omega,H,F,W)$,
where the Hamiltonian $H:T^{\ast}Q\rightarrow\mathbb{R}$, the fiber-preserving
map $F:T^{\ast}Q\rightarrow T^{\ast}Q$ and the fiber submanifold $W$ of
$T^{\ast}Q$ are all $G$-invariant, is called a regular point reducible RCH
system, if there exists a point $\mu\in\mathfrak{g}^{\ast}$, which is a
regular value of the momentum map $\mathbf{J}$, such that the regular point
reduced system, that is, the 5-tuple
$((T^{\ast}Q)_{\mu},\omega_{\mu},h_{\mu},f_{\mu},W_{\mu})$, where
$(T^{\ast}Q)_{\mu}=\mathbf{J}^{-1}(\mu)/G_{\mu}$,
$\pi_{\mu}^{\ast}\omega_{\mu}=i_{\mu}^{\ast}\omega$,
$h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$, $f_{\mu}\cdot\pi_{\mu}=\pi_{\mu}\cdot
F\cdot i_{\mu}$, $W\subset\mathbf{J}^{-1}(\mu)$, $W_{\mu}=\pi_{\mu}(W)$, is a
RCH system, which is simply written as $R_{P}$-reduced RCH system. Where
$((T^{\ast}Q)_{\mu},\omega_{\mu})$ is the $R_{P}$-reduced space, the function
$h_{\mu}:(T^{\ast}Q)_{\mu}\rightarrow\mathbb{R}$ is called the reduced
Hamiltonian, the fiber-preserving map
$f_{\mu}:(T^{\ast}Q)_{\mu}\rightarrow(T^{\ast}Q)_{\mu}$ is called the reduced
(external) force map, $W_{\mu}$ is a fiber submanifold of $(T^{\ast}Q)_{\mu}$
and is called the reduced control subset.
Denote by $X_{(T^{\ast}Q,G,\omega,H,F,u)}$ the vector field of regular point
reducible RCH system $(T^{\ast}Q,G,\omega,H,\\\ F,W)$ with a control law $u$,
then
$X_{(T^{\ast}Q,G,\omega,H,F,u)}=(\mathbf{d}H)^{\sharp}+\textnormal{vlift}(F)+\textnormal{vlift}(u).$
(4)
Moreover, for the regular point reducible RCH system we can also introduce the
regular point reduced controlled Hamiltonian equivalence (RpCH-equivalence) as
follows.
###### Definition 4.2
(RpCH-equivalence) Suppose that we have two regular point reducible RCH
systems $(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2$, we say
them to be RpCH-equivalent, or simply,
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RpCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2})$, if there
exists a diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$ such that the
following Hamiltonian matching conditions hold:
RpHM-1: The cotangent lift map $\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow
T^{\ast}Q_{1}$ is symplectic.
RpHM-2: For $\mu_{i}\in\mathfrak{g}^{\ast}_{i}$, the regular reducible points
of RCH systems $(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2$,
the map $\varphi_{\mu}^{\ast}=i_{\mu_{1}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mu_{2}}:\mathbf{J}_{2}^{-1}(\mu_{2})\rightarrow\mathbf{J}_{1}^{-1}(\mu_{1})$
is $(G_{2\mu_{2}},G_{1\mu_{1}})$-equivariant and
$W_{1}=\varphi_{\mu}^{\ast}(W_{2})$, where $\mu=(\mu_{1},\mu_{2})$, and denote
by $i_{\mu_{1}}^{-1}(S)$ the preimage of a subset $S\subset T^{\ast}Q_{1}$ for
the map $i_{\mu_{1}}:\mathbf{J}_{1}^{-1}(\mu_{1})\rightarrow T^{\ast}Q_{1}$.
RpHM-3:
$Im[(\mathbf{d}H_{1})^{\sharp}+\textnormal{vlift}(F_{1})-((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}-\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})]\subset\textnormal{vlift}(W_{1})$.
It is worthy of note that for the regular point reducible RCH system, the
induced equivalent map $\varphi^{*}$ not only keeps the symplectic structure,
but also keeps the equivariance of $G$-action at the regular point. If a
feedback control law $u_{\mu}:(T^{\ast}Q)_{\mu}\rightarrow W_{\mu}$ is chosen,
the $R_{P}$-reduced RCH system
$((T^{\ast}Q)_{\mu},\omega_{\mu},h_{\mu},f_{\mu},u_{\mu})$ is a closed-loop
regular dynamic system with a control law $u_{\mu}$. Assume that its vector
field $X_{((T^{\ast}Q)_{\mu},\omega_{\mu},h_{\mu},f_{\mu},u_{\mu})}$ can be
expressed by
$X_{((T^{\ast}Q)_{\mu},\omega_{\mu},h_{\mu},f_{\mu},u_{\mu})}=(\mathbf{d}h_{\mu})^{\sharp}+\textnormal{vlift}(f_{\mu})+\textnormal{vlift}(u_{\mu}),$
(5)
and satisfies the condition
$X_{((T^{\ast}Q)_{\mu},\omega_{\mu},h_{\mu},f_{\mu},u_{\mu})}\cdot\pi_{\mu}=T\pi_{\mu}\cdot
X_{(T^{\ast}Q,G,\omega,H,F,u)}\cdot i_{\mu}.$ (6)
Then we can obtain the following regular point reduction theorem for RCH
system, which explains the relationship between the RpCH-equivalence for
regular point reducible RCH systems with symmetry and the RCH-equivalence for
associated $R_{P}$-reduced RCH systems. This theorem can be regarded as an
extension of regular point reduction theorem of Hamiltonian systems under
regular controlled Hamiltonian equivalence conditions.
###### Theorem 4.3
Two regular point reducible RCH systems
$(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i})$, $i=1,2,$ are RpCH-
equivalent if and only if the associated $R_{P}$-reduced RCH systems
$((T^{\ast}Q_{i})_{\mu_{i}},\omega_{i\mu_{i}},h_{i\mu_{i}},f_{i\mu_{i}},\\\
W_{i\mu_{i}}),i=1,2,$ are RCH-equivalent.
Proof: If
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RpCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2})$, then there
exists a diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$ such that
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is symplectic and for
$\mu_{i}\in\mathfrak{g}^{\ast}_{i},i=1,2$,
$\varphi_{\mu}^{\ast}=i_{\mu_{1}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mu_{2}}:\mathbf{J}_{2}^{-1}(\mu_{2})\rightarrow\mathbf{J}_{1}^{-1}(\mu_{1})$
is $(G_{2\mu_{2}},G_{1\mu_{1}})$-equivariant,
$W_{1}=\varphi_{\mu}^{\ast}(W_{2})$ and RpHM-3 holds. From the following
commutative Diagram-3:
$\begin{CD}T^{\ast}Q_{2}@<{i_{\mu_{2}}}<{}<\mathbf{J}_{2}^{-1}(\mu_{2})@>{\pi_{\mu_{2}}}>{}>(T^{\ast}Q_{2})_{\mu_{2}}\\\
@V{\varphi^{\ast}}V{}V@V{\varphi^{\ast}_{\mu}}V{}V@V{\varphi^{\ast}_{\mu/G}}V{}V\\\
T^{\ast}Q_{1}@<{i_{\mu_{1}}}<{}<\mathbf{J}_{1}^{-1}(\mu_{1})@>{\pi_{\mu_{1}}}>{}>(T^{\ast}Q_{1})_{\mu_{1}}\end{CD}$
Diagram-3
We can define a map
$\varphi_{\mu/G}^{\ast}:(T^{\ast}Q_{2})_{\mu_{2}}\rightarrow(T^{\ast}Q_{1})_{\mu_{1}}$
such that
$\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}}=\pi_{\mu_{1}}\cdot\varphi^{\ast}_{\mu}$.
Because
$\varphi_{\mu}^{\ast}:\mathbf{J}_{2}^{-1}(\mu_{2})\rightarrow\mathbf{J}_{1}^{-1}(\mu_{1})$
is $(G_{2\mu_{2}},G_{1\mu_{1}})$-equivariant, $\varphi_{\mu/G}^{\ast}$ is
well-defined. We shall show that $\varphi_{\mu/G}^{\ast}$ is symplectic and
$W_{1\mu_{1}}=\varphi_{\mu/G}^{\ast}(W_{2\mu_{2}})$. In fact, since
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is symplectic, the map
$(\varphi^{\ast})^{\ast}:\Omega^{2}(T^{\ast}Q_{1})\rightarrow\Omega^{2}(T^{\ast}Q_{2})$
satisfies $(\varphi^{\ast})^{\ast}\omega_{1}=\omega_{2}$. By (3),
$i_{\mu_{i}}^{\ast}\omega_{i}=\pi_{\mu_{i}}^{\ast}\omega_{i\mu_{i}},i=1,2$,
from the following commutative Diagram-4,
$\begin{CD}\Omega^{2}(T^{\ast}Q_{1})@
>i_{\mu_{1}}^{\ast}>>\Omega^{2}(\mathbf{J}_{1}^{-1}(\mu_{1}))@
<\pi_{\mu_{1}}^{\ast}<<\Omega^{2}((T^{\ast}Q_{1})_{\mu_{1}})\\\
@V{(\varphi^{\ast})^{\ast}}V{}V@V{(\varphi^{\ast}_{\mu})^{\ast}}V{}V@V{(\varphi^{\ast}_{\mu/G})^{\ast}}V{}V\\\
\Omega^{2}(T^{\ast}Q_{2})@>{i_{\mu_{2}}^{\ast}}>{}>\Omega^{2}(\mathbf{J}_{2}^{-1}(\mu_{2}))@<{\pi_{\mu_{2}}^{\ast}}<{}<\Omega^{2}((T^{\ast}Q_{2})_{\mu_{2}})\end{CD}$
Diagram-4
we have that
$\displaystyle\pi_{\mu_{2}}^{\ast}\cdot(\varphi_{\mu/G}^{\ast})^{\ast}\omega_{1\mu_{1}}$
$\displaystyle=(\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}})^{\ast}\omega_{1\mu_{1}}=(\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast})^{\ast}\omega_{1\mu_{1}}=(i_{\mu_{1}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mu_{2}})^{\ast}\cdot\pi_{\mu_{1}}^{\ast}\omega_{1\mu_{1}}$
$\displaystyle=i_{\mu_{2}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\cdot(i_{\mu_{1}}^{-1})^{\ast}\cdot
i_{\mu_{1}}^{\ast}\omega_{1}=i_{\mu_{2}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\omega_{1}=i_{\mu_{2}}^{\ast}\omega_{2}=\pi_{\mu_{2}}^{\ast}\omega_{2\mu_{2}}.$
Notice that $\pi_{\mu_{2}}^{\ast}$ is a surjective, thus,
$(\varphi_{\mu/G}^{\ast})^{\ast}\omega_{1\mu_{1}}=\omega_{2\mu_{2}}$. Because
by hypothesis $W_{i}\subset\mathbf{J}_{i}^{-1}(\mu_{i})$,
$W_{i\mu_{i}}=\pi_{\mu_{i}}(W_{i}),\;i=1,2$ and
$W_{1}=\varphi_{\mu}^{\ast}(W_{2})$, we have that
$W_{1\mu_{1}}=\pi_{\mu_{1}}(W_{1})=\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast}(W_{2})=\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}}(W_{2})=\varphi_{\mu/G}^{\ast}(W_{2\mu_{2}}).$
Next, from (4) and (5), we know that for $i=1,2$,
$X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}=(\mathbf{d}H_{i})^{\sharp}+\textnormal{vlift}(F_{i})+\textnormal{vlift}(u_{i}),$
$X_{((T^{\ast}Q_{i})_{\mu_{i}},\omega_{i\mu_{i}},h_{i\mu_{i}},f_{i\mu_{i}},u_{i\mu_{i}})}=(\mathbf{d}h_{i\mu_{i}})^{\sharp}+\textnormal{vlift}(f_{i\mu_{i}})+\textnormal{vlift}(u_{i\mu_{i}}),$
and from (6), we have that
$X_{((T^{\ast}Q_{i})_{\mu_{i}},\omega_{i\mu_{i}},h_{i\mu_{i}},f_{i\mu_{i}},u_{i\mu_{i}})}\cdot\pi_{\mu_{i}}=T\pi_{\mu_{i}}\cdot
X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}\cdot i_{\mu_{i}}.$
Since $H_{i},F_{i}$ and $W_{i}$ are all $G_{i}$-invariant, $i=1,2$ and
$h_{i\mu_{i}}\cdot\pi_{\mu_{i}}=H_{i}\cdot
i_{\mu_{i}},\;\;f_{i\mu_{i}}\cdot\pi_{\mu_{i}}=\pi_{\mu_{i}}\cdot F_{i}\cdot
i_{\mu_{i}},\;\;u_{i\mu_{i}}\cdot\pi_{\mu_{i}}=\pi_{\mu_{i}}\cdot u_{i}\cdot
i_{\mu_{i}},\;\;i=1,2.$
From the following commutative Diagram-5,
$\begin{CD}T^{\ast}T^{\ast}Q_{2}@>{i_{\mu_{2}}^{\ast}}>{}>T^{\ast}\mathbf{J}_{2}^{-1}(\mu_{2})@<{\pi_{\mu_{2}}^{\ast}}<{}<T^{\ast}((T^{\ast}Q_{2})_{\mu_{2}})\\\
@V{(\varphi^{\ast})_{\ast}}V{}V@V{(\varphi^{\ast}_{\mu})_{\ast}}V{}V@V{(\varphi^{\ast}_{\mu/G})_{\ast}}V{}V\\\
T^{\ast}T^{\ast}Q_{1}@>{i_{\mu_{1}}^{\ast}}>{}>T^{\ast}\mathbf{J}_{1}^{-1}(\mu_{1})@<{\pi_{\mu_{1}}^{\ast}}<{}<T^{\ast}((T^{\ast}Q_{1})_{\mu_{1}})\end{CD}$
Diagram-5
we have that
$\pi_{\mu_{1}}^{\ast}\cdot(\varphi_{\mu/G}^{\ast})_{\ast}\mathbf{d}h_{2\mu_{2}}=i_{\mu_{1}}^{\ast}\cdot(\varphi^{\ast})_{\ast}\mathbf{d}H_{2}$,
then
$((\varphi_{\mu/G}^{\ast})_{\ast}\mathbf{d}h_{2\mu_{2}})^{\sharp}\cdot\pi_{\mu_{1}}=T\pi_{\mu_{1}}\cdot((\varphi^{\ast})_{\ast}\mathbf{d}H_{2})^{\sharp}\cdot
i_{\mu_{1}},$ $\textnormal{vlift}(\varphi_{\mu/G}^{\ast}\cdot
f_{2\mu_{2}}\cdot\varphi_{\mu/G\ast})\cdot\pi_{\mu_{1}}=T\pi_{\mu_{1}}\cdot\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})\cdot
i_{\mu_{1}},$ $\textnormal{vlift}(\varphi_{\mu/G}^{\ast}\cdot
u_{2\mu_{2}}\cdot\varphi_{\mu/G\ast})\cdot\pi_{\mu_{1}}=T\pi_{\mu_{1}}\cdot\textnormal{vlift}(\varphi^{\ast}u_{2}\varphi_{\ast})\cdot
i_{\mu_{1}},$
where
$\varphi_{\mu/G\ast}=(\varphi^{-1})^{\ast}_{\mu/G}:(T^{\ast}Q_{1})_{\mu_{1}}\rightarrow(T^{\ast}Q_{2})_{\mu_{2}}$
and
$(\varphi_{\mu/G}^{\ast})_{\ast}=(\varphi_{\mu/G\ast})^{\ast}:T^{\ast}((T^{\ast}Q_{2})_{\mu_{2}})\rightarrow
T^{\ast}((T^{\ast}Q_{1})_{\mu_{1}})$. From Hamiltonian matching condition
RpHM-3 we have that
$\displaystyle
Im[(\mathrm{d}h_{1\mu_{1}})^{\sharp}+\textnormal{vlift}(f_{1\mu_{1}})-((\varphi_{\mu/G}^{\ast})_{\ast}\mathrm{d}h_{2\mu_{2}})^{\sharp}-\textnormal{vlift}(\varphi_{\mu/G}^{\ast}\cdot
f_{2\mu_{2}}\cdot\varphi_{\mu/G\ast})]$ (7) $\displaystyle\hskip
211.26027pt\subset\textnormal{vlift}(W_{1\mu_{1}}).$
So,
$((T^{\ast}Q_{1})_{\mu_{1}},\omega_{1\mu_{1}},h_{1\mu_{1}},f_{1\mu_{1}},W_{1\mu_{1}})\stackrel{{\scriptstyle
RCH}}{{\sim}}((T^{\ast}Q_{2})_{\mu_{2}},\omega_{2\mu_{2}},h_{2\mu_{2}},f_{2\mu_{2}},W_{2\mu_{2}}).$
Conversely, assume that $R_{P}$-reduced RCH systems
$((T^{\ast}Q_{i})_{\mu_{i}},\omega_{i\mu_{i}},h_{i\mu_{i}},f_{i\mu_{i}},W_{i\mu_{i}})$,
$i=1,2,$ are RCH-equivalent. Then there exists a diffeomorphism
$\varphi_{\mu/G}^{\ast}:(T^{\ast}Q_{2})_{\mu_{2}}\rightarrow(T^{\ast}Q_{1})_{\mu_{1}}$,
which is symplectic,
$W_{1\mu_{1}}=\varphi_{\mu/G}^{\ast}(W_{2\mu_{2}}),\;\mu_{i}\in\mathfrak{g}_{i}^{\ast},\;i=1,2$
and (7) holds. We can define a map
$\varphi_{\mu}^{\ast}:\mathbf{J}^{-1}_{2}(\mu_{2})\rightarrow\mathbf{J}^{-1}_{1}(\mu_{1})$
such that
$\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast}=\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}};$
and the map $\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ such that
$\varphi^{\ast}\cdot i_{\mu_{2}}=i_{\mu_{1}}\cdot\varphi_{\mu}^{\ast};$ see
the commutative Diagram-3, as well as a diffeomorphism
$\varphi:Q_{1}\rightarrow Q_{2},$ whose cotangent lift is just
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$. From definition of
$\varphi_{\mu}^{\ast}$, we know that $\varphi_{\mu}^{\ast}$ is
$(G_{2\mu_{2}},G_{1\mu_{1}})$-equivariant. In fact, for any
$z_{i}\in\mathbf{J}_{i}^{-1}(\mu_{i})$, $g_{i}\in G_{i\mu_{i}}$, $i=1,2$ such
that $z_{1}=\varphi_{\mu}^{\ast}(z_{2})$,
$[z_{1}]=\varphi^{\ast}_{\mu/G}[z_{2}]$, then we have that
$\displaystyle\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast}(\Phi_{2g_{2}}(z_{2}))$
$\displaystyle=\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast}(g_{2}z_{2})=\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}}(g_{2}z_{2})=\varphi_{\mu/G}^{\ast}[z_{2}]=[z_{1}]$
$\displaystyle=\pi_{\mu_{1}}(g_{1}z_{1})=\pi_{\mu_{1}}(\Phi_{1g_{1}}(z_{1}))=\pi_{\mu_{1}}\cdot\Phi_{1g_{1}}\cdot\varphi_{\mu}^{\ast}(z_{2}).$
Since $\pi_{\mu_{1}}$ is surjective, so,
$\varphi_{\mu}^{\ast}\cdot\Phi_{2g_{2}}=\Phi_{1g_{1}}\cdot\varphi_{\mu}^{\ast}$.
Moreover,
$\pi_{\mu_{1}}(W_{1})=W_{1\mu_{1}}=\varphi_{\mu/G}^{\ast}(W_{2\mu_{2}})=\varphi_{\mu/G}^{\ast}\cdot\pi_{2\mu_{2}}(W_{2})=\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast}(W_{2})$.
Since $W_{i}\subset\mathbf{J}_{i}^{-1}(\mu_{i}),i=1,2$ and $\pi_{\mu_{1}}$ is
surjective, then $W_{1}=\varphi_{\mu}^{\ast}(W_{2})$. We shall show that
$\varphi^{\ast}$ is symplectic. Because
$\varphi_{\mu/G}^{\ast}:(T^{\ast}Q_{2})_{\mu_{2}}\rightarrow(T^{\ast}Q_{1})_{\mu_{1}}$
is symplectic, the map
$(\varphi_{\mu/G}^{\ast})^{\ast}:\Omega^{2}((T^{\ast}Q_{1})_{\mu_{1}})\rightarrow\Omega^{2}((T^{\ast}Q_{2})_{\mu_{2}})$
satisfies
$(\varphi_{\mu/G}^{\ast})^{\ast}\omega_{1\mu_{1}}=\omega_{2\mu_{2}}$. By (3),
$i_{\mu_{i}}^{\ast}\omega_{i}=\pi_{\mu_{i}}^{\ast}\omega_{i\mu_{i}},i=1,2$,
from the commutative Diagram-4, we have that
$\displaystyle i_{\mu_{2}}^{\ast}\omega_{2}$
$\displaystyle=\pi_{\mu_{2}}^{\ast}\omega_{2\mu_{2}}=\pi_{\mu_{2}}^{\ast}\cdot(\varphi_{\mu/G}^{\ast})^{\ast}\omega_{1\mu_{1}}=(\varphi_{\mu/G}^{\ast}\cdot\pi_{\mu_{2}})^{\ast}\omega_{1\mu_{1}}=(\pi_{\mu_{1}}\cdot\varphi_{\mu}^{\ast})^{\ast}\omega_{1\mu_{1}}$
$\displaystyle=(i_{\mu_{1}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mu_{2}})^{\ast}\cdot\pi_{\mu_{1}}^{\ast}\omega_{1\mu_{1}}=i_{\mu_{2}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\cdot(i^{-1}_{\mu_{1}})^{\ast}\cdot
i_{\mu_{1}}^{\ast}\omega_{1}=i_{\mu_{2}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\omega_{1}.$
Notice that $i_{\mu_{2}}^{\ast}$ is injective, thus,
$\omega_{2}=(\varphi^{\ast})^{\ast}\omega_{1}$. Since the vector field
$X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}$ and
$X_{((T^{\ast}Q_{i})_{\mu_{i}},\omega_{i\mu_{i}},h_{i\mu_{i}},f_{i\mu_{i}},u_{i\mu_{i}})}$
is $\pi_{\mu_{i}}$-related, $i=1,2,$ and $H_{i},F_{i}$ and $W_{i}$ are all
$G_{i}$-invariant, $i=1,2$, in the same way, from (7), we have that
$Im[(\mathbf{d}H_{1})^{\sharp}+\textnormal{vlift}(F_{1})-((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}-\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})]\subset\textnormal{vlift}(W_{1}),$
that is, Hamiltonian matching condition RpHM-3 holds. Thus,
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RpCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2}).\hskip
28.45274pt\blacksquare$
## 5 Regular Orbit Reduction of RCH Systems
Let $\mu\in\mathfrak{g}^{\ast}$ be a regular value of the momentum map
$\mathbf{J}$ and $\mathcal{O}_{\mu}=G\cdot\mu\subset\mathfrak{g}^{\ast}$ be
the $G$-orbit of the coadjoint $G$-action through the point $\mu$. Since $G$
acts freely, properly and symplectically on $T^{\ast}Q$, then the quotient
space $(T^{\ast}Q)_{\mathcal{O}_{\mu}}=\mathbf{J}^{-1}(\mathcal{O}_{\mu})/G$
is a regular quotient symplectic manifold with the symplectic form
$\omega_{\mathcal{O}_{\mu}}$ uniquely characterized by the relation
$i_{\mathcal{O}_{\mu}}^{\ast}\omega=\pi_{\mathcal{O}_{\mu}}^{\ast}\omega_{\mathcal{O}_{\mu}}+\mathbf{J}_{\mathcal{O}_{\mu}}^{\ast}\omega_{\mathcal{O}_{\mu}}^{+},$
(8)
where $\mathbf{J}_{\mathcal{O}_{\mu}}$ is the restriction of the momentum map
$\mathbf{J}$ to $\mathbf{J}^{-1}(\mathcal{O}_{\mu})$, that is,
$\mathbf{J}_{\mathcal{O}_{\mu}}=\mathbf{J}\cdot i_{\mathcal{O}_{\mu}}$ and
$\omega_{\mathcal{O}_{\mu}}^{+}$ is the $(+)$-symplectic structure on the
orbit $\mathcal{O}_{\mu}$ given by
$\omega_{\mathcal{O}_{\mu}}^{+}(\nu)(\xi_{\mathfrak{g}^{\ast}}(\nu),\eta_{\mathfrak{g}^{\ast}}(\nu))=<\nu,[\xi,\eta]>,\;\;\forall\;\nu\in\mathcal{O}_{\mu},\;\xi,\eta\in\mathfrak{g}.$
(9)
The maps $i_{\mathcal{O}_{\mu}}:\mathbf{J}^{-1}(\mathcal{O}_{\mu})\rightarrow
T^{\ast}Q$ and
$\pi_{\mathcal{O}_{\mu}}:\mathbf{J}^{-1}(\mathcal{O}_{\mu})\rightarrow(T^{\ast}Q)_{\mathcal{O}_{\mu}}$
are natural injection and the projection, respectively. The pair
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$ is called the
symplectic orbit reduced space of $(T^{\ast}Q,\omega)$. If
$H:T^{\ast}Q\rightarrow\mathbb{R}$ is a $G$-invariant Hamiltonian, the flow
$F_{t}$ of the Hamiltonian vector field $X_{H}$ leaves the connected
components of $\mathbf{J}^{-1}(\mathcal{O}_{\mu})$ invariant and commutes with
the $G$-action, then it induces a flow $f_{t}^{\mathcal{O}_{\mu}}$ on
$(T^{\ast}Q)_{\mathcal{O}_{\mu}}$, defined by
$f_{t}^{\mathcal{O}_{\mu}}\cdot\pi_{\mathcal{O}_{\mu}}=\pi_{\mathcal{O}_{\mu}}\cdot
F_{t}\cdot i_{\mathcal{O}_{\mu}}$, and the vector field
$X_{h_{\mathcal{O}_{\mu}}}$ generated by the flow $f_{t}^{\mathcal{O}_{\mu}}$
on $((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$ is
Hamiltonian with the associated regular orbit reduced Hamiltonian function
$h_{\mathcal{O}_{\mu}}:(T^{\ast}Q)_{\mathcal{O}_{\mu}}\rightarrow\mathbb{R}$
defined by $h_{\mathcal{O}_{\mu}}\cdot\pi_{\mathcal{O}_{\mu}}=H\cdot
i_{\mathcal{O}_{\mu}}$ and the Hamiltonian vector fields $X_{H}$ and
$X_{h_{\mathcal{O}_{\mu}}}$ are $\pi_{\mathcal{O}_{\mu}}$-related. See Ortega
and Ratiu [26].
When $Q=G$ is a Lie group with Lie algebra $\mathfrak{g}$, and the $G$-action
is the cotangent lift of left translation, then the associated momentum map
$\mathbf{J}_{L}:T^{\ast}G\rightarrow\mathfrak{g}^{\ast}$ is right invariant.
In the same way, the momentum map
$\mathbf{J}_{R}:T^{\ast}G\rightarrow\mathfrak{g}^{\ast}$ for the cotangent
lift of right translation is left invariant. For regular value
$\mu\in\mathfrak{g}^{\ast}$,
$\mathcal{O}_{\mu}=G\cdot\mu=\\{\operatorname{Ad}^{\ast}_{g^{-1}}\mu|g\in
G\\}$ and the Kostant-Kirilllov-Sourian (KKS) symplectic forms on coadjoint
orbit $\mathcal{O}_{\mu}(\subset\mathfrak{g}^{\ast})$ are given by
$\omega_{\mathcal{O}_{\mu}}^{-}(\nu)(\operatorname{ad}_{\xi}^{\ast}(\nu),\operatorname{ad}_{\eta}^{\ast}(\nu))=-<\nu,[\xi,\eta]>,\;\;\forall\;\nu\in\mathcal{O}_{\mu},\;\xi,\eta\in\mathfrak{g}.$
From Ortega and Ratiu [26], we know that by using the momentum map
$\mathbf{J}_{R}$ one can induce a symplectic diffeomorphism from the
symplectic point reduced space $((T^{\ast}G)_{\mu},\omega_{\mu})$ to the
symplectic orbit space $(\mathcal{O}_{\mu},\omega_{\mathcal{O}_{\mu}}^{-})$.
In general case, we maybe thought that the structure of the symplectic orbit
reduced space $((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$
is more complex than that of the symplectic point reduced space
$((T^{\ast}Q)_{\mu},\omega_{\mu})$, but, from the regular reduction diagram,
we know that the regular orbit reduced space
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$ is symplectic
diffeomorphic to the regular point reduced space
$((T^{*}Q)_{\mu},\omega_{\mu})$, and hence is also symplectic diffeomorphic to
a symplectic fiber bundle. Thus, we can introduce a kind of the regular orbit
reducible RCH systems as follows.
###### Definition 5.1
(Regular Orbit Reducible RCH System) A 6-tuple $(T^{\ast}Q,G,\omega,H,F,W)$,
where the Hamiltonian $H:T^{\ast}Q\rightarrow\mathbb{R}$, the fiber-preserving
map $F:T^{\ast}Q\rightarrow T^{\ast}Q$ and the fiber submanifold $W$ of
$T^{\ast}Q$ are all $G$-invariant, is called a regular orbit reducible RCH
system, if there exists a orbit
$\mathcal{O}_{\mu},\;\mu\in\mathfrak{g}^{\ast}$, where $\mu$ is a regular
value of the momentum map $\mathbf{J}$, such that the regular orbit reduced
system, that is, the 5-tuple
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}},h_{\mathcal{O}_{\mu}},f_{\mathcal{O}_{\mu}},W_{\mathcal{O}_{\mu}})$,
where $(T^{\ast}Q)_{\mathcal{O}_{\mu}}=\mathbf{J}^{-1}(\mathcal{O}_{\mu})/G$,
$\pi_{\mathcal{O}_{\mu}}^{\ast}\omega_{\mathcal{O}_{\mu}}=i_{\mathcal{O}_{\mu}}^{\ast}\omega-\mathbf{J}_{\mathcal{O}_{\mu}}^{\ast}\omega_{\mathcal{O}_{\mu}}^{+}$,
$h_{\mathcal{O}_{\mu}}\cdot\pi_{\mathcal{O}_{\mu}}=H\cdot
i_{\mathcal{O}_{\mu}}$,
$f_{\mathcal{O}_{\mu}}\cdot\pi_{\mathcal{O}_{\mu}}=\pi_{\mathcal{O}_{\mu}}\cdot
F\cdot i_{\mathcal{O}_{\mu}}$, $W\subset\mathbf{J}^{-1}(\mathcal{O}_{\mu})$,
$W_{\mathcal{O}_{\mu}}=\pi_{\mathcal{O}_{\mu}}(W)$, is a RCH system, which is
simply written as $R_{O}$-reduced RCH system. Where
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}})$ is the
$R_{O}$-reduced space, the function
$h_{\mathcal{O}_{\mu}}:(T^{\ast}Q)_{\mathcal{O}_{\mu}}\rightarrow\mathbb{R}$
is called the reduced Hamiltonian, the fiber-preserving map
$f_{\mathcal{O}_{\mu}}:(T^{\ast}Q)_{\mathcal{O}_{\mu}}\rightarrow(T^{\ast}Q)_{\mathcal{O}_{\mu}}$
is called the reduced (external) force map, $W_{\mathcal{O}_{\mu}}$ is a fiber
submanifold of $(T^{\ast}Q)_{\mathcal{O}_{\mu}}$, and is called the reduced
control subset.
Denote by $X_{(T^{\ast}Q,G,\omega,H,F,u)}$ the vector field of the regular
orbit reducible RCH system $(T^{\ast}Q,G,\omega,\\\ H,F,W)$ with a control law
$u$, then
$X_{(T^{\ast}Q,G,\omega,H,F,u)}=(\mathbf{d}H)^{\sharp}+\textnormal{vlift}(F)+\textnormal{vlift}(u).$
(10)
Moreover, for the regular orbit reducible RCH system we can also introduce the
regular orbit reduced controlled Hamiltonian equivalence (RoCH-equivalence) as
follows.
###### Definition 5.2
(RoCH-equivalence) Suppose that we have two regular orbit reducible RCH
systems $(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i})$, $i=1,2$, we say
them to be RoCH-equivalent, or simply,
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RoCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2})$, if there
exists a diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$ such that the
following Hamiltonian matching conditions hold:
RoHM-1: The cotangent lift map $\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow
T^{\ast}Q_{1}$ is symplectic.
RoHM-2: For $\mathcal{O}_{\mu_{i}},\;\mu_{i}\in\mathfrak{g}^{\ast}_{i}$, the
regular reducible orbits of RCH systems
$(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i})$, $i=1,2$, the map
$\varphi^{\ast}_{\mathcal{O}_{\mu}}=i_{\mathcal{O}_{\mu_{1}}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mathcal{O}_{\mu_{2}}}:\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})\rightarrow\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})$
is $(G_{2},G_{1})$-equivariant,
$W_{1}=\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})$, and
$\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+},$
where $\mu=(\mu_{1},\mu_{2})$, and denote by
$i_{\mathcal{O}_{\mu_{1}}}^{-1}(S)$ the preimage of a subset $S\subset
T^{\ast}Q_{1}$ for the map
$i_{\mathcal{O}_{\mu_{1}}}:\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})\rightarrow
T^{\ast}Q_{1}$.
RoHM-3:
$Im[(\mathbf{d}H_{1})^{\sharp}+\textnormal{vlift}(F_{1})-((\varphi_{\ast})^{\ast}\mathbf{d}H_{2})^{\sharp}-\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})]\subset\textnormal{vlift}(W_{1}).$
It is worthy of note that for the regular orbit reducible RCH system, the
induced equivalent map $\varphi^{*}$ not only keeps the symplectic structure
and the restriction of the $(+)$-symplectic structure on the regular orbit to
$\mathbf{J}^{-1}(\mathcal{O}_{\mu})$, but also keeps the equivariance of
$G$-action on the regular orbit. If a feedback control law
$u_{\mathcal{O}_{\mu}}:(T^{\ast}Q)_{\mathcal{O}_{\mu}}\rightarrow
W_{\mathcal{O}_{\mu}}$ is chosen, the $R_{O}$-reduced RCH system
$((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}},h_{\mathcal{O}_{\mu}},f_{\mathcal{O}_{\mu}},u_{\mathcal{O}_{\mu}})$
is a closed-loop regular dynamic system with a control law
$u_{\mathcal{O}_{\mu}}$. Assume that its vector field
$X_{((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}},h_{\mathcal{O}_{\mu}},f_{\mathcal{O}_{\mu}},u_{\mathcal{O}_{\mu}})}$
can be expressed by
$X_{((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}},h_{\mathcal{O}_{\mu}},f_{\mathcal{O}_{\mu}},u_{\mathcal{O}_{\mu}})}=(\mathbf{d}h_{\mathcal{O}_{\mu}})^{\sharp}+\textnormal{vlift}(f_{\mathcal{O}_{\mu}})+\textnormal{vlift}(u_{\mathcal{O}_{\mu}}),$
(11)
and satisfies the condition
$X_{((T^{\ast}Q)_{\mathcal{O}_{\mu}},\omega_{\mathcal{O}_{\mu}},h_{\mathcal{O}_{\mu}},f_{\mathcal{O}_{\mu}},u_{\mathcal{O}_{\mu}})}\cdot\pi_{\mathcal{O}_{\mu}}=T\pi_{\mathcal{O}_{\mu}}\cdot
X_{(T^{\ast}Q,G,\omega,H,F,u)}\cdot i_{\mathcal{O}_{\mu}}.$ (12)
Then we can obtain the following regular orbit reduction theorem for RCH
system, which explains the relationship between the RoCH-equivalence for the
regular orbit reducible RCH systems with symmetry and the RCH-equivalence for
associated $R_{O}$-reduced RCH systems. This theorem can be regarded as an
extension of regular orbit reduction theorem of Hamiltonian systems under
regular controlled Hamiltonian equivalence conditions.
###### Theorem 5.3
If two regular orbit reducible RCH systems
$(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},W_{i})$, $i=1,2,$ are RoCH-
equivalent, then their associated $R_{O}$-reduced RCH systems
$((T^{\ast}Q)_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},f_{i\mathcal{O}_{\mu_{i}}},W_{i\mathcal{O}_{\mu_{i}}})$,
$i=1,2,$ must be RCH-equivalent. Conversely, if $R_{O}$-reduced RCH systems
$((T^{\ast}Q)_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},\\\
f_{i\mathcal{O}_{\mu_{i}}},W_{i\mathcal{O}_{\mu_{i}}})$, $i=1,2,$ are RCH-
equivalent and the induced map
$\varphi^{\ast}_{\mathcal{O}_{\mu}}:\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})\rightarrow\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})$,
such that
$\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+},$
then the regular orbit reducible RCH systems
$(T^{\ast}Q_{i},G_{i},\omega_{i},\\\ H_{i},F_{i},W_{i})$, $i=1,2,$ are RoCH-
equivalent.
Proof: If
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RoCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2})$, then there
exists a diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$, such that
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is symplectic and for
$\mu_{i}\in\mathfrak{g}_{i}^{\ast},i=1,2$,
$\varphi_{\mathcal{O}_{\mu}}^{\ast}=i_{\mathcal{O}_{\mu_{1}}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mathcal{O}_{\mu_{2}}}:\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})\rightarrow\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})$
is $(G_{2},G_{1})$-equivariant,
$W_{1}=\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})$,
$\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}$,
and RoHM-3 holds. From the following commutative Diagram-6,
$\begin{CD}T^{\ast}Q_{2}@<{i_{\mathcal{O}_{\mu_{2}}}}<{}<\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})@>{\pi_{\mathcal{O}_{\mu_{2}}}}>{}>(T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}}\\\
@V{\varphi^{\ast}}V{}V@V{\varphi^{\ast}_{\mathcal{O}_{\mu}}}V{}V@V{\varphi^{\ast}_{\mathcal{O}_{\mu/G}}}V{}V\\\
T^{\ast}Q_{1}@<{i_{\mathcal{O}_{\mu_{1}}}}<{}<\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})@>{\pi_{\mathcal{O}_{\mu_{1}}}}>{}>(T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}}\end{CD}$
Diagram-6
we can define a map
$\varphi_{\mathcal{O}_{\mu}/G}^{\ast}:(T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}}\rightarrow(T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}}$,
such that
$\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}}=\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}$.
Because
$\varphi_{\mathcal{O}_{\mu}}^{\ast}:\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})\rightarrow\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})$
is $(G_{2},G_{1})$-equivariant, $\varphi_{\mathcal{O}_{\mu}/G}^{\ast}$ is
well-defined. We can prove that $\varphi_{\mathcal{O}_{\mu}/G}^{\ast}$ is
symplectic, that is,
$(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=\omega_{2\mathcal{O}_{\mu_{2}}}$
and
$W_{1\mathcal{O}_{\mu_{1}}}=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}(W_{2\mathcal{O}_{\mu_{2}}})$.
In fact, since $\varphi^{\ast}:T^{\ast}Q_{1}\to T^{\ast}Q_{2}$ is symplectic,
the map
$(\varphi^{\ast})^{\ast}:\Omega^{2}(T^{\ast}Q_{1})\rightarrow\Omega^{2}(T^{\ast}Q_{2})$
satisfies $(\varphi^{\ast})^{\ast}\omega_{1}=\omega_{2}$. By (8),
$i_{\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i}=\pi_{\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i\mathcal{O}_{\mu_{i}}}+\mathbf{J}_{i\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i\mathcal{O}_{\mu_{i}}}^{+}$,
$i=1,2,$ and
$\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}$,
from the following commutative Diagram-7,
$\begin{CD}\Omega^{2}(T^{\ast}Q_{1})@>{i_{\mathcal{O}_{\mu_{1}}}^{\ast}}>{}>\Omega^{2}(\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}}))@<{\pi_{\mathcal{O}_{\mu_{1}}}^{\ast}}<{}<\Omega^{2}((T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}})\\\
@V{(\varphi^{\ast})^{\ast}}V{}V@V{(\varphi^{\ast}_{\mathcal{O}_{\mu}})^{\ast}}V{}V@V{(\varphi^{\ast}_{\mathcal{O}_{\mu}/G})^{\ast}}V{}V\\\
\Omega^{2}(T^{\ast}Q_{2})@>{i_{\mathcal{O}_{\mu_{2}}}^{\ast}}>{}>\Omega^{2}(\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}}))@<{\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}}<{}<\Omega^{2}((T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}})\end{CD}$
Diagram-7
we have that
$\displaystyle\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}\cdot(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=(\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=(\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}$
$\displaystyle=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=(i_{\mathcal{O}_{\mu_{1}}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mathcal{O}_{\mu_{2}}})^{\ast}\cdot
i_{\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1}-(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}$
$\displaystyle=i_{\mathcal{O}_{\mu_{2}}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\omega_{1}-\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=i_{\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2}-\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}.$
Because $\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}$ is surjective, thus
$(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=\omega_{2\mathcal{O}_{\mu_{2}}}$.
Notice that $W_{i}\subset\mathbf{J}_{i}^{-1}(\mathcal{O}_{\mu_{i}})$,
$W_{i\mathcal{O}_{\mu_{i}}}=\pi_{\mathcal{O}_{\mu_{i}}}(W_{i})$, $i=1,2,$ and
$W_{1}=\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})$, we have that
$W_{1\mathcal{O}_{\mu_{1}}}=\pi_{\mathcal{O}_{\mu_{1}}}(W_{1})=\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}}(W_{2})=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}(W_{2\mathcal{O}_{\mu_{2}}}).$
Next, from (10) and (11), we know that for $i=1,2,$
$X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}=(\mathbf{d}H_{i})^{\sharp}+\textnormal{vlift}(F_{i})+\textnormal{vlift}(u_{i}),$
$X_{((T^{\ast}Q_{i})_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},f_{i\mathcal{O}_{\mu_{i}}},u_{i\mathcal{O}_{\mu_{i}}})}=(\mathbf{d}h_{i\mathcal{O}_{\mu_{i}}})^{\sharp}+\textnormal{vlift}(f_{i\mathcal{O}_{\mu_{i}}})+\textnormal{vlift}(u_{i\mathcal{O}_{\mu_{i}}}),$
and from (12), we have that
$X_{((T^{\ast}Q_{i})_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},f_{i\mathcal{O}_{\mu_{i}}},u_{i\mathcal{O}_{\mu_{i}}})}\cdot\pi_{\mathcal{O}_{\mu_{i}}}=T\pi_{\mathcal{O}_{\mu_{i}}}\cdot
X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}\cdot
i_{\mathcal{O}_{\mu_{i}}}.$
Since $H_{i}$, $F_{i}$ and $W_{i}$ are all $G_{i}$-invariant, $i=1,2,$ and
$\displaystyle h_{i\mathcal{O}_{\mu_{i}}}\cdot\pi_{\mathcal{O}_{\mu_{i}}}$
$\displaystyle=H_{i}\cdot
i_{\mathcal{O}_{\mu_{i}}},\;\;\;f_{i\mathcal{O}_{\mu_{i}}}\cdot\pi_{\mathcal{O}_{\mu_{i}}}=\pi_{\mathcal{O}_{\mu_{i}}}\cdot
F_{i}\cdot i_{\mathcal{O}_{\mu_{i}}},$ $\displaystyle
u_{i\mathcal{O}_{\mu_{i}}}\cdot\pi_{\mathcal{O}_{\mu_{i}}}$
$\displaystyle=\pi_{\mathcal{O}_{\mu_{i}}}\cdot u_{i}\cdot
i_{\mathcal{O}_{\mu_{i}}},\qquad i=1,2.$
From the following commutative Diagram-8,
$\begin{CD}T^{\ast}T^{\ast}Q_{2}@>{i_{\mathcal{O}_{\mu_{2}}}^{\ast}}>{}>T^{\ast}\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})@<{\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}}<{}<T^{\ast}((T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}})\\\
@V{(\varphi^{\ast})_{\ast}}V{}V@V{(\varphi^{\ast}_{\mathcal{O}_{\mu}})_{\ast}}V{}V@V{(\varphi^{\ast}_{\mathcal{O}_{\mu}/G})_{\ast}}V{}V\\\
T^{\ast}T^{\ast}Q_{1}@>{i_{\mathcal{O}_{\mu_{1}}}^{\ast}}>{}>T^{\ast}\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})@<{\pi_{\mathcal{O}_{\mu_{1}}}^{\ast}}<{}<T^{\ast}((T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}})\end{CD}$
Diagram-8
we have that
$\pi_{\mathcal{O}_{\mu_{1}}}^{\ast}\cdot(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})_{\ast}\mathbf{d}h_{2\mathcal{O}_{\mu_{2}}}=i_{\mathcal{O}_{\mu_{1}}}^{\ast}\cdot(\varphi^{\ast})_{\ast}\mathbf{d}H_{2}$,
then
$((\varphi_{\mathcal{O}_{\mu}/G}^{\ast})_{\ast}\mathbf{d}h_{2\mathcal{O}_{\mu_{2}}})^{\sharp}\cdot\pi_{\mathcal{O}_{\mu_{1}}}=T\pi_{\mathcal{O}_{\mu_{1}}}\cdot((\varphi^{\ast})_{\ast}\mathbf{d}H_{2})^{\sharp}\cdot
i_{\mathcal{O}_{\mu_{1}}},$
$\textnormal{vlift}(\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot
f_{2\mathcal{O}_{\mu_{2}}}\cdot\varphi_{\mathcal{O}_{\mu}/G\ast})\cdot\pi_{\mathcal{O}_{\mu_{1}}}=T\pi_{\mathcal{O}_{\mu_{1}}}\cdot\textnormal{vlift}(\varphi^{\ast}F_{2}\varphi_{\ast})\cdot
i_{\mathcal{O}_{\mu_{1}}},$
$\textnormal{vlift}(\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot
u_{2\mathcal{O}_{\mu_{2}}}\cdot\varphi_{\mathcal{O}_{\mu}/G\ast})\cdot\pi_{\mathcal{O}_{\mu_{1}}}=T\pi_{\mathcal{O}_{\mu_{1}}}\cdot\textnormal{vlift}(\varphi^{\ast}u_{2}\varphi_{\ast})\cdot
i_{\mathcal{O}_{\mu_{1}}},$
where the map
$\varphi_{\mathcal{O}_{\mu}/G\ast}=(\varphi^{-1})_{\mathcal{O}_{\mu}/G}^{\ast}:(T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}}\rightarrow(T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}}$
and
$(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})_{\ast}=(\varphi_{\mathcal{O}_{\mu}/G\ast})^{\ast}:T^{\ast}((T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}})\rightarrow
T^{\ast}((T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}})$. From the Hamiltonian
matching condition RoHM-3 we have that
$Im[(\mathbf{d}h_{1\mathcal{O}_{\mu_{1}}})^{\sharp}+\textnormal{vlift}(f_{1\mathcal{O}_{\mu_{1}}})-((\varphi_{\mathcal{O}_{\mu}/G}^{\ast})_{\ast}\mathbf{d}h_{2\mathcal{O}_{\mu_{2}}})^{\sharp}-\textnormal{vlift}(\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot
f_{2\mathcal{O}_{\mu_{2}}}\cdot\varphi_{\mathcal{O}_{\mu}/G\ast})]$
$\subset\textnormal{vlift}(W_{1\mathcal{O}_{\mu_{1}}}).$ (13)
So,
$\displaystyle((T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}},\omega_{1\mathcal{O}_{\mu_{1}}},h_{1\mathcal{O}_{\mu_{1}}},f_{1\mathcal{O}_{\mu_{1}}},W_{1\mathcal{O}_{\mu_{1}}})$
$\displaystyle\hskip 140.84256pt\stackrel{{\scriptstyle
RCH}}{{\sim}}((T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}},\omega_{2\mathcal{O}_{\mu_{2}}},h_{2\mathcal{O}_{\mu_{2}}},f_{2\mathcal{O}_{\mu_{2}}},W_{2\mathcal{O}_{\mu_{2}}}).$
Conversely, assume that $R_{O}$-reduced RCH systems
$((T^{\ast}Q_{i})_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},f_{i\mathcal{O}_{\mu_{i}}},W_{i\mathcal{O}_{\mu_{i}}})$,
$i=1,2,$ are RCH-equivalent, then there exists a diffeomorphism
$\varphi_{\mathcal{O}_{\mu}/G}^{\ast}:(T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}}\rightarrow(T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}}$,
which is symplectic,
$W_{1\mathcal{O}_{\mu_{1}}}=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}(W_{2\mathcal{O}_{\mu}})$
and (13) hold. Thus, we can define a map
$\varphi_{\mathcal{O}_{\mu}}^{\ast}:\mathbf{J}_{2}^{-1}(\mathcal{O}_{\mu_{2}})\rightarrow\mathbf{J}_{1}^{-1}(\mathcal{O}_{\mu_{1}})$
such that
$\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}};$
and map $\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ such that
$i_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}=\varphi^{\ast}\cdot
i_{\mathcal{O}_{\mu_{2}}};$ see the commutative Diagram-6, as well as a
diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$, whose cotangent lift is just
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$. At first, from
definition of $\varphi_{\mathcal{O}_{\mu}}^{\ast}$ we know that
$\varphi_{\mathcal{O}_{\mu}}^{\ast}$ is $(G_{2},G_{1})$-equivariant. In fact,
for any $z_{i}\in\mathbf{J}_{i}^{-1}(\mathcal{O}_{\mu_{i}})$, $g_{i}\in
G_{i}$, $i=1,2$ such that $z_{1}=\varphi_{\mathcal{O}_{\mu}}^{\ast}(z_{2})$,
$[z_{1}]=\varphi^{\ast}_{\mathcal{O}_{\mu}/G}[z_{2}]$, then we have that
$\displaystyle\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}(\Phi_{2g_{2}}(z_{2}))=\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}(g_{2}z_{2})=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}}(g_{2}z_{2})=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}[z_{2}]$
$\displaystyle=[z_{1}]=\pi_{\mathcal{O}_{\mu_{1}}}(g_{1}z_{1})=\pi_{\mathcal{O}_{\mu_{1}}}(\Phi_{1g_{1}}(z_{1}))=\pi_{\mathcal{O}_{\mu_{1}}}\cdot\Phi_{1g_{1}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}(z_{2}).$
Since $\pi_{\mathcal{O}_{\mu_{1}}}$ is surjective, so,
$\varphi_{\mathcal{O}_{\mu}}^{\ast}\cdot\Phi_{2g_{2}}=\Phi_{1g_{1}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}$.
Moreover,
$\pi_{\mathcal{O}_{\mu_{1}}}(W_{1})=W_{1\mathcal{O}_{\mu_{1}}}=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}(W_{2\mathcal{O}_{\mu_{2}}})=\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}}(W_{2})=\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})$.
Since $W_{i}\subset\mathbf{J}_{i}^{-1}(\mathcal{O}_{\mu_{i}}),\;i=1,2,$ and
$\pi_{\mathcal{O}_{\mu_{1}}}$ is surjective, then
$W_{1}=\varphi_{\mathcal{O}_{\mu}}^{\ast}(W_{2})$. Now we shall show that
$\varphi^{\ast}$ is symplectic, that is,
$\omega_{2}=(\varphi^{\ast})^{\ast}\omega_{1}$. In fact, since
$\varphi_{\mathcal{O}_{\mu}/G}^{\ast}:(T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}}\rightarrow(T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}}$
is symplectic, the map
$(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}:\Omega^{2}((T^{\ast}Q_{1})_{\mathcal{O}_{\mu_{1}}})\rightarrow\Omega^{2}((T^{\ast}Q_{2})_{\mathcal{O}_{\mu_{2}}})$
satisfies
$(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}=\omega_{2\mathcal{O}_{\mu_{2}}}$.
By (8),
$i_{\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i}=\pi_{\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i\mathcal{O}_{\mu_{i}}}+\mathbf{J}_{i\mathcal{O}_{\mu_{i}}}^{\ast}\omega_{i\mathcal{O}_{\mu_{i}}}^{+}$,
$i=1,2$, from the commutative Diagram-7, we have that
$\displaystyle
i_{\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2}=\pi_{\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=\pi_{2\mathcal{O}_{\mu_{2}}}^{\ast}\cdot(\varphi_{\mathcal{O}_{\mu}/G}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}$
$\displaystyle=(\varphi_{\mathcal{O}_{\mu}/G}^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{2}}})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\pi_{\mathcal{O}_{\mu_{1}}}\cdot\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}$
$\displaystyle=(i_{\mathcal{O}_{\mu_{1}}}^{-1}\cdot\varphi^{\ast}\cdot
i_{\mathcal{O}_{\mu_{2}}})^{\ast}\cdot\pi_{\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}$
$\displaystyle=i_{\mathcal{O}_{\mu_{2}}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\cdot(i_{\mathcal{O}_{\mu_{1}}}^{-1})^{\ast}\cdot[i_{\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1}-\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}]+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}$
$\displaystyle=i_{\mathcal{O}_{\mu_{2}}}^{\ast}\cdot(\varphi^{\ast})^{\ast}\omega_{1}-(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}+\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}$
Notice that $i_{\mathcal{O}_{\mu_{2}}}^{\ast}$ is injective, and by our
hypothesis,
$\mathbf{J}_{2\mathcal{O}_{\mu_{2}}}^{\ast}\omega_{2\mathcal{O}_{\mu_{2}}}^{+}=(\varphi_{\mathcal{O}_{\mu}}^{\ast})^{\ast}\cdot\mathbf{J}_{1\mathcal{O}_{\mu_{1}}}^{\ast}\omega_{1\mathcal{O}_{\mu_{1}}}^{+}$,
then $\omega_{2}=(\varphi^{\ast})^{\ast}\omega_{1}$, that is, $\varphi^{\ast}$
is symplectic. Since the vector fields
$X_{(T^{\ast}Q_{i},G_{i},\omega_{i},H_{i},F_{i},u_{i})}$ and
$X_{((T^{\ast}Q_{i})_{\mathcal{O}_{\mu_{i}}},\omega_{i\mathcal{O}_{\mu_{i}}},h_{i\mathcal{O}_{\mu_{i}}},f_{i\mathcal{O}_{\mu_{i}}},u_{i\mathcal{O}_{\mu_{i}}})}$
is $\pi_{\mathcal{O}_{\mu_{i}}}$-related, $i=1,2,$ and $H_{i},F_{i}$ and
$W_{i}$ are all $G_{i}$-invariant, $i=1,2$, in the same way, from (13) we have
that Hamiltonian matching condition RoHM-3 holds. Thus,
$(T^{\ast}Q_{1},G_{1},\omega_{1},H_{1},F_{1},W_{1})\stackrel{{\scriptstyle
RoCH}}{{\sim}}(T^{\ast}Q_{2},G_{2},\omega_{2},H_{2},F_{2},W_{2}).\hskip
28.45274pt\blacksquare$
## 6 Applications
As the applications of regular point reduction theory of RCH system with
symmetry, in this section, we first study the regular point reducible RCH
system on a Lie group, and its $R_{P}$-reduced RCH system is a RCH system on a
coadjoint orbit
$\mathcal{O}_{\mu}\subset\mathfrak{g}^{\ast},\;\mu\in\mathfrak{g}^{\ast}$,
where $\mathfrak{g}$ is a Lie algebra of $G$ and $\mathfrak{g}^{\ast}$ is the
dual of $\mathfrak{g}$. Next, we regard the rigid body and heavy top as well
as them with internal rotors (or the external force torques) as the regular
point reducible RCH systems on the rotation group $\textmd{SO}(3)$ and on the
Euclidean group $\textmd{SE}(3)$, respectively, and give their $R_{P}$-reduced
RCH systems and discuss their RCH-equivalence. Moreover, in order to
understand well the abstract definition of RCH system and the significance of
Theorem 3.3, we describe the RCH system from the viewpoint of port Hamiltonian
system with a symplectic structure, and state the relationship between RCH-
equivalence and equivalence of port Hamiltonian system.
### 6.1 Regular Point Reducible RCH System on a Lie Group
Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $T^{\ast}G$ its
cotangent bundle with the canonical symplectic form $\omega_{0}$. A RCH system
on $G$ is a 5-tuple $(T^{\ast}G,\omega_{0},H,F,W)$, where
$(T^{\ast}G,\omega_{0},H)$ is a Hamiltonian system and
$H:T^{\ast}G\rightarrow\mathbb{R}$ is a Hamiltonian, the fiber-preserving map
$F:T^{\ast}G\rightarrow T^{\ast}G$ is a (external) force map and the fiber
submanifold $W$ of $T^{\ast}G$ is a control subset.
At first, for the Lie group $G$, the left and right translation on $G$,
defined by the map $L_{g}:G\rightarrow G,\;h\mapsto gh$ and
$R_{g}:G\rightarrow G,\;h\mapsto hg$, for someone $g\in G$, induce the left
and right action of $G$ on itself. Let $I_{g}:G\to G$;
$I_{g}(h)=ghg^{-1}=L_{g}\cdot R_{g^{-1}}(h)$, for $g,h\in G$, be the inner
automorphism on $G$. The adjoint representation of a Lie group $G$ is defined
by $\operatorname{Ad}_{g}=T_{e}I_{g}=T_{g^{-1}}L_{g}\cdot
T_{e}R_{g^{-1}}:\mathfrak{g}\to\mathfrak{g}$. The coadjoint representation is
given by
$\operatorname{Ad}_{g^{-1}}^{\ast}:\mathfrak{g}^{\ast}\to\mathfrak{g}^{\ast}$,
where $\operatorname{Ad}_{g^{-1}}^{\ast}$ is the dual of the linear map
$\operatorname{Ad}_{g^{-1}}$, defined by
$\langle\operatorname{Ad}_{g^{-1}}^{\ast}(\mu),\xi\rangle=\langle\mu,\operatorname{Ad}_{g^{-1}}(\xi)\rangle$,
where $\mu\in\mathfrak{g}^{\ast}$, $\xi\in\mathfrak{g}$ and $\langle,\rangle$
denotes the pairing between $\mathfrak{g}^{\ast}$ and $\mathfrak{g}$. Since
the coadjoint representation
$\operatorname{Ad}_{g^{-1}}^{\ast}:\mathfrak{g}^{\ast}\to\mathfrak{g}^{\ast}$
can induce a left coadjoint action of $G$ on $\mathfrak{g}^{\ast}$, the
coadjoint orbit $\mathcal{O}_{\mu}$ of this action through
$\mu\in\mathfrak{g}^{\ast}$ is the subset of $\mathfrak{g}^{\ast}$ defined by
$\mathcal{O}_{\mu}:=\\{\operatorname{Ad}_{g^{-1}}^{\ast}(\mu)\in\mathfrak{g}^{\ast}|g\in
G\\}$, and $\mathcal{O}_{\mu}$ is an immersed submanifold of
$\mathfrak{g}^{\ast}$. We know that $\mathfrak{g}^{\ast}$ is a Poisson
manifold with respect to the $(\pm)$-Lie-Poisson bracket
$\\{\cdot,\cdot\\}_{\pm}$ defined by
$\\{f,g\\}_{\pm}(\mu):=\pm<\mu,[\frac{\delta f}{\delta\mu},\frac{\delta
g}{\delta\mu}]>,\;\;\forall f,g\in
C^{\infty}(\mathfrak{g}^{\ast}),\;\;\mu\in\mathfrak{g}^{\ast},$ (14)
where the element $\frac{\delta f}{\delta\mu}\in\mathfrak{g}$ is defined by
the equality $<v,\frac{\delta f}{\delta\mu}>:=Df(\mu)\cdot v$, for any
$v\in\mathfrak{g}^{\ast}$, see Marsden and Ratiu [22]. Thus, for the coadjoint
orbit $\mathcal{O}_{\mu},\;\mu\in\mathfrak{g}^{\ast}$, the orbit symplectic
structure can be defined by
$\omega_{\mathcal{O}_{\mu}}^{\pm}(\nu)(\operatorname{ad}_{\xi}^{\ast}(\nu),\operatorname{ad}_{\eta}^{\ast}(\nu))=\pm\langle\nu,[\xi,\eta]\rangle,\qquad\forall\;\xi,\eta\in\mathfrak{g},\;\;\nu\in\mathcal{O}_{\mu}\subset\mathfrak{g}^{\ast},$
(15)
which are coincide with the restriction of the Lie-Poisson brackets on
$\mathfrak{g}^{\ast}$ to the coadjoint orbit $\mathcal{O}_{\mu}$. From the
Symplectic Stratification theorem we know that a finite dimensional Poisson
manifold is the disjoint union of its symplectic leaves, and its each
symplectic leaf is an injectively immersed Poisson submanifold whose induced
Poisson structure is symplectic. When $\mathfrak{g}^{\ast}$ is endowed one of
the Lie Poisson structures $\\{\cdot,\cdot\\}_{\pm}$, the symplectic leaves of
the Poisson manifolds $(\mathfrak{g}^{\ast},\\{\cdot,\cdot\\}_{\pm})$ coincide
with the connected components of the orbits of the elements in
$\mathfrak{g}^{\ast}$ under the coadjoint action. From Abraham and Marsden
[1], we know that
###### Proposition 6.1
The coadjoint orbit
$(\mathcal{O}_{\mu},\omega_{\mathcal{O}_{\mu}}^{-}),\;\mu\in\mathfrak{g}^{\ast},$
is symplectic diffeomorphic to a regular point reduced space
$((T^{\ast}G)_{\mu},\omega_{\mu})$ of $T^{*}G$.
We now identify $T^{\ast}G$ and $G\times\mathfrak{g}^{\ast}$ by using the left
translation. In fact, the map $\lambda:T^{\ast}G\rightarrow
G\times\mathfrak{g}^{\ast},\;\lambda(\alpha_{g}):=(g,(T_{e}L_{g})^{\ast}\alpha_{g})$,
for any $\alpha_{g}\in T^{\ast}_{g}G$, which defines a vector bundle
isomorphism usually referred to as the left trivialization of $T^{\ast}G$. In
the same way, we can also identify tangent bundle $TG$ and
$G\times\mathfrak{g}$ by using the left translation. In consequence, we can
consider the Lagrangian $L(g,\xi):TG\cong G\times\mathfrak{g}\to\mathbb{R}$,
which is usual the kinetic minus potential energy of the system, where
$(g,\xi)\in G\times\mathfrak{g}$, and $\xi\in\mathfrak{g}$, regarded as the
velocity of system. If we introduce the conjugate momentum
$p_{i}=\frac{\partial L}{\partial\xi^{i}}$, $i=1,\cdots,n,\;n=dimG$, and by
the Legendre transformation $FL:TG\cong G\times\mathfrak{g}\to T^{\ast}G\cong
G\times\mathfrak{g}^{\ast}$, $(g^{i},\xi^{i})\to(g^{i},p_{i})$, we have the
Hamiltonian $H(g,p):T^{\ast}G\cong G\times\mathfrak{g}^{\ast}\to\mathbb{R}$
given by
$H(g^{i},p_{i})=\sum_{i=1}^{n}p_{i}\xi^{i}-L(g^{i},\xi^{i}).$ (16)
If the Hamiltonian $H(g,p):T^{\ast}G\cong G\times\mathfrak{g}\to\mathbb{R}$ is
left cotangent lifted $G$-action invariant, for $\mu\in\mathfrak{g}^{\ast}$ we
have the associated reduced Hamiltonian
$h_{\mu}:(T^{\ast}G)_{\mu}\cong\mathcal{O}_{\mu}\to\mathbb{R}$, defined by
$h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$. By the $(\pm)$-Lie-Poisson brackets on
$\mathfrak{g}^{\ast}$ and the symplecitic structure on the coadjoint orbit
$\mathcal{O}_{\mu}$, we have the associated Hamiltonian vector field
$X_{h_{\mu}}$ given by
$X_{h_{\mu}}(\nu)=\mp\operatorname{ad}^{\ast}_{\delta
h_{\mu}/\delta\nu}\nu,\quad\forall\nu\in\mathcal{O}_{\mu}.$ (17)
See Marsden and Ratiu [22]. Thus, if the Hamiltonian
$H:T^{\ast}G\to\mathbb{R}$, the fiber-preserving map $F:T^{\ast}G\to
T^{\ast}G$ and the fiber submanifold $W$ of $T^{\ast}G$ are all left cotangent
lifted $G$-action invariant, we may define the RCH system with symmetry on
$G$, and give its $R_{P}$-reduced RCH system as follows.
###### Theorem 6.2
The 6-tuple $(T^{\ast}G,G,\omega_{0},H,F,W)$ is a regular point reducible RCH
system on Lie group $G$, where the Hamiltonian $H:T^{\ast}G\to\mathbb{R}$, the
fiber-preserving map $F:T^{\ast}G\to T^{\ast}G$ and the fiber submanifold $W$
of $T^{\ast}G$ are all left cotangent lifted $G$-action invariant. For a point
$\mu\in\mathfrak{g}^{\ast}$, the regular value of the momentum map
$\mathbf{J}_{L}:T^{\ast}G\rightarrow\mathfrak{g}^{\ast}$, the $R_{P}$-reduced
system, that is, the 5-tuple
$(\mathcal{O}_{\mu},\omega_{\mathcal{O}_{\mu}}^{-},h_{\mu},f_{\mu},W_{\mu})$,
is a RCH system, where $\mathcal{O}_{\mu}\subset\mathfrak{g}^{\ast}$ is the
coadjoint orbit, $\omega_{\mathcal{O}_{\mu}}^{-}$ is orbit symplectic form,
$h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$, $f_{\mu}\cdot\pi_{\mu}=\pi_{\mu}\cdot
F\cdot i_{\mu}$, $W\subset\mathbf{J}_{L}^{-1}(\mu)$, and
$W_{\mu}=\pi_{\mu}(W)\subset\mathcal{O}_{\mu}$. Moreover, two regular point
reducible RCH system $(T^{\ast}G_{i},G_{i},\omega_{i0},H_{i},F_{i},W_{i}),$
$i=1,2,$ are RpCH-equivalent if and only if the associated $R_{P}$-reduced RCH
systems
$(\mathcal{O}_{i\mu_{i}},\omega_{\mathcal{O}_{i\mu_{i}}}^{-},h_{i\mu_{i}},f_{i\mu_{i}},W_{i\mu_{i}}),\;i=1,2,$
are RCH-equivalent.
Next, in order to study the regular reduction of rigid body and heavy top with
internal rotors, we need the regular symplectic reduction theory of the
cotangent bundle $T^{\ast}Q$, where the configuration space $Q=G\times V$, and
$G$ is a Lie group and $V$ is a $k$-dimensional vector space. Defined the left
$G$-action $\Phi:G\times Q\rightarrow Q,\;\Phi(g,(h,\theta)):=(gh,\theta)$,
for any $g,h\in G,\;\theta\in V$, that is , the $G$-action on $Q$ is the left
translation on the first factor $G$, and $G$ acts trivially on the second
factor $V$. Because $T^{\ast}Q=T^{\ast}G\times T^{\ast}V$, and
$T^{\ast}V=V\times V^{\ast}$, by using the left trivialization of $T^{\ast}G$,
we have that $T^{\ast}Q=G\times\mathfrak{g}^{\ast}\times V\times V^{\ast}$. If
the left $G$-action $\Phi:G\times Q\rightarrow Q$ is free and proper, then the
cotangent lift of the action to its cotangent bundle $T^{\ast}Q$, given by
$\Phi^{T^{*}}:G\times T^{*}Q\rightarrow
T^{*}Q,\;\Phi^{T^{*}}(g,(h,\mu,\theta,\lambda)):=(gh,\mu,\theta,\lambda)$, for
any $g,h\in G,\;\mu\in\mathfrak{g}^{\ast},\;\theta\in V,\;\lambda\in
V^{\ast}$, is also a free and proper action, and the orbit space
$(T^{\ast}Q)/G$ is a smooth manifold and $\pi:T^{*}Q\rightarrow(T^{*}Q)/G$ is
a smooth submersion. Since $G$ acts trivially on $\mathfrak{g}^{\ast}$, $V$
and $V^{\ast}$, it follows that $(T^{\ast}Q)/G$ is diffeomorphic to
$\mathfrak{g}^{\ast}\times V\times V^{\ast}$.
For $\mu\in\mathfrak{g}^{\ast}$, the coadjoint orbit
$\mathcal{O}_{\mu}\subset\mathfrak{g}^{\ast}$ has the orbit symplectic forms
$\omega^{\pm}_{\mathcal{O}_{\mu}}$. Let $\omega_{V}$ be the canonical
symplectic form on $T^{\ast}V\cong V\times V^{\ast}$ given by
$\omega_{V}((\theta_{1},\lambda_{1}),(\theta_{2},\lambda_{2}))=<\lambda_{2},\theta_{1}>-<\lambda_{1},\theta_{2}>,$
where $(\theta_{i},\lambda_{i})\in V\times V^{\ast},\;i=1,2$, $<\cdot,\cdot>$
is the natural pairing between $V^{\ast}$ and $V$. Thus, we can induce a
symplectic forms $\tilde{\omega}^{\pm}_{\mathcal{O}_{\mu}\times V\times
V^{\ast}}=\pi_{\mathcal{O}_{\mu}}^{\ast}\omega^{\pm}_{\mathcal{O}_{\mu}}+\pi_{V}^{\ast}\omega_{V}$
on the smooth manifold $\mathcal{O}_{\mu}\times V\times V^{\ast}$, where the
maps $\pi_{\mathcal{O}_{\mu}}:\mathcal{O}_{\mu}\times V\times
V^{\ast}\to\mathcal{O}_{\mu}$ and $\pi_{V}:\mathcal{O}_{\mu}\times V\times
V^{\ast}\to V\times V^{\ast}$ are canonical projections. On the other hand,
from $T^{\ast}Q=T^{\ast}G\times T^{\ast}V$ we know that there is a canonical
symplectic form $\omega_{Q}=\pi^{\ast}_{1}\omega_{0}+\pi^{\ast}_{2}\omega_{V}$
on $T^{\ast}Q$, where $\omega_{0}$ is the canonical symplectic form on
$T^{\ast}G$ and the maps $\pi_{1}:Q=G\times V\to G$ and $\pi_{2}:Q=G\times
V\to V$ are canonical projections. Then the cotangent lift of the left
$G$-action $\Phi^{T^{*}}:G\times T^{\ast}Q\to T^{\ast}Q$ is also symplectic,
and admits an associated $\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}_{Q}:T^{\ast}Q\to\mathfrak{g}^{\ast}$ such that
$\mathbf{J}_{Q}\cdot\pi^{\ast}_{1}=\mathbf{J}_{L}$, where
$\mathbf{J}_{L}:T^{\ast}G\rightarrow\mathfrak{g}^{\ast}$ is a momentum map of
left G-action on $T^{\ast}G$, and $\pi^{\ast}_{1}:T^{\ast}G\to T^{\ast}Q$. If
$\mu\in\mathfrak{g}^{\ast}$ is a regular value of $\mathbf{J}_{Q}$, then
$\mu\in\mathfrak{g}^{\ast}$ is also a regular value of $\mathbf{J}_{L}$ and
$\mathbf{J}_{Q}^{-1}(\mu)\cong\mathbf{J}_{L}^{-1}(\mu)\times V\times
V^{\ast}$. Denote by $G_{\mu}$ the isotropy subgroup of the coadjoint action
of $G$ at the point $\mu\in\mathfrak{g}^{\ast}$, which is defined by
$G_{\mu}=\\{g\in G|\operatorname{Ad}_{g}^{\ast}\mu=\mu\\}$. It follows that
$G_{\mu}$ acts also freely and properly on $\mathbf{J}_{Q}^{-1}(\mu)$, the
regular point reduced space
$(T^{\ast}Q)_{\mu}=\mathbf{J}_{Q}^{-1}(\mu)/G_{\mu}\cong(T^{\ast}G)_{\mu}\times
V\times V^{\ast}$ of $(T^{\ast}Q,\omega_{Q})$ at $\mu$, is a symplectic
manifold with symplectic form $\omega_{\mu}$ uniquely characterized by the
relation
$\pi_{\mu}^{\ast}\omega_{\mu}=i_{\mu}^{\ast}\omega_{Q}=i_{\mu}^{\ast}\pi^{\ast}_{1}\omega_{0}+i_{\mu}^{\ast}\pi^{\ast}_{2}\omega_{V}$,
where the map $i_{\mu}:\mathbf{J}_{Q}^{-1}(\mu)\rightarrow T^{\ast}Q$ is the
inclusion and $\pi_{\mu}:\mathbf{J}_{Q}^{-1}(\mu)\rightarrow(T^{\ast}Q)_{\mu}$
is the projection. Because $((T^{\ast}G)_{\mu},\omega_{\mu})$ is symplectic
diffeomorphic to $(\mathcal{O}_{\mu},\omega_{\mathcal{O}_{\mu}}^{-})$, we have
that $((T^{\ast}Q)_{\mu},\omega_{\mu})$ is symplectic diffeomorphic to
$(\mathcal{O}_{\mu}\times V\times
V^{\ast},\tilde{\omega}_{\mathcal{O}_{\mu}\times V\times V^{\ast}}^{-})$.
We now consider the Lagrangian $L(g,\xi,\theta,\dot{\theta}):TQ\cong
G\times\mathfrak{g}\times TV\to\mathbb{R}$, which is usual the total kinetic
minus potential energy of the system, where $(g,\xi)\in G\times\mathfrak{g}$,
and $\theta\in V$, $\xi^{i}$ and
$\dot{\theta}^{j}=\frac{\mathrm{d}\theta^{j}}{\mathrm{d}t}$,
($i=1,\cdots,n,\;j=1,\cdots,k$, $n=\dim G$, $k=\dim V$), regarded as the
velocity of system. If we introduce the conjugate momentum
$p_{i}=\frac{\partial L}{\partial\xi^{i}},\;l_{j}=\frac{\partial
L}{\partial\dot{\theta}^{j}}$, $i=1,\cdots,n,\;j=1,\cdots,k,$ and by the
Legendre transformation $FL:TQ\cong G\times\mathfrak{g}\times V\times V\to
T^{\ast}Q\cong G\times\mathfrak{g}^{\ast}\times V\times V^{\ast}$,
$(g^{i},\xi^{i},\theta^{j},\dot{\theta}^{j})\to(g^{i},p_{i},\theta^{j},l_{j})$,
we have the Hamiltonian $H(g,p,\theta,l):T^{\ast}Q\cong
G\times\mathfrak{g}^{\ast}\times V\times V^{\ast}\to\mathbb{R}$ given by
$H(g^{i},p_{i},\theta^{j},l_{j})=\sum_{i=1}^{n}p_{i}\xi^{i}+\sum_{j=1}^{k}l_{j}\dot{\theta}^{j}-L(g^{i},\xi^{i},\theta^{j},\dot{\theta}^{j}).$
(18)
If the Hamiltonian $H(g,p,\theta,l):T^{\ast}Q\cong
G\times\mathfrak{g}^{\ast}\times V\times V^{\ast}\to\mathbb{R}$ is left
cotangent lifted $G$-action $\Phi^{T^{*}}$ invariant, for
$\mu\in\mathfrak{g}^{\ast}$ we have the associated reduced Hamiltonian
$h_{\mu}(\nu,\theta,l):(T^{\ast}Q)_{\mu}\cong\mathcal{O}_{\mu}\times V\times
V^{\ast}\to\mathbb{R}$, defined by $h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$.
Note that for $F,K:T^{\ast}V\cong V\times V^{\ast}\to\mathbb{R}$, by using the
canonical symplectic form $\omega_{V}$ on $T^{\ast}V\cong V\times V^{\ast}$,
we can define the Poisson bracket $\\{\cdot,\cdot\\}_{V}$ on $T^{\ast}V$ as
follows
$\\{F,K\\}_{V}(\theta,\lambda)=<\frac{\delta F}{\delta\theta},\frac{\delta
K}{\delta\lambda}>-<\frac{\delta K}{\delta\theta},\frac{\delta
F}{\delta\lambda}>$
If $\theta_{i},\;i=1,\cdots,k,$ is a base of $V$, and
$\lambda_{i},\;i=1,\cdots,k,$ a base of $V^{\ast}$, then we have that
$\\{F,K\\}_{V}(\theta,\lambda)=\sum_{i=1}^{k}(\frac{\partial
F}{\partial\theta_{i}}\frac{\partial K}{\partial\lambda_{i}}-\frac{\partial
K}{\partial\theta_{i}}\frac{\partial F}{\partial\lambda_{i}}).$ (19)
Thus, by the $(\pm)$-Lie-Poisson brackets on $\mathfrak{g}^{\ast}$ and the
Poisson bracket $\\{\cdot,\cdot\\}_{V}$ on $T^{\ast}V$, for
$F,K:\mathfrak{g}^{\ast}\times V\times V^{\ast}\to\mathbb{R}$, we can define
the Poisson bracket on $\mathfrak{g}^{\ast}\times V\times V^{\ast}$ as follows
$\displaystyle\\{F,K\\}_{\pm}(\mu,\theta,\lambda)=\\{F,K\\}_{\pm}(\mu)+\\{F,K\\}_{V}(\theta,\lambda)$
$\displaystyle=\pm<\mu,[\frac{\delta F}{\delta\mu},\frac{\delta
K}{\delta\mu}]>+\sum_{i=1}^{k}(\frac{\partial
F}{\partial\theta_{i}}\frac{\partial K}{\partial\lambda_{i}}-\frac{\partial
K}{\partial\theta_{i}}\frac{\partial F}{\partial\lambda_{i}}).$
See Krishnaprasad and Marsden [17]. In particular, for
$F_{\mu},K_{\mu}:\mathcal{O}_{\mu}\times V\times V^{\ast}\to\mathbb{R}$, we
have that $\tilde{\omega}_{\mathcal{O}_{\mu}\times V\times
V^{\ast}}^{-}(X_{F_{\mu}},X_{K_{\mu}})=\\{F_{\mu},K_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}\times
V\times V^{\ast}}$. Moreover, for reduced Hamiltonian
$h_{\mu}(\nu,\theta,l):\mathcal{O}_{\mu}\times V\times V^{\ast}\to\mathbb{R}$,
we have the Hamiltonian vector field
$X_{h_{\mu}}(K_{\mu})=\\{K_{\mu},h_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}\times
V\times V^{\ast}}.$ Thus, if the Hamiltonian $H:T^{\ast}Q\to\mathbb{R}$, the
fiber-preserving map $F:T^{\ast}Q\to T^{\ast}Q$ and the fiber submanifold $W$
of $T^{\ast}Q$ are all left cotangent lifted $G$-action $\Phi^{T^{*}}$
invariant, then we have the following theorem.
###### Theorem 6.3
The 6-tuple $(T^{\ast}Q,G,\omega_{0},H,F,W)$ is a regular point reducible RCH
system, where $Q=G\times V$, and $G$ is a Lie group and $V$ is a
$k$-dimensional vector space, and the Hamiltonian $H:T^{\ast}Q\to\mathbb{R}$,
the fiber-preserving map $F:T^{\ast}Q\to T^{\ast}Q$ and the fiber submanifold
$W$ of $T^{\ast}Q$ are all left cotangent lifted $G$-action $\Phi^{T^{*}}$
invariant. For a point $\mu\in\mathfrak{g}^{\ast}$, the regular value of the
momentum map $\mathbf{J}_{Q}:T^{\ast}Q\rightarrow\mathfrak{g}^{\ast}$, the
$R_{P}$-reduced system, that is, the 5-tuple $(\mathcal{O}_{\mu}\times V\times
V^{\ast},\tilde{\omega}_{\mathcal{O}_{\mu}\times V\times
V^{\ast}}^{-},h_{\mu},f_{\mu},W_{\mu})$, is a RCH system, where
$\mathcal{O}_{\mu}\subset\mathfrak{g}^{\ast}$ is the coadjoint orbit,
$\tilde{\omega}_{\mathcal{O}_{\mu}\times V\times V^{\ast}}^{-}$ is orbit
symplectic form on $\mathcal{O}_{\mu}\times V\times V^{\ast}$,
$h_{\mu}\cdot\pi_{\mu}=H\cdot i_{\mu}$, $f_{\mu}\cdot\pi_{\mu}=\pi_{\mu}\cdot
F\cdot i_{\mu}$, $W\subset\mathbf{J}_{Q}^{-1}(\mu)$, and
$W_{\mu}=\pi_{\mu}(W)\subset\mathcal{O}_{\mu}\times V\times V^{\ast}$.
Moreover, two regular point reducible RCH system
$(T^{\ast}Q_{i},G_{i},\omega_{i0},H_{i},F_{i},W_{i}),$ $i=1,2,$ are RpCH-
equivalent if and only if the associated $R_{P}$-reduced RCH systems
$(\mathcal{O}_{i\mu_{i}}\times V_{i}\times
V_{i}^{\ast},\tilde{\omega}_{\mathcal{O}_{i\mu_{i}}}^{-},h_{i\mu_{i}},f_{i\mu_{i}},W_{i\mu_{i}}),\;i=1,2,$
are RCH-equivalent.
The third, in order to study the regular reduction of heavy top we need to the
theory of Hamiltonian reduction by stages for semidirect product Lie group.
See Marsden et al [21]. Assume that $S=G\circledS V$ is a semidirect product
Lie group, where $V$ is a vector space and $V^{\ast}$ its dual space, $G$ is a
Lie group acting on the left by linear maps on $V$, and $\mathfrak{g}$ its Lie
algebra and $\mathfrak{g}^{\ast}$ the dual of $\mathfrak{g}$. Note that $G$
also acts on the left on the dual space $V^{\ast}$ of $V$, and the action by
an element $g$ on $V^{\ast}$ is the transpose of the action of $g^{-1}$ on
$V$. As a set, the underlying manifold of $S$ is $G\times V$ and the
multiplication on $S$ is given by
$(g_{1},v_{1})(g_{2},v_{2}):=(g_{1}g_{2},v_{1}+\sigma(g_{1})v_{2}),\quad
g_{1},g_{1}\in G,\quad v_{1},v_{2}\in V$ (20)
where $\sigma:G\to\operatorname{Aut}(V)$ is a representation of the Lie group
$G$ on $V$, $\operatorname{Aut}(V)$ denotes the Lie group of linear
isomorphisms of $V$ onto itself whose Lie algebra is $\operatorname{End}(V)$,
the space of all linear maps of $V$ to itself.
The Lie algebra of $S$ is the semidirect product of Lie algebras
$\mathfrak{s}=\mathfrak{g}\circledS V$, $\mathfrak{s}^{\ast}$ is the dual of
$\mathfrak{s}$, that is, $\mathfrak{s}^{\ast}=(\mathfrak{g}\circledS
V)^{\ast}$. The underlying vector space of $\mathfrak{s}$ is
$\mathfrak{g}\times V$ and the Lie bracket on $\mathfrak{s}$ is given by
$[(\xi_{1},v_{1}),(\xi_{2},v_{2})]=([\xi_{1},\xi_{2}],\sigma^{\prime}(\xi_{1})v_{2}-\sigma^{\prime}(\xi_{2})v_{1}),\quad\forall\xi_{1},\xi_{2}\in\mathfrak{g},\quad
v_{1},v_{2}\in V$ (21)
where $\sigma^{\prime}:\mathfrak{g}\to\operatorname{End}(V)$ is the induced
Lie algebra representation given by
$\sigma^{\prime}(\xi)v:=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\sigma(\exp
t\xi)v,\quad\xi\in\mathfrak{g},\quad v\in V$ (22)
Identify the underlying vector space of $\mathfrak{s}^{\ast}$ with
$\mathfrak{g}^{\ast}\times V^{\ast}$ by using the duality pairing on each
factor. We can give the formula for the $(\pm)$-Lie-Poisson bracket on the
semidirect product $\mathfrak{s}^{\ast}$ as follows, that is, for
$F,K:\mathfrak{s}^{\ast}\to\mathbb{R}$, their semidirect product bracket is
given by
$\\{F,K\\}_{\pm}(\mu,a)=\pm\langle\mu,[\frac{\delta F}{\delta\mu},\frac{\delta
K}{\delta\mu}]\rangle\pm\langle a,\frac{\delta F}{\delta\mu}\cdot\frac{\delta
K}{\delta a}-\frac{\delta K}{\delta\mu}\cdot\frac{\delta F}{\delta a}\rangle$
(23)
where $(\mu,a)\in\mathfrak{s}^{\ast}$ and $\dfrac{\delta
F}{\delta\mu}\in\mathfrak{g}$, $\dfrac{\delta F}{\delta a}\in V$ are the
functional derivatives. Moreover, the Hamiltonian vector field of a smooth
function $H:\mathfrak{s}^{\ast}\to\mathbb{R}$ is given by
$X_{H}(\mu,a)=\mp(\operatorname{ad}_{\delta
H/\delta\mu}^{\ast}\mu-\rho_{\delta H/\delta a}^{\ast}a,\;\frac{\delta
H}{\delta\mu}\cdot a),$ (24)
where the infinitesimal action of $\mathfrak{g}$ on $V$ can be denoted by
$\xi\cdot v=\rho_{v}(\xi)$, for any $\xi\in\mathfrak{g}$, $v\in V$ and the map
$\rho_{v}:\mathfrak{g}\to V$ is the derivative of the map $g\mapsto gv$ at the
identity and $\rho_{v}^{\ast}:V^{\ast}\to\mathfrak{g}^{\ast}$ is its dual.
We consider a symplectic action of $S$ on a symplectic manifold $P$ and assume
that this action has an $\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}_{S}:P\to\mathfrak{s}^{\ast}$. On the one hand, we can regard $V$
as a normal subgroup of $S$, it also acts on $P$ and has a momentum map
$\mathbf{J}_{V}:P\to V^{\ast}$ given by
$\mathbf{J}_{V}=i_{V}^{\ast}\cdot\mathbf{J}_{S}$, where
$i_{V}:V\to\mathfrak{s};\;v\mapsto(0,v)$ is the inclusion, and
$i_{V}^{\ast}:\mathfrak{s}^{\ast}\to V^{\ast}$ is its dual. $\mathbf{J}_{V}$
is called the second component of $\mathbf{J}_{S}$. On the other hand, we can
also regard $G$ as a subgroup of $S$ by the inclusion $i_{G}:G\to S$,
$g\mapsto(g,0)$. Thus, $G$ also has a momentum map
$\mathbf{J}_{G}:P\to\mathfrak{g}^{\ast}$ given by
$\mathbf{J}_{G}=i_{G}^{\ast}\cdot\mathbf{J}_{S}$, which is called the first
component of $\mathbf{J}_{S}$. Moreover, from the
$\operatorname{Ad}^{\ast}$-equivariance of $\mathbf{J}_{S}$ under $G$-action,
we know that $\mathbf{J}_{V}$ is also $\operatorname{Ad}^{\ast}$-equivariant
under $G$-action. Thus, we can carry out reduction of $P$ by $S$ at a regular
value $\sigma=(\mu,a)\in\mathfrak{s}^{\ast}$ of the momentum map
$\mathbf{J}_{S}$ in two stages using the following procedure.
(i)First reduce $P$ by $V$ at the value $a\in V^{\ast}$, and get the reduced
space $P_{a}=\mathbf{J}_{V}^{-1}(a)/V$. Since the reduction is by the abelian
group $V$, so the quotient is done using the whole of $V$.
(ii)The isometry subgroup $G_{a}\subset G$, consists of elements of $G$ that
leave the point $a\in V^{\ast}$ fixed using the action of $G$ on $V^{\ast}$.
We can prove that the group $G_{a}$ leaves the set
$\mathbf{J}_{V}^{-1}(a)\subset P$ invariant, and acts symplectically on the
reduced space $P_{a}$ and has a naturally induced momentum map
$\mathbf{J}_{a}:P_{a}\to\mathfrak{g}_{a}^{\ast}$, where $\mathfrak{g}_{a}$ is
the Lie algebra of the isometric subgroup $G_{a}$ and
$\mathfrak{g}_{a}^{\ast}$ is its dual.
(iii)Reduce the first reduced space $P_{a}$ at the point
$\mu_{a}=\mu|_{\mathfrak{g}^{\ast}_{a}}\in\mathfrak{g}_{a}^{\ast}$, we can get
the second reduced space
$(P_{a})_{\mu_{a}}=\mathbf{J}_{a}^{-1}(\mu_{a})/(G_{a})_{\mu_{a}}$.
Thus, we can give the theorem on the reduction by stages for semidirect
products as follows.
###### Proposition 6.4
The reduced space $(P_{a})_{\mu_{a}}$ is symplectically diffeomorphic to the
reduced space $P_{\sigma}$ obtained by reducing $P$ by $S$ at the regular
point $\sigma=(\mu,a)\in\mathfrak{s}^{\ast}$.
In particular, we can choose that $P=T^{\ast}S$, where $S=G\circledS V$ is a
semidirect product Lie group, with the cotangent lift action of $S$ on
$T^{\ast}S$ induced by left translations of $S$ on itself. Since the reduction
of $T^{\ast}S$ by the action of $V$ can give a space which is isomorphic to
$T^{\ast}G$, from the above reduction by stages theorem for semidirect
products we can get the following semidirect product reduction theorem.
###### Proposition 6.5
The reduction of $T^{\ast}G$ by $G_{a}$ at the regular values
$\mu_{a}=\mu|_{\mathfrak{g}^{\ast}_{a}}$ gives a space which is isomorphic to
the coadjoint orbit $\mathcal{O}_{\sigma}\subset\mathfrak{s}^{\ast}$ through
the point $\sigma=(\mu,a)\in\mathfrak{s}^{\ast}$, where $\mathfrak{s}^{\ast}$
is the dual of the Lie algebra $\mathfrak{s}$ of $S$.
### 6.2 Rigid Body and Heavy Top
In this subsection, we regard the rigid body and heavy top as well as them
with internal rotors (or external force torques) as the regular point
reducible RCH systems on the rotation group $\textmd{SO}(3)$ and on the
Euclidean group $\textmd{SE}(3)$, respectively, and give their $R_{P}$-reduced
RCH systems and discuss their RCH-equivalence. Note that our description of
the motion and the equations of rigid body and heavy top follows some of the
notations and conventions in Marsden and Ratiu [22], Marsden [20].
(1). Rigid Body with External Force Torque.
In the following we take Lie group $G=\textmd{SO}(3),$ and state the rigid
body with external force torque to be a regular point reducible RCH system. It
is well known that, usually, the configuration space for a $3$-dimensional
rigid body moving freely in space is $\textmd{SE}(3)$, the six dimension group
of Euclidean (rigid) transformations of three dimentional space
$\mathbb{R}^{3}$, that is, all possible rotations and translations. If
translation are ignored and only rotations are considered, then the
configuration space $Q$ is $\textmd{SO}(3)$, consists of all orthogonal linear
transformations of Euclidean three space to itself, which have determinant
one. Its Lie algebra, denoted $\mathfrak{so}(3)$, consists of all $3\times 3$
skew matrices. By using the isomorphism
$\hat{}:\mathbb{R}^{3}\to\mathfrak{so}(3)$ defined by
$(\Omega_{1},\Omega_{2},\Omega_{3})=\Omega\to\hat{\Omega}=\begin{bmatrix}0&-\Omega_{3}&\Omega_{2}\\\
\Omega_{3}&0&-\Omega_{1}\\\ -\Omega_{2}&\Omega_{1}&0\end{bmatrix},$
we can identify the Lie algebra $(\mathfrak{so}(3),[,])$ with
$(\mathbb{R}^{3},\times)$ and the Lie algebra bracket $[,]$ on
$\mathfrak{so}(3)$ with the cross product $\times$ of vectors in
$\mathbb{R}^{3}$. Denote by $\mathfrak{so}^{\ast}(3)$ the dual of the Lie
algebra $\mathfrak{so}(3)$, and we also identity $\mathfrak{so}^{\ast}(3)$
with $\mathbb{R}^{3}$ by pairing the Euclidean inner product. Since the
functional derivative of a function defined on $\mathbb{R}^{3}$ is equal to
the usual gradient of the function, from (14) we know that the Lie-Poisson
bracket on $\mathfrak{so}^{\ast}(3)$ take the form
$\\{f,g\\}_{\pm}(\Pi)=\pm\Pi\cdot(\nabla_{\Pi}f\times\nabla_{\Pi}g),\;\;\forall
f,g\in C^{\infty}(\mathfrak{so}^{\ast}(3)),\;\;\Pi\in\mathfrak{so}^{\ast}(3).$
(25)
The phase space of a rigid body is the cotangent bundle
$T^{\ast}G=T^{\ast}\textmd{SO}(3)\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)$,
with the canonical symplectic form. Assume that Lie group $G=\textmd{SO}(3)$
acts freely and properly by the left translations on $\textmd{SO}(3)$, then
the action of $\textmd{SO}(3)$ on the phase space $T^{\ast}\textmd{SO}(3)$ is
by cotangent lift of left translations at the identity, that is,
$\Phi:\textmd{SO}(3)\times
T^{\ast}\textmd{SO}(3)\cong\textmd{SO}(3)\times\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\to\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3),$
given by $\Phi(B,(A,\Pi))=(BA,\Pi)$, for any
$A,B\in\textmd{SO}(3),\;\Pi\in\mathfrak{so}^{\ast}(3)$, which is also free and
proper, and admits an associated $\operatorname{Ad}^{\ast}$-equivariant
momentum map $\mathbf{J}:T^{\ast}\textmd{SO}(3)\to\mathfrak{so}^{\ast}(3)$ for
the left $\textmd{SO}(3)$ action. If $\Pi\in\mathfrak{so}^{\ast}(3)$ is a
regular value of $\mathbf{J}$, then the regular point reduced space
$(T^{\ast}\textmd{SO}(3))_{\Pi}=\mathbf{J}^{-1}(\Pi)/\textmd{SO}(3)_{\Pi}$ is
symplectically diffeomorphic to the coadjoint orbit
$\mathcal{O}_{\Pi}\subset\mathfrak{so}^{\ast}(3)$.
Let $I$ be the moment of inertia tensor computed with respect to a body fixed
frame, which, in a principal body frame, we may represent by the diagonal
matrix diag $(I_{1},I_{2},I_{3})$. Let
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})$ be the vector of angular
velocities computed with respect to the axes fixed in the body and
$(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$. Consider the
Lagrangian
$L(A,\Omega):\textmd{TSO}(3)\cong\textmd{SO}(3)\times\mathfrak{so}(3)\to\mathbb{R}$,
which is the total kinetic energy of the rigid body, given by
$L(A,\Omega)=\dfrac{1}{2}\langle\Omega,\Omega\rangle=\dfrac{1}{2}(I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2}),$
where $A\in\textmd{SO}(3)$,
$(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$. If we introduce the
conjugate angular momentum $\Pi_{i}=\dfrac{\partial
L}{\partial\Omega_{i}}=I_{i}\Omega_{i}$, $i=1,2,3$, which is also computed
with respect to a body fixed frame, and by the Legendre transformation
$FL:\textmd{TSO}(3)\cong\textmd{SO}(3)\times\mathfrak{so}(3)\to
T^{\ast}\textmd{SO}(3)\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3),\;(A,\Omega)\to(A,\Pi)$,
where $\Pi=(\Pi_{1},\Pi_{2},\Pi_{3})\in\mathfrak{so}^{\ast}(3)$, we have the
Hamiltonian
$H(A,\Pi):T^{\ast}\textmd{SO}(3)\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\to\mathbb{R}$
given by
$\displaystyle H(A,\Pi)$ $\displaystyle=\Omega\cdot\Pi-L(A,\Omega)$
$\displaystyle=I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2}-\frac{1}{2}(I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2})$
$\displaystyle=\frac{1}{2}(\frac{\Pi_{1}^{2}}{I_{1}}+\frac{\Pi_{2}^{2}}{I_{2}}+\frac{\Pi_{3}^{2}}{I_{3}}).$
From the above expression of the Hamiltonian, we know that $H(A,\Pi)$ is
invariant under the left $\textmd{SO}(3)$-action $\Phi:\textmd{SO}(3)\times
T^{\ast}\textmd{SO}(3)\to T^{\ast}\textmd{SO}(3)$. For the case
$\Pi_{0}=\mu\in\mathfrak{so}^{\ast}(3)$ is a regular value of $\mathbf{J}$, we
have the reduced Hamiltonian
$h_{\mu}(\Pi):\mathcal{O}_{\mu}\subset\mathfrak{so}^{\ast}(3)\to\mathbb{R}$
given by $h_{\mu}(\Pi)=H(A,\Pi)|_{\mathcal{O}_{\mu}}$. From the Lie-Poisson
bracket on $\mathfrak{g}^{\ast}$, we can get the rigid body Poisson bracket on
$\mathfrak{so}^{\ast}(3)$, that is, for
$F,K:\mathfrak{so}^{\ast}(3)\to\mathbb{R},$ we have that
$\\{F,K\\}_{-}(\Pi)=-\Pi\cdot(\nabla_{\Pi}F\times\nabla_{\Pi}K)$. In
particular, for $F_{\mu},K_{\mu}:\mathcal{O}_{\mu}\to\mathbb{R}$, we have that
$\omega_{\mathcal{O}_{\mu}}^{-}(X_{F_{\mu}},X_{K_{\mu}})=\\{F_{\mu},K_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}}$.
Moreover, for reduced Hamiltonian
$h_{\mu}(\Pi):\mathcal{O}_{\mu}\to\mathbb{R}$, we have the Hamiltonian vector
field $X_{h_{\mu}}(K_{\mu})=\\{K_{\mu},h_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}},$
and hence we have that
$\displaystyle\frac{\mathrm{d}\Pi}{\mathrm{d}t}$
$\displaystyle=X_{h_{\mu}}(\Pi)=\\{\Pi,h_{\mu}(\Pi)\\}_{-}|_{\mathcal{O}_{\mu}}$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Pi}h_{\mu})=-\nabla_{\Pi}\Pi\cdot(\nabla_{\Pi}h_{\mu}\times\Pi)=\Pi\times\Omega,$
since $\nabla_{\Pi}\Pi=1$ and $\nabla_{\Pi}h_{\mu}=\Omega$. Thus, the
equations of motion for rigid body is given by
$\dfrac{\mathrm{d}\Pi}{\mathrm{d}t}=\Pi\times\Omega.$ (26)
If we consider the rigid body with a external force torque
$u:T^{\ast}\textmd{SO}(3)\to T^{\ast}\textmd{SO}(3)$, and $u$ is invariant
under the left $\textmd{SO}(3)$-action, then the external force torque $u$ can
be regarded as a control of the rigid body, and its reduced control
$u_{\mu}:\mathcal{O}_{\mu}\to\mathcal{O}_{\mu}$ is given by
$u_{\mu}(\Pi)=u(A,\Pi)|_{\mathcal{O}_{\mu}}.$ Thus, the equations of motion
for the rigid body with external force torques $u:T^{\ast}\textmd{SO}(3)\to
T^{\ast}\textmd{SO}(3)$ are given by
$\dfrac{\mathrm{d}\Pi}{\mathrm{d}t}=\Pi\times\Omega+\mbox{vlift}(u_{\mu}),$
(27)
where $\mbox{vlift}(u_{\mu})\in T\mathcal{O}_{\mu}.$ To sum up the above
discussion, we have the following proposition.
###### Proposition 6.6
The 5-tuple $(T^{\ast}\textmd{SO}(3),\textmd{SO}(3),\omega_{0},H,u)$ is a
regular point reducible RCH system. For a point
$\mu\in\mathfrak{so}^{\ast}(3)$, the regular value of the momentum map
$\mathbf{J}:T^{\ast}\textmd{SO}(3)\to\mathfrak{so}^{\ast}(3)$, the
$R_{P}$-reduced system is the 4-tuple
$(\mathcal{O}_{\mu},\omega_{\mathcal{O}_{\mu}}^{-},h_{\mu},u_{\mu}),$ where
$\mathcal{O}_{\mu}\subset\mathfrak{so}^{\ast}(3)$ is the coadjoint orbit,
$\omega_{\mathcal{O}_{\mu}}^{-}$ is orbit symplectic form on
$\mathcal{O}_{\mu}$, $h_{\mu}(\Pi)=H(A,\Pi)|_{\mathcal{O}_{\mu}}$,
$u_{\mu}(\Pi)=u(A,\Pi)|_{\mathcal{O}_{\mu}}$, and its equation of motion is
given by (27).
(2). The Rigid Body with Internal Rotors.
In the following we take Lie group $G=\textmd{SO}(3),\;V=S^{1}\times
S^{1}\times S^{1},\;Q=G\times V$ and state the rigid body with three symmetric
internal rotors to be a regular point reducible RCH system. We consider a
rigid body (to be called the carrier body) carrying three symmetric rotors.
Denote the system center of mass by $O$ in the body frame and at $O$ place a
set of (orthonormal) body axes. Assume that the rotor and the body coordinate
axes are aligned with principal axes of the carrier body. The rotor spins
under the influence of a torque $u$ acting on the rotor. The configuration
space is $Q=\textmd{SO}(3)\times V$, where $V=S^{1}\times S^{1}\times S^{1}$,
with the first factor being rigid body attitude and the second factor being
the angles of rotors. The corresponding phase space is the cotangent bundle
$T^{\ast}Q=T^{\ast}\textmd{SO}(3)\times T^{\ast}V$, where
$T^{\ast}V=T^{\ast}(S^{1}\times S^{1}\times S^{1})\cong
T^{\ast}\mathbb{R}^{3}$, with the canonical symplectic form. Assume that Lie
group $G=\textmd{SO}(3)$ acts freely and properly on $Q$ by the left
translations on $\textmd{SO}(3)$, then the action of $\textmd{SO}(3)$ on the
phase space $T^{\ast}Q$ is by cotangent lift of left translations on
$\textmd{SO}(3)$ at the identity, that is, $\Phi:\textmd{SO}(3)\times
T^{\ast}\textmd{SO}(3)\times
T^{\ast}V\cong\textmd{SO}(3)\times\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3},$
given by $\Phi(B,(A,\Pi,\alpha,l))=(BA,\Pi,\alpha,l)$, for any
$A,B\in\textmd{SO}(3),\;\Pi\in\mathfrak{so}^{\ast}(3),\;\alpha,l\in\mathbb{R}^{3}$,
which is also free and proper, and admits an associated
$\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}_{Q}:T^{\ast}Q\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathfrak{so}^{\ast}(3)$
for the left $\textmd{SO}(3)$ action. If $\Pi\in\mathfrak{so}^{\ast}(3)$ is a
regular value of $\mathbf{J}_{Q}$, then the regular point reduced space
$(T^{\ast}Q)_{\Pi}=\mathbf{J}^{-1}_{Q}(\Pi)/\textmd{SO}(3)_{\Pi}$ is
symplectically diffeomorphic to the coadjoint orbit
$\mathcal{O}_{\Pi}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\subset\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}$.
Let $I=diag(I_{1},I_{2},I_{3})$ be the moment of inertia of the carrier body
in the principal body-fixed frame, and $J_{i},\;i=1,2,3$ be the moments of
inertia of rotors around their rotation axes. Let
$J_{ik},\;i=1,2,3,\;k=1,2,3,$ be the moments of inertia of the $i$th rotor
with $i=1,2,3,$ around the $k$th principal axis with $k=1,2,3,$ respectively,
and denote by $\bar{I}_{i}=I_{i}+J_{1i}+J_{2i}+J_{3i}-J_{ii},\;i=1,2,3$. Let
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})$ be the vector of body angular
velocities computed with respect to the axes fixed in the body and
$(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$. Let
$\alpha_{i},\;i=1,2,3,$ be the relative angles of rotors and
$\dot{\alpha}=(\dot{\alpha_{1}},\dot{\alpha_{2}},\dot{\alpha_{3}})$ the vector
of rotor relative angular velocities about the principal axes with respect to
a carrier body fixed frame.
Consider the Lagrangian of the system
$L(A,\Omega,\alpha,\dot{\alpha}):TQ\cong\textmd{SO}(3)\times\mathfrak{so}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$,
which is the total kinetic energy of the rigid body plus the total kinetic
energy of rotors, given by
$L(A,\Omega,\alpha,\dot{\alpha})=\dfrac{1}{2}[\bar{I}_{1}\Omega_{1}^{2}+\bar{I}_{2}\Omega_{2}^{2}+\bar{I}_{3}\Omega_{3}^{2}+J_{1}(\Omega_{1}+\dot{\alpha}_{1})^{2}+J_{2}(\Omega_{2}+\dot{\alpha}_{2})^{2}+J_{3}(\Omega_{3}+\dot{\alpha}_{3})^{2}],$
where $A\in\textmd{SO}(3)$,
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$,
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})\in\mathbb{R}^{3}$,
$\dot{\alpha}=(\dot{\alpha}_{1},\dot{\alpha}_{2},\dot{\alpha}_{3})\in\mathbb{R}^{3}$.
If we introduce the conjugate angular momentum, which is given by
$\Pi_{i}=\dfrac{\partial
L}{\partial\Omega_{i}}=\bar{I}_{i}\Omega_{i}+J_{i}(\Omega_{i}+\dot{\alpha}_{i}),\quad
l_{i}=\dfrac{\partial
L}{\partial\dot{\alpha}_{i}}=J_{i}(\Omega_{i}+\dot{\alpha}_{i}),\quad
i=1,2,3,$
and by the Legendre transformation
$FL:TQ\cong\textmd{SO}(3)\times\mathfrak{so}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to
T^{\ast}Q\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3},\quad(A,\Omega,\alpha,\dot{\alpha})\to(A,\Pi,\alpha,l)$,
where $\Pi=(\Pi_{1},\Pi_{2},\Pi_{3})\in\mathfrak{so}^{\ast}(3)$,
$l=(l_{1},l_{2},l_{3})\in\mathbb{R}^{3}$, we have the Hamiltonian
$H(A,\Pi,\alpha,l):T^{\ast}Q\cong\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$
given by
$\displaystyle H(A,\Pi,\alpha,l)$
$\displaystyle=\Omega\cdot\Pi+\dot{\alpha}\cdot
l-L(A,\Omega,\alpha,\dot{\alpha})$
$\displaystyle=\bar{I}_{1}\Omega_{1}^{2}+J_{1}(\Omega_{1}^{2}+\Omega_{1}\dot{\alpha}_{1})+\bar{I}_{2}\Omega_{2}^{2}+J_{2}(\Omega_{2}^{2}+\Omega_{2}\dot{\alpha}_{2})+\bar{I}_{3}\Omega_{3}^{2}+J_{3}(\Omega_{3}^{2}$
$\displaystyle\quad+\Omega_{3}\dot{\alpha}_{3})+J_{1}(\dot{\alpha}_{1}\Omega_{1}+\dot{\alpha}_{1}^{2})+J_{2}(\dot{\alpha}_{2}\Omega_{2}+\dot{\alpha}_{2}^{2})+J_{3}(\dot{\alpha}_{3}\Omega_{3}+\dot{\alpha}_{3}^{2})$
$\displaystyle\quad-\frac{1}{2}[\bar{I}_{1}\Omega_{1}^{2}+\bar{I}_{2}\Omega_{2}^{2}+\bar{I}_{3}\Omega_{3}^{2}+J_{1}(\Omega_{1}+\dot{\alpha}_{1})^{2}+J_{2}(\Omega_{2}+\dot{\alpha}_{2})^{2}+J_{3}(\Omega_{3}+\dot{\alpha}_{3})^{2}]$
$\displaystyle=\frac{1}{2}[\frac{(\Pi_{1}-l_{1})^{2}}{\bar{I}_{1}}+\frac{(\Pi_{2}-l_{2})^{2}}{\bar{I}_{2}}+\frac{(\Pi_{3}-l_{3})^{2}}{\bar{I}_{3}}+\frac{l_{1}^{2}}{J_{1}}+\frac{l_{2}^{2}}{J_{2}}+\frac{l_{3}^{2}}{J_{3}}].$
From the above expression of the Hamiltonian, we know that $H(A,\Pi,\alpha,l)$
is invariant under the left $\textmd{SO}(3)$-action $\Phi:\textmd{SO}(3)\times
T^{\ast}Q\to T^{\ast}Q$. For the case $\Pi_{0}=\mu\in\mathfrak{so}^{\ast}(3)$
is the regular value of $\mathbf{J}_{Q}$, we have the reduced Hamiltonian
$h_{\mu}(\Pi,\alpha,l):\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}(\subset\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3})\to\mathbb{R}$
given by
$h_{\mu}(\Pi,\alpha,l)=H(A,\Pi,\alpha,l)|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}.$
From the rigid body Poisson bracket on $\mathfrak{so}^{\ast}(3)$ and the
Poisson bracket on $T^{\ast}\mathbb{R}^{3}$, we can get the Poisson bracket on
$T^{\ast}Q$, that is, for
$F,K:\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R},$
we have that
$\\{F,K\\}_{-}(\Pi,\alpha,l)=-\Pi\cdot(\nabla_{\Pi}F\times\nabla_{\Pi}K)+\\{F,K\\}_{V}(\alpha,l)$.
In particular, for
$F_{\mu},K_{\mu}:\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$,
we have that
$\tilde{\omega}_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}^{-}(X_{F_{\mu}},X_{K_{\mu}})=\\{F_{\mu},K_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}$.
Moreover, for reduced Hamiltonian
$h_{\mu}(\Pi,\alpha,l):\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathbb{R}$,
we have the Hamiltonian vector field
$X_{h_{\mu}}(K_{\mu})=\\{K_{\mu},h_{\mu}\\}_{-}|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}},$
and hence we have that
$\displaystyle\frac{\mathrm{d}\Pi}{\mathrm{d}t}$
$\displaystyle=X_{h_{\mu}}(\Pi)(\Pi,\alpha,l)=\\{\Pi,h_{\mu}\\}_{-}(\Pi,\alpha,l)$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Pi}h_{\mu})+\sum_{i=1}^{3}(\frac{\partial\Pi}{\partial\alpha_{i}}\frac{\partial
h_{\mu}}{\partial l_{i}}-\frac{\partial
h_{\mu}}{\partial\alpha_{i}}\frac{\partial\Pi}{\partial l_{i}})$
$\displaystyle=-\nabla_{\Pi}\Pi\cdot(\nabla_{\Pi}h_{\mu}\times\Pi)=\Pi\times\Omega,$
since $\nabla_{\Pi}\Pi=1$, $\nabla_{\Pi}h_{\mu}=\Omega$ and
$\frac{\partial\Pi}{\partial\alpha_{i}}=\frac{\partial
h_{\mu}}{\partial\alpha_{i}}=0,\;i=1,2,3.$ If we consider the rigid body-rotor
system with a control torque $u:T^{\ast}Q\to T^{\ast}Q$ acting on the rotors,
and $u$ is invariant under the left $\textmd{SO}(3)$-action, and its reduced
control torque
$u_{\mu}:\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$
is given by
$u_{\mu}(\Pi,\alpha,l)=u(A,\Pi,\alpha,l)|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}.$
Thus, the equations of motion for rigid body-rotor system with the control
torque $u$ acting on the rotors are given by
$\left\\{\begin{aligned} \frac{\mathrm{d}\Pi}{\mathrm{d}t}&=\Pi\times\Omega\\\
\frac{\mathrm{d}l}{\mathrm{d}t}&=\mbox{vlift}(u_{\mu})\end{aligned}\right.$
(28)
where $\mbox{vlift}(u_{\mu})\in
T(\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}).$ To sum up the
above discussion, we have the following proposition.
###### Proposition 6.7
The 5-tuple
$(T^{\ast}(\textmd{SO}(3)\times\mathbb{R}^{3}),\textmd{SO}(3),\omega_{0},H,u)$
is a regular point reducible RCH system. For a point
$\mu\in\mathfrak{so}^{\ast}(3)$, the regular value of the momentum map
$\mathbf{J}:\textmd{SO}(3)\times\mathfrak{so}^{\ast}(3)\times\mathbb{R}^{3}\times\mathbb{R}^{3}\to\mathfrak{so}^{\ast}(3)$,
the $R_{P}$-reduced system is the 4-tuple
$(\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3},\tilde{\omega}_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}^{-},h_{\mu},u_{\mu}),$
where $\mathcal{O}_{\mu}\subset\mathfrak{so}^{\ast}(3)$ is the coadjoint
orbit,
$\tilde{\omega}_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}^{-}$
is orbit symplectic form on
$\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$,
$h_{\mu}(\Pi,\alpha,l)=H(A,\Pi,\alpha,l)|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}$,
$u_{\mu}(\Pi,\alpha,l)=u(A,\Pi,\alpha,l)|_{\mathcal{O}_{\mu}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}$,
and its equations of motion are given by (28).
(3). Heavy Top.
In the following we take Lie group $G=\textmd{SE}(3)$ and state the heavy top
to be a regular point reducible Hamiltonian system, and hence also to be a
regular point reducible RCH system without the external force and control. We
know that a heavy top is by definition a rigid body with a fixed point in
$\mathbb{R}^{3}$ and moving in gravitational field. Usually, exception of the
singular point, its physical phase space is $T^{\ast}\textmd{SO}(3)$ and the
symmetry group is $S^{1}$, regarded as rotations about the z-axis, the axis of
gravity, this is because gravity breaks the symmetry and the system is no
longer $\textmd{SO}(3)$ invariant. By the semidirect product reduction theorem
(See Proposition 6.5 ), we show that the reduction of $T^{\ast}\textmd{SO}(3)$
by $S^{1}$ gives a space which is symplectically diffeomorphic to the reduced
space obtained by the reduction of $T^{\ast}\textmd{SE}(3)$ by left action of
$\textmd{SE}(3)$, that is the coadjoint orbit
$\mathcal{O}_{(\mu,a)}\subset\mathfrak{se}^{\ast}(3)\cong
T^{\ast}\textmd{SE}(3)/\textmd{SE}(3)$. In fact, in this case, we can identify
the phase space $T^{\ast}\textmd{SO}(3)$ with the reduction of the cotangent
bundle of the special Euclidean group
$\textmd{SE}(3)=\textmd{SO}(3)\circledS\mathbb{R}^{3}$ by the Euclidean
translation subgroup $\mathbb{R}^{3}$ and identifies the symmetry group
$S^{1}$ with isotropy group $G_{a}=\\{A\in\textmd{SO}(3)\mid Aa=a\\}=S^{1}$,
which is abelian and
$(G_{a})_{\mu_{a}}=G_{a}=S^{1},\;\forall\mu_{a}\in\mathfrak{g}^{\ast}_{a}$,
where $a$ is a vector aligned with the direction of gravity and where
$\textmd{SO}(3)$ acts on $\mathbb{R}^{3}$ in the standard way.
Now we consider the cotangent bundle
$T^{\ast}G=T^{\ast}\textmd{SE}(3)\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)$,
with the canonical symplectic form. Assume that Lie group $G=\textmd{SE}(3)$
acts freely and properly by the left translations on $\textmd{SE}(3)$, then
the action of $\textmd{SE}(3)$ on the phase space $T^{\ast}\textmd{SE}(3)$ is
by cotangent lift of left translations at the identity, that is,
$\Phi:\textmd{SE}(3)\times
T^{\ast}\textmd{SE}(3)\cong\textmd{SE}(3)\times\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\to\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3),$
given by $\Phi((B,u),(A,v,\Pi,w))=(BA,v,\Pi,w)$, for any
$A,B\in\textmd{SO}(3),\;\Pi\in\mathfrak{so}^{\ast}(3),\;u,v,w\in\mathbb{R}^{3}$,
which is also free and proper, and admits an associated
$\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}:T^{\ast}\textmd{SE}(3)\to\mathfrak{se}^{\ast}(3)$ for the left
$\textmd{SE}(3)$ action. If $(\Pi,w)\in\mathfrak{se}^{\ast}(3)$ is a regular
value of $\mathbf{J}$, then the regular point reduced space
$(T^{\ast}\textmd{SE}(3))_{(\Pi,w)}=\mathbf{J}^{-1}(\Pi,w)/\textmd{SE}(3)_{(\Pi,w)}$
is symplectically diffeomorphic to the coadjoint orbit
$\mathcal{O}_{(\Pi,w)}\subset\mathfrak{se}^{\ast}(3)$.
Let $I=diag(I_{1},I_{2},I_{3})$ be the moment of inertia of the heavy top in
the body-fixed frame, which in principal body frame. Let
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})$ be the vector of heavy top angular
velocities computed with respect to the axes fixed in the body and
$(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$. Let $\Gamma$ be the
unit vector viewed by an observer moving with the body, $m$ be that total mass
of the system, $g$ be the magnitude of the gravitational acceleration, $\chi$
be the unit vector on the line connecting the origin $O$ to the center of mass
of the system, and $h$ be the length of this segment.
Consider the Lagrangian
$L(A,v,\Omega,\Gamma):\textmd{TSE}(3)\cong\textmd{SE}(3)\times\mathfrak{se}(3)\to\mathbb{R}$
, which is the total kinetic minus potential energy of the heavy top, given by
$L(A,v,\Omega,\Gamma)=\dfrac{1}{2}\langle\Omega,\Omega\rangle-
mgh\Gamma\cdot\chi=\dfrac{1}{2}(I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2})-mgh\Gamma\cdot\chi,$
where $(A,v)\in\textmd{SE}(3)$,
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$,
$\Gamma\in\mathbb{R}^{3}$. If we introduce the conjugate angular momentum
$\Pi_{i}=\dfrac{\partial L}{\partial\Omega_{i}}=I_{i}\Omega_{i},\;i=1,2,3,$
and by the Legendre transformation
$FL:\textmd{TSE}(3)\cong\textmd{SE}(3)\times\mathfrak{se}(3)\to
T^{\ast}\textmd{SE}(3)\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3),\quad(A,v,\Omega,\Gamma)\to(A,v,\Pi,\Gamma)$,
where $\Pi=(\Pi_{1},\Pi_{2},\Pi_{3})\in\mathfrak{so}^{\ast}(3)$, we have the
Hamiltonian
$H(A,v,\Pi,\Gamma):T^{\ast}\textmd{SE}(3)\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\to\mathbb{R}$
given by
$\displaystyle H(A,v,\Pi,\Gamma)$
$\displaystyle=\Omega\cdot\Pi-L(A,\Omega,,\Gamma)$
$\displaystyle=I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2}-\frac{1}{2}(I_{1}\Omega_{1}^{2}+I_{2}\Omega_{2}^{2}+I_{3}\Omega_{3}^{2})+mgh\Gamma\cdot\chi$
$\displaystyle=\frac{1}{2}(\frac{\Pi_{1}^{2}}{I_{1}}+\frac{\Pi_{2}^{2}}{I_{2}}+\frac{\Pi_{3}^{2}}{I_{3}})+mgh\Gamma\cdot\chi.$
From the above expression of the Hamiltonian, we know that $H(A,v,\Pi,\Gamma)$
is invariant under the left $\textmd{SE}(3)$-action $\Phi:\textmd{SE}(3)\times
T^{\ast}\textmd{SE}(3)\to T^{\ast}\textmd{SE}(3)$. For the case
$(\Pi_{0},\Gamma_{0})=(\mu,a)\in\mathfrak{se}^{\ast}(3)$ is a regular value of
$\mathbf{J}$, we have the reduced Hamiltonian
$h_{(\mu,a)}(\Pi,,\Gamma):\mathcal{O}_{(\mu,a)}(\subset\mathfrak{se}^{\ast}(3))\to\mathbb{R}$
given by $h_{(\mu,a)}(\Pi,\Gamma)=H(A,v,\Pi,\Gamma)|_{\mathcal{O}_{(\mu,a)}}$.
From the semidirect product bracket (23), we can get the heavy top Poisson
bracket on $\mathfrak{se}^{\ast}(3)$, that is, for
$F,K:\mathfrak{se}^{\ast}(3)\to\mathbb{R},$ we have that
$\\{F,K\\}_{-}(\Pi,\Gamma)=-\Pi\cdot(\nabla_{\Pi}F\times\nabla_{\Pi}K)-\Gamma\cdot(\nabla_{\Pi}F\times\nabla_{\Gamma}K-\nabla_{\Pi}K\times\nabla_{\Gamma}F).$
In particular, for
$F_{(\mu,a)},K_{(\mu,a)}:\mathcal{O}_{(\mu,a)}\to\mathbb{R}$, we have that
$\omega_{\mathcal{O}_{(\mu,a)}}^{-}(X_{F_{(\mu,a)}},X_{K_{(\mu,a)}})=\\{F_{(\mu,a)},K_{(\mu,a)}\\}_{-}|_{\mathcal{O}_{(\mu,a)}}.$
Moreover, for reduced Hamiltonian
$h_{(\mu,a)}(\Pi,\Gamma):\mathcal{O}_{(\mu,a)}\to\mathbb{R}$, we have the
Hamiltonian vector field
$X_{h_{(\mu,a)}}(K_{(\mu,a)})=\\{K_{(\mu,a)},h_{(\mu,a)}\\}_{-}|_{\mathcal{O}_{(\mu,a)}},$
and hence we have that
$\displaystyle\frac{\mathrm{d}\Pi}{\mathrm{d}t}$
$\displaystyle=X_{h_{(\mu,a)}}(\Pi)=\\{\Pi,h_{(\mu,a)}(\Pi,\Gamma)\\}_{-}$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Pi}h_{(\mu,a)})-\Gamma\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Gamma}h_{(\mu,a)}-\nabla_{\Pi}h_{(\mu,a)}\times\nabla_{\Gamma}\Pi)$
$\displaystyle=\Pi\times\Omega-
mgh\chi\times\Gamma=\Pi\times\Omega+mgh\Gamma\times\chi,$
$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}t}$
$\displaystyle=X_{h_{(\mu,a)}}(\Gamma)=\\{\Gamma,h_{(\mu,a)}(\Pi,\Gamma)\\}_{-}$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Gamma\times\nabla_{\Pi}h_{(\mu,a)})-\Gamma\cdot(\nabla_{\Pi}\Gamma\times\nabla_{\Gamma}h_{(\mu,a)}-\nabla_{\Pi}h_{(\mu,a)}\times\nabla_{\Gamma}\Gamma)$
$\displaystyle=\nabla_{\Gamma}\Gamma\cdot(\Gamma\times\nabla_{\Pi}h_{(\mu,a)})=\Gamma\times\Omega,$
since
$\nabla_{\Pi}\Pi=1,\;\nabla_{\Gamma}\Gamma=1,\;\nabla_{\Gamma}\Pi=\nabla_{\Pi}\Gamma=0,$
and $\nabla_{\Pi}h_{(\mu,a)}=\Omega$. Thus, the equations of motion for heavy
top is given by
$\left\\{\begin{aligned}
&\frac{\mathrm{d}\Pi}{\mathrm{d}t}=\Pi\times\Omega+mgh\Gamma\times\chi,\\\
&\frac{\mathrm{d}\Gamma}{\mathrm{d}t}=\Gamma\times\Omega.\end{aligned}\right.$
(29)
To sum up the above discussion, we have the following proposition.
###### Proposition 6.8
The 4-tuple $(T^{\ast}\textmd{SE}(3),\textmd{SE}(3),\omega_{0},H)$ is a
regular point reducible Hamiltonian system. For a point
$(\mu,a)\in\mathfrak{se}^{\ast}(3)$, the regular value of the momentum map
$\mathbf{J}:T^{\ast}\textmd{SE}(3)\to\mathfrak{se}^{\ast}(3)$, the
$R_{P}$-reduced system is the 3-tuple
$(\mathcal{O}_{(\mu,a)},\omega_{\mathcal{O}_{(\mu,a)}},h_{(\mu,a)})$, where
$\mathcal{O}_{(\mu,a)}\subset\mathfrak{se}^{\ast}(3)$ is the coadjoint orbit,
$\omega_{\mathcal{O}_{(\mu,a)}}$ is orbit symplectic form on
$\mathcal{O}_{(\mu,a)}$,
$h_{(\mu,a)}(\Pi,\Gamma)=H(A,v,\Pi,\Gamma)|_{\mathcal{O}_{(\mu,a)}}$, and its
equations of motion are given by (29).
(4). The Heavy Top with Internal Rotors.
In the following we take Lie group $G=\textmd{SE}(3),\;V=S^{1}\times
S^{1},\;Q=G\times V$ and state the heavy top with two pairs of symmetric
internal rotors to be a regular point reducible RCH system. We shall first
describe a heavy top with two pairs of symmetric rotors. We mount two pairs of
rotors within the top so that each pair’s rotation axis is parallel to the
first and the second principal axes of the top; see Chang and Marsden [10].
The rotor spins under the influence of a torque $u$ acting on the rotor. The
configuration space is $Q=\textmd{SE}(3)\times V$, where $V=S^{1}\times
S^{1}$, with the first factor being the position of the heavy top and the
second factor being the angles of rotors. The corresponding phase space is the
cotangent bundle $T^{\ast}Q=T^{\ast}\textmd{SE}(3)\times T^{\ast}V$, where
$T^{\ast}V=T^{\ast}(S^{1}\times S^{1})\cong T^{\ast}\mathbb{R}^{2}$, with the
canonical symplectic form. Assume that Lie group $G=\textmd{SE}(3)$ acts
freely and properly on $Q$ by the left translations on $\textmd{SE}(3)$, then
the action of $\textmd{SE}(3)$ on the phase space $T^{\ast}Q$ is by cotangent
lift of left translations on $\textmd{SE}(3)$ at the identity, that is,
$\Phi:\textmd{SE}(3)\times T^{\ast}\textmd{SE}(3)\times
T^{\ast}V\cong\textmd{SE}(3)\times\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2},$
given by $\Phi((B,u)((A,v),(\Pi,w),\alpha,l))=((BA,v),(\Pi,w),\alpha,l)$, for
any
$A,B\in\textmd{SO}(3),\;\Pi\in\mathfrak{so}^{\ast}(3),\;u,v,w\in\mathbb{R}^{3},\;\alpha,l\in\mathbb{R}^{2}$,
which is also free and proper, and admits an associated
$\operatorname{Ad}^{\ast}$-equivariant momentum map
$\mathbf{J}_{Q}:T^{\ast}Q\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathfrak{se}^{\ast}(3)$
for the left $\textmd{SE}(3)$ action. If $(\Pi,w)\in\mathfrak{se}^{\ast}(3)$
is a regular value of $\mathbf{J}_{Q}$, then the regular point reduced space
$(T^{\ast}Q)_{(\Pi,w)}=\mathbf{J}^{-1}_{Q}(\Pi,w)/\textmd{SE}(3)_{(\Pi,w)}$ is
symplectically diffeomorphic to the coadjoint orbit
$\mathcal{O}_{(\Pi,w)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\subset\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}$.
Let $I=diag(I_{1},I_{2},I_{3})$ be the moment of inertia of the heavy top in
the body-fixed frame. Let $J_{i},i=1,2$ be the moments of inertia of rotors
around their rotation axes. Let $J_{ik},\;i=1,2,\;k=1,2,3,$ be the moments of
inertia of the $i$-th rotor with $i=1,2$ around the $k$-th principal axis with
$k=1,2,3,$ respectively, and denote by
$\bar{I}_{i}=I_{i}+J_{1i}+J_{2i}-J_{ii},\;i=1,2$, and
$\bar{I}_{3}=I_{3}+J_{13}+J_{23}$. Let
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})$ be the vector of heavy top angular
velocities computed with respect to the axes fixed in the body and
$(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$. Let
$\theta_{i},\;i=1,2,$ be the relative angles of rotors and
$\dot{\theta}=(\dot{\theta_{1}},\dot{\theta_{2}})$ the vector of rotor
relative angular velocities about the principal axes with respect to the body
fixed frame of heavy top. Let $m$ be that total mass of the system, $g$ be the
magnitude of the gravitational acceleration and $h$ be the distance from the
origin $O$ to the center of mass of the system.
Consider the Lagrangian
$L(A,v,\Omega,\Gamma,\theta,\dot{\theta}):TQ\cong\textmd{SE}(3)\times\mathfrak{se}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$,
which is the total kinetic energy of the heavy top plus the total kinetic
energy of rotors minus potential energy of the system, given by
$L(A,v,\Omega,\Gamma,\theta,\dot{\theta})=\dfrac{1}{2}[\bar{I}_{1}\Omega_{1}^{2}+\bar{I}_{2}\Omega_{2}^{2}+\bar{I}_{3}\Omega_{3}^{2}+J_{1}(\Omega_{1}+\dot{\theta}_{1})^{2}+J_{2}(\Omega_{2}+\dot{\theta}_{2})^{2}]-mgh\Gamma\cdot\chi,$
where $(A,v)\in\textmd{SE}(3)$, $(\Omega,\Gamma)\in\mathfrak{se}(3)$ and
$\Omega=(\Omega_{1},\Omega_{2},\Omega_{3})\in\mathfrak{so}(3)$,
$\Gamma\in\mathbb{R}^{3}$, $\theta=(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$,
$\dot{\theta}=(\dot{\theta}_{1},\dot{\theta}_{2})\in\mathbb{R}^{2}$. If we
introduce the conjugate angular momentum, which is given by
$\Pi_{i}=\dfrac{\partial
L}{\partial\Omega_{i}}=\bar{I}_{i}\Omega_{i}+J_{i}(\Omega_{i}+\dot{\theta}_{i}),\;i=1,2,$
$\Pi_{3}=\dfrac{\partial L}{\partial\Omega_{3}}=\bar{I}_{3}\Omega_{3},\quad
l_{i}=\dfrac{\partial
L}{\partial\dot{\theta}_{i}}=J_{i}(\Omega_{i}+\dot{\theta}_{i}),\;i=1,2,$
and by the Legendre transformation
$FL:TQ\cong\textmd{SE}(3)\times\mathfrak{se}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to
T^{\ast}Q\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2},\quad(A,v,\Omega,\Gamma,\theta,\dot{\theta})\to(A,v,\Pi,\Gamma,\theta,l),$
where $\Pi=(\Pi_{1},\Pi_{2},\Pi_{3})\in\mathfrak{so}^{\ast}(3)$,
$l=(l_{1},l_{2})\in\mathbb{R}^{2}$, we have the Hamiltonian
$H(A,v,\Pi,\Gamma,\theta,l):T^{\ast}Q\cong\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$
given by
$\displaystyle H(A,v,\Pi,\Gamma,\theta,l)=\Omega\cdot\Pi+\dot{\theta}\cdot
l-L(A,v,\Omega,\Gamma,\theta,\dot{\theta})$
$\displaystyle=\bar{I}_{1}\Omega_{1}^{2}+J_{1}(\Omega_{1}^{2}+\Omega_{1}\dot{\theta}_{1})+\bar{I}_{2}\Omega_{2}^{2}+J_{2}(\Omega_{2}^{2}+\Omega_{2}\dot{\theta}_{2})+\bar{I}_{3}\Omega_{3}^{2}+J_{1}(\dot{\theta}_{1}\Omega_{1}+\dot{\theta}_{1}^{2})$
$\displaystyle\quad+J_{2}(\dot{\theta}_{2}\Omega_{2}+\dot{\theta}_{2}^{2})-\frac{1}{2}[\bar{I}_{1}\Omega_{1}^{2}+\bar{I}_{2}\Omega_{2}^{2}+\bar{I}_{3}\Omega_{3}^{2}+J_{1}(\Omega_{1}+\dot{\theta}_{1})^{2}+J_{2}(\Omega_{2}+\dot{\theta}_{2})^{2}]+mgh\Gamma\cdot\chi$
$\displaystyle=\frac{1}{2}[\frac{(\Pi_{1}-l_{1})^{2}}{\bar{I}_{1}}+\frac{(\Pi_{2}-l_{2})^{2}}{\bar{I}_{2}}+\frac{\Pi_{3}^{2}}{\bar{I}_{3}}+\frac{l_{1}^{2}}{J_{1}}+\frac{l_{2}^{2}}{J_{2}}]+mgh\Gamma\cdot\chi.$
From the above expression of the Hamiltonian, we know that
$H(A,v,\Pi,\Gamma,\theta,l)$ is invariant under the left
$\textmd{SE}(3)$-action $\Phi:\textmd{SE}(3)\times T^{\ast}Q\to T^{\ast}Q$.
For the case $(\Pi_{0},\Gamma_{0})=(\mu,a)\in\mathfrak{se}^{\ast}(3)$ is the
regular value of $\mathbf{J}_{Q}$, we have the reduced Hamiltonian
$h_{(\mu,a)}(\Pi,\Gamma,\theta,l):\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}(\subset\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2})\to\mathbb{R}$
given by
$h_{(\mu,a)}(\Pi,\Gamma,\theta,l)=H(A,v,\Pi,\Gamma,\theta,l)|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}$.
From the heavy top Poisson bracket on $\mathfrak{se}^{\ast}(3)$ and the
Poisson bracket on $T^{\ast}\mathbb{R}^{2}$, we can get the Poisson bracket on
$T^{\ast}Q$, that is, for
$F,K:\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R},$
we have that
$\displaystyle\\{F,K\\}_{-}(\Pi,\Gamma,\theta,l)$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}F\times\nabla_{\Pi}K)-\Gamma\cdot(\nabla_{\Pi}F\times\nabla_{\Gamma}K-\nabla_{\Pi}K\times\nabla_{\Gamma}F)$
$\displaystyle\quad+\\{F,K\\}_{V}(\theta,l).$
In particular, for
$F_{(\mu,a)},K_{(\mu,a)}:\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$,
we have that
$\tilde{\omega}_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}^{-}(X_{F_{(\mu,a)}},X_{K_{(\mu,a)}})=\\{F_{(\mu,a)},K_{(\mu,a)}\\}_{-}|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}.$
Moreover, for reduced Hamiltonian
$h_{(\mu,a)}(\Pi,\Gamma):\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$,
we have the Hamiltonian vector field
$X_{h_{(\mu,a)}}(K_{(\mu,a)})=\\{K_{(\mu,a)},h_{(\mu,a)}\\}_{-}|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}},$
and hence we have that
$\displaystyle\frac{\mathrm{d}\Pi}{\mathrm{d}t}$
$\displaystyle=X_{h_{(\mu,a)}}(\Pi)(\Pi,\Gamma,\theta,l)=\\{\Pi,h_{(\mu,a)}\\}_{-}(\Pi,\Gamma,\theta,l)$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Pi}h_{(\mu,a)})-\Gamma\cdot(\nabla_{\Pi}\Pi\times\nabla_{\Gamma}h_{(\mu,a)}-\nabla_{\Pi}h_{(\mu,a)}\times\nabla_{\Gamma}\Pi)$
$\displaystyle\quad+\sum_{i=1}^{2}(\frac{\partial\Pi}{\partial\theta_{i}}\frac{\partial
h_{(\mu,a)}}{\partial l_{i}}-\frac{\partial
h_{(\mu,a)}}{\partial\theta_{i}}\frac{\partial\Pi}{\partial l_{i}})$
$\displaystyle=\Pi\times\Omega-
mgh\chi\times\Gamma=\Pi\times\Omega+mgh\Gamma\times\chi,$
$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}t}$
$\displaystyle=X_{h_{(\mu,a)}}(\Gamma)(\Pi,\Gamma,\theta,l)=\\{\Gamma,h_{(\mu,a)}\\}_{-}(\Pi,\Gamma,\theta,l)$
$\displaystyle=-\Pi\cdot(\nabla_{\Pi}\Gamma\times\nabla_{\Pi}h_{(\mu,a)})-\Gamma\cdot(\nabla_{\Pi}\Gamma\times\nabla_{\Gamma}h_{(\mu,a)}-\nabla_{\Pi}h_{(\mu,a)}\times\nabla_{\Gamma}\Gamma)$
$\displaystyle\quad+\sum_{i=1}^{2}(\frac{\partial\Gamma}{\partial\theta_{i}}\frac{\partial
h_{(\mu,a)}}{\partial l_{i}}-\frac{\partial
h_{(\mu,a)}}{\partial\theta_{i}}\frac{\partial\Gamma}{\partial l_{i}})$
$\displaystyle=\nabla_{\Gamma}\Gamma\cdot(\Gamma\times\nabla_{\Pi}h_{(\mu,a)})=\Gamma\times\Omega,$
since
$\nabla_{\Pi}\Pi=1,\;\nabla_{\Gamma}\Gamma=1,\;\nabla_{\Gamma}\Pi=\nabla_{\Pi}\Gamma=0$,
$\nabla_{\Pi}h_{(\mu,a)}=\Omega$, and
$\frac{\partial\Pi}{\partial\theta_{i}}=\frac{\partial\Gamma}{\partial\theta_{i}}=\frac{\partial
h_{(\mu,a)}}{\partial\theta_{i}}=0,\;i=1,2$. If we consider the heavy top-
rotor system with a control torque $u:T^{\ast}Q\to T^{\ast}Q$ acting on the
rotors, and $u$ is invariant under the left $\textmd{SE}(3)$-action, and its
reduced control torque
$u_{(\mu,a)}:\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}$
is given by
$u_{(\mu,a)}(\Pi,\Gamma,\theta,l)=u(A,v,\Pi,\Gamma,\theta,l)|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}.$
Thus, the equations of motion for heavy top-rotor system with the control
torque $u$ acting on the rotors are given by
$\left\\{\begin{aligned}
&\frac{\mathrm{d}\Pi}{\mathrm{d}t}=\Pi\times\Omega+mgh\Gamma\times\chi,\\\
&\frac{\mathrm{d}\Gamma}{\mathrm{d}t}=\Gamma\times\Omega,\\\
&\frac{\mathrm{d}l}{\mathrm{d}t}=\mbox{vlift}(u_{(\mu,a)}).\end{aligned}\right.$
(30)
where $\mbox{vlift}(u_{(\mu,a)})\in
T(\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}).$ To sum up
the above discussion, we have the following proposition.
###### Proposition 6.9
The 5-tuple
$(T^{\ast}(\textmd{SE}(3)\times\mathbb{R}^{2}),\textmd{SE}(3),\omega_{0},H,u)$
is a regular point reducible RCH system. For a point
$(\mu,a)\in\mathfrak{se}^{\ast}(3)$, the regular value of the momentum map
$\mathbf{J}:\textmd{SE}(3)\times\mathfrak{se}^{\ast}(3)\times\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathfrak{se}^{\ast}(3)$,
the $R_{P}$-reduced system is the 4-tuple
$(\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2},\\\
\tilde{\omega}^{-}_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}},\;h_{(\mu,a)},\;u_{(\mu,a)})$,
where $\mathcal{O}_{(\mu,a)}\subset\mathfrak{se}^{\ast}(3)$ is the coadjoint
orbit,
$\tilde{\omega}^{-}_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}}$
is orbit symplectic form on
$\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}$,
$h_{(\mu,a)}(\Pi,\Gamma,\theta,l)=H(A,v,\Pi,\Gamma,\theta,l)|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}},$
and
$u_{(\mu,a)}(\Pi,\Gamma,\theta,l)=u(A,v,\Pi,\Gamma,\theta,l)|_{\mathcal{O}_{(\mu,a)}\times\mathbb{R}^{2}\times\mathbb{R}^{2}},$
and its equations of motion are given by (30).
(5). Regular Controlled Hamiltonian Equivalence.
In the following we may consider the RCH-equivalences of the rigid body with
external force torques and that with internal rotors, as well as the heavy top
and that with internal rotors. We can choose the feedback control law such
that the equivalent RCH systems produce the same equations of motion (up to a
diffeomorphism ).
At first, we consider the RCH-equivalence between the rigid body with external
force torques and that with internal rotors. Now let us choose the feedback
control laws such that the closed-loop systems are Hamiltonian and retains the
symmetry. If we choose the feedback control law $u$, such that
$\mbox{vlift}(u_{\mu})=p\times\Omega$, where $p$ is a constant vector, from
the equations (27) of motion for the rigid body with the
$\textmd{SO}(3)$-invariant external force torque $u$, we have that
$\dfrac{\mathrm{d}\Pi}{\mathrm{d}t}=\Pi\times\Omega+p\times\Omega.$ (31)
On the other hand, for the rigid body with internal rotors, we choose the
feedback control law $u$, such that
$\mbox{vlift}(u_{\mu})=k(\Pi\times\Omega)$, where $k$ is a gain parameter.
From the equations (28) of motion for the rigid body with internal rotors, we
have that
$\frac{\mathrm{d}l}{\mathrm{d}t}=\mbox{vlift}(u_{\mu})=k\frac{\mathrm{d}\Pi}{\mathrm{d}t}$,
and by solving the integrable equation, we get that $l-k\Pi=p$, where $p$ is a
constant vector. Assuming that $N=\Pi-l=\Pi-k\Pi-p=(1-k)\Pi-p$, then we have
that
$\frac{\mathrm{d}N}{\mathrm{d}t}=\frac{\mathrm{d}\Pi}{\mathrm{d}t}-\frac{\mathrm{d}l}{\mathrm{d}t}=(1-k)\Pi\times\Omega=N\times\Omega+p\times\Omega.$
(32)
By comparing (31) and (32) we know that the rigid body with external force
torque and that with internal rotors are RCH-equivalent by a diffeomorphism
$\varphi:\mathfrak{so}^{\ast}(3)\rightarrow\mathfrak{so}^{\ast}(3),\Pi\rightarrow
N$. In particular, if we take that $\mbox{vlift}(u_{\mu})=(u_{\mu 1},u_{\mu
2},u_{\mu
3})=(0,0,-\varepsilon\frac{I_{1}-I_{2}}{I_{1}I_{2}}\Pi_{1}\Pi_{2})\in\mathbb{R}^{3}$,
we recover the result in Bloch et al. [6], also see Marsden [20].
Next, we consider the RCH-equivalence between the rigid body with internal
rotors and heavy top. If assuming that $N=\Pi+\Gamma$, from the equations (29)
of motion for the heavy top, we have that
$\frac{\mathrm{d}N}{\mathrm{d}t}=N\times\Omega+mgh\Gamma\times\chi=N\times\Omega-
mgh\chi\times\Gamma$
Thus, take that $\Gamma=\lambda\Omega$ and $p=-mgh\lambda\chi$, where
$\lambda$ is a constant, then
$\frac{\mathrm{d}N}{\mathrm{d}t}=N\times\Omega+p\times\Omega.$ (33)
In this case, by comparing (32) and (33) we know that the heavy top and the
rigid body with internal rotors are RCH-equivalent. In the same way, from (31)
we know that the rigid body with the external force torques and the heavy top
are also RCH-equivalent. Also see Holm and Marsden [15].
At last, we consider the RCH-equivalence between the rigid body with internal
rotors and heavy top with internal rotors. For the heavy top with internal
rotors, we choose the feedback control law $u$, such that
$\mbox{vlift}(u_{(\mu,a)})=k(\Gamma\times\Omega)$, where $k$ is a gain
parameter. From the equations (30) of motion for the heavy top with internal
rotors, we have that
$\frac{\mathrm{d}\bar{l}}{\mathrm{d}t}=\mbox{vlift}(u_{(\mu,a)})=k\frac{\mathrm{d}\Gamma}{\mathrm{d}t}$,
where $\bar{l}=(l_{1},l_{2},0)$, and by solving the integrable equation, we
get that $\bar{l}-k\Gamma=p_{0}$, where $p_{0}$ is a constant vector. Assuming
that $N=\Pi+\Gamma-\bar{l}=\Pi+(1-k)\Gamma-p_{0}$, then we have that
$\frac{\mathrm{d}N}{\mathrm{d}t}=\frac{\mathrm{d}\Pi}{\mathrm{d}t}+\frac{\mathrm{d}\Gamma}{\mathrm{d}t}-\frac{\mathrm{d}\bar{l}}{\mathrm{d}t}=\Pi\times\Omega+(1-k)\Gamma\times\Omega-
mgh\chi\times\Gamma=N\times\Omega+p_{0}\times\Omega-mgh\chi\times\Gamma.$
Thus, take that $\Gamma=\lambda\Omega$ and $p=p_{0}-mgh\lambda\chi$, where
$\lambda$ is a constant, then
$\frac{\mathrm{d}N}{\mathrm{d}t}=N\times\Omega+p\times\Omega.$ (34)
In this case, by comparing (32) and (34) we know that the rigid body with
internal rotors and the heavy top with internal rotors are RCH-equivalent.
To sum up, we have the following theorem.
###### Theorem 6.10
As two $R_{P}$-reduced RCH systems,
(i) the rigid body with external force torque and that with internal rotors
are RCH-equivalent;
(ii) the rigid body with internal rotors (or external force torque) and the
heavy top are RCH-equivalent;
(iii) the rigid body with internal rotors and the heavy top with internal
rotors are RCH-equivalent.
### 6.3 Port Hamiltonian System with a Symplectic Structure
In order to understand well the abstract definition of RCH system and the RCH-
equivalence, in this subsection we will describe the RCH system and RCH-
equivalence from the viewpoint of port Hamiltonian system with a symplectic
structure. Recently years, the study of stability analysis and control of port
Hamiltonian systems and their applications have become more and more
important, and there have been a lot of beautiful results; see Dalsmo and van
der Schaft [13], van der Schaft [30, 31]. To describe the RCH systems well
from the viewpoint of port Hamiltonian system, in the following we first give
some relevant definitions and basic facts about the port Hamiltonian systems.
###### Definition 6.11
Let $(T^{\ast}Q,\omega)$ be a symplectic manifold and $\omega$ be the
canonical symplectic form on $T^{\ast}Q$. Assume that
$H:T^{\ast}Q\rightarrow\mathbb{R}$ is a Hamiltonian, and there exists a subset
$U\subset T^{\ast}Q$ and a vector field $X_{H}\in TT^{\ast}Q$ on $T^{\ast}Q$
such that $i_{X_{H}}\omega(z)=\mathbf{d}H(z),\;\forall z\in U$, then the
triple $(T^{\ast}Q,\omega,H)$ is a Hamiltonian system defined on the set $U$.
Assume that $V\subset T^{\ast}Q$ is a subset of $T^{\ast}Q$, and
$P=(Y,\alpha)$, where for any $z\in V$, $Y(z)\in T_{z}T^{\ast}Q$ and
$\alpha(z)\in T^{\ast}_{z}T^{\ast}Q$. If $U\cap V\neq\emptyset$, and
$i_{(X_{H}+Y)}\omega(z)=(\mathbf{d}H+\alpha)(z),\;\forall z\in U\cap V$, then
$P=(Y,\alpha)$ is called a port of the Hamiltonian system
$(T^{\ast}Q,\omega,H)$ defined on the set $U$. The 4-tuple
$(T^{\ast}Q,\omega,H,P)$ is called a port Hamiltonian system.
For the port Hamiltonian system $(T^{\ast}Q,\omega,H,P)$, since
$i_{X_{H}}\omega(z)=\mathbf{d}H(z),\;\forall z\in U$, from
$i_{(X_{H}+Y)}\omega(z)\\\ =(\mathbf{d}H+\alpha)(z),\;\forall z\in U\cap V$,
we have that $i_{X_{H}}\omega(z)+i_{Y}\omega(z)=\mathbf{d}H(z)+\alpha(z).$
Thus, we can get the port balance condition that $P=(Y,\alpha)$ is a port of
the Hamiltonian system $(T^{\ast}Q,\omega,H)$ as follows
$i_{Y}\omega(z)=\alpha(z),\;\;\;\;\forall z\in U\cap V.$ (35)
In particular, for $U=V=T^{\ast}Q$, from the port balance condition (35) we
know that $P=(X_{H},\mathbf{d}H)$ is a trivial port of the Hamiltonian system
$(T^{\ast}Q,\omega,H)$.
Assume that $(T^{\ast}Q,\omega,H,F,u)$ is a RCH system with a control law $u$.
We can take that $Y=\textnormal{vlift}(F+u)\in TT^{\ast}Q$, from the port
balance condition (35) we take that $\alpha=i_{Y}\omega\in T^{\ast}T^{\ast}Q$,
then $P=(Y,\alpha)$ is a force-controlled port of the Hamiltonian system
$(T^{\ast}Q,\omega,H)$, and $(T^{\ast}Q,\omega,H,P)$ is a port Hamiltonian
system with a symplectic structure. Thus, we have the following proposition.
###### Proposition 6.12
Any RCH system $(T^{\ast}Q,\omega,H,F,u)$ with control law $u$, is a port
Hamiltonian system with symplectic structure.
If we consider the canonical coordinates $z=(q,p)$ of the phase space
$T^{\ast}Q$, then $X_{H}=(\dot{q},\dot{p})$, and the local expression of the
RCH system is given by
$\dot{q}=\frac{\partial H}{\partial
p}(q,p),\;\;\;\;\;\;\dot{p}=-\frac{\partial H}{\partial
q}(q,p)+\textnormal{vlift}(F+u)(q,p).$ (36)
We can derive the energy balance condition, that is,
$\frac{dH}{dt}=(\frac{\partial H}{\partial q})^{T}(q,p)\dot{q}+(\frac{\partial
H}{\partial p})^{T}(q,p)\dot{p}=(\frac{\partial H}{\partial
p})^{T}\textnormal{vlift}(F+u)(q,p)=\dot{q}^{T}\textnormal{vlift}(F+u)(q,p),$
(37)
which expresses that the increase in energy of the system is equal to the
supplied work (that is, conservation of energy). This motivates to define the
output of the system as $e=\dot{q}$, which is considered as the vector of
generalized velocities, and the local expression of the port controlled
Hamiltonian system is given by
$\dot{q}=\frac{\partial H}{\partial p}(q,p),\;\;\;\;\dot{p}=-\frac{\partial
H}{\partial q}(q,p)+B(q)f,\;\;\;\;e=B^{T}(q)\dot{q}.$ (38)
where $\textnormal{vlift}(F+u)=B(q)f$, and $f$ is a input of system; see van
der Schaft [30, 31].
In the following we shall state the relationships between RCH-equivalence of
RCH systems and the equivalence of port Hamiltonian systems. We first give the
definitions of equivalence of Hamiltonian systems, port-equivalence of port
Hamiltonian systems and equivalence of port Hamiltonian systems as follows.
Assume that $(T^{\ast}Q_{i},\omega_{i}),\;i=1,2,$ are two symplectic
manifolds, and $\psi:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$ is a symplectic
diffeomorphism. Let $T\psi:TT^{\ast}Q_{1}\rightarrow TT^{\ast}Q_{2}$ be the
tangent map of $\psi:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$, and
$\psi_{\ast}=(\psi^{-1})^{\ast}:T^{\ast}T^{\ast}Q_{1}\rightarrow
T^{\ast}T^{\ast}Q_{2}$ be the cotangent map of
$\psi^{-1}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$. Then we can describe the
equivalence of the Hamiltonian systems as follows.
###### Definition 6.13
Assume that $(T^{\ast}Q_{i},\omega_{i},H_{i}),\;i=1,2,$ are two Hamiltonian
systems. We say them to be equivalent, if there exists a symplectic
diffeomorphism $\psi:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$, such that
$T\psi(X_{H_{1}})=X_{H_{2}}\cdot\psi,\;\psi_{\ast}(\mathbf{d}H_{1})=\mathbf{d}H_{2}\cdot\psi$,
where $i_{X_{H_{i}}}\omega=\mathbf{d}H_{i},\;i=1,2.$
Moreover, we can describe the port-equivalence of port Hamiltonian systems and
the equivalence of port Hamiltonian systems as follows.
###### Definition 6.14
Assume that $(T^{\ast}Q_{i},\omega_{i},H_{i},P_{i}),\;i=1,2,$ are two port
Hamiltonian systems. We say them to be port -equivalent, if there exists a
diffeomorphism $\psi:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$, such that
$T\psi(Y_{1})=Y_{2}\cdot\psi,\;\psi_{\ast}(\alpha_{1})=\alpha_{2}\cdot\psi$,
where $P_{i}=(Y_{i},\alpha_{i})$, and for any $z_{i}\in V_{i}(\subset
T^{\ast}Q_{i})$, $Y_{i}(z_{i})\in T_{z_{i}}T^{\ast}Q_{i}$ and
$\alpha_{i}(z_{i})\in T^{\ast}_{z_{i}}T^{\ast}Q_{i}$, $i=1,2.$ Furthermore, we
say two port Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},P_{i}),\;i=1,2,$ to be equivalent, if there
exists a diffeomorphism $\psi:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$, such
that not only two Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i}),\;i=1,2,$ are equivalent, but also their
ports are equivalent.
Thus, we can obtain the following theorem.
###### Theorem 6.15
(i) If two RCH systems $(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2,$
are RCH-equivalent and their associated Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i})$, $i=1,2,$ are also equivalent, then they
must be equivalent for port Hamiltonian systems.
(ii) If two RCH systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2,$ are RCH-equivalent,
but the associated Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i}),\;i=1,2,$ are not equivalent, then we can
choose the control law $u_{i}$, such that they are port-equivalent for port
Hamiltonian systems.
Proof. (i) In fact, assume that two RCH systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2,$ are RCH-equivalent,
then there exists a diffeomorphism $\varphi:Q_{1}\rightarrow Q_{2}$, such that
$\varphi^{\ast}:T^{\ast}Q_{2}\rightarrow T^{\ast}Q_{1}$ is symplectic, and
from Theorem 3.3 there exist two control laws $u_{i}:T^{\ast}Q_{i}\rightarrow
W_{i},\;i=1,2,$ such that the two associated closed-loop systems produce the
same equations of motion, that is,
$X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}\cdot\varphi^{\ast}=T\varphi^{\ast}X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}$.
If the associated Hamiltonian systems $(T^{\ast}Q_{i},\omega_{i},H_{i}),$
$i=1,2$ are also equivalent, from
$\varphi_{\ast}=(\varphi^{-1})^{\ast}:T^{\ast}Q_{1}\rightarrow T^{\ast}Q_{2}$
is symplectic, and $T\varphi_{\ast}(X_{H_{1}})=X_{H_{2}}\cdot\varphi_{\ast}$,
and $X_{H_{i}}=(\mathbf{d}H_{i})^{\sharp},\;i=1,2,$ we have that
$T\varphi^{\ast}(\mathbf{d}H_{2})^{\sharp}=(\mathbf{d}H_{1})^{\sharp}\cdot\varphi^{\ast}$.
Note that
$X_{(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},u_{i})}=(\mathbf{d}H_{i})^{\sharp}+\textnormal{vlift}(F_{i})+\textnormal{vlift}(u_{i}),\;i=1,2,$
then,
$T\varphi^{\ast}(\textnormal{vlift}(F_{2})+\textnormal{vlift}(u_{2}))=(\textnormal{vlift}(F_{1})+\textnormal{vlift}(u_{1}))\cdot\varphi^{\ast}$.
We can first take that $Y_{i}=\textnormal{vlift}(F_{i}+u_{i})\in
TT^{\ast}Q_{i},\;i=1,2,$ then we have that
$T\varphi^{\ast}(Y_{2})=Y_{1}\cdot\varphi^{\ast}$, and hence
$T\varphi_{\ast}(Y_{1})=Y_{2}\cdot\varphi_{\ast}$. Then we take that
$\alpha_{i}=i_{Y_{i}}\omega_{i}\in T^{\ast}T^{\ast}Q_{i},\;i=1,2.$ Since the
map
$(\varphi_{\ast})_{\ast}=(\varphi_{\ast}^{-1})^{\ast}:T^{\ast}T^{\ast}Q_{1}\rightarrow
T^{\ast}T^{\ast}Q_{2}$, such that
$(\varphi_{\ast})_{\ast}(i_{Y_{1}}\omega_{1})=i_{T\varphi_{\ast}(Y_{1})}(\varphi_{\ast})_{\ast}(\omega_{1})=i_{Y_{2}}\omega_{2}\cdot\varphi_{\ast}$,
we have that
$(\varphi_{\ast})_{\ast}(\alpha_{1})=\alpha_{2}\cdot\varphi_{\ast}.$ Thus, the
ports $P_{i}=(Y_{i},\alpha_{i})$, satisfying
$T\varphi_{\ast}(Y_{1})=Y_{2}\cdot\varphi_{\ast}$, and
$(\varphi_{\ast})_{\ast}(\alpha_{1})=\alpha_{2}\cdot\varphi_{\ast}$, are
equivalent, and hence the port Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},P_{i}),\;i=1,2,$ are equivalent.
(ii) Assume that two RCH systems
$(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},W_{i}),\;i=1,2,$ are RCH-equivalent,
but the associated Hamiltonian systems
$(T^{\ast}Q_{i},\omega_{i},H_{i}),\;i=1,2,$ are not equivalent, from Theorem
3.3 we can choose the control law $u_{i}:T^{\ast}Q_{i}\rightarrow
W_{i},\;i=1,2,$ such that $T(\varphi^{\ast})\cdot
X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}=X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}\cdot\varphi^{\ast}$,
and hence $T(\varphi_{\ast})\cdot
X_{(T^{\ast}Q_{1},\omega_{1},H_{1},F_{1},u_{1})}=X_{(T^{\ast}Q_{2},\omega_{2},H_{2},F_{2},u_{2})}\cdot\varphi_{\ast}$.
We can take that
$Y_{i}=X_{(T^{\ast}Q_{i},\omega_{i},H_{i},F_{i},u_{i})}=(\mathbf{d}H_{i})^{\sharp}+\textnormal{vlift}(F_{i})+\textnormal{vlift}(u_{i})\in
TT^{\ast}Q_{i},$ and $\alpha_{i}=i_{Y_{i}}\omega_{i}\in
T^{\ast}T^{\ast}Q_{i},$ $i=1,2$. Then the ports
$P_{i}=(Y_{i},\alpha_{i}),\;i=1,2,$ satisfy that
$T\varphi_{\ast}(Y_{1})=Y_{2}\cdot\varphi_{\ast}$, and
$(\varphi_{\ast})_{\ast}(\alpha_{1})=\alpha_{2}\cdot\varphi_{\ast}$, and hence
the port Hamiltonian systems $(T^{\ast}Q_{i},\omega_{i},H_{i},P_{i}),$
$i=1,2,$ are port-equivalent. $\blacksquare$
The mechanical control system theory is a very important subject. In this
paper, we study the regular reduction theory of controlled Hamiltonian systems
with the symplectic structure and symmetry. It is a natural problem what and
how we could do, if we define a controlled Hamiltonian system on the cotangent
bundle $T^{*}Q$ by using a Poisson structure, and if symplectic reduction
procedure does not work or is not efficient enough. Wang and Zhang in [32]
study the optimal reduction theory of controlled Hamiltonian systems with
Poisson structure and symmetry by using the optimal momentum map.
Acknowledgments: J.E. Marsden’s research was partially supported by NSF Grant.
H. Wang’s research was partially supported by the Natural Science Foundation
of Tianjin (05YFJMJC01200) and the Key Laboratory of Pure Mathematics and
Combinatorics, Ministry of Education, China.
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jerrold E. Marsden (California Institute of Technology), Hong Wang\n (Nankai University), Zhen-Xing Zhang (Nankai University)",
"submitter": "Hong Wang",
"url": "https://arxiv.org/abs/1202.3564"
}
|
1202.3606
|
11institutetext: Fehmi Ekmekçi, Lale Çelik, H. Volkan Şenavcı 22institutetext:
Ankara University, Faculty of Science, Department of Astronomy and Space
Sciences, 06100, Tandoğan, Ankara, Turkey, 22email:
fekmekci@science.ankara.edu.tr
# RR Lyrae type stars, ST Boo and RR Leo: 2007 Observations and the
preliminary results of the frequency analaysis
Fehmi EKMEKÇİ Lale ÇELİK H. Volkan ŞENAVCI
###### Abstract
We present BVR light curves of pulsating stars, ST Boo and RR Leo, obtained
between March and September 2007 at the Ankara University Observatory (AUG)
and the TÜBİTAK National Observatory (TUG). Although these observational data
are insufficient to obtain the reliable results for a frequency analysis of ST
Boo and RR Leo stars, in this study, we tried to investigate the pulsation
phenomena of these two stars, as an overview, using the Period04 software
package. As preliminary results, we present the possible frequencies for ST
Boo and RR Leo.
## 1 Observations and Results
CCD observations of ST Boo and RR Leo were carried out by using an Apogee ALTA
$U47+CCD$ camera ($1024\times 1024$ pixels) with BVR filters mounted on both
40 cm Schmidt-Cassegrain telescopes of the Ankara University Observatory (AUG)
and the TÜBİTAK National Observatory (TUG) between March and September 2007.
BVR light curves of both ST Boo and RR Leo were normalized to maximum light
level to construct the data set for simultaneous multiple-frequency analysis
using Period04 (V 1.0) (Lenz05 ) which has a Fourier analysis definition of
$f(t)=Z+\sum_{i}A_{i}sin(2\pi(\Omega_{i}t+\phi_{i})).$ (1)
The results of multi-frequency solutions, with their errors calculated based
on Monte Carlo Simulation, for ST Boo and RR Leo are given in Table 1. Fig. 1
shows some of the light curve data with the fit curve of multi-frequency
solutions for ST Boo and RR Leo. Clearly, it must be included more and more
photometric data in the frequency analysis to have more definite and reliable
results for both of these pulsating stars.
Table 1: The results of multiple-frequency analysis of ST Boo and RR Leo
| ST Boo | | | |
---|---|---|---|---|---
f($cd^{-1}$) | Amp.(mag.) | S/N | f($cd^{-1}$) | Amp.(mag.) | S/N
$4.201\pm 0.079$ | $0.553\pm 0.664$ | 2.84 | $18.925\pm 0.663$ | $0.012\pm 0.022$ | 32.02
$4.221\pm 0.668$ | $0.171\pm 0.754$ | 617.14 | $28.315\pm 0.845$ | $0.011\pm 0.012$ | 19.93
$0.187\pm 0.005$ | $0.127\pm 0.209$ | 455.01 | $16.236\pm 0.342$ | $0.010\pm 0.033$ | 19.81
$7.311\pm 0.034$ | $0.119\pm 0.072$ | 538.73 | $20.333\pm 0.695$ | $0.008\pm 0.026$ | 25.20
$6.337\pm 0.027$ | $0.097\pm 0.071$ | 419.63 | $33.717\pm 0.370$ | $0.008\pm 0.004$ | 9.53
$10.247\pm 2.389$ | $0.088\pm 0.143$ | 398.35 | $28.724\pm 0.176$ | $0.007\pm 0.009$ | 12.22
$10.649\pm 0.504$ | $0.085\pm 0.178$ | 365.89 | $23.532\pm 0.418$ | $0.006\pm 0.014$ | 10.29
$7.500\pm 0.082$ | $0.076\pm 0.114$ | 348.95 | $602.962\pm 0.856$ | $0.005\pm 0.002$ | 5.96
$17.073\pm 0.093$ | $0.043\pm 0.055$ | 87.02 | $35.863\pm 0.108$ | $0.005\pm 0.006$ | 5.00
$0.227\pm 0.060$ | $0.037\pm 0.560$ | 133.81 | $604.857\pm 0.239$ | $0.004\pm 0.002$ | 5.22
$10.875\pm 1.406$ | $0.033\pm 0.139$ | 138.95 | $510.716\pm 0.157$ | $0.004\pm 0.002$ | 5.53
$3.113\pm 0.540$ | $0.028\pm 0.099$ | 96.61 | $30.484\pm 1.174$ | $0.003\pm 0.006$ | 6.18
$21.491\pm 0.105$ | $0.024\pm 0.020$ | 65.53 | $568.959\pm 7.439$ | $0.003\pm 0.002$ | 4.91
$11.980\pm 0.313$ | $0.024\pm 0.043$ | 83.07 | $599.575\pm 0.112$ | $0.003\pm 0.003$ | 4.21
$14.403\pm 0.093$ | $0.024\pm 0.043$ | 49.74 | $566.738\pm 0.198$ | $0.003\pm 0.002$ | 5.00
$16.399\pm 3.804$ | $0.024\pm 0.055$ | 45.08 | $559.057\pm 0.150$ | $0.003\pm 0.002$ | 4.03
$6.465\pm 0.255$ | $0.021\pm 0.135$ | 90.25 | $47.909\pm 0.366$ | $0.003\pm 0.002$ | 4.28
$24.493\pm 2.307$ | $0.016\pm 0.020$ | 23.59 | $562.309\pm 7.929$ | $0.002\pm 0.002$ | 4.29
$21.501\pm 0.250$ | $0.012\pm 0.016$ | 31.06 | - | - | -
| RR Leo | | | |
f($cd^{-1}$) | Amp.(mag.) | S/N | f($cd^{-1}$) | Amp.(mag.) | S/N
$3.418\pm 0.001$ | $0.672\pm 0.041$ | 2.88 | $10.788\pm 0.003$ | $0.055\pm 0.016$ | 10.81
$8.634\pm 0.001$ | $0.229\pm 0.026$ | 35.15 | $19.478\pm 0.001$ | $0.049\pm 0.010$ | 14.42
$4.541\pm 0.206$ | $0.166\pm 0.060$ | 17.06 | $16.500\pm 0.239$ | $0.031\pm 0.017$ | 7.18
$13.469\pm 0.001$ | $0.132\pm 0.022$ | 28.57 | $21.518\pm 0.037$ | $0.031\pm 0.006$ | 9.97
$6.962\pm 0.057$ | $0.075\pm 0.038$ | 10.08 | $24.141\pm 0.001$ | $0.030\pm 0.005$ | 9.44
$2.259\pm 0.246$ | $0.062\pm 0.049$ | 4.04 | $27.494\pm 0.159$ | $0.016\pm 0.004$ | 4.48
Figure 1: An example of the light curve measurements of ST Boo (left panel)
and RR Leo (right panel) together with the fit curve of multi-frequency
solution. The axis of the time is in unit of HJD(2450000+…) and observed BVR
is in normalized values to maximum level of the light curve
## References
* (1) Lenz, P., Breger, M.: Period04 User Guide. Comm. in Asteroseismol. 146, 53–136 (2005)
|
arxiv-papers
| 2012-02-16T14:30:35 |
2024-09-04T02:49:27.489963
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fehmi Ekmek\\c{C}{\\.i}, Lale \\c{C}el{\\.i}k, H. Volkan \\c{S}enavci",
"submitter": "Fehmi Ekmek\\c{c}i",
"url": "https://arxiv.org/abs/1202.3606"
}
|
1202.3607
|
11institutetext: L. Çelik, F. Ekmekçi, H. V. Şenavcı 22institutetext: Ankara
Univ., Faculty of Science, Dept. of Astronomy and Space Sciences, 06100,
Tandoğan, Ankara, Turkey, 22email: lalecelik81@gmail.com, 33institutetext: J.
Nemec 44institutetext: Dept. of Physics & Astronomy, Camosun College,
Victoria, British Columbia, V8P 5J2, Canada, 55institutetext: K. Kolenberg
66institutetext: Harvard-Smithsonian Center for Astrophysics, 60 Garden St.,
Cambridge MA 02138 USA,
Instituut voor Sterrenkunde, Celestijnenlaan 200D, 3001 Heverlee, Belgium
77institutetext: J. Benkő, R. Szabó 88institutetext: Konkoly Obs. of the
Hungarian Academy of Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest,
Hungary, 99institutetext: D. KURTZ 1010institutetext: D. Kurtz
1111institutetext: Jeremiah Horrocks Institute, Univ. of Central Lancashire,
Preston PR1 2HE, 1212institutetext: K. Kinemuchi 1313institutetext: Bay Area
Environmental Research Inst./NASA Ames Research Center, MS 244-30, Moffet
Field, CA 94035, USA
# How to Correctly Stitch Together Kepler Data of a Blazhko Star
L. Çeli̇k F. Ekmekçi̇ J. Nemec K. Kolenberg J. M. Benkő R. Szabó D. W.
Kurtz K. Kinemuchi H. V. Şenavcı
###### Abstract
One of the most challenging difficulties that precedes the frequency analysis
of Kepler data for a Blazhko star is stitching together the data from
different seasons (quarters). We discuss the preliminary steps in the
stitching, detrending and rescaling process using the data for long-term
Blazhko stars. We present the process on Kepler data of a Blazhko star with a
variable Blazhko cycle and some first results of our analysis.
## 1 Stitching, Detrending and The Rescaling Process for Kepler
Several models have been proposed to explain the Blazhko effect (see e.g.,
Kolenberg2010 ) but it still remains a problem to be solved. An additional
difficulty for the analysis of Blazhko stars is that the data obtained in
subsequent quarters display some discrepancies in their flux values.
Therefore, when stitching together the light curves from different quarters,
these discrepancies must be removed.
To overcome the problems originating from Kepler itself and/or from the
“Automated Pipeline” routine, the users of the Kepler archive can use the
PyKEPyKE software. In this study, we applied the rescaling process to five
Blazhko stars. After stitching the data for all quarters, the most notable
difference is the flux offset between subsequent quarters originating mainly
from the instrumental effects (see left panel of Fig. 1). Our rescaling
process matches the light curves from consecutive quarters. This matching is
based on the assumption that phase-ordered light curves with a few cycles
closest to each other between two consecutive quarters must have nearly the
same flux values at the same phases. The first parameter is the period of the
star. Another parameter, the folding epoch, must be determined to carry out
the phase ordering process. During the matching process, the corresponding
flux values for the same phases between phase-ordered light curves of two
consecutive quarters are determined and proportioned. Therefore, the phase
scaling factors are determined for, and applied to short ranges of phase. The
right panel of Fig. 1 represents a simple diagram of this approach for the
rescaling process.
Figure 1: The light curve of a Blazhko star, using the data from Q1 up to Q7
quarters without rescaling procedure (left panel), and with rescaling
procedure (right panel)
###### Acknowledgements.
We thank M. E. TÖRÜN (MSc) for his assistance during the software improvements
and thank the entire Kepler team for the efforts which have made these results
possible.
## References
* (1) Kolenberg, K., Szabó, R., Kurtz, D. W., Gilliland, R. L. et al: First Kepler Results on RR Lyrae Stars. ApJL. 713, 198–203 (2010)
* (2) http://keplergo.arc.nasa.gov/ContributedSoftwarePyKEP.shtml.Cited15Aug2011
|
arxiv-papers
| 2012-02-16T14:39:20 |
2024-09-04T02:49:27.494653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. \\c{C}el\\.ik, F. Ekmek\\c{c}\\.i, J. Nemec, K. Kolenberg, J. M.\n Benk\\H{o}, R. Szab\\'o, D. W. Kurtz, K. Kinemuchi, H. V. \\c{S}enavc{\\i}",
"submitter": "Fehmi Ekmek\\c{c}i",
"url": "https://arxiv.org/abs/1202.3607"
}
|
1202.3833
|
# Observation of a pseudogap in the optical conductivity of underdoped
Ba1-xKxFe2As2
Y. M. Dai LPEM, ESPCI-ParisTech, CNRS, UPMC, 10 rue Vauquelin, F-75231 Paris
Cedex 5, France Beijing National Laboratory for Condensed Matter Physics,
National Laboratory for Superconductivity, Institute of Physics, Chinese
Academy of Sciences, P.O. Box 603, Beijing 100190, China B. Xu B. Shen
Beijing National Laboratory for Condensed Matter Physics, National Laboratory
for Superconductivity, Institute of Physics, Chinese Academy of Sciences, P.O.
Box 603, Beijing 100190, China H. H. Wen Beijing National Laboratory for
Condensed Matter Physics, National Laboratory for Superconductivity, Institute
of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
National Laboratory of Solid State Microstructures and Department of Physics,
Nanjing University, Nanjing 210093, China J. P. Hu Beijing National
Laboratory for Condensed Matter Physics, National Laboratory for
Superconductivity, Institute of Physics, Chinese Academy of Sciences, P.O. Box
603, Beijing 100190, China Department of Physics, Purdue University, West
Lafayette, Indiana 47907, USA X. G. Qiu xgqiu@aphy.iphy.ac.cn Beijing
National Laboratory for Condensed Matter Physics, National Laboratory for
Superconductivity, Institute of Physics, Chinese Academy of Sciences, P.O. Box
603, Beijing 100190, China R. P. S. M. Lobo lobo@espci.fr LPEM, ESPCI-
ParisTech, CNRS, UPMC, 10 rue Vauquelin, F-75231 Paris Cedex 5, France
(August 27, 2024)
###### Abstract
We report the observation of a pseudogap in the _ab_ -plane optical
conductivity of underdoped Ba1-xKxFe2As2 ($x=0.2$ and 0.12) single crystals.
Both samples show prominent gaps opened by a spin density wave (SDW) order and
superconductivity at the transition temperatures $T_{\it SDW}$ and $T_{c}$,
respectively. In addition, we observe an evident pseudogap below
$T^{\ast}\sim$ 75 K, a temperature much lower than $T_{\it SDW}$ but much
higher than $T_{c}$. A spectral weight analysis shows that the pseudogap is
closely connected to the superconducting gap, indicating the possibility of
its being a precursor of superconductivity. The doping dependence of the gaps
is also supportive of such a scenario.
###### pacs:
74.25.Gz, 74.70.Xa, 78.30.-j
Among all the families of iron-pnictide superconductors discovered to date,
Kamihara et al. (2008); Rotter et al. (2008a); Sefat et al. (2008); Li et al.
(2009); Torikachvili et al. (2008); Tapp et al. (2008); Hsu et al. (2008);
Sales et al. (2009) the BaFe2As2 (Ba122) family is one of the most studied.
The parent BaFe2As2 composition is a poor Pauli-paramagnetic metal with a
structural and magnetic phase transition at 140 K. Rotter et al. (2008b)
Superconductivity arises with the suppression of magnetism which can be
achieved by applying pressure Torikachvili et al. (2008) or chemical
substitution. Rotter et al. (2008a); Sefat et al. (2008); Li et al. (2009) The
substitution of Ba with K atoms yields hole-doping Rotter et al. (2008a) with
a maximum $T_{c}\approx 39$ K and the substitution of Fe atoms by Co or Ni
atoms results in electron-doping Sefat et al. (2008); Li et al. (2009) with a
maximum $T_{c}\approx 25$ K. Extensive studies have been carried out in the
parent BaFe2As2, Hu et al. (2008); Akrap et al. (2009) electron-doped
BaFe${}_{2-x}A_{x}$As2 ($A$ = Co, Ni), Lobo et al. (2010); Tu et al. (2010);
Teague et al. (2011); Terashima et al. (2009) as well as optimally hole-doped
Ba0.6K0.4Fe2As2 Li et al. (2008); Shan et al. (2011); Ding et al. (2008)
compounds. However, the hole-underdoped regime of the phase diagram is
relatively unexplored.
This hole-underdoped region is arguably the most important regime because of
the following two reasons. First, the superconducting mechanism is deeply tied
with magnetism. The interplay between magnetism and superconductivity is
manifest in this regime. In a considerably large portion of the underdoped
regime, the SDW phase and superconductivity coexist. Park et al. (2009); Goko
et al. (2009); Aczel et al. (2008); Massee et al. (2009); Chia et al. (2010)
Second, in cuprates, the most exciting, yet puzzling, physics takes place in
the hole-underdoped regime. This regime thus is pivotal to the comparison
between iron-pnictides and cuprates.
Xu _et al._ have performed the surface sensitive angle-resolved photoemission
(ARPES) measurements on underdoped Ba1-xKxFe2As2. Xu et al. (2011) Their data
showed a distinct pseudogap coexisting with the superconducting gap and
suggested that both the pseudogap and superconductivity are driven by
antiferromagnetic fluctuations. However, one key issue in understanding the
origin of the pseudogap and, in particular, its relation to superconductivity
is the question of whether it shares electronic states with the
superconducting condensation. Yu et al. (2008) Infrared spectroscopy probes
the charge dynamics of bulk materials and the spectral weight analysis is a
powerful tool to address this issue.
We present broadband infrared spectroscopy measurements on two underdoped
Ba1-xKxFe2As2 ($x=0.2$ and 0.12) single crystals. In both samples, the opening
of the SDW gap and the superconducting gap was clearly observed on the optical
conductivity. In addition, another small gap opens below $T^{\ast}\sim$ 75 K,
closely resembling the famous pseudogap in the hole-underdoped cuprates. We
find that the SDW gap depletes the spectral weight available for the
superconducting condensate, which suggests that the SDW order competes with
superconductivity. However, both doping and temperature dependence of the
spectral weight inside the pseudogap indicate that it shares the same
electronic origin with the superconducting gap.
High quality Ba1-xKxFe2As2 single crystals were grown by the self-flux method
using FeAs as the flux. Luo et al. (2008)
Figure 1: (color online) Left panel: Temperature dependence of the
resistivity of Ba1-xKxFe2As2 ($x=0.2$) single crystal (red solid line). A
steep superconducting transition can be seen at $T_{c}=19$ K. The blue solid
squares are values from the zero frequency extrapolation of the optical
conductivity. The inset shows the derivative of the resistivity $d\rho/dT$ as
a function of temperature. The sharp peak at 104 K in $d\rho/dT$ is associated
with the SDW transition. The right panel depicts the same curves for $x=0.12$
sample with $T_{c}=11$ K, and $T_{\it SDW}=121$ K.
The left panel of Fig. 1 shows the temperature dependence of the DC
resistivity [$\rho(T)$] for the Ba1-xKxFe2As2 ($x=0.2$) sample. The $\rho(T)$
curve is characterized by a steep superconducting transition at $T_{c}$ = 19
K. The inset shows the derivative of the resistivity $d\rho/dT$ as a function
of temperature. The SDW transition manifests itself as a sharp peak in
$d\rho/dT$ at $T_{\it SDW}=104$ K, which corresponds to a small kink on the
$\rho(T)$ curve. The right panel displays the same curves for the $x=0.12$
sample, which has $T_{c}=11$ K, and $T_{\it SDW}=121$ K.
The _ab_ -plane reflectivity [$R(\omega)$] was measured at a near-normal angle
of incidence on Bruker IFS113v and IFS66v/s spectrometers. An _in situ_ gold
overfilling technique Homes et al. (1993) was used to obtain the absolute
reflectivity of the samples. Data from 20 to 12000$\leavevmode\nobreak\
\textrm{cm}^{-1}$ were collected at 18 different temperatures from 5 to 300 K
on freshly cleaved surfaces. In order to use Kramers-Kronig analysis, we
extended the data to the visible and UV range (10000 to
55000$\leavevmode\nobreak\ \textrm{cm}^{-1}$) at room temperature with an
AvaSpec-2048 $\times$ 14 model fiber optic spectrometer.
Figure 2 shows the infrared reflectivity at selected temperatures for both
samples up to 1200$\leavevmode\nobreak\ \textrm{cm}^{-1}$. The inset in each
panel displays the reflectivity for the full measured range at 300 K. For the
$x=0.2$ sample, shown in the top panel, the reflectivity exhibits a metallic
response and approaches unity at zero frequency. Below $T_{\it SDW}=104$ K, a
substantial suppression of $R(\omega)$ at about 650$\leavevmode\nobreak\
\textrm{cm}^{-1}$ sets in and intensifies with the decreasing temperature.
Figure 2: (color online) Reflectivity of Ba1-xKxFe2As2 single crystals below
1200$\leavevmode\nobreak\ \textrm{cm}^{-1}$ at various temperatures. Top
panel: $x=0.2$; Bottom panel: $x=0.12$. Inset: Reflectivity of full measured
range at 300 K.
Simultaneously, the low frequency reflectivity continues increasing towards
unity. This is a signature of a partial SDW gap on the Fermi surface. Below 75
K, defined as $T^{\ast}$ here, another suppression of $R(\omega)$ appears in a
lower energy scale ($\sim 150\leavevmode\nobreak\ \textrm{cm}^{-1}$) signaling
the opening of a second partial gap (pseudogap) with a smaller value. Upon
crossing the superconducting transition, which occurs at $T_{c}$ = 19 K, the
reflectivity below $\sim 150\leavevmode\nobreak\ \textrm{cm}^{-1}$ increases
indicating the opening of a superconducting gap. Similar features are observed
on $R(\omega)$ for the $x=0.12$ sample as shown in the bottom panel of Fig. 2.
The real part of the optical conductivity $\sigma_{1}(\omega)$ was determined
by Kramers-Kronig analysis of the measured reflectivity.
Figure 3: (color online) Top panel: Optical conductivity of Ba1-xKxFe2As2
($x=0.2$) at selected temperatures. The inset shows the enlarged view of the
optical conductivity at low frequencies. The bottom panel displays the same
spectra for the $x=0.12$ sample.
Figure 3 shows $\sigma_{1}(\omega)$ at different temperatures for the two
samples. The zero frequency extrapolations of $\sigma_{1}(\omega)$ represent
the inverse dc resistivity of the sample, shown as blue solid squares in Fig.
1, which are in good agreement with the transport measurement. The top panel
of Fig. 3 shows $\sigma_{1}(\omega)$ for Ba1-xKxFe2As2 ($x=0.2$) below
1700$\leavevmode\nobreak\ \textrm{cm}^{-1}$. At 150 K and 125 K, hence above
$T_{\it SDW}$, a Drude-like metallic response dominates the low frequency
optical conductivity. Below $T_{\it SDW}$, $\sigma_{1}(\omega)$ below about
650 $\leavevmode\nobreak\ \textrm{cm}^{-1}$ is severely suppressed. Meanwhile,
it increases in a higher energy scale from 650 $\leavevmode\nobreak\
\textrm{cm}^{-1}$ to 1700 $\leavevmode\nobreak\ \textrm{cm}^{-1}$. The optical
conductivity for the normal state and that for the SDW state just below
$T_{\it SDW}$ show an intersection point at about 650 $\leavevmode\nobreak\
\textrm{cm}^{-1}$. As the temperature decreases, both the low energy spectral
suppression and the high energy bulge become stronger; and the intersection
point moves to a higher energy scale. This spectral evolution manifests the
behavior of the SDW gap in this material: transfer of low frequency spectral
weight to high frequencies. If we take the intersection points as an
estimative of the gap values, we can see that the gap increases with
decreasing temperature. Below $T^{\ast}\sim 75$ K, a second suppression in the
optical conductivity below roughly 110$\leavevmode\nobreak\ \textrm{cm}^{-1}$
with a bulge extending from about 110$\leavevmode\nobreak\ \textrm{cm}^{-1}$
to 250$\leavevmode\nobreak\ \textrm{cm}^{-1}$ sets in and develops with the
temperature decrease, implying the opening of the pseudogap on the Fermi
surface. The inset of Fig. 3 shows the enlarged view of the low temperature
optical conductivity at low frequencies, where the pseudogap is seen more
clearly. This pseudogap is unlikely due to the SDW transition as (i) it opens
at 75 K, well below $T_{\it SDW}$ and (ii) it redistributes spectral weight at
a different, smaller energy scale.
The superconducting transition at $T_{c}$ = 19 K implies the opening of a
superconducting gap. As shown in the inset of Fig. 3, this leads to the
reduction of the optical conductivity at low frequencies between 20 K and 5 K.
The spectral weight lost in the transition is recovered by the
$\delta(\omega)$ function at zero frequency representing the infinite DC
conductivity in the superconducting state. This $\delta(\omega)$ function is
not visible in the $\sigma_{1}(\omega)$ spectra, because only finite
frequencies are experimentally measured. Nevertheless, its weight can be
calculated from the imaginary part of the optical conductivity. Zimmers et al.
(2004); Dordevic et al. (2002) Note that, the spectral depletion in
$\sigma_{1}(\omega)$ due to the superconducting condensate extends up to
180$\leavevmode\nobreak\ \textrm{cm}^{-1}$. This is the same energy scale of
the pseudogap, hinting that the superconducting gap and the pseudogap share
the same electronic states, and may have the same origin.
In the $x=0.12$ sample, very similar features are observed, as shown in the
bottom panel of Fig. 3, but remarkable differences exist: (i) the SDW gap
opens at a higher temperature ($T_{\it SDW}=121$ K) and the gap value shifts
to a higher energy scale ($\sim 750\leavevmode\nobreak\ \textrm{cm}^{-1}$);
(ii) The low frequency spectral suppression due to the SDW gap is stronger,
indicating that a larger part of the Fermi surface is removed in the SDW
state; (iii) In contrast to the SDW gap, both the pseudogap and the
superconducting gap features are weaker in the more underdoped sample. The
evolution of the three gaps (SDW, pseudogap and superconducting) with doping
also suggests that the pseudogap and the superconducting gap may have a common
origin while the SDW is a competitive order to superconductivity.
In order to investigate the origin of the pseudogap and the relationship among
all gaps, we analyzed the data utilizing a restricted spectral weight, defined
as:
$SW_{\omega_{a}}^{\omega_{b}}=\int_{\omega_{a}}^{\omega_{b}}\sigma_{1}(\omega)d\omega,$
(1)
where $\omega_{a}$ and $\omega_{b}$ are lower and upper cut-off frequencies,
respectively. By choosing appropriate values for $\omega_{a}$ and
$\omega_{b}$, one can study the relations among different phase transitions.
Yu et al. (2008) When replacing $\omega_{a}$ by 0 and $\omega_{b}$ by
$\infty$, we fall back to the standard $f$-sum rule and the spectral weight is
conserved.
Figure 4: (color online) Temperature dependence of the spectral weight,
SW${}_{\omega_{a}}^{\omega_{b}}$ =
$\int_{\omega_{a}}^{\omega_{b}}\sigma_{1}(\omega)d\omega$, between different
lower and upper cutoff frequencies for the $x=0.2$ sample. The vertical dashed
lines denote $T_{c}$, $T^{\ast}$ and $T_{\it SDW}$.
Figure 4 shows the temperature dependence of the $x=0.2$ sample spectral
weight, normalized by its value at 300 K, at different cut-off frequencies.
The vertical dashed lines denote $T_{c}$, $T^{\ast}$ and $T_{\it SDW}$. The
blue solid circles in the top panel of Fig. 4 are the normalized spectral
weight with cut-off frequencies $\omega_{a}=0$ and
$\omega_{b}=12000\leavevmode\nobreak\ \textrm{cm}^{-1}$ as a function of
temperature. Here, the weight of the zero frequency $\delta$-function is
included below $T_{c}$. Moreover, since the optical conductivity is measured
only down to 20 $\leavevmode\nobreak\ \textrm{cm}^{-1}$, we estimate the
spectral weight below that energy by fitting the low frequency normal state
optical conductivity to a Drude model. The upper cut-off frequency
($\omega_{b}=12000\leavevmode\nobreak\ \textrm{cm}^{-1}$) is high enough to
cover the whole spectrum responsible for the phase transitions in this
material. Hence the blue solid circles form a flat line at about unity,
indicating that the spectral weight is conserved.
The red solid circles in the top panel show the temperature dependence of the
normalized spectral weight with cut-off frequencies $\omega_{a}=0^{+}$ and
$\omega_{b}=650\leavevmode\nobreak\ \textrm{cm}^{-1}$. Here $0^{+}$ means that
the superfluid weight is not included. Above $T_{\it SDW}$, the continuous
increase of the normalized SW${}_{0^{+}}^{650}$ with decreasing $T$ is related
to the narrowing of the Drude band. This is the typical optical response of a
metallic material. A strong spectral weight suppression occurs at $T_{\it
SDW}$, which is the consequence of the SDW gap opening. At $T_{c}$, another
sharp drop of the spectral weight breaks in, indicating the superconducting
gap opening.
The temperature dependence of the normalized SW${}_{650}^{1700}$, shown as
green solid circles in the bottom panel of Fig. 4, provides clues about the
relation between the superconducting and the SDW gaps. Above $T_{\it SDW}$,
the material shows a metallic response which can be described by a Drude peak
centered at zero frequency. With the temperature decrease, the DC conductivity
increases and the scattering rate reduces. The continuous narrowing of the
Drude band induces a transfer of spectral weight from the mid-infrared to the
far infrared, resulting in the continuous decrease of the spectral weight
observed in the 650–1700$\leavevmode\nobreak\ \textrm{cm}^{-1}$ range. Below
$T_{\it SDW}$, the opposite behavior dominates the optical conductivity. The
SDW gap depletes the spectral weight below 650$\leavevmode\nobreak\
\textrm{cm}^{-1}$ and transfers it to the 650–1700$\leavevmode\nobreak\
\textrm{cm}^{-1}$ range, leading to the continuous increase of
SW${}_{650}^{1700}$ with decreasing $T$. This behavior continues into the
superconducting state and does not show any feature at $T_{c}$. These
observations indicate that the SDW and superconducting gaps are separate and
even act as competitive orders in this material.
If a partial gap is due to a precursor order of superconductivity, for example
preformed pairs without phase coherence, once the long range superconductivity
is established, a significant part of the spectral weight transferred to high
frequencies by the partial gap should be transferred back to low energies and
join the superconducting condensate. Ioffe and Millis (1999, 2000) Whereas, a
partial gap due to a competitive order to superconductivity depletes the low-
energy spectral weight and holds it in a high energy scale without
transferring it back to the superfluid weight below $T_{c}$. Yu et al. (2008)
From the normalized SW${}_{650}^{1700}$ _vs_ $T$ curve (green solid circles)
we note that no loss of spectral weight is observed at $T_{c}$. This means
that the spectral weight transferred to high frequencies by the SDW gap
remains in the high frequency scale and does not contribute to the
superconducting condensate. Therefore, the SDW acts as a competitive order to
superconductivity in this material.
Along these lines, the origin of the pseudogap and its relationship to
superconductivity can be revealed by a close inspection of the temperature
dependence of the normalized SW${}_{110}^{250}$, shown as pink solid circles
in the bottom panel. Above $T^{*}$, this curve shows the same feature as the
normalized SW${}_{0^{+}}^{650}$ _vs_ $T$ curve, _i.e._ , continuous increase
upon cooling down followed by a suppression at $T_{\it SDW}$ due to the SDW
gap opening. At $T^{*}$, the spectral weight in the
110–250$\leavevmode\nobreak\ \textrm{cm}^{-1}$ range reaches a minimum and
starts to increase with decreasing temperature. This is due to the opening of
the pseudogap. The pseudogap, opening at $T^{*}$, depletes the spectral weight
below 110$\leavevmode\nobreak\ \textrm{cm}^{-1}$ and retrieves it in the
110–250$\leavevmode\nobreak\ \textrm{cm}^{-1}$ frequency range, leading to the
increase of SW${}_{110}^{250}$ below $T^{*}$. An interesting phenomenom
happens to the pseudogap when the material undergoes the superconducting
transition. In contrast to the case of the SDW gap, a significant loss of
spectral weight in the 110–250$\leavevmode\nobreak\ \textrm{cm}^{-1}$
frequency range is observed below $T_{c}$. This observation indicates that the
spectral weight transferred to the 110–250$\leavevmode\nobreak\
\textrm{cm}^{-1}$ range by the pseudogap joins the superconducting condensate
when superconductivity is established. Hence, the pseudogap is likely a
precursor order with respect to superconductivity.
In summary, we measured the optical conductivity of two underdoped
Ba1-xKxFe2As2 ($x=0.2$ and 0.12) single crystals. In both samples, besides the
SDW gap and superconducting gap, the optical conductivity reveals another
small partial gap (pseudogap) opening below $T^{\ast}\sim$ 75 K an
intermediate temperature between $T_{SDW}$ and $T_{c}$. A spectral weight
analysis shows that the SDW gap diminishes the low energy spectral weight
available for the superconducting condensate while the pseudogap shares the
same electronic states with the superconducting gap. These observations,
together with the doping dependence of these gaps, suggest the SDW as a
competitive order and the pseudogap as a precursor to superconductivity.
We thank Li Yu, Lei Shan, Cong Ren and Zhiguo Chen for helpful discussion.
Work in Paris was supported by the ANR under Grant No. BLAN07-1-183876
GAPSUPRA. Work in Beijing was supported by the National Science Foundation of
China (No. 91121004) and the Ministry of Science and Technology of China (973
Projects No. 2011CBA00107, No. 2012CB821400 and No. 2009CB929102). We
acknowledge the financial support from the Science and Technology Service of
the French Embassy in China.
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|
arxiv-papers
| 2012-02-17T03:00:49 |
2024-09-04T02:49:27.504079
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. M. Dai, B. Xu, B. Shen, H. H. Wen, J. P. Hu, X. G. Qiu and R. P. S.\n M. Lobo",
"submitter": "Yaomin Dai",
"url": "https://arxiv.org/abs/1202.3833"
}
|
1202.3846
|
Supplementary material for:
Multiscale non-adiabatic dynamics with radiative decay, case study on the
post-ionization fragmentation of rare-gas tetramers.
Ivan Janeček
Institute of Geonics of the AS CR, v.v.i., & Institute of Clean Technologies
for Mining and Utilization of Raw Materials for Energy Use, Studentská 1768,
708 00 Ostrava, Czech Republic
Tomáš Janča, Pavel Naar
Department of Physics, Faculty of Sciences, University of Ostrava, 30. dubna
22, 701 03 Ostrava, Czech Republic
Frederic Renard
Faculté des Sciences, Université du Maine, 72085 Le Mans Cedex 9, France
René Kalus
Centre of Excellence IT4Innovations & Department of Applied Mathematics, VŠB -
Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech
Republic
Florent X. Gadéa
LCPQ and UMR5626 du CNRS, IRSAMC, Université de Toulouse, 118 route de
Narbonne, 31062 Toulouse Cedex, France
Abstract
In this supplementary material, we recollect, for reader’s convenience, the
general scheme of suggested multiscale model (Sec. 1), and basic informations
about approaches used for pilot study: a detailed description of the
interaction model (Sec. 2) and dynamical methods used for the dark dynamics
step (Sec. 3) reported previously in two preceding studies [1, 2]. In
addition, a detailed description of the treatment of radiative processes is
also given (Sec. 4).
Last update:
## 1 General model scheme
Figure: General scheme of suggested multiscale model.
## 2 Interaction model
The intra-cluster interactions are described within an extended diatomics-in-
molecules (DIM) model with the spin-orbit interaction included. The DIM
approach was developed by Ellison [3] and later on applied to singly ionized
rare-gas cluster cations by Kuntz and Valldorf [4]. How the spin-orbit (SO)
interaction can be included in the DIM model was proposed by Amarouche et al.
[5]. Note that, in addition to the SO term, further extensions to the original
DIM approach can also be considered, e.g., the inclusion of the leading three-
body polarization forces [5]. However, we do not use these extensions in the
present study as they do not contribute much for small clusters [1], and,
consequently, we omit them from the following explanation.
The original DIM approach consists in a) re-writing the electronic hamiltonian
as a sum of diatomic and atomic contributions,
$\mathrm{\hat{H}}=\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}\mathrm{\hat{H}}_{jk}-(n-2)\sum_{k=1}^{n}\mathrm{\hat{H}}_{k},$
(1)
where $n$ denotes the number of atoms, and b) designing an appropriate basis
set of wave functions for which the elements of the corresponding hamiltonian
matrix can be calculated by means of the electronic energies of atomic and
diatomic fragments. If the SO coupling is not considered, the atomic
contributions of Eq. 1 are constant and their sum can be identified with the
zero energy level.111In our work, the zero of energy is identified with the
energy of fully dissociate state, $\mathrm{Rg}^{+}+(n-1)\mathrm{Rg}$ ,
calculated with the SO interaction _not_ included and the atomic contributions
of Eq. 1 can be omitted. If, on the other hand, the SO coupling is considered,
the atomic contribution corresponds either to the energy of
$\mathrm{Rg}^{+}(^{2}\mathrm{P}_{3/2})$ or
$\mathrm{Rg}^{+}(^{2}\mathrm{P}_{1/2})$ measured from the SO-free
$\mathrm{Rg}^{+}$ level. The diatomic energies are to be supplied from
independent sources (usually from ab initio calculations).
The basis set proposed for an $n$-atom rare-gas cluster cation,
Rg${}_{n}^{+}$, consists for the SO-free model of $3n$ valence-bond Slater
determinants, $|\Phi_{k,p_{m}}\rangle$ (where $k=1,\cdots,n$ and $m=x,y,z$)
and represents states with the positive charge localized in a valence
$p_{m}$-orbital of atom $k$. The corresponding $3n\times 3n$ hamiltonian
matrix,
$H_{k,p_{m};k^{\prime},p_{m}^{\prime}}\equiv\langle\Phi_{k,p_{m}}|\mathrm{\hat{H}}|\Phi_{k^{\prime},p^{\prime}_{m}}\rangle,$
(2)
is constructed as described in Ref. [4] from diatomic potential energy curves
for the electronic ground state of the neutral dimer, Rg2, and the electronic
ground and three lowest excited states of the ionic dimer, Rg${}_{2}^{+}$. In
our calculations, we have used semiempirical curves for neutral dimers [6, 7]
and accurate ab initio curves for ionic diatoms [8, 9].
This simple model is easily augmented [5] with SO coupling terms via a semi-
empirical atoms-in-molecules scheme [10]. If the SO coupling is taken into
account, the number of the basis set wave functions doubles, as there are two
possible orientations of the spin of the electron removed from the valence
shell of a particular atom, $s_{z}=\pm 1/2$, as well as the dimension of the
electronic hamiltonian matrix. In addition, the matrix of Eq. 2 must be
replaced by ($\delta$ denotes the Kronecker delta) [5]
$H_{k,p_{m},s_{z};k^{\prime},p_{m}^{\prime},s^{\prime}_{z}}^{\mathrm{(SO)}}=H_{k,p_{m};k^{\prime},p_{m}^{\prime}}\delta_{s_{z};s_{z}^{\prime}}+h_{p_{m},s_{z};p_{m}^{\prime},s_{z}^{\prime}}^{\mathrm{(SO)}}\delta_{k;k^{\prime}},$
(3)
where
$h_{p_{m}\sigma,l\sigma}^{\mathrm{(SO)}}\delta_{k;k^{\prime}}=\xi\langle\phi_{k,p_{m},s_{z}}|\widehat{L}_{k}\widehat{s}_{k}|\phi_{k^{\prime},p_{m}^{\prime},s_{z}^{\prime}}\rangle,$
(4)
$\xi$ is the SO coupling constant, and $\widehat{L}_{k}$ and $\widehat{s}_{k}$
are angular and spin operators for the $k$-th atom, respectively. The SO
constants are independent inputs to the model and have been extracted here
from experiments reporting on the SO splitting between the ${}^{2}P_{1/2}$ and
${}^{2}P_{3/2}$ states of atomic monomers [11]. Hereafter, we denote this
extended DIM model by DIM+SO.
In the following text we use a simplified indices for the electronic wave
function and hamiltonian matrix components, e.g., $\alpha=[k,p_{m},s_{z}]$
etc.
## 3 Non-radiative dynamics
The semi-classical dynamical method (classical nuclei and quantum electrons)
we use in our work for the non-radiative stage of our calculations, the MFQ-
AMP/S method of Ref. [2], combines a) the well known Ehrenfest mean-field
approach [12], detailed for the rare-gas cluster cations in [1], with b) the
inclusion of quantum decoherence as introduced in Ref. [2].
### 3.1 Mean-field method
The equations of motion for a system of classical nuclei surrounded by a cloud
of electrons can be written within the mean-field approximation as coupled
classical Hamilton equations for the nuclei
$\dot{q}_{i}=\frac{p_{i}}{m_{i}},$ (5)
$\dot{p_{i}}=\langle\psi|-\frac{\partial\mathrm{\hat{H}}}{\partial
q_{i}}|\psi\rangle$ (6)
and time dependent Schrödinger equation for the electrons
$i\hbar\frac{\partial|\psi\rangle}{\partial t}=\mathrm{\hat{H}}|\psi\rangle.$
(7)
In Eqs. 5 – 7, $q_{i}$ and $p_{i}$ denote respectively generalized nuclear
coordinates and momenta, $\mathrm{\hat{H}}$ denotes the electronic
hamiltonian, which depends parametrically on the nuclear coordinates, and
$|\psi\rangle$ is a time dependent wave function representing the current
electronic state. Small latin indices are used to label nuclear degrees of
freedom and range between 1 through $3n$.
Within the DIM+SO approach, the electronic wave function, $|\psi\rangle$, can
be expanded using basis set wave functions of Sec. 2, $|\Phi_{\alpha}\rangle$,
also parametrically dependent on nuclear coordinates and, consequently, on
time as well,
$|\psi(t)\rangle=\sum_{\alpha}a_{\alpha}(t)|\Phi_{\alpha}(q_{i}(t))\rangle,$
(8)
with $\alpha$ introduced above, $\alpha=[k,p_{m},s_{z}]$. The electronic
hamiltonian can also be expressed in an expanded form
$\mathrm{\hat{H}}=\sum_{\beta,\gamma}\tilde{H}_{\beta\gamma}|\Phi_{\beta}\rangle\langle\Phi_{\gamma}|,$
(9)
where
$\tilde{H}_{\beta\gamma}=S_{\beta\kappa}H_{\kappa\lambda}S_{\lambda\gamma}$
(with $H_{\kappa\lambda}$ being the DIM+SO hamiltonian matrix given by Eq. 3,
for simplicity we omit the (SO) upper index), and
$S_{\alpha\beta}\equiv\langle\Phi_{\alpha}|\Phi_{\beta}\rangle$ are overlap
matrix elements. Note that matrix $\tilde{H}_{\beta\gamma}$ is equal to the
DIM+SO hamiltonian matrix, $H_{\beta\gamma}$, if the overlaps are neglected
($S_{\alpha\beta}=0$ for $\alpha\neq\beta$) and wavefunctions
$|\Phi_{\alpha}\rangle$ are normalized ($S_{\alpha\alpha}=1$).222This is a
usual and sufficiently accurate approximation adopted in all DIM models as yet
developed for the rare-gas ionic clusters.
After inserting the expanded forms of the electronic hamiltonian and time-
dependent electronic wave function into Eq. 6, one obtains
$\dot{p}_{i}=-\sum_{\alpha,\beta,\gamma,\delta}\left[S_{\alpha\beta}S_{\gamma\delta}\frac{\partial\tilde{H}_{\beta\gamma}}{\partial
q_{i}}+D_{\alpha\beta}^{(i)}S_{\gamma\delta}\tilde{H}_{\beta\gamma}+S_{\alpha\beta}D_{\delta\gamma}^{(i)*}\tilde{H}_{\beta\gamma}\right]a_{\alpha}^{*}a_{\delta},$
(10)
where
$D_{\alpha\beta}^{(i)}\equiv\langle\Phi_{\alpha}|\frac{\partial\Phi_{\beta}}{\partial
q_{i}}\rangle$ are non-diabatic coupling coefficients, and asterisks denote
complex conjugation. The overlaps and non-diabatic couplings are usually
neglected in DIM approaches and, consequently, Eq. 10 can be further
simplified by setting $S_{\alpha\beta}\approx\delta_{\alpha\beta}$ and
$D_{\alpha\beta}^{(i)}\approx 0$, (with $\delta_{\alpha\beta}$ being the
Kronecker delta),
$\dot{p}_{i}=-\sum_{\alpha,\beta}a_{\alpha}^{*}a_{\beta}\frac{\partial
H_{\alpha\beta}}{\partial q_{i}},$ (11)
where, after neglecting the overlaps, $\tilde{H}_{\alpha\beta}$ is replaced
with $H_{\alpha\beta}$.
Further simplification of Eq. 11 is possible if coefficients $a_{\alpha}$ and
matrix elements $H_{\alpha\beta}$, which are in general complex, are rewritten
using their real and imaginary parts,
$a_{\alpha}=a_{\alpha}^{(\mathrm{re})}+ia_{\alpha}^{(\mathrm{im})}$ and
$H_{\alpha\beta}=H_{\alpha\beta}^{(\mathrm{re})}+iH_{\alpha\beta}^{(\mathrm{im})}$,333
It directly follows from hermicity of the electronic hamiltonian matrix that
its real part, $H_{\alpha\beta}^{(\mathrm{re})}$, is symmetric and the
imaginary part, $H_{\alpha\beta}^{(\mathrm{im})}$, is antisymmetric. After
using this property, we obtain immediately Eq. 12.
$\dot{p}_{i}=-\sum_{\alpha,\beta}\left[\left(a_{\alpha}^{(\mathrm{re})}a_{\beta}^{(\mathrm{re})}+a_{\alpha}^{(\mathrm{im})}a_{\beta}^{(\mathrm{im})}\right)\frac{\partial
H_{\alpha\beta}^{(\mathrm{re})}}{\partial
q_{i}}+\left(a_{\alpha}^{(\mathrm{re})}a_{\beta}^{(\mathrm{im})}-a_{\alpha}^{(\mathrm{im})}a_{\beta}^{(\mathrm{re})}\right)\frac{\partial
H_{\alpha\beta}^{(\mathrm{im})}}{\partial q_{i}}\right].$ (12)
The imaginary part of the electronic hamiltonian matrix is non-zero only if
the SO coupling is included. If it is done using the Cohen-Schneider, atoms-
in-molecules scheme [10], all the imaginary terms are constant as they do not
depend on the nuclear positions, and, consequently, $\frac{\partial
H_{\alpha\beta}^{(\mathrm{im})}}{\partial q_{i}}=0$. The second term on the
right-hand-side of Eq. 12 thus vanishes and the equation can be written in the
final form
$\dot{p}_{i}=-\sum_{\alpha,\beta}\left(a_{\alpha}^{(\mathrm{re})}a_{\beta}^{(\mathrm{re})}+a_{\alpha}^{(\mathrm{im})}a_{\beta}^{(\mathrm{im})}\right)\frac{\partial
H_{\alpha\beta}^{(\mathrm{re})}}{\partial q_{i}}.$ (13)
Similarly, the electronic Schrödinger equation, Eq. 7, can be rewritten after
inserting the expansion of Eq. 8 to
$i\hbar\sum_{\beta}\dot{a}_{\beta}|\Phi_{\beta}\rangle+i\hbar\sum_{\beta,j}a_{\beta}\dot{q}_{j}\frac{\partial|\Phi_{\beta}\rangle}{\partial
q_{j}}=\sum_{\beta}a_{\beta}\mathrm{\hat{H}}|\Phi_{\beta}\rangle,$ (14)
and after multiplying by $\langle\Phi_{\alpha}|$ from the left to
$i\hbar\sum_{\beta}S_{\alpha\beta}\dot{a}_{\beta}+i\hbar\sum_{\beta,j}D_{\alpha\beta}^{(j)}a_{\beta}\dot{q}_{j}=\sum_{\beta,\gamma,\delta}H_{\gamma\delta}a_{\beta}.$
(15)
A significant simplification of Eq. 15 is further possible if the overlap
matrix is replaced with the Kronecker delta,
$S_{\alpha\beta}=\delta_{\alpha\beta}$, and basis set $|\Phi_{\alpha}\rangle$
is considered diabatic, $D_{\alpha\beta}^{(i)}=0$,
$i\hbar\dot{a}_{\alpha}=\sum_{\beta}H_{\alpha\beta}a_{\beta}.$ (16)
Eq. 16 must be treated with care, however, namely due to rapid oscillations
occurring in the electronic wave function and, consequently, also in expansion
coefficients $a_{\alpha}$. A special scheme has been developed for tackling
this problem in the previous work. Since it is of technical rather than
methodological importance, it is not discussed here and the reader is directed
to Ref. [1] for details.
### 3.2 Inclusion of quantum decoherence
As shown elsewhere [2], quantum decoherence is important, particularly for the
heavy rare gases, krypton and xenon. It is introduced into the mean-field
approach by periodically quenching the electronic wave function.444Proper
settings of the quenching period was thoroughly discussed in [2]. In this work
we use quenching period $t_{\mathrm{quench}}=100$ fs. We denote this extended
dynamical approach by MFQ (Mean Field with Quenchings). The quenching
algorithm comprises basically two steps. Firstly, the probabilities for
collapsing the current electronic wavefunction into one of adiabatic states is
calculated and a wave function collapse is proposed according to these
probabilities. Secondly, in case the proposed collapse has been accepted, the
kinetic energy of nuclei is adjusted so that the total energy of the system
remains unchanged. The proposed jump can be, in general, rejected in both
steps of the present algorithm and, in that case, the system resumes the
coherent evolution until the next hop attempt.555For the algorithm used in
this work, AMP (see below), the proposed electronic jump is always accepted in
the first step and rejection can occur only during the second step, namely, if
there is not enough kinetic energy to cover expenses of an upward electronic
jump.
Several quenchings schemes have been developed previously [2]. In this work we
use computationally cheap, but several times successfully tested MFQ-AMP/S
algorithm. The procedure starts with calculating the adiabatic amplitudes of
the current electronic wave function (hence the acronym AMP). More
specifically, the normalized probability for collapsing the current electronic
state, $\psi$, to a particular adiabatic state, $\phi_{\mu}$, is calculated as
$g_{\psi\rightarrow\mu}^{\mathrm{AMP}}=\rho_{\mu\mu},$ (17)
where $\rho_{\mu\mu}$ represents the diagonal element of the electronic
density matrix ($\rho_{\mu\nu}\equiv c_{\mu}c_{\nu}^{*}$ and $c_{\mu}$ are
amplitudes of the current electronic wave function, $\psi$, expanded in the
adiabatic basis set, $\psi=\sum_{\mu}c_{\mu}\phi_{\mu}$). After the electronic
jump is complete, the kinetic energy of nuclei is adjusted so that the total
energy of the system is conserved. In the MFQ-AMP/S model, this is achieved by
scaling (hence the third acronym, S) nuclear velocities, as rationalized in
[2].
## 4 Radiative dynamics
After the non-radiative dynamics is stopped at time $t_{\mathrm{DD}}$, each
trajectory is evaluated as an ensemble undergoing first-order decay due to
radiative transitions in the electronic subsystem. In principle, many decay
processes may occur in such an ensemble, both parallel and serial, which may
lead to a complex system of coupled first-order equations governing the time
evolution of this ensemble. In principle, such equations can be derived and
solved. Nevertheless, since in our case a) transitions are expected only from
the upper family of states of the charged fragment, an excited state resulting
for the particular trajectory from the non-radiative dynamics at
$t_{\mathrm{DD}}$, to the lower family of states and b) the fragments undergo,
after the radiative transition, a rapid non-radiative decay, the radiative
processes can be assumed parallel and a simplified set of decay equations can
be used. In particular, if a population of $n_{I0}$ identical initial states
from the upper family of states (e.g., all being state $I$) is considered for
a particular trajectory and assumed to decay to the lower family of states
($J$), the corresponding population numbers will change with time $\Delta
t=t-t_{\mathrm{DD}}$ according to (dot denotes the time derivative)
$\dot{n}_{I}=-\sum_{J}\Gamma_{IJ}n_{I},\quad\dot{n}_{J}=\Gamma_{IJ}n_{I},$
(18)
with initial conditions $n_{I}(\Delta t=0)=n_{I0}$ (=1 for one particular
trajectory) and $n_{J}(\Delta t=0)=0$. It is easy to find, that the only
solution to these equations is given by Eq. (2) of the letter, namely,
$n_{J}(\Delta t)=n_{I0}(1-e^{-\Gamma\Delta t})\Gamma_{IJ}/\Gamma,\quad
n_{I}(\Delta t)=n_{I0}e^{-\Gamma\Delta t},$ (19)
where $\Gamma=\sum_{K=1}^{I-1}\Gamma_{IK}$ and $\Delta t\approx t$ since $t\gg
t_{\mathrm{DD}}$.
Note also that $n_{J}(\Delta t)|_{n_{I0}=1}$ gives the probability that, at
time $\Delta t$, the system will be found in state $J$, and $n_{I}(\Delta
t)|_{n_{I0}=1}$ is the probability of surviving the system in excited state
$I$. The evaluation of fragments at time $\Delta t$ consists then in a cycle
repeated for all trajectories and comprising the following steps:
1. 1.
identify the fragmentation channel corresponding to the particular trajectory,
2. 2.
subtract from the total number of trajectories leading to the same
fragmentation channel at $t_{\mathrm{DD}}$ value of $1-n_{I}(\Delta
t)|_{n_{I0}=1}$,
3. 3.
identify the fragmentation channel for each state $J$ (this can be done either
by running additional non-radiative dynamical simulation or by simple
energetic considerations),
4. 4.
add $n_{J}(\Delta t)|_{n_{I0}=1}$ to the number of trajectories leading at
$t_{\mathrm{DD}}$ to the same fragmentation channel.
After this cycle is complete, one gets updated abundances of fragments as
should be detected at time $\Delta t\approx t$ of radiative decay.
The decay rates of Eq. 18 are calculated from a standard formula for
spontaneous radiation (Eq. 1 of the letter),
$\Gamma_{IJ}=\frac{1}{{3\pi\varepsilon_{0}\hbar^{4}c^{3}}}\left({E_{I}-E_{J}}\right)^{3}\left|{\mu_{IJ}}\right|^{2},$
(20)
where the transition dipole moment is obtained for a particular charged
fragment geometry, ${\bf R}$, within the point-charge approximation [13],
${\mu}_{IJ}({\bf R})\approx
e\sum_{k=1}^{n}\sum_{m=x}^{z}\sum_{s_{z}=-1/2}^{+1/2}{c_{kp_{m}s_{z}}^{(I)}}^{\ast}c_{kp_{m}s_{z}}^{(J)}{\bf
R}_{k}.$ (21)
The first sum of Eq, 21 runs over all atoms in the charged fragment and
$c_{kp_{m}s_{z}}^{(I)}$ and $c_{kp_{m}s_{z}}^{(J)}$ are amplitudes of
adiabatic states $I$ and $J$, respectively, expressed in the DIM+SO basis set
introduced in Sec. 2. Alike in our earlier work, the point-charge
approximation has been further improved in the present work by including
damped polarization effects [14] consisting in a replacement
${\bf R}_{k}\rightarrow{\bf R}_{k}\sum_{i\neq
k}\alpha^{*}_{\mathrm{eff}}(R_{ik})\frac{{\bf R}_{ik}}{{R_{ik}}^{3}},$ (22)
where ${\bf R}_{ik}={\bf R}_{i}-{\bf R}_{j}$, $R_{ik}=|{\bf R}_{ik}|$, and
$\alpha^{*}_{\mathrm{eff}}(R_{ik})$ is a damped effective polarizibility
expressed in atomic units [14],
$\alpha_{\mathrm{eff}}^{*}(R)=\frac{N_{e}}{(\sqrt{N_{e}/\alpha^{*}}+1/R)^{2}}.$
(23)
## References
* [1] I. Janeček, D. Hrivňák, R. Kalus, and F. X. Gadea. J. Chem. Phys., 125:Art. No. 104315, 2006.
* [2] I. Janeček, S. Cintavá, D. Hrivňák, R. Kalus, M. Fárník, and F. X. Gadea. J. Chem. Phys., 131:Art. No. 114306, 2009.
* [3] F. O. Ellison. J. Am. Chem. Soc., 85:3540, 1963.
* [4] P. J. Kuntz and J. Valldorf. Z. Phys. D, 8:195, 1988.
* [5] M. Amarouche, G. Durand, and J. P. Malrieu. J. Chem. Phys., 88:1010, 1988.
* [6] A. K. Dham, A. R. Allnatt, W. J. Meath, and R. A. Aziz. Mol. Phys., 67:1291, 1989.
* [7] A. K. Dham, W. J. Meath, A. R. Allnatt, R. A. Aziz, and M. J. Slaman. Chem. Phys., 142:173, 1990.
* [8] R. Kalus, I. Paidarová, D. Hrivňák, P. Paška, and F. X. Gadea. Chem. Phys, 294:141, 2003.
* [9] I. Paidarová and F. X. Gadéa. Chem. Phys., 274:1, 2001.
* [10] J. S. Cohen and B. I. Schneider. J. Chem. Phys., 61:3230, 1974.
* [11] Yu. Ralchenko, A. E. Kramida, J. Reader, and team. Nist atomic spectra database (ver. 4.1.0). http://physics.nist.gov/asd3.
* [12] P. Ehrenfest. Z. Phys., 45:455, 1927.
* [13] T. Ikegami, T. Kondow, and S. Iwata. J. Chem. Phys., 98:3038, 1993.
* [14] F. Y. Naumkin. Chem. Phys., 252:301, 2000.
|
arxiv-papers
| 2012-02-17T06:26:23 |
2024-09-04T02:49:27.510850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ivan Jane\\v{c}ek, Tom\\'a\\v{s} Jan\\v{c}a, Pavel Naar, Frederic Renard,\n Ren\\'e Kalus and Florent X. Gad\\'ea",
"submitter": "Ivan Janecek",
"url": "https://arxiv.org/abs/1202.3846"
}
|
1202.3949
|
On the complexity of solving linear congruences and computing nullspaces
modulo a constant Niel de Beaudrap [1ex] DAMTP, Centre for Mathematical
Sciences, University of Cambridge, [-0.5ex] Wilberforce Road, Cambridge CB3
0WA, UK 5 March, 2012
###### Abstract
We consider the problems of determining the feasibility of a linear
congruence, producing a solution to a linear congruence, and finding a
spanning set for the nullspace of an integer matrix, where each of these
problems are considered modulo an arbitrary constant $k\geqslant 2$. These
problems are known to be complete for the logspace modular counting classes
$\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ in special case that $k$ is prime [4].
By considering relaxed modular variants of standard logspace function classes,
related to $\textbf{\\#}{\mathsf{L}}$ and functions computable by
$\mathsf{UL}$ machines but only characterizing the number of accepting paths
mod $k$, we show that these problems of linear algebra are also complete for
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ for any constant $k\geqslant 2$.
## 1 Introduction
Solving a system of linear equations, or determining that it has none, is the
definitive elementary problem of linear algebra over any ring. This problem is
the practical motivator of the notions of matrix products, inverses, and
determinants, among other concepts; and relates to other computational
problems of abelian groups, such as testing membership in a subgroup [1].
Characterizing the complexity of this problem for common number systems, such
as the integers, finite fields, or the integers modulo $k$ is therefore
naturally of interest.
We are interested in the difficulty of _deciding feasibility of linear
congruences modulo $k$_ (or LCONk) and _computing solutions to linear
congruences modulo $k$_ (or LCONXk) for an arbitrary constant $k\geqslant 2$.
This is a special case of the problem LCON defined by McKenzie and Cook [1],
in which $k$ is taken as part of the input, but represented by its prime-power
factors $p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$; where
$e_{j}\in O(\log n)$ for each $j$ (one says that each factor $p_{j}^{e_{j}}$
is _tiny_). Setting $k$ to a constant is a natural, if slightly restrictive,
special case.
Arvind and Vijayaraghavan [2] recently defined $\mathsf{Mod}$$\mathsf{L}$ (a
logspace analogue the class $\mathsf{Mod}$$\mathsf{P}$ defined by Köbler and
Toda [3]), which is contained in $\mathsf{NC}^{2}$. They show that LCON is
hard for $\mathsf{Mod}$$\mathsf{L}$ under $\mathsf{P}$-uniform
$\mathsf{NC}^{1}$ reductions, and contained in
$\mathsf{L}^{\mathsf{Mod}\mathsf{L}}/\mathsf{poly}=\mathsf{L}^{\textbf{\\#}{\mathsf{L}}}/\mathsf{poly}$.
This is of course in contrast to the problem of determining integer-
feasibility of integer matrix equations, which is at least as hard as
computing greatest common divisors over $\mathbb{Z}$; the latter problem is
not known to be in $\mathsf{NC}^{j}$ for any $j\geqslant 0$. Furthermore,
Buntrock _et al._ [4] show — for the special case of $k$ prime — that
determining the feasibility of systems of linear equations is complete for
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$; where these are the complementary
classes to the better known classes $\mathsf{Mod}_{k}$$\mathsf{L}$ which
generalize $\oplus\mathsf{L}$, corresponding to logspace nondeterministic
Turing machines which can distinguish between having a number of accepting
paths which are either zero or nonzero _modulo $k$_.
The above results suggest that the difficulty of solving linear equations over
integer matrices is strongly governed by the presence and the prime-power
factorization of the modulus involved, and indicates that LCONk may be
particularly tractable. Also implicit in Ref. [4] is that LCONk is
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$-hard for all $k\geqslant 2$. This
suggests the question: for an _arbitrary_ modulus $k$, what is the precise
relationship of the problem LCONk of deciding the feasibility of linear
congruences modulo $k$, to the classes
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$?
We show how the analysis of McKenzie and Cook [1] for the problem LCON may be
adapted to exhibit a $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ algorithm for
LCONk. Using techniques similar to those used by Hertrampf, Reif, and Vollmer
[5] to show closure of the class $\mathsf{Mod}_{p}\mathsf{L}$ under oracle
reductions for $p$ prime, we describe a function class $\mathsf{FUL}_{p}$
which is well-suited for describing oracles which may be simulated in mod-
logspace computations. We describe a recursive construction for a
$\mathsf{FUL}_{p^{e}}$ algorithm (for any fixed prime power $p^{e}$) to solve
the problem LCONNULL${}_{p^{e}}$ of computing a spanning set for a basis of
the nullspace of a matrix modulo $p^{e}$. This allows us to demonstrate that
LCONk is $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$-complete, and both LCONXk
and LCONNULLk are
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$-complete,
for any constant $k\geqslant 2$.
## 2 Preliminaries
Throughout the following, $k\geqslant 2$ is a constant modulus, with a
factorization into powers of distinct primes
$k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$. When we consider the
case of a modulus which is a prime power, we will write $p^{e}$ rather than
$k$, for $p$ some prime and $e\geqslant 1$ some positive integer which are
independent of the input.
We consider the complexity of the following problems, which are named in
analogy to problems considered by McKenzie and Cook [1]:
###### Problems.
Fix $k\geqslant 2$. For an $m\times n$ integer matrix $A$ and vector
$\mathbf{y}\in\mathbb{Z}^{m}$ provided as input, we define the following
problems:
* •
LCONk : determine whether $A\mathbf{x}\equiv\mathbf{y}\pmod{k}$ has solutions
for $\mathbf{x}\in\mathbb{Z}^{n}$.
* •
LCONXk : output a solution to the congruence
$A\mathbf{x}\equiv\mathbf{y}\pmod{k}$, or indicate that no solutions exist.
* •
LCONNULLk : output a set $\mathbf{x}_{1},\ldots,\mathbf{x}_{N}$ of vectors
spanning the solution space of the congruence
$A\mathbf{x}\equiv\mathbf{0}\pmod{k}$.
Without loss of generality, we may suppose $m=n$ by padding the matrix $A$.
We wish to describe the relationship of these problems to the classes
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ for $k\geqslant 2$, which are the
complements of the better known classes $\mathsf{Mod}_{k}$$\mathsf{L}$ defined
by Buntrock _et al._ [4].
###### Definition I.
The class $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ (respectively
$\mathsf{Mod}_{k}$$\mathsf{L}$) is the set of languages $L$ for which there
exists $\varphi\in\textbf{\\#}{\mathsf{L}}$ such that $x\in L$ if and only if
$\varphi(x)\equiv 0\pmod{k}$ (respectively, $\varphi(x)\not\equiv 0\pmod{k}$).
The following results are a synopsis of Ref. [4, Theorem 9]:
###### Proposition 1.
We may characterize $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ as the class
of decision problems which are log-space reducible to verifying matrix
determinants mod $k$, or coefficients of integer matrix products or matrix
inverses mod $k$.
###### Proposition 2.
For $p$ prime, LCONp is $\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$-complete.
Buntrock _et al._ also characterize the classes
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ in terms of the prime factors of
$k$, and show closure results which will prove useful. The following are
implicit in Lemma 5, Theorem 6, and Corollary 7 of Ref. [4]:
###### Proposition 3 (normal form).
Let $k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$ be the
factorization of $k\geqslant 2$ into prime powers $p_{j}^{e_{j}}$. Then
$L\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ if and only if there are languages
$L_{j}\in\mathsf{co}\mathsf{Mod}_{p_{j}}\mathsf{L}$ such that
$L=L_{1}\cap\cdots\cap L_{\ell}$. In particular,
$\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}=\mathsf{co}\mathsf{Mod}_{p_{1}p_{2}\cdots
p_{\ell}}\mathsf{L}$.
###### Proposition 4 (closure under intersections).
For any $k\geqslant 2$ and languages
$L,L^{\prime}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$, we have $L\cap
L^{\prime}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
###### Proposition 5 (limited closure under complements).
For any prime $p$ and $e\geqslant 1$, we have
$\mathsf{co}\mathsf{Mod}_{p^{e}}\mathsf{L}=\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}=\mathsf{Mod}_{p}\mathsf{L}=\mathsf{Mod}_{p^{e}}\mathsf{L}$.
A system of linear congruences mod $k$ has solutions if and only if it has
solutions modulo each prime power divisor $p_{j}^{e_{j}}$ of $k$. We then have
$\mbox{{{LCON${}_{k}$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ if and only
if
$\mbox{{{LCON${}_{p^{e}}$}}}\in\mathsf{co}\mathsf{Mod}_{p^{e}}\mathsf{L}=\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$
by Proposition 3. (In fact, this suffices to show that
$\mbox{{{LCON${}_{k}$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ for all
square-free integers $k\geqslant 2$.)
We see from Propositions 2 and 5 that the case of a prime modulus is special.
For $p$ prime, Buntrock _et al._ also implicitly characterize the complexity
of LCONXp and LCONNULLp. We may describe the complexity of these function
problems as follows. For a function $f(x):\Sigma^{\ast}\to\Sigma^{\ast}$ and
$x\in\Sigma^{\ast}$, let $|f(x)|$ denote the length of the representation of
$f(x)$; and let $f(x)_{j}$ denote the $j\textsuperscript{th}$ symbol in that
representation. Following Hertrampf, Reif, and Vollmer [5], for a function
$f:\Sigma^{\ast}\to\Sigma^{\ast}$ on some alphabet $\Sigma$, and for some
symbol $\bullet\notin\Sigma$, we may define the decision problem
$\mbox{{{bits$(f)$}}}=\left\\{(x,j,b)\;\left|\;\begin{array}[]{r@{}l@{~\text{and}~}r@{}l}\text{either}~{}j&{}\leqslant|f(x)|\hfil~{}\text{and}&b&{}=f(x)_{j}\\\
\text{or}~{}j&{}>|f(x)|\hfil~{}\text{and}&b&{}=\bullet\end{array}\right\\}\right..$
(1)
Abusing notation, we write $f(x)_{j}=\bullet$ in case $|f(x)|<j$. We extend
this definition to _partial_ functions $f$ by asserting
$(x,j,b)\in\mbox{{{bits$(f)$}}}$ only if $x\in\operatorname{dom}(f)$.
###### Definition II.
The class $\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ is the
set of (partial) functions $f$ such that $|f(x)|\in\operatorname{poly}(|x|)$
for all $x\in\Sigma^{\ast}$, and for which
$\mbox{{{bits$(f)$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$. (We define the
class $\mathsf{F}$$\mathsf{Mod}_{k}$$\mathsf{L}$ similarly.)
Then Ref. [4, Theorem 9] also implicitly shows:
###### Proposition 6.
For $p$ prime, the problems LCONXp and LCONNULLp are
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$-complete.
In Section 3, we describe two additional function classes which are natural
when considering modular logspace computation. Relationships between these
classes in the case of prime-power modulus will allow us to easily show in
Section 4 that in fact
$\mbox{{{LCONX${}_{p^{e}}$}}},\mbox{{{LCONNULL${}_{p^{e}}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$
for all prime powers $p^{e}$. These results then naturally extend to all
moduli $k\geqslant 2$, so that
$\mbox{{{LCONX${}_{k}$}}},\mbox{{{LCONNULL${}_{k}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$,
with $\mbox{{{LCON${}_{k}$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$
following as a corollary.
## 3 Natural function classes for modular logspace
We now introduce two classes for counting classes in logarithmic space: a
modular analogue of $\textbf{\\#}{\mathsf{L}}$, and a class of function
problems which is naturally low for $\mathsf{Mod}_{k}$$\mathsf{L}$ and
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$. We describe the relationships of
these classes to $\mathsf{F}$$\mathsf{Mod}_{k}$$\mathsf{L}$ and
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$, and to
each other in the case of a prime modulus.
###### Definition III.
The class $\textbf{\\#}{\mathsf{L}}_{k}$ is the set of functions
$f:\Sigma^{\ast}\to\mathbb{Z}/k\mathbb{Z}$ such that
$f(x)=\varphi(x)+k\mathbb{Z}$ for some function
$\varphi\in\textbf{\\#}{\mathsf{L}}$.
Note that $\textbf{\\#}{\mathsf{L}}_{k}$ inherits closure under addition,
multiplication, and constant powers from $\textbf{\\#}{\mathsf{L}}$; it is
closed under subtraction as well, as $M-N\equiv M+(k-1)N\pmod{k}$. We may then
rephrase Proposition 1 as follows:
###### Proposition 7.
Evaluating matrix determinants modulo $k$, coefficients of products of integer
matrices modulo $k$, and coefficients of inverses modulo $k$ of integer
matrices, are complete problems for $\textbf{\\#}{\mathsf{L}}_{k}$.
Similar containments hold for each of the problems listed in Ref. [4, Theorem
9]: any decision problem in $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ (such
as the complete problems listed by Buntrock _et al._) consists of comparing
some function $f\in\textbf{\\#}{\mathsf{L}}_{k}$ to a constant or an input
value. Thus we trivially have:
###### Lemma 8.
For any $k\geqslant 2$,
$\textbf{\\#}{\mathsf{L}}_{k}\subseteq\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
We may adopt the common conflation between equivalence classes
$a+k\mathbb{Z}\in\mathbb{Z}/k\mathbb{Z}$ and integers $0\leqslant a<k$, in
which case we may instead require $f\in\textbf{\\#}{\mathsf{L}}_{k}$ to
satisfy $0\leqslant f(x)<k$ and $f(x)\equiv\varphi(x)\bmod{k}$ for some
$\varphi\in\textbf{\\#}{\mathsf{L}}$. This will allow us to consider logspace
machines which compute $\textbf{\\#}{\mathsf{L}}_{k}$ functions on their
output tapes. We will be interested in a particular sort of nondeterministic
logspace machine which is suitable for performing computations as subroutines
of $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ machines: the main result of
this section is to describe conditions under which it can compute functions in
$\textbf{\\#}{\mathsf{L}}_{k}$.
###### Definition IV.
A _$\mathsf{FUL}_{k}$ machine computing a (partial) function $f$_ is a
nondeterministic logspace Turing machine which (a) for inputs
$x\in\operatorname{dom}(f)$, computes $f(x)$ on its output tape in some number
$\varphi(x,f(x))\equiv 1\pmod{k}$ of its accepting branches, and (b) for each
$y\neq f(x)$ (or for any string $y$, in the case
$x\notin\operatorname{dom}(f)$), computes $y$ on its output tape on some
number $\varphi(x,y)\equiv 0\pmod{k}$ of its accepting branches. We say that
$f\in\mathsf{FUL}_{k}$ if there exists a $\mathsf{FUL}_{k}$ machine which
computes $f$.
If we replace the relation of equivalence modulo $k$ with equality in the
definition of $\mathsf{FUL}_{k}$ above, we obtain a class $\mathsf{FUL}$ of
functions computable by nondeterministic logspace machines with a single
accepting branch. This latter class is analogous to the class $\mathsf{UPF}$
described in Ref. [6], which is in effect a class of functions which may be
computed by a nondeterministic polynomial time Turing machine as a subroutine
without affecting the number of accepting branches of that machine. Modulo $k$
and in logarithmic space, this is the significance of the class
$\mathsf{FUL}_{k}$. Note that in many branches (perhaps even the vast majority
of them), what is written on the output tape of a $\mathsf{FUL}_{k}$ machine
$\mathbf{U}$ may not be the function $f(x)$ which it “computes”; but any
result other than $f(x)$ which $\mathbf{U}$ is meant to compute, cannot affect
the number of accepting branches modulo $k$ of any machine which simulates
$\mathbf{U}$ directly, _e.g._ as a subroutine. These “incorrect results” may
therefore be neglected for the purpose of counting accepting branches modulo
$k$, just as if all accepting branches of $\mathbf{U}$ (of which there are not
a multiple of $k$) computed the result $f(x)$ on the output tape.
In this sense, the closure result
$\mathsf{Mod}_{p}\mathsf{L}^{\mathsf{Mod}_{p}\mathsf{L}}=\mathsf{Mod}_{p}\mathsf{L}$
for $p$ prime shown by Hertrampf, Reif, and Vollmer [5] may be interpreted as
saying that the characteristic function of any
$L\in\mathsf{Mod}_{p}\mathsf{L}$ may be computed by a $\mathsf{FUL}_{p}$
machine; and so a $\mathsf{Mod}_{p}$$\mathsf{L}$ oracle can be directly
simulated by a $\mathsf{Mod}_{p}$$\mathsf{L}$ machine, by simulating the
corresponding $\mathsf{FUL}_{p}$ machine as a subroutine. Our interest in the
function class $\mathsf{FUL}_{k}$ is for essentially the same reason, _i.e._
an oracle for computing any function $f\in\mathsf{FUL}_{k}$ can be substituted
with a simulation of the $\mathsf{FUL}_{k}$ machine itself in the same manner:
###### Lemma 9.
$\mathsf{Mod}_{k}\mathsf{L}^{\mathsf{FUL}_{k}}=\mathsf{Mod}_{k}\mathsf{L}$,
$\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}^{\mathsf{FUL}_{k}}=\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$,
and $\mathsf{FUL}_{k}^{\mathsf{FUL}_{k}}=\mathsf{FUL}_{k}$ for all $k\geqslant
2$.
The proof is essentially the same as that for the oracle closure result of
Ref. [5], of which this Lemma is a natural extension. From simple number-
theoretic considerations, the classes $\mathsf{FUL}_{k}$ have other properties
which are similar to those of $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$:
###### Theorem 10.
Let $k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$ be the
factorization of $k\geqslant 2$ into prime power factors $p_{j}^{e_{j}}$. Then
$\mathsf{FUL}_{k}=\mathsf{FUL}_{p_{1}}\cap\mathsf{FUL}_{p_{2}}\cap\cdots\cap\mathsf{FUL}_{p_{\ell}}$,
and in particular $\mathsf{FUL}_{k}=\mathsf{FUL}_{p_{1}p_{2}\cdots p_{\ell}}$.
###### Proof.
Throughout the following, let $\kappa=p_{1}p_{2}\cdots p_{\ell}$ be the
largest square-free factor of $k$. We first show
$\mathsf{FUL}_{\kappa}=\mathsf{FUL}_{p_{1}}\\!\cap\cdots\cap\mathsf{FUL}_{p_{\ell}}$.
Suppose $f\in\mathsf{FUL}_{p_{j}}$ for each $1\leqslant j\leqslant\ell$, and
is computed by some $\mathsf{FUL}_{p_{j}}$ machine $\mathbf{U}_{j}$ in each
case. Let
$\gamma\;=\;\kappa/p_{1}+\kappa/p_{2}+\cdots+\kappa/p_{\ell}\;.$ (2)
For each prime $p_{j}$, all of the terms in the right-hand sum are divisible
by $p_{j}$ except for the $j\textsuperscript{th}$ term. Then $\gamma$ is
coprime to $p_{j}$ for each $j$, and so is also coprime to $\kappa$. Let
$\beta\equiv\gamma^{-1}\pmod{\kappa}$, and consider the machine
$\mathbf{U}^{\prime}$ which performs the following:
1. 1.
Nondeterministically write some index $1\leqslant j\leqslant\ell$ on the work
tape.
2. 2.
For each such $j$, nondeterministically select some integer $0\leqslant
q<\beta\kappa/p_{j}$.
3. 3.
In each branch, simulate $\mathbf{U}_{j}$ on the input $x$, accepting if and
only if $\mathbf{U}_{j}$ accepts.
For any string $y\in\Sigma^{\ast}$ different from $f(x)$, the number of
branches in which $\mathbf{U}_{j}$ accepts is $m_{j}p_{j}$ for some
$m_{j}\in\mathbb{N}$; and so $\mathbf{U}^{\prime}$ has $m_{j}\beta\kappa$
branches where $j$ is written on the work tape and $y$ is written on the
output tape. Summing over all $j$, we find that any $y\neq f(x)$ is written on
the output tape in a number of branches which is a multiple of $\kappa$.
Similarly, for the case $y=f(x)$, the number of branches in which
$\mathbf{U}_{j}$ accepts is $m_{j}p_{j}+1$ for some $m_{j}\in\mathbb{N}$; and
so $\mathbf{U}^{\prime}$ has $m_{j}\beta\kappa+\beta\kappa/p_{j}$ branches
where $j$ is written on the work tape and $f(x)$ is written on the output
tape. Summing over all $j$ and neglecting multiples of $\kappa$, we have
$\beta\bigl{(}\kappa/p_{1}+\cdots+\kappa/p_{\ell})=\beta\gamma\equiv
1\pmod{\kappa}$ branches in which $f(x)$ is written on the output tape; thus
$\mathbf{U}^{\prime}$ is an $\mathsf{FUL}_{\kappa}$ machine computing $f$. The
converse containment $\mathsf{FUL}_{\kappa}\subseteq\mathsf{FUL}_{p_{j}}$ for
each $1\leqslant j\leqslant\ell$ is trivial.
It remains to show that $\mathsf{FUL}_{\kappa}\subseteq\mathsf{FUL}_{k}$, the
reverse containment again being easy. Let $\mathbf{U}^{\prime}$ be a
$\mathsf{FUL}_{\kappa}$ machine computing a function
$f:\Sigma^{\ast}\to\Sigma^{\ast}$ with length bounded above by
$|f(x)|\leqslant N(x)\in\operatorname{poly}(|x|)$. Suppose $N(x)\in
O(\log|x|)$: we may then construct a $\mathsf{FUL}_{k}$ machine
${\mathbf{U}}^{\prime\prime}$ which computes $f$ by simply performing
$k/\kappa$ consecutive independent simulations of $\mathbf{U}^{\prime}$,
recording the outcome of each simulation on the work tape. For each
$1\leqslant j\leqslant k/\kappa$, in any given computational branch, let
$\varphi_{j}(x)$ be the string computed by the $j\textsuperscript{th}$
simulation of $\mathbf{U}^{\prime}$. If any of the simulations produce a
different output (_i.e._ if $\varphi_{h}(x)\neq\varphi_{j}(x)$ for any
$1\leqslant h,j\leqslant k/\kappa$) or if any of the simulations rejected the
input, $\bar{\mathbf{U}}$ rejects. Otherwise, $\bar{\mathbf{U}}$ writes the
string $\varphi_{1}(x)$ agreed upon by the simulations to the output tape.
More generally, if $N(x)\in\omega(\log|x|)$, then fix some $L\in O(\log|x|)$,
and define for each $1\leqslant m\leqslant N(x)/L$ a machine
$\mathbf{U}^{\prime}_{m}$ which writes the $m\textsuperscript{th}$ block of
$L$ consecutive characters from $f(x)$, padding the end of $f(x)$ with a
symbol $\bullet\notin\Sigma$ if necessary. Rather than perform $k/\kappa$
simulations of $\mathbf{U}^{\prime}$, the machine $\mathbf{U}^{\prime\prime}$
performs $k/\kappa$ simulations of each such $\mathbf{U}^{\prime}_{m}$, again
writing their outcomes (excluding any instance of the symbol
$\bullet\notin\Sigma$) to the work tape if and only if each simulation accepts
and agrees on their output. Once $N(x)$ symbols have been written to the
output tape, $\mathbf{U}^{\prime\prime}$ accepts unconditionally.
Let $\varphi(x,y)$ be the number of computational branches in which
$\mathbf{U}$ accepts with the string $y\in\Sigma^{\ast}$ written on the tape:
by hypothesis, $\varphi(x,y)\equiv 0\pmod{\kappa}$ for each $y\neq f(x)$, and
$\varphi(x,f(x))\equiv 1\pmod{\kappa}$. Similarly, let
$\varphi_{m}(x,y^{(m)})$ be the number of branches in which
$\mathbf{U}^{\prime}_{m}$ accepts with $y^{(m)}\in\Sigma^{L}$ written on the
tape for each $1\leqslant r\leqslant N(x)/L$, and $\Phi(x,y)$ be the number of
branches in which $\mathbf{U}^{\prime\prime}$ accepts with $y\in\Sigma^{\ast}$
written on the tape. Let $M=N(x)/L$ for the sake of brevity. If
$y=y^{(1)}y^{(2)}\cdots y^{(M)}\in\Sigma^{\ast}$, then
$\Phi(x,y)\;=\;\varphi_{1}\bigl{(}x,y^{(1)}\bigr{)}^{k\\!\\!\;/\\!\\!\>\kappa}\;\varphi_{2}\bigl{(}x,y^{(2)}\bigr{)}^{k\\!\\!\;/\\!\\!\>\kappa}\;\cdots\;\varphi_{M}\bigl{(}x,y^{(M)}\bigr{)}^{k\\!\\!\;/\\!\\!\>\kappa},$
(3)
as for each $y^{(m)}$, the number of branches in which
$\mathbf{U}^{\prime\prime}$ accepts with $y^{(m)}$ written on the output tape
is independent of the other substrings $y^{(j)}$ for $j\neq m$, and results
from $k/\kappa$ simulations of $\mathbf{U}^{\prime}_{m}$ which each produce
the substring $y^{(m)}$ as output.
Note that $\varphi_{m}(x,\lambda_{m})$ is equal to the number of computational
branches in which $\mathbf{U}^{\prime}$ writes a string
$\sigma\in\Sigma^{\ast}$ on the output tape in which the
$m\textsuperscript{th}$ block is equal to $y^{(m)}$, which is the sum of
$\varphi(x,\sigma)$ over all strings $\sigma$ consistent with the substring
$y^{(m)}$. By hypothesis, $\varphi(x,\sigma)$ is a multiple of $\kappa$ except
for the single case where $\sigma=f(x)$, in which case
$\varphi(x,\sigma)\equiv 1\pmod{\kappa}$. Thus $\varphi_{m}(x,y^{(m)})\equiv
1\pmod{\kappa}$ if $y^{(m)}\in\Sigma^{L}$ is consistent with the
$m\textsuperscript{th}$ block of $f(x)$; otherwise,
$\varphi_{m}(x,y^{(m)})\equiv 0\pmod{\kappa}$. We then observe the following:
* •
Let $E=\max_{j}\\{e_{j}\\}$ be the largest power of a prime $p_{j}$ dividing
$k$; then $E\leqslant p_{j}^{E-1}\leqslant k/\kappa$ for any $1\leqslant
j\leqslant\ell$. As $k$ divides $\kappa^{E}=p_{1}^{E}\cdots
p_{\ell}^{E}\leqslant\kappa^{k/\kappa}$, we then have
$\varphi_{m}(x,y^{(m)})^{k/\kappa}\equiv 0\pmod{k}$ if
$\varphi_{m}(x,y^{(m)})\equiv 0\pmod{\kappa}$.
* •
The integers which are congruent to $1$ modulo $\kappa$ form a subgroup of
order $k/\kappa$ within the integers modulo $k$; it then follows that
$\varphi_{m}(x,y^{(m)})^{k/\kappa}\equiv 1\pmod{k}$ if
$\varphi(x,y^{(m)})\equiv 1\pmod{\kappa}$.
Taking the product over $1\leqslant m\leqslant M$, we have $\Phi(x,y)\equiv
0\pmod{k}$ unless each substring $y^{(m)}$ is consistent with the
$m\textsuperscript{th}$ block of $f(x)$, in which case $y=f(x)$ and
$\Phi(x,y)\equiv 1\pmod{k}$. Thus $\mathbf{U}^{\prime\prime}$ is an
$\mathsf{FUL}_{k}$ machine computing $f$. ∎
The requirement that an $\mathsf{FUL}_{k}$ machine have one accepting branch
modulo $k$ allows us to easily relate $\mathsf{FUL}_{k}$ to the classes
$\mathsf{F}$$\mathsf{Mod}_{k}$$\mathsf{L}$ and
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$:
###### Lemma 11.
For all $k\geqslant 2$, we have
$\mathsf{FUL}_{k}\subseteq\mathsf{F}\mathsf{Mod}_{k}\mathsf{L}\,\cap\,\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
###### Proof.
Let $\mathbf{U}$ be a $\mathsf{FUL}_{k}$ machine computing
$f:\Sigma^{\ast}\to\Sigma^{\ast}$. Consider a nondeterministic logspace
machine $\mathbf{T}$ taking inputs
$(x,j,b)\in{\Sigma^{\ast}\times\mathbb{N}\times\bigl{(}\Sigma\cup\\{\bullet\\}\bigr{)}}$,
and which simulates $\mathbf{U}$, albeit ignoring all instructions to write to
the output tape, except for the $j\textsuperscript{th}$ symbol which it writes
to the work-tape. (If $j>|f(x)|$, $\mathbf{T}$ instead writes “$\bullet$” to
the work-tape.) Then $\mathbf{T}$ compares the resulting symbol $f(x)_{j}$
against $b$, accepting if they are equal and rejecting otherwise. Then the
number of accepting branches is equivalent to $1$ modulo $k$ if $f(x)_{j}=b$,
and is a multiple of $p$ otherwise, so that
$\mbox{{{bits$(f)$}}}\in\mathsf{Mod}_{k}\mathsf{L}$. To show
$\mbox{{{bits$(f)$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$, we may
consider a machine $\mathbf{T}^{\prime}$ which differs from $\mathbf{T}$ only
in that it rejects if $f(x)_{j}=b$, and accepts otherwise. Thus
$\mathsf{FUL}_{k}\subseteq\mathsf{F}\mathsf{Mod}_{k}\mathsf{L}\,\cap\,\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
∎
This identifies $\mathsf{FUL}_{k}$ as an important subclass of the existing
logspace-modular function classes. For prime-power moduli, we may sharpen
Lemma 11 to obtain a useful identity:
###### Lemma 12.
For any prime $p$ and $e\geqslant 1$,
$\mathsf{FUL}_{p^{e}}=\mathsf{F}\mathsf{Mod}_{p}\mathsf{L}=\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$.
###### Proof.
By Proposition 5 and Lemma 10, it suffices to prove
$\mathsf{FUL}_{p}=\mathsf{F}\mathsf{Mod}_{p}\mathsf{L}$ for $p$ prime. For
$f\in\mathsf{F}\mathsf{Mod}_{p}\mathsf{L}$, we may construct from the
$\mathsf{Mod}_{p}$$\mathsf{L}$ machine $\mathbf{T}$ which decides bits$(f)$ a
family of machines $\mathbf{T}_{j,b}$ (for each $j\in\mathbb{N}$ and
$b\in\Sigma\cup\\{\bullet\\}$), each of which writes $b$ on its output tape
and deciding whether $(x,j,b)\in\mbox{{{bits$(f)$}}}$ on an input
$x\in\Sigma^{\ast}$. Without loss of generality, as in [5, Corollary 3.2] each
machine $\mathbf{T}_{j,b}$ accepts on a number of branches
$\varphi(x,j,b)\equiv 1\pmod{p}$ if case $f(x)_{j}=b$, and
$\varphi(x,j,b)\equiv 0\pmod{p}$ otherwise.
We form a $\mathsf{FUL}_{p}$ machine $\mathbf{U}_{j}$ computing $f(x)_{j}$ by
taking the “disjunction” of the machines $\mathbf{T}_{j,b}$ over all
$b\in\Sigma\cup\\{\bullet\\}$: _i.e._ $\mathbf{U}_{j}$ branches
nondeterministically by selecting $b\in\Sigma\cup\\{\bullet\\}$ to write on
the work-tape and simulates $\mathbf{T}_{j,b}$, accepting with one branch mod
$p$ if and only if $b=f(x)_{j}$ and accepting with zero branches mod $p$
otherwise. Given some upper bound $|f(x)|\leqslant
N(x)\in\operatorname{poly}(|x|)$, we then construct a $\mathsf{FUL}_{p}$
machine $\mathbf{U}$ to compute $f(x)$ by simply simulating $\mathbf{U}_{j}$
for each $1\leqslant j\leqslant N(x)$ in sequence, writing the symbols
$f(x)_{j}$ individually on the output tape; accepting once it either computes
a symbol $f(x)_{j}=\bullet$ (without writing $\bullet$ to the output) or the
final iteration has been carried out. ∎
Lemma 12, together with Lemma 9, may be taken as re-iterating the closure
result of Ref. [5] explicitly in terms of function classes. The importance of
this result to us is in the following consequences, which follow from
Proposition 8 and Lemma 9:
###### Corollary 13.
For any prime $p$ and $e\geqslant 1$,
$\textbf{\\#}{\mathsf{L}}_{p^{e}}\subseteq\mathsf{FUL}_{p}$.
###### Corollary 14.
For any prime $p$ and $e\geqslant 1$,
$\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}\big{.}^{\textbf{\\#}{\mathsf{L}}_{p^{\\!\\!\;e}}}=\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$.
The former result states that we may explicitly compute functions in
$\textbf{\\#}{\mathsf{L}}$ (albeit up to equivalence modulo $p^{e}$) on the
work tape, as subroutines in decision algorithms for
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$; this allows us to simulate
logspace counting oracles modulo $p^{e}$ in
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$. In the following section, we use
this to describe an algorithm for LCONNULL${}_{p^{e}}$ in
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ by a
similar analysis to McKenzie and Cook [1]. By standard techniques, we may then
demonstrate containments for LCONk, LCONNULLk, and LCONXk in terms of
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$.
## 4 Solving congruences and nullspaces mod $k$
We return to the motivating problems of this article. We let $A$ be an
$n\times n$ integer matrix and $\mathbf{y}\in\mathbb{Z}^{n}$ which are
provided as the input to LCONk or LCONXk; and for LCONNULLk, we consider an
$n\times n$ matrix $B$. Without loss of generality, the coefficients of $A$
and $\mathbf{y}$, or of $B$, are non-negative and bounded strictly above by
$k$ (as reducing the input modulo $k$ can be performed in $\mathsf{NC}^{1}$).
We essentially follow the analysis of Ref. [1, Section 8], which reduces
solving linear congruences to computing generating sets for nullspaces modulo
the primes $p_{j}$ dividing $k$. The technical contribution of this section is
to show that the latter problem can be solved for prime powers via a reduction
to matrix multiplication together with modular counting oracles from
$\textbf{\\#}{\mathsf{L}}_{p^{e}}$ for prime powers $p^{e}$.
### 4.1 Computing nullspaces modulo prime powers
We consider an $\mathsf{NC}^{1}$ reduction to matrix inversion and iterated
matrix products modulo $p^{e}$, in a machine equipped with a
$\textbf{\\#}{\mathsf{L}}_{p^{e}}$ oracle to compute certain matrix
coefficients. As we note in Proposition 7, computing individual coefficients
of matrix inverses and matrix products are complete problems for
$\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$, and Corollary
14 implies that this class can simulate $\textbf{\\#}{\mathsf{L}}_{p^{e}}$
oracles. The $\mathsf{NC}^{1}$ reduction itself is essentially the same as
that of McKenzie and Cook [1], which we may summarize as follows.
#### The prime modulus case.
First, consider the case $e=1$, which as we note in Section 2 is solved by
Buntrock _et al._ [4, Theorem 9]. For an $n\times n$ integer matrix $B$, we
may reduce the problem of computing a basis of $\operatorname{null}(B)$ mod
$p$ to rank computations and matrix inversion using the techniques of Borodin,
von zur Gathen, and Hopcroft [7, Theorem 5]. This involves testing the ranks
of a nested collection of sub-matrices of $B$, to determine a subset of
columns forming a basis for $\operatorname{img}(B)$; the reduction from
nullspaces is a truth-table reduction, which for the ultimate reduction to
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ means that we must use Proposition
4 (to enable conjunctive reductions) and Proposition 5 (to allow disjunctive
reductions). Computing the rank of a matrix modulo a prime (_i.e._ in the
field $\mathbb{F}_{p}$) may be reduced to computing characteristic polynomials
of matrices in $\mathbb{F}_{p}(\tau)$ for a formal indeterminate $\tau$ using
a result of Mulmuley [8]; this may be reduced to iterated matrix products over
$\mathbb{F}_{p}(\tau)$ by a construction of Berkowitz [9], where the
coefficients of the matrices are all either constants or drawn from the
coefficients of the matrix $M$. By deriving a suitable bound on the degrees of
the polynomials over $\tau$ involved in these iterated matrix products, one
may substitute the polynomial coefficients by polynomial-size Toeplitz matrix
blocks [10], thereby reducing the iterated matrix product over
$\mathbb{F}_{p}(\tau)$ to one over $\mathbb{F}_{p}$.
#### Recursive reduction for higher powers of primes.
The remainder of the $\mathsf{NC}^{1}$ reduction consists essentially of Ref.
[1, Lemma 8.1] which put LCONNULL (for a variable modulus with magnitude at
most linear in $n$) in $\mathsf{NC}^{3}$: in our case, we reduce
LCONNULL${}_{p^{e}}$ to LCONNULLp together with matrix products and access to
oracles for computing coefficients of certain matrices. Inducting on
$1\leqslant t\leqslant e$, suppose that we have a generating set
$\smash{\mathbf{V}^{(t)}_{\\!1},\ldots,\mathbf{V}^{(t)}_{\\!N_{t}}}$ over
$\mathbb{Z}/p^{e}\mathbb{Z}$ for the nullspace of $B$ modulo $p^{t}$.
Certainly any solution to $B\mathbf{w}\equiv 0\pmod{p^{t+1}}$ must also be a
solution to $B\mathbf{w}\equiv 0\pmod{p^{t}}$ as well;
then we may decompose such $\mathbf{w}$ modulo $p^{e}$ as a linear combination
of the vectors $\smash{\mathbf{V}^{(t)}_{\\!j}}$,
$\mathbf{w}=u_{1}\mathbf{V}^{(t)}_{\\!1}+\cdots+u_{N_{t}}\mathbf{V}^{(t)}_{\\!N_{t}}+p^{t}\mathbf{\hat{w}}$
(4a) for some $\mathbf{\hat{w}}\in\mathbb{Z}^{n}$; or more concisely,
$\mathbf{w}=V^{(t)}\mathbf{z}\;,$ (4b) where we define the block matrices
$V^{(t)}=\smash{\bigl{[}\,\mathbf{V}^{(t)}_{\\!1}\,\,\mathbf{V}^{(t)}_{\\!2}\;\cdots\;\mathbf{V}^{(t)}_{\\!N_{t}}\;\big{|}\;p^{t}I\;\bigr{]}}$
and
$\mathbf{z}=\smash{\bigl{[}\,u_{1}\,\,u_{2}\;\cdots\;u_{N_{t}}\,\big{|}\>\mathbf{\hat{w}}\>\bigr{]}}^{\mathsf{T}}\\!\\!\in\mathbb{Z}^{N_{t}+n}$.
To consider the additional constraints imposed by $B\mathbf{w}\equiv
0\pmod{p^{t+1}}$, consider a decomposition $B=B_{t}+p^{t}\hat{B}_{t}$, where
the coefficients of $B_{t}$ are bounded between $0$ and $p^{t}$.
We then have
$\displaystyle\Biggl{(}\sum_{j=1}^{N_{t}}u_{j}\Bigl{[}B_{t}\mathbf{V}^{(t)}_{\\!j}\,$
$\displaystyle+\;p^{t}\\!\hat{B}_{t}\mathbf{V}^{(t)}_{\\!j}\Bigr{]}\Biggr{)}+p^{t}B_{t}\mathbf{\hat{w}}$
$\displaystyle\equiv\,B\Biggl{(}\sum_{j=1}^{N_{t}}u_{j}\mathbf{V}^{(t)}_{\\!j}\\!\Biggr{)}+p^{t}\mathbf{\hat{w}}\equiv
0\pmod{p^{t+1}}\,.$ (5a) As the vectors $B_{t}\mathbf{V}^{(t)}_{\\!j}$ have
coefficients divisible by $p^{t}$ by construction, we may simplify to
$\Biggl{(}\sum_{j=1}^{N_{t}}u_{j}\Bigl{[}B_{t}\mathbf{V}^{(t)}_{\\!j}\\!\big{/}p^{t}\,+\;\\!\hat{B}_{t}\mathbf{V}^{(t)}_{\\!j}\Bigr{]}\Biggr{)}+B_{t}\mathbf{\hat{w}}\equiv
0\pmod{p}\,,$ (5b) or somewhat more concisely, $\bar{B}^{(t)}\mathbf{z}\equiv
0\pmod{p},$ (5c)
where we define
$\displaystyle\bar{B}^{(t)}$
$\displaystyle=\,\bigl{[}\,\mathbf{b}^{(t)}_{1}\,\,\mathbf{b}^{(t)}_{2}\;\cdots\;\mathbf{b}^{(t)}_{N_{t}}\,\bigr{|}\;B_{t}\,\bigr{]},$
$\displaystyle\quad\text{for}\;\;\mathbf{b}^{(t)}_{j}$
$\displaystyle=\,B_{t}\mathbf{V}^{(t)}_{\\!j}\\!\big{/}p^{t}\;+\;\hat{B}_{t}\mathbf{V}^{(t)}_{\\!j},$
(6)
and where $\mathbf{z}$ is as we defined it above. To find not just one vector
$\mathbf{w}$ but a set of generators
$\smash{\mathbf{V}^{(t+1)}_{\\!1}\\!},\,\ldots,\smash{\mathbf{V}^{(t+1)}_{\\!N_{t+1}}}$
over $\mathbb{Z}/p^{e}\mathbb{Z}$ for $\operatorname{null}(B)$ mod $p^{t+1}$,
it suffices to find a generating set
$\mathbf{z}_{1},\ldots,\mathbf{z}_{N_{t+1}}\\!$ for the nullspace of
$\bar{B}^{(t)}$ mod $p$, and then set
$\smash{\mathbf{V}^{(t+1)}_{\\!h}=V^{(t)}\mathbf{z}_{h}}$. Note that the
nullspace of $\bar{B}^{(t)}$ modulo $p$ over $\mathbb{Z}/p^{e}\mathbb{Z}$ will
contain many vectors which are equivalent mod $p$, but at most $N_{t}+n$
equivalence classes; we may then without loss of generality select vectors
$\mathbf{z}_{1}=p\mathbf{\hat{e}}_{1}$,
$\mathbf{z}_{2}=p\mathbf{\hat{e}}_{2}$, …,
$\mathbf{z}_{N_{t}}=p\mathbf{\hat{e}}_{N_{t}}$, and choose the remaining
vectors $\mathbf{z}_{h}$ representing non-trivial vectors in
$\operatorname{null}(\bar{B}^{(t)})$ mod $p$ to have coefficients bounded
between $0$ and $p$. We thus obtain $N_{t+1}\leqslant 2N_{t}+n$ vectors over
$\mathbb{Z}/p^{e}\mathbb{Z}$ which span $\operatorname{null}(B)$ modulo
$p^{t+1}$. Because $\mbox{{{LCONNULL${}_{p}$}}}\in\mathsf{FUL}_{p^{e}}$, which
is low for $\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$, we have reduced to
computing matrix products involving the matrix $\bar{B}^{(t)}$ in a
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machine.
#### Matrix products in oracle models.
The natural approach outlined in Buntrock _et al._ [4] for evaluating the
coefficients of an iterated matrix product $M_{1}M_{2}\cdots
M_{\operatorname{poly}(n)}$ modulo $k$ — _i.e._ as a
$\textbf{\\#}{\mathsf{L}}_{k}$ function — requires access to individual
coefficients at any given step of the algorithm. One simulates a branching
program with nondeterministic choices, in which the matrices act as transition
functions on the row-positions of a vector
$\mathbf{v}_{\tau}\in(\mathbb{Z}/kZ)^{n}$, to obtain a new vector
$\mathbf{v}_{\tau+1}$. To evaluate the $(h,j)$-coefficient of the matrix
product, we count the number of computational branches which end at a the
$h\textsuperscript{th}$ row, given an initial vector
$\mathbf{v}_{0}=\mathbf{\hat{e}}_{j}$: we do this by accepting all branches
which end with the row position $h$, and rejecting all others. This approach
requires only non-deterministic selection of row-positions, logarithmic space
to record the row-positions, and the ability query individual coefficients of
the matrices being multiplied. When the matrices $M_{j}$ are specified as part
of the input, or more generally for any problem reduced _projectively_ to
matrix products (meaning that the matrices involved have coefficients which
are either constants or taken from the input tape), the algorithm to evaluate
the matrix products is straightforward; more generally, for any class $C$
which is low for $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ (_e.g._
$C=\mathsf{FUL}_{k}$), we may compute any matrix product in
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ where the coefficients are
obtained from the input by may be obtained by queries to $C$ oracles.
We may use these observations to reduce LCONNULL${}_{p^{e}}$ to matrix
products modulo $p^{e}$. In the recursive reduction for prime powers outlined
above to LCONNULLp, every step is projective except for the matrix
multiplications, and the problem of finding null spaces modulo $p$ for the
matrices $\bar{B}^{(t)}$ (which are not themselves part of the input). The
columns of $\bar{B}^{(t)}$ are either columns of $\hat{B}_{t}$ (which are
themselves the result of integer division of columns of $B$ by $p^{t}$, this
dividend being bounded by a constant) or are integer vectors of the form
$B\mathbf{V}^{(t)}/p^{t}$. The coefficients of $B\mathbf{V}^{(t)}$ are
computable as a matrix product, and thus may be computed in
$\textbf{\\#}{\mathsf{L}}_{p^{e}}$ from $B$ itself and $\mathbf{V}^{(t)}$;
provided a $\textbf{\\#}{\mathsf{L}}_{p^{e}}$ oracle, we may then obtain those
coefficients and divide them by $p^{t}$ in $\mathsf{NC}^{1}$. By Corollary 13,
we have $\textbf{\\#}{\mathsf{L}}_{p^{e}}\subseteq\mathsf{FUL}_{p}$, which is
low for $\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$. We therefore have a
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$-reduction from computing a basis
for $\operatorname{null}(B)$ modulo $p^{t+1}$ to computing the basis
$\smash{\mathbf{V}^{(t)}_{1}},\ldots,\smash{\mathbf{V}^{(t)}_{N_{t}}}$ modulo
$p^{t}$. We may then carry out the recursive reduction to obtain a
$\mathsf{FUL}_{p^{e}}$-reduction from LCONNULL${}_{p^{e}}$ to iterated matrix
products, via LCONNULLp; the number of vectors $\mathbf{V}^{(e)}_{j}$ in the
generating set will, by induction, be $N_{e}\leqslant
n+2n+\cdots+2^{e-1}n\leqslant p^{e}n\in O(n)$.
An important feature the recursive reduction described above is that the
exponent $e$ is itself a constant. The $\textbf{\\#}{\mathsf{L}}_{p^{e}}$
oracles to compute coefficients of $\bar{B}^{(e-1)}$ require access to the
coefficients of vectors $\smash{V^{(e)}_{j}}$, which in turn will require
$\textbf{\\#}{\mathsf{L}}_{p^{e}}$ oracles to compute coefficients of
$\bar{B}^{(e-2)}$, and so on. This is a sequential reduction, and the space
resources can be described straightforwardly using a stack model of the work
tape: each nested $\textbf{\\#}{\mathsf{L}}_{p^{e}}$ oracle is simulated as a
$\mathsf{FUL}_{p^{e}}$ subroutine which is allocated $O(\log|B|)=O(\log(n))$
space on the tape (where $|B|\in O(n^{2})$ is the size of the input matrix
after reduction modulo $p^{e}$), and which makes further recursive calls to
$\mathsf{FUL}_{p^{e}}$ subroutines which do likewise, down depth at most $e$.
The space resources then scale as $O(e\log(n))$; in our setting of a constant
modulus, the space requirements are then $O(\log(n))$.
Consider a nondeterministic logspace machine with alphabet
$\bar{\Sigma}=\Sigma\cup\\{\bullet\\}$ for $\Sigma=\\{0,\ldots,p^{e}-1\\}$.
Using a $\mathsf{FUL}_{p^{e}}$-reduction to reduce LCONNULL${}_{p^{e}}$ for
prime powers $p^{e}$ to computing coefficients of matrix products, we may test
equality of individual coefficients against some reference value
$b\in\bar{\Sigma}$ provided as input. Therefore:
###### Lemma 15.
$\mbox{{{LCONNULL${}_{p^{e}}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$.
### 4.2 Completeness results for arbitrary constant moduli
The above suffices to show that LCONk, LCONXk, and LCONNULLk are complete
problems for $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ and
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$, as we now
show. We consider nondeterministic logspace machines operating on an alphabet
$\bar{\Sigma}_{k}=\Sigma_{k}\cup\\{\bullet\\}$, where $\Sigma_{k}$ is the set
of integers $0\leqslant r<k$. For the function problems LCONNULLk and LCONXk,
we wish respectively to compute
* •
a function $\mathcal{N}_{k}:\Sigma_{k}^{n^{2}}\to\Sigma_{k}^{Nn}$ for $N\in
O(n)$ such that $\mathcal{N}_{k}(B)$ is a sequence of vectors
$(\mathbf{Z}_{0},\mathbf{Z}_{1},\ldots,\mathbf{Z}_{N-1})$ which generate
$\operatorname{null}(B)$ in $\mathbb{Z}/k\mathbb{Z}$; and
* •
a partial function $\mathcal{S}_{k}:\Sigma^{n^{2}+n}\rightharpoonup\Sigma^{n}$
such that $(A,\mathbf{y})\in\operatorname{dom}(\mathcal{S}_{k})$ if and only
if there exists a solution $\mathbf{x}$ to the system
$A\mathbf{x}\equiv\mathbf{y}\pmod{k}$, in which case
$\mathcal{S}_{k}(A,\mathbf{y})$ is such a solution.
Following [1, Lemma 5.3], we may reduce LCONk and LCONXk for $k\geqslant 2$ to
LCONNULLk, as follows. Suppose $A\mathbf{x}\equiv\mathbf{y}\pmod{k}$ has
solutions. Consider $B=[\,A\,|\,\mathbf{y}\,]$: then there are solutions to
the equation $B\bar{\mathbf{x}}\equiv 0\pmod{k}$. In particular, there will be
a solutions $\bar{\mathbf{x}}=\mathbf{x}\oplus x_{n+1}$ in which $x_{n+1}=-1$,
and more generally in which $x_{n+1}$ is coprime to $k$. Conversely, if there
is such a solution $\bar{\mathbf{x}}$ to $B\bar{\mathbf{x}}\equiv 0\pmod{k}$,
we may take $\alpha\equiv-x_{n+1}^{-1}\pmod{k}$ and obtain
$A(\alpha\mathbf{x})\equiv-\alpha x_{n+1}\mathbf{y}\equiv\mathbf{y}\pmod{k}$.
To determine whether $A\mathbf{x}\equiv\mathbf{y}\pmod{k}$ has solutions, or
to construct a solution, it thus suffices to compute a basis for the nullspace
of $B$, and determine from this basis whether any of the vectors
$\bar{\mathbf{x}}\in\operatorname{null}(B)$ have a final coefficient coprime
to $k$; if so, the remainder of the coefficients of $\bar{\mathbf{x}}$ may be
used to compute a solution to the original system.
In the special case $k=p^{e}$ of a prime power, coprimality to $k$ simply
entails that $k$ is not divisible by $p$. To solve LCON${}_{p^{e}}$ and
LCONX${}_{p^{e}}$, we compute individually the final coefficients of the
vectors
$(\mathbf{Z}_{0},\mathbf{Z}_{1},\mathbf{Z}_{2},\ldots)=\mathcal{N}_{p^{e}}(B)$
for $B=[\,A\,|\,\mathbf{y}\,]$, searching for an index $1\leqslant h\leqslant
N_{e}$ for which the dot product $\mathbf{\hat{e}}_{n+1}\cdot\mathbf{Z}_{h}$
is not divisible by $p$. Without loss of generality, we select the minimum
such $h$: the search problem can be formulated as a truth-table reduction on
divisibility tests of these coefficients by $p$. Both the reduction and the
divisibility test are feasible for
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$; we may suppose that this
reduction and test are performed by a $\mathsf{FUL}_{p}$ oracle so that the
outcome is explicitly recorded on the work tape in a single branch mod $p$. If
there is no such index $h$, we indicate that no solution exists by accepting
unconditionally, indicating either a _no_ instance of bits$(\mathcal{S}_{k})$
or of LCONk on a $\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machine.
Otherwise, there exists a solution to the linear congruence. To indicate for
LCON${}_{p^{e}}$ that $(A,\mathbf{y})$ is a _yes_ instance on a
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machine, we reject on all
computational branches to make the number of accepting branches zero modulo
$p$. To solve bits$(\mathcal{S}_{k})$, we compute the minimum index $h$ and
the coefficient $\mathbf{\hat{e}}_{n+1}^{\mathsf{T}}\mathbf{Z}_{h}$, which we
store on the work tape in binary. We then compute $\alpha\equiv-
x_{n+1}^{-1}\pmod{p^{e}}$, and then obtain the coefficients of
$\smash{\alpha\mathbf{Z}_{h}}$, which we compare to input coefficients,
rejecting (to indicate a _yes_ instance) if the coefficients match, and
accepting (to indicate a _no_ instance) otherwise. Therefore:
###### Lemma 16.
$\mbox{{{LCON${}_{p^{e}}$}}}\in\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$ and
$\mbox{{{LCONX${}_{p^{e}}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p}\mathsf{L}$.
As we remarked in Section 2, we may solve LCONk for arbitrary moduli
$k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$ by reduction to the
problems LCON${}_{\smash{p_{\\!\\!\;j}}^{\\!e_{\\!\\!\;j}}}$ for $1\leqslant
j\leqslant\ell$; the same is true for LCONXk and LCONNULLk. Let
$q_{j}=p_{j}^{e_{j}}$ for the sake of brevity. For LCONk, we simply have
$\mbox{{{LCON${}_{k}$}}}=\mbox{{{LCON${}_{q_{\\!\\!\;1}}$}}}\cap\cdots\cap\mbox{{{LCON${}_{q_{\\!\\!\;\ell}}$}}}$.
For LCONNULLk and LCONXk, let congbits$(f,q_{j})$ be the decision problem of
determining for inputs
$(x,h,b)\in\Sigma_{k}^{\ast}\times\mathbb{N}\times\bar{\Sigma}_{k}$ whether
$x\in\operatorname{dom}(f)$, and (this being granted) whether either
$f(x)_{h}\equiv b\pmod{q_{j}}$ for $b\neq\bullet$ or $f(x)_{h}=\bullet=b$.
* •
Clearly bits$(\mathcal{S}_{k})$ is the intersection of the problems
congbits$(\mathcal{S}_{k},q_{j})$ for ${1\leqslant j\leqslant\ell}$. We may
show
$\mbox{{{congbits$(\mathcal{S}_{k},q_{j})$}}}\in\mathsf{co}\mathsf{Mod}_{q_{j}}\mathsf{L}$
for each ${1\leqslant j\leqslant\ell}$, as follows. For $b\in\Sigma_{k}$, we
may expand $b$ in binary on the work tape and evaluate its reduction
$0\leqslant b^{\prime}<q_{j}$ modulo a given prime power $q_{j}$; for
$b=\bullet$ we simply let $b^{\prime}=\bullet$ as well, so that
$b^{\prime}\in\bar{\Sigma}_{q_{j}}$. We perform a similar reduction for each
coefficient in $(A,\mathbf{y})$ to obtain an input
$(A^{\prime},\mathbf{y}^{\prime})$ with coefficients in $\Sigma_{q_{j}}$. Then
we may simulate a $\mathsf{co}$$\mathsf{Mod}_{p_{j}}$$\mathsf{L}$ machine to
decide whether
$((A^{\prime},\mathbf{y}^{\prime}),h,b^{\prime})\in\mbox{{{bits$(\mathcal{S}_{q_{j}})$}}}$.
Thus
$\mbox{{{bits$(\mathcal{S}_{k})$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
* •
To show
$\mbox{{{bits$(\mathcal{N}_{k})$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$,
we follow the reduction of McKenzie and Cook in Ref. [1, Theorem 8.3]. Given
vectors $\smash{\mathbf{X}^{(q_{j})}_{1},\ldots,\mathbf{X}^{(q_{j})}_{N_{j}}}$
spanning the nullspace of $B$ modulo $q_{j}$ for each $1\leqslant
j\leqslant\ell$, the nullspace of $B$ modulo $k$ is spanned over the integers
modulo $k$ by the vectors
$\tfrac{k}{q_{1}}\mathbf{X}^{(q_{1})}_{1},\;\ldots\,,\;\tfrac{k}{q_{1}}\mathbf{X}^{(q_{1})}_{N_{1}},\;\tfrac{k}{q_{2}}\mathbf{X}^{(q_{2})}_{1},\;\ldots\,,\tfrac{k}{q_{j}}\mathbf{X}^{(q_{j})}_{h}\,,\;\ldots\,,\;\tfrac{k}{q_{\ell}}\mathbf{X}^{(q_{\ell})}_{N_{\ell}}\;.$
(7)
(We omit the vectors $k\mathbf{\hat{e}}_{h}$ included by Ref. [1], as these
are congruent to $\mathbf{0}$ in $\mathbb{Z}/k\mathbb{Z}$.) Let
$\mathbf{Z}_{h}$ be the list of such vectors, for $0\leqslant
h<N_{1}+\cdots+N_{\ell}$: we define $\mathcal{N}_{k}$ for $k$ divisible by
more than one prime to produce this sequence of vectors as output. Notice that
each $\mathbf{Z}_{h}$ is
* –
congruent to $\mathbf{0}$ modulo $q_{j}$ for every $j\neq 1$ for $0\leqslant
h<N_{1}$,
* –
congruent to $\mathbf{0}$ modulo $q_{j}$ for every $j\neq 2$ for
$N_{1}\leqslant h<N_{1}+N_{2}$, and
* –
generally, congruent to $\mathbf{0}$ modulo $q_{j}$ for every $j\geqslant 1$,
except for the index $j$ for which $M_{j-1}\leqslant h<M_{j}$, where for the
sake of brevity we write $M_{j}=\sum_{t=1}^{j}N_{t}$ .
We may then reduce bits$(\mathcal{N}_{k})$ to testing the congruence of
coefficients of $\mathbf{Z}_{h}$ with $0$ modulo $q_{j}$ for all prime powers
for which $h<M_{j-1}$ or $h\geqslant M_{j}$, and testing congruence with the
coefficients of $\smash{\frac{k}{q_{j}}\mathbf{X}^{(q_{j})}_{h-M_{j}+1}}$
otherwise. These congruences modulo each prime power $q_{j}$ can again be
evaluated in $\mathsf{co}$$\mathsf{Mod}_{q_{j}}$$\mathsf{L}$ algorithm for
congbits${}_{j}(\mathcal{N}_{k})$, using the $\mathsf{NC}^{1}$ reduction to
bits$(\mathcal{N}_{q_{j}})$ as above.
The above reductions suffice to show:
###### Theorem 17.
For all $k\geqslant 2$, we have
$\mbox{{{LCONNULL${}_{k}$}}},\,\mbox{{{LCONX${}_{k}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$
and $\mbox{{{LCON${}_{k}$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
Finally, note that one may also LCON${}_{p_{j}}$ to LCONk, for any prime
$p_{j}$ dividing $k$, by considering the feasibility of the congruence
$(kA/p_{j})\,\mathbf{x}\;\equiv\;k\mathbf{y}/p_{j}\pmod{k},$ (8)
which is equivalent to $A\mathbf{x}\equiv\mathbf{y}\pmod{p_{j}}$. By
Propositions 2–4, all problems in LCONk may be reduced to solving some
instances of LCON${}_{p_{j}}$ for each ${1\leqslant j\leqslant\ell}$: then
LCONk is $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$-hard. Similar remarks
apply to LCONXk and LCONNULLk. Therefore:
###### Theorem 18.
For all $k\geqslant 2$, LCONk is
$\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$-complete, and LCONNULLk and LCONXk
are $\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$-complete.
## 5 Further Remarks
The above analysis was motivated by observing that the reduction of McKenzie
and Cook [1] for LCONX and LCONNULL (which take the modulus $k$ as input, as a
product of prime powers $p_{j}^{e_{j}}\in O(n)$) was very nearly a projective
reduction to matrix multiplication, and that it remained only to find a way to
realize the division by prime powers $p^{t}$ involved in the reduction to
LCONNULLp. By showing that logspace counting oracles modulo $p^{e}$ could be
simulated by a $\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machine, using the
function class $\mathsf{FUL}_{k}$ as a notion of naturally simulatable oracles
for the classes $\mathsf{Mod}_{k}$$\mathsf{L}$ and
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$, the containments of Theorem 17
became feasible.
Extending the definition of bits$(f)$ to accomodate partial functions in the
is crucial to our result that
$\mbox{{{LCONX${}_{k}$}}}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$,
in the sense that there is no obvious way to extend the algorithm to decide
bits$(\bar{S}_{k})$ for any unambiguous extension of $\mathcal{S}_{k}$ to
infesible systems of equations, _e.g._ by accepting on some symbol “!” if and
only if there is no solution to a congruence provided as input. Such an
algorithm would be a signficant result, as it would follow that
$\mbox{{{LCON${}_{k}$}}}\in\mathsf{Mod}_{k}\mathsf{L}$, thereby showing that
this class is closed under complements.
In the recursive reduction for LCONNULL${}_{p^{e}}$, the fact that $e\in O(1)$
is essential not only for the logarithmic bound on the work tape, but also for
the running time on a $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ machine to
be polynomial. The $\mathsf{FUL}_{p}$ machines used to implement the
$\textbf{\\#}{\mathsf{L}}_{p^{e}}$ oracles, from the constructions of Theorem
10 and Lemma 12, implicitly involve many repeated simulations of
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machines ($p^{e}/p=p^{e-1}$ times
each) to decide equality of counting functions with residues $0\leqslant
r<p^{e}$: this contributes to a factor of overhead growing quickly with $e$.
Therefore our results are mainly of theoretical interest, characterizing the
complexity of these problems with respect to logspace reductions. It is
reasonable to ask if there is an algorithm on a
$\mathsf{co}$$\mathsf{Mod}_{p}$$\mathsf{L}$ machine for LCONNULL${}_{p^{e}}$,
whose running time grows slowly with $e$.
Given the natural role of the class $\mathsf{FUL}_{p^{e}}$ in simulating of
$\textbf{\\#}{\mathsf{L}}_{k}$ oracles, one might ask _e.g._ whether the
characteristic function of LCONk is contained in $\mathsf{FUL}_{k}$. It is
interesting to consider the difference between such potential containments,
and those proven as Theorem 17. We first note an alternative characterization
of $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$:
###### Proposition 19.
For every $k\geqslant 2$, $L\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ if and
only if there exists $\varphi\in\textbf{\\#}{\mathsf{L}}$ such that $x\in L$
if and only if $\varphi(x)$ is _coprime_ to $k$.
###### Proof.
For $k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$ as usual, we have
$L\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ if and only if $L=L_{1}\cap
L_{2}\cap\cdots\cap L_{\ell}$ for languages
$L_{j}\in\mathsf{co}\mathsf{Mod}_{p_{j}}\mathsf{L}=\mathsf{Mod}_{p_{j}}\mathsf{L}$
by Propositions 3 and 5. Let $\mathbf{T}_{1},\ldots,\mathbf{T}_{\ell}$ be
nondeterministic logspace machines such that $\mathbf{T}_{j}$ accepts on input
$x$ with a number of branches not divisible by $p_{j}$ if $x\in L_{j}$, and
with zero branches modulo $p_{j}^{e_{j}}$ otherwise. Using a similar
construction to that of Lemma 10 for the square-free case, we may obtain a
_single_ nondeterministic logspace machine $\mathbf{T}$ which accepts on a
number of branches not divisible by $p_{j}$ if $x\in L_{j}$, and on a number
of branches equivalent to $0$ mod $p_{j}$ otherwise. If $x\in L$, then the
number of branches on which $\mathbf{T}$ accepts is not divisible by any prime
$p_{j}$, which means that it is coprime to $k$; otherwise, there exists some
prime $p_{j}$ which divides the number of accepting branches, so that the
number of branches is not coprime to $k$. ∎
This, in turn, suggests a characterization for
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ in the
same vein as the definition of $\mathsf{FUL}_{k}$:
###### Proposition 20.
For all $k\geqslant 2$,
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ consists
of those (partial) functions $f$ computable by a nondeterministic logspace
machine which (a) for inputs $x\in\operatorname{dom}(f)$, computes $f(x)$ on
its output tape in some number $\varphi(x,f(x))$ of its accepting branches
which is coprime to $k$, and (b) for each output string $y\neq f(x)$ (or for
any string $y$, in the case $x\notin\operatorname{dom}(f)$), computes $y$ on
its output tape on some number $\varphi(x,y)$ such that
$\gcd\bigl{(}\varphi(x,y),k\bigr{)}>1$. Furthermore, we may require without
loss of generality for all $y\in\Sigma^{\ast}$ that either $\varphi(x,y)\equiv
1\pmod{k}$, or $\gcd\bigl{(}\varphi(x,y),k\bigr{)}$ is a product of some of
the maximal prime powers $p_{j}^{e_{j}}$ which divide $k$.
###### Proof.
Let $k=p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\ell}^{e_{\ell}}$ be the
factorization of $k$ into its prime power factors. For
$f\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$, consider
the $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ machine $\mathbf{T}$ for
deciding bits$(f)$ with the characteristics described in Proposition 19. Using
a construction similar to that of Lemma 12, we may construct machines
$\mathbf{C}_{h}$ which simulate $\mathbf{T}$ on inputs $(x,h,b)$ for each
$b\in\Sigma\cup\\{\bullet\\}$, writing the symbol $b$ on the output tape. For
$x\notin\operatorname{dom}(f)$, each possible output is written to the output
tape in some number of branches which has a non-trivial common divisor with
$k$. Otherwise, for $x\in\operatorname{dom}(f)$, this machine writes
$f(x)_{h}$ on the tape in some number of branches coprime to $k$, and every
$b\neq f(x)_{h}$ on the tape some number of branches which has prime divisors
in common with $k$, by hypothesis. Still following Lemma 12, consider a
machine $\mathbf{C}$ simulating each $\mathbf{C}_{h}$ in turn for $1\leqslant
h\leqslant N(x)$ up to some upper bound $|f(x)|\leqslant
N(x)\in\operatorname{poly}(|x|)$ or until we encounter symbols
$f(x)_{h}=\bullet$. A string $y\in\Sigma^{\ast}$ written to the output tape
occurs in a number of accepting branches which is coprime to $k$ if and only
if each character of $y$ occurs a number of times coprime to $k$, which is to
say if $y=f(x)$.
The stricter characterization of the number of branches in which each $y=f(x)$
or $y\neq f(x)$ is accepted may be obtained as follows. By Proposition 3,
consider functions
$f_{j}\in\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{p_{j}}\mathsf{L}$ such
that
$\mbox{{{bits$(f)$}}}=\mbox{{{bits$(f_{1})$}}}\cap\cdots\cap\mbox{{{bits$(f_{\ell})$}}}$.
Using Theorem 10 and Lemma 12, consider
$\mathsf{FUL}_{\smash{\smash{p_{\\!\\!\;j}}^{\\!e_{\\!\\!\;j}}}}$ machines
$\mathbf{U}_{j}$ which compute $f_{j}$ for each $1\leqslant j\leqslant\ell$.
By a similar construction to Theorem 10, we may obtain a machine
$\tilde{\mathbf{U}}$ which writes $f(x)$ on the tape in a number of branches
which is equivalent to $1$ modulo every prime power $p_{j}^{e_{j}}$, and which
writes any $y\neq f(x)$ on the output tape a number of times which is
equivalent to $0$ modulo one or more powers $p_{j}^{e_{j}}$ (but equivalent to
$1$ for the other prime powers $p_{h}^{e_{h}}$). Then $f(x)$ is written on the
output tape on one branch modulo $k$; the other strings $y\neq f(x)$ occur on
the output tape a number of times which is divisible by some maximal prime
power divisors $p_{j}^{e_{j}}$, but which is coprime to the other maximal
prime power divisors $p_{h}^{e_{h}}$.
To show the converse, _i.e._ that the functions $f$ computable by such
logspace nondeterministic machines $\mathbf{C}$ are indeed in
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$, simply
consider a machine $\tilde{\mathbf{T}}$ which takes a tuple $(x,h,b)$ as
input, and simulates a machine $\mathbf{C}$ as described above long enough to
compute $f(x)_{h}$, accepting unconditionally. Then the number of branches on
which $\tilde{\mathbf{T}}$ accepts is coprime to $k$ if and only if
$x\in\operatorname{dom}(f)$ and $f(x)_{j}=b$, by construction. By Proposition
19, it follows that
$\mbox{{{bits$(f)$}}}\in\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$. ∎
The definition of $\mathsf{FUL}_{k}$ differs from the above characterization
of $\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ by the
further requirement that, for a $\mathsf{FUL}_{k}$ machine computing some
function $f$, output strings $y\neq f(x)$ must occur in _zero_ branches mod
$k$ and not just in a number of branches which has maximal prime power factors
in common with $k$. Thus, we see that Definition IV does not result in a class
which is entirely different in signficance from
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$, even for
$k$ composite.
There is no obvious way to bridge the gap between the definition of
$\mathsf{FUL}_{k}$, and the characterization of
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ given by
Proposition 19. Of course, LCONk can be solved in $\mathsf{FUL}_{k}$ if and
only if
$\mathsf{FUL}_{k}=\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
As $\mathsf{FUL}_{k}$ is low for $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$,
this would imply $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ is closed under
logspace Turing reductions, and that therefore
$\mathsf{Mod}_{k}\mathsf{L}=\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
Furthermore, by Proposition 3 and Theorem 10,
$\mathsf{FUL}_{k}=\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$
would imply a surprising collapse of logspace mod classes beneath
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$: for any distinct prime divisors
$p_{h},p_{j}$ of $k$ we would have
$\mathsf{F}\mathsf{Mod}_{p_{\\!\\!\;h}\\!}\mathsf{L}\subseteq\mathsf{F}\mathsf{Mod}_{k}\mathsf{L}=\mathsf{FUL}_{k}\subseteq\mathsf{FUL}_{p_{\\!\\!\;j}\\!}\\!=\mathsf{F}\mathsf{Mod}_{p_{\\!\\!\;j}\\!}\mathsf{L}$,
and in particular
$\mathsf{Mod}_{p_{\\!\\!\;h}\\!}\mathsf{L}=\mathsf{Mod}_{p_{\\!\\!\;j}\\!}\mathsf{L}$.
The converse, that
$\mathsf{Mod}_{p_{h}\\!}\mathsf{L}=\mathsf{Mod}_{p_{j}\\!}\mathsf{L}$ for all
primes dividing $k$ only if
$\mathsf{FUL}_{k}=\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$,
is trivial. A similar collapse would occur even if the characteristic function
of LCONk could be computed in
$\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$; not only would
this indicate that $\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ is closed under
containment, but also under oracles, as it would allow simulation of
$\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$ oracles in a way
much similar to the simulation of $\mathsf{FUL}_{k}$ oracles by
$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ machines (where a collection of
branches having the same tape-conents are insignificant if the number of
branches has prime power divisors in common with $k$, although not necessarily
divisible by $k$). If we suppose that $\mathsf{FUL}_{k}$,
$\mathsf{F}$$\mathsf{Mod}_{k}$$\mathsf{L}$, and
$\mathsf{F}{\cdot\>\\!}$$\mathsf{co}$$\mathsf{Mod}_{k}$$\mathsf{L}$ are
distinct for any $k\geqslant 2$ divisible by two or more primes, it would be
interesting to characterize $\mathsf{FUL}_{k}$ as a subclass of
$\mathsf{F}\mathsf{Mod}_{k}\mathsf{L}\cap\mathsf{F}{\cdot\>\\!}\mathsf{co}\mathsf{Mod}_{k}\mathsf{L}$.
### Acknowledgements
This work was supported by the EC project QCS. I would like to thank Bjarki
Holm for feedback in the early stages of work on this problem, and for
indicating helpful references in the literature on the variable modulus
problem LCON.
### Contact
Please send questions or feedback to [niel.debeaudrap@gmail.com].
## References
* [1] P. McKenzie, S. Cook. _The parallel complexity of abelian permutation group problems_. SIAM Journal of Computing 16 (pp. 880–909), 1987.
* [2] V. Arvind, T. C. Vijayaraghavan. _Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes_. Computational Complexity 19 (pp. 57–98), 2010.
* [3] J. Köbler, S. Toda. _On the Power of Generalized MOD-Classes_. Mathematical Systems Theory 29 (pp. 33–46), 1996.
* [4] G. Buntrock, C. Damm, U. Hertrampf, C. Meinel. _Structure and importance of logspace-MOD classes_. Theory of Computing Systems 25 (pp. 223-237), 1992.
* [5] U. Hertrampf, S. Reith, H. Vollmer. _A note on closure properties of logspace MOD classes_. Information Processing Letters 75 (pp. 91–93), 2000.
* [6] R. Beigel, J. Gill, U. Hertrampf. _Counting classes: Thresholds, parity, mods, and fewness_. Proc. STACS 90, Lecture Notes in Computer Science 415 (pp. 49–57), 1990.
* [7] A. Borodin, J. von zur Gathen, J. Hopcroft. _Fast parallel matrix and GCD computations_. 23rd Annual Symposium on Foundations of Computer Science (pp. 65–71), 1982.
* [8] K. Mulmuley. _A fast parallel algorithm to compute the rank of a matrix over an arbitrary field_. Combinatorica 7 (pp. 101–104), 1987.
* [9] S. J. Berkowitz. _On computing the determinant in small parallel time using a small number of processors_. Information Processing Letters 18 (pp. 147–150), 1984.
* [10] J. von zur Gathen. “Parallel linear algebra”. In J. H. Reif, editor, _Synthesis of parallel algorithms_ (pp. 573–617). Morgan Kaufmann, 1993.
|
arxiv-papers
| 2012-02-17T16:18:46 |
2024-09-04T02:49:27.525962
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Niel de Beaudrap (DAMTP, Centre for Mathematical Studies, University\n of Cambridge)",
"submitter": "Niel de Beaudrap",
"url": "https://arxiv.org/abs/1202.3949"
}
|
1202.4027
|
# Determinant of pseudo-laplacians
Tayeb Aissiou aissiou@math.mcgill.ca Department of Mathematics and
Statistics, Concordia University
1455 de Maisonneuve Blvd. West
Montreal, Quebec H3G 1M8 Canada , Luc Hillairet Luc.Hillairet@math.univ-
nantes.fr UMR CNRS 6629-Université de Nantes, 2 rue de la Houssinière
BP 92 208, F-44 322 Nantes Cedex 3, France and Alexey Kokotov
alexey@mathstat.concordia.ca Department of Mathematics and Statistics,
Concordia University
1455 de Maisonneuve Blvd. West
Montreal, Quebec H3G 1M8 Canada
Abstract. We derive comparison formulas relating the zeta-regularized
determinant of an arbitrary self-adjoint extension of the Laplace operator
with domain $C^{\infty}_{c}(X\setminus\\{P\\})\subset L_{2}(X)$ to the zeta-
regularized determinant of the Laplace operator on $X$. Here $X$ is a compact
Riemannian manifold of dimension $2$ or $3$; $P\in X$.
## 1\. Introduction
Let $X_{d}$ be a complete Riemannian manifold of dimension $d\geq 2$ and let
$\Delta$ be the (positive) Laplace operator on $X_{d}$. Choose a point $P\in
X_{d}$ and consider $\Delta$ as an unbounded symmetric operator in the space
$L_{2}(X_{d})$ with domain $C^{\infty}_{c}(X_{d}\setminus\\{P\\})$. It is
well-known that thus obtained operator is essentially self-adjoint if and only
if $d\geq 4$. In case $d=2,3$ it has deficiency indices $(1,1)$ and there
exists a one-parameter family $\Delta_{\alpha,P}$ of its self-adjoint
extensions (called pseudo-laplacians; see [3]). One of these extensions (the
Friedrichs extension $\Delta_{0,P}$) coincides with the self-adjoint operator
$\Delta$ on $X_{d}$. In case $X_{d}=R^{d}$, $d=2,3$ the scattering theory for
the pair $(\Delta_{\alpha,P},\Delta)$ was extensively studied in the
literature (see e. g., [1]). The spectral theory of the operator
$\Delta_{\alpha,P}$ on compact manifolds $X_{d}$ $(d=2,3)$ was studied in [3],
notice also a recent paper [15] devoted to the case, where $X_{d}$ is a
compact Riemann surface equipped with Poincaré metric.
The zeta-regularized determinant of Laplacian on a compact Riemannian manifold
was introduced in [11] and since then was studied and used in an immense
number of papers in string theory and geometric analysis, for our future
purposes we mention here the memoir [5], where the determinant of Laplacian is
studied as a functional on the space of smooth Riemannian metrics on a compact
two-dimensional manifold, and the papers [6] and [13], where the reader may
find explicit calculation of the determinant of Laplacian for three-
dimensional flat tori and for the sphere $S^{3}$ (respectively).
The main result of the present paper is a comparison formula relating ${\rm
det}(\Delta_{\alpha,P}-\lambda)$ to ${\rm det}(\Delta-\lambda)$, for
$\lambda\in{\mathbb{C}}\setminus\left({\rm Spectrum}(\Delta)\cup{\rm
Spectrum}(\Delta_{\alpha,P})\right)$.
It should be mentioned that in case of two-dimensional manifold the zeta-
regularization of ${\rm det}(\Delta_{\alpha,P}-\lambda)$ is not that standard,
since the corresponding operator zeta-function has logarithmic singularity at
$0$.
It should be also mentioned that in the case when the manifold $X_{d}$ is flat
in a vicinity of the point $P$ we deal with a very special case of the
situation (Laplacian on a manifold with conical singularity) considered in
[10], [8], [9] and, via other method, in [7]. The general scheme of the
present work is close to that of [7], although some calculations from [9] also
appear very useful for us.
Acknowledgements.The work of T. A. was supported by FQRNT. Research of A. K.
was supported by NSERC.
## 2\. Pseudo-laplacians, Krein formula and scattering coefficient
Let $X_{d}$ be a compact manifold of dimension $d=2$ or $d=3$; $P\in X_{d}$
and $\alpha\in[0,\pi)$. Following Colin de Verdière [3], introduce the set
${\mathcal{D}}(\Delta_{\alpha,P})=\\{f\in H^{2}(X_{d}\setminus\\{P\\}):\exists
c\in{\mathbb{C}}:{\text{\ }in\ a\ vicinity\ of\ }P{\text{\ }one\ has}$ (1)
$f(x)=c(\sin\alpha\cdot G_{d}(r)+\cos\alpha)+o(1){\text{\ }as\ }r\to 0\\}\,,$
where
$H^{2}(X_{d}\setminus\\{P\\})=\\{f\in L_{2}(X_{d}):\exists
C\in{\mathbb{C}}:\Delta f-C\delta_{P}\in L_{2}(X_{d})\\}\,,$
$r$ is the geodesic distance between $x$ and $P$ and
$G_{d}(r)=\begin{cases}\frac{1}{2\pi}\log r,\ \ d=2\\\ -\frac{1}{4\pi r},\ \
d=3.\end{cases}$
Then (see [3]) the self-adjoint extensions of symmetric operator $\Delta$ with
domain $C^{\infty}_{c}(X_{d}\setminus\\{P\\})$ are the operators
$\Delta_{\alpha,P}$ with domains ${\mathcal{D}}(\Delta_{\alpha,P})$ acting via
$u\mapsto\Delta u$. The extension $\Delta_{0,P}$ coincides with the Friedrichs
extension and is nothing but the self-adjoint Laplacian on $X_{d}$.
Let $R(x,y;\lambda)$ be the resolvent kernel of the self-adjoint Laplacian
$\Delta$ on $X_{d}$.
Following [3] define the scattering coefficient $F(\lambda;P)$ via
(2) $-R(x,P;\lambda)=G_{d}(r)+F(\lambda;P)+o(1)$
as $x\to P$. (Notice that in [3] the resolvent is defined as
$(\lambda-\Delta)^{-1}$, whereas for us it is $(\Delta-\lambda)^{-1}$. This
results in the minus sign in (2).)
As it was already mentioned the deficiency indices of the symmetric operator
$\Delta$ with domain $C^{\infty}_{c}(X_{d}\setminus\\{P\\})$ are $(1,1)$,
therefore, one has the following Krein formula (see, e. g., [1], p. 357) for
the resolvent kernel, $R_{\alpha}(x,y;\lambda)$, of the self-adjoint extension
$\Delta_{\alpha,P}$:
(3)
$R_{\alpha}(x,y;\lambda)=R(x,y;\lambda)+k(\lambda;P)R(x,P;\lambda)R(P,y;\lambda)$
with some $k(\lambda;P)\in{\mathbb{C}}$.
The following Lemma relates $k(\lambda;P)$ to the scattering coefficient
$F(\lambda;P)$.
###### Lemma 1.
One has the relation
(4) $k(\lambda;P)=\frac{\sin\alpha}{F(\lambda;P)\sin\alpha-\cos\alpha}\,.$
Proof. Send $x\to P$ in (3), observing that $R_{\alpha}(\,\cdot\,,y;\lambda)$
belongs to ${\mathcal{D}}_{\alpha,P}$, make use of (1) and (2), and then
compare the coefficients near $G_{d}(r)$ and the constant terms in the
asymptotical expansions at the left and at the right. $\square$
It follows in particular from the Krein formula that the difference of the
resolvents $(\Delta_{\alpha,P}-\lambda)^{-1}-(\Delta-\lambda)^{-1}$ is a rank
one operator. The following simple Lemma is the key observation of the present
work.
###### Lemma 2.
One has the relation
(5) ${\rm
Tr}\,\left((\Delta_{\alpha,P}-\lambda)^{-1}-(\Delta-\lambda)^{-1}\right)=\frac{F_{\lambda}^{\prime}(\lambda;P)\sin\alpha}{\cos\alpha-F(\lambda;P)\sin\alpha}\,.$
Proof. One has
$-F_{\lambda}^{\prime}(\lambda;P)=\frac{\partial
R(y,P;\lambda)}{\partial\lambda}\Big{|}_{y=P}=\lim_{\mu\to\lambda}\frac{R(y,P;\mu)-R(y,P;\lambda)}{\mu-\lambda}$
Using resolvent identity we rewrite the last expression as
$\lim_{\mu\to\lambda}\int_{X_{d}}R(y,z;\mu)R(P,z;\lambda)dz\Big{|}_{y=P}=\int_{X_{d}}[R(P,z;\lambda)]^{2}dz$
From (3) it follows that
$[R(P,z;\lambda)]^{2}=\frac{1}{k(\lambda;P)}\left(R_{\alpha,P}(x,z;\lambda)-R(x,z;\lambda)\right)\Big{|}_{x=z}\,.$
This implies
$-F_{\lambda}^{\prime}(\lambda;P)=\frac{1}{k(\lambda,P)}{\rm
Tr}\,\left((\Delta_{\alpha,P}-\lambda)^{-1}-(\Delta-\lambda)^{-1}\right)$
which, together with Lemma 1, imply (5).$\square$
Introduce the domain
$\Omega_{\alpha,P}={\mathbb{C}}\setminus\\{\lambda-it,\lambda\in{\rm
Spectrum}\,(\Delta)\cup{\rm
Spectrum}\,(\Delta_{\alpha,P});t\in(-\infty,0]\\}\,.$
Then in $\Omega_{\alpha,P}$ one can introduce the function
(6) $\tilde{\xi}(\lambda)=-\frac{1}{2\pi
i}\log(\cos\alpha-F(\lambda;P)\sin\alpha)$
(It should be noted that the function $\xi=\Re(\tilde{\xi})$ is the spectral
shift function of $\Delta$ and $\Delta_{\alpha,P}$.) One can rewrite (5) as
(7) ${\rm
Tr}\,\left((\Delta_{\alpha,P}-\lambda)^{-1}-(\Delta-\lambda)^{-1}\right)=2\pi
i\tilde{\xi}^{\prime}(\lambda)$
## 3\. Operator zeta-function of $\Delta_{\alpha,P}$
Denote by $\zeta(s,A)$ the zeta-function
$\zeta(s,A)=\sum_{\mu_{k}\in{\rm Spectrum}\,(A)}\frac{1}{\mu_{k}^{s}}$
of the operator $A$. (We assume that the spectrum of $A$ is discrete and does
not contain $0$.)
Take any $\tilde{\lambda}$ from ${\mathbb{C}}\setminus({\rm
Spectrum}\,(\Delta_{\alpha,P})\cup{\rm Spectrum}\,(\Delta)))$. From the
results of [3] it follows that the function
$\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})$ is defined for sufficiently large
$\Re s$. It is well-known that $\zeta(s,\Delta-\tilde{\lambda})$ is
meromorphic in ${\mathbb{C}}$.
The proof of the following lemma coincides verbatim with the proof of
Proposition 5.9 from [7].
###### Lemma 3.
Suppose that the function $\tilde{\xi}^{\prime}(\lambda)$ from (7) is
$O(|\lambda|^{-1})$ as $\lambda\to-\infty$. Let $-C$ be a sufficiently large
negative number and let $c_{\tilde{\lambda},\epsilon}$ be a contour encircling
the cut $c_{\tilde{\lambda}}$ which starts from $-\infty+0i$, follows the real
line till $-C$ and then goes to $\tilde{\lambda}$ remaining in
$\Omega_{\alpha,P}$. Assume that ${\rm
dist}\,(z,c_{\tilde{\lambda}})=\epsilon$ for any $z\in
c_{\tilde{\lambda},\epsilon}$. Let also
$\zeta_{2}(s)=\int_{c_{\tilde{\lambda},\epsilon;2}}(\lambda-\tilde{\lambda})^{-s}\tilde{\xi}^{\prime}(\lambda)d\lambda,$
where the the integral at the right hand side is taken over the part
$c_{\tilde{\lambda},\epsilon;2}$ of the contour $c_{\tilde{\lambda},\epsilon}$
lying in the half-plane $\\{\lambda:\Re\lambda>-C\\}$. Let
$\hat{\zeta}_{2}(s)=\lim_{\epsilon\to 0}\zeta_{2}(s)=2i\sin(\pi
s)\int_{-C}^{\tilde{\lambda}}(\lambda-\tilde{\lambda})_{0}^{-s}\tilde{\xi}^{\prime}(\lambda)\,d\lambda\,,$
where $(\lambda-\tilde{\lambda})_{0}^{-s}=e^{-i\pi s}\lim_{\lambda\downarrow
c_{\tilde{\lambda}}}(\lambda-\tilde{\lambda})^{-s}$. Then the function
(8)
$R(s,\tilde{\lambda})=\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda}))-\zeta(s,\Delta-\tilde{\lambda})-2i\sin(\pi
s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde{\xi}^{\prime}(\lambda)d\lambda-\hat{\zeta}_{2}(s)$
can be analitically continued to $\Re s>-1$ with
$R(0,\tilde{\lambda})=R^{\prime}_{s}(0,\tilde{\lambda})=0$.
For completeness we give a sketch of proof. Using (7), one has for
sufficiently large $\Re s$
$\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})-\zeta(s,\Delta-\tilde{\lambda})=\frac{1}{2\pi
i}\int_{c_{\tilde{\lambda},\epsilon}}(\lambda-\tilde{\lambda})^{-s}{\rm
Tr}((\Delta_{\alpha,P}-\lambda)^{-1}-(\Delta-\lambda)^{-1})d\lambda=$
$=\int_{c_{\tilde{\lambda},\epsilon}}(\lambda-\tilde{\lambda})^{-s}\tilde{\xi}^{\prime}(\lambda)\,d\lambda=\zeta_{1}(s)+\zeta_{2}(s)\,,$
where
$\zeta_{1}(s)=\left\\{\int_{-\infty+i\epsilon}^{-C+i\epsilon}-\int_{-\infty-i\epsilon}^{-C-i\epsilon}\right\\}(\lambda-\tilde{\lambda})^{-s}\tilde{\xi}^{\prime}(\lambda)d\lambda\,.$
It is easy to show (see Lemma 5. 8 in [7]) that in the limit $\epsilon\to 0$
$\zeta_{1}(s)$ gives
(9) $2i\sin(\pi
s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde{\xi}^{\prime}(\lambda)\,d\lambda+2i\sin(\pi
s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde{\xi}^{\prime}(\lambda)\rho(s,\tilde{\lambda}/\lambda)d\lambda\,,$
where $\rho(s,z)=(1+z)^{-s}-1$ and
$\rho(s,\tilde{\lambda}/\lambda)=O(|\lambda|^{-1})$
as $\lambda\to-\infty$. Using the assumption on the asymptotics of
$\tilde{\xi}(\lambda)$ as $\lambda\to-\infty$ and the obvious relation
$\rho(0,z)=0$ one can see that the last term in (9) can be analytically
continued to $\Re s>-1$ and vanishes together with its first derivative w. r.
t. $s$ at $s=0$. Denoting it by $R(s,\tilde{\lambda})$ one gets the Lemma.
$\square$
As it is stated in the introduction the main object we are to study in the
present paper is the zeta-regularized determinant of the operator
$\Delta_{\alpha,P}-\lambda$. Let us remind the reader that the usual
definition of the zeta-regularized determinant of an operator $A$
(10) ${\rm det}\,A=\exp{(-\zeta^{\prime}(0,A))}$
requires analyticity of $\zeta(s,A)$ at $s=0$.
Since the operator zeta-function $\zeta(s,\Delta-\tilde{\lambda})$ is regular
at $s=0$ (in fact, it is true in case of $\Delta$ being an arbitrary elliptic
differential operator on any compact manifold) and the function
$\hat{\zeta}_{2}(s)$ is entire, Lemma 3 shows that the behavior of the
function $\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})$ at $s=0$ is determined
by the properties of the analytic continuation of the term
(11) $2i\sin(\pi
s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde{\xi}^{\prime}(\lambda)d\lambda$
in (8). These properties in their turn are determined by the asymptotical
behavior of the function $\tilde{\xi}^{\prime}(\lambda)$ as
$\lambda\to-\infty$.
It turns out that the latter behavior depends on dimension $d$. In particular,
in the next section we will find out that in case $d=2$ the function
$\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})$ is not regular at $s=0$,
therefore, in order to define ${\rm det}(\Delta_{\alpha,P}-\tilde{\lambda})$
one has to use a modified version of (10) .
## 4\. Determinant of pseudo-laplacian on two-dimensional compact manifold
Let $X$ be a two-dimensional Riemannian manifold, then introducing isothermal
local coordinates $(x,y)$ and setting $z=x+iy$, one can write the area element
on $X$ as
$\rho^{-2}(z)|dz|^{2}$
The following estimate of the resolvent kernel, $R(z^{\prime},z;\lambda)$, of
the Laplacian on $X$ was found by J. Fay (see [5]; Theorem 2.7 on page 38 and
the formula preceding Corollary 2.8 on page 39; notice that Fay works with
negative Laplacian, so one has to take care of signs when using his formulas).
###### Lemma 4.
(J. Fay) The following equality holds true
(12)
$-R(z,z^{\prime};\lambda)=G_{2}(r)+\frac{1}{2\pi}\left[\gamma+\log\frac{\sqrt{|\lambda|+1}}{2}\right.$
$\left.-\frac{1}{2(|\lambda|+1)}(1+\frac{4}{3}\rho^{2}(z)\partial^{2}_{z\bar{z}}\rho(z))+\hat{R}(z^{\prime},z;\lambda)\right]\,,$
where $\hat{R}(z^{\prime},z;\lambda)$ is continuous for $z^{\prime}$ near $z$,
$\hat{R}(z,z;\lambda)=O(|\lambda|^{-2})$
uniformly w. r. t. $z\in X$ as $\lambda\to-\infty$; $r={\rm
dist}(z,z^{\prime})$, $\gamma$ is the Euler constant.
Using (12), we immediately get the following asymptotics of the scattering
coefficient $F(\lambda,P)$ as $\lambda\to-\infty$:
(13) $F(\lambda,P)=$ $\frac{1}{4\pi}\log(|\lambda|+1)+\frac{\gamma-\log
2}{2\pi}-\frac{1}{4\pi(|\lambda|+1)}\left[1+\frac{4}{3}\rho^{2}(z)\partial^{2}_{z\bar{z}}\rho(z)\Big{|}_{z=z(P)}\right]+O(|\lambda|^{-2})\,.$
###### Remark 1.
It is easy to check that the expression
$\rho^{2}(z)\partial^{2}_{z\bar{z}}\rho(z)\Big{|}_{z=z(P)}$ is independent of
the choice of conformal local parameter $z$ near $P$.
Now from (6) and (13) it follows that
$2\pi
i\tilde{\xi}^{\prime}(\lambda)=-\frac{\frac{1}{4\pi(|\lambda|+1)}-\frac{b}{(|\lambda|+1)^{2}}+O(|\lambda|^{-3})}{\cot\alpha-a-\frac{1}{4\pi}\log(|\lambda|+1)+\frac{b}{|\lambda|+1}+O(|\lambda|^{-2})},$
where $a=\frac{1}{2\pi}(\gamma-\log 2)$ and
$b=\frac{1}{4\pi}(1+\frac{4}{3}\rho^{2}\partial^{2}_{z\bar{z}}\rho)$. This
implies that for $-\infty<\lambda\leq-C$ one has
(14) $2\pi
i\tilde{\xi}^{\prime}(\lambda)=\frac{1}{|\lambda|(\log|\lambda|-4\pi\cot\alpha+4\pi
a)}+f(\lambda)\,,$
with $f(\lambda)=O(|\lambda|^{-2})$ as $\lambda\to-\infty$. Now knowing (14),
one can study the behaviour of the term (11) in (8). We have
(15) $2i\sin(\pi
s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde{\xi}^{\prime}(\lambda)d\lambda=$
$\frac{\sin(\pi
s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot\alpha+4\pi
a)}+\frac{\sin(\pi
s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s}f(\lambda)\,d\lambda\,.$
The first integral in the right hand side of (15) appeared in ([9], p. 15),
where it was observed that it can be easily rewritten through the function
${\rm
Ei}(z)=-\int_{-z}^{\infty}e^{-y}\frac{dy}{y}=\gamma+\log(-z)+\sum_{k=1}^{\infty}\frac{z^{k}}{k\cdot
k!}\,$
which leads to the representation
(16) $\frac{\sin(\pi
s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot\alpha+4\pi
a)}=$ $-\frac{\sin(\pi s)}{\pi}e^{-s\kappa}\left[\gamma+\log(s(\log
C-\kappa))+e(s)\right]\,$
where $e(s)$ is an entire function such that $e(0)=0$;
$\kappa=4\pi\cot\alpha-4\pi a$. From this we conclude that
(17) $\frac{\sin(\pi
s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot\alpha+4\pi
a)}=-s\log s+g(s)\,$
where $g(s)$ is differentiable at $s=0$.
Now (8) and (17) justify the following definition.
###### Definition 1.
Let $\Delta_{\alpha,P}$ be the pseudo-laplacian on a two-dimensional compact
Riemannian manifold. Then the zeta-regularized determinant of the operator
$\Delta_{\alpha,P}-\tilde{\lambda}$ with
$\tilde{\lambda}\in{\mathbb{C}}\setminus{\rm Spectrum}(\Delta_{\alpha,P})$ is
defined as
(18) ${\rm
det}(\Delta_{\alpha,P}-\tilde{\lambda})=\exp\left\\{-\frac{d}{ds}\left[\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})+s\log
s\right]\Big{|}_{s=0}\right\\}$
We are ready to get our main result: the formula relating ${\rm
det}(\Delta_{\alpha,P}-\tilde{\lambda})$ to ${\rm
det}(\Delta-\tilde{\lambda})$.
From (8, 11) it follows that
$\frac{d}{ds}\left[\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})+s\log
s-\zeta(s,\Delta-\tilde{\lambda})\right]\Big{|}_{s=0}=$
$\frac{d}{ds}\hat{\zeta}_{2}(s)\Big{|}_{s=0}+\int_{-\infty}^{-C}f(\lambda)\,d\lambda+$
$-\frac{d}{ds}\left\\{\frac{\sin\pi
s}{\pi}e^{-s\kappa}\left[\gamma+\log(s(\log C-\kappa))+e(s)\right]+s\log
s\right\\}\Big{|}_{s=0}=$ $2\pi
i\left(\tilde{\xi}(\tilde{\lambda})-\tilde{\xi}(-C)\right)+\int_{-\infty}^{-C}f(\lambda)\,d\lambda-\gamma-\log(\log
C-\kappa)=$ (19) $2\pi i\tilde{\xi}(\tilde{\lambda})-\gamma+$
$\int_{-\infty}^{-C}f(\lambda)\,d\lambda-2\pi i\tilde{\xi}(-C)-\log(\log
C-4\pi\cot\alpha+2\gamma-\log 4)\,.$
Notice that the expression in the second line of (19) should not depend on
$C$, so one can send $C$ to $+\infty$ there. Together with (13) this gives
(20) $\frac{d}{ds}\left[\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})+s\log
s-\zeta(s,\Delta-\tilde{\lambda})\right]\Big{|}_{s=0}=$ $2\pi
i\tilde{\xi}(\tilde{\lambda})-\gamma+\log(\sin\alpha/(4\pi))-i\pi\,$
which implies the comparison formula for the determinants stated in the
following theorem.
###### Theorem 1.
Let $\tilde{\lambda}$ do not belong to the union of spectra of $\Delta$ and
$\Delta_{\alpha,P}$ and let the zeta-regularized determinant of
$\Delta_{\alpha,P}$ be defined as in (18). Then one has the relation
(21) ${\rm det}(\Delta_{\alpha,P}-\tilde{\lambda})=-4\pi
e^{\gamma}(\cot\alpha-F(\tilde{\lambda},P)){\rm
det}(\Delta-\tilde{\lambda})\,.$
Observe now that $0$ is the simple eigenvalue of $\Delta$ and, therefore, it
follows from Theorem 2 in [3] that $0$ does not belong to the spectrum of the
operator $\Delta_{\alpha,P}$ and that $\Delta_{\alpha,P}$ has one strictly
negative simple eigenvalue. Thus, the determinant in the left hand side of
(21) is well defined for $\tilde{\lambda}=0$, whereas the determinant at the
right hand side has the asymtotics
(22) ${\rm det}(\Delta-\tilde{\lambda})\sim(-\tilde{\lambda}){\rm
det}^{*}\Delta\,$
as $\tilde{\lambda}\to 0-$. Here ${\rm det}^{*}\Delta$ is the modified
determinant of an operator with zero mode.
From the standard asymptotics
$-R(x,y;\lambda)=\frac{1}{{\rm Vol}(X)}\frac{1}{\lambda}+G_{2}(r)+O(1)$
as $\lambda\to 0$ and $x\to y$ one gets the asymptotics
(23) $F(\lambda,P)=\frac{1}{{\rm Vol}(X)}\frac{1}{\lambda}+O(1)$
as $\lambda\to 0$. Now sending $\tilde{\lambda}\to 0-$ in (21) and using 22
and 23 we get the following corollary of the Theorem 1.
###### Corollary 1.
The following relation holds true
(24) ${\rm det}\Delta_{\alpha,P}=-\frac{4\pi e^{\gamma}}{{\rm Vol}(X)}{\rm
det}^{*}\Delta\,.$
## 5\. Determinant of pseudo-laplacian on three-dimensional manifolds
Let $X$ be a three-dimensional compact Riemannian manifold. We start with the
Lemma describing the asymptotical behavior of the scattering coefficient as
$\lambda\to-\infty$.
###### Lemma 5.
One has the asymptotics
(25)
$F(\lambda;P)=\frac{1}{4\pi}\sqrt{-\lambda}+c_{1}(P)\frac{1}{\sqrt{-\lambda}}+O(|\lambda|^{-1})$
as $\lambda\to-\infty$
Proof. Consider Minakshisundaram-Pleijel asymptotic expansion ([12])
(26) $H(x,P;t)=(4\pi
t)^{-3/2}e^{-d(x,P)^{2}/(4t)}\sum_{k=0}^{\infty}u_{k}(x,P)t^{k}$
for the heat kernel in a small vicinity of $P$, here $d(x,P)$ is the geodesic
distance from $x$ to $P$, functions $u_{k}(\cdot,P)$ are smooth in a vicinity
of P, the equality is understood in the sense of asymptotic expansions. We
will make use of the standard relation
(27) $R(x,y;\lambda)=\int_{0}^{+\infty}H(x,y;t)e^{\lambda t}\,dt\,.$
Let us first truncate the sum (26) at some fixed $k=N+1$ so that the
remainder, $r_{n}$, is $O(t^{N})$. Defining
$\tilde{R}_{N}(x,P;-\lambda):=\int_{0}^{\infty}r_{n}(t,x,P)e^{t\lambda}dt\,,$
we see that
$\tilde{R}_{N}(x,P;\lambda)=O(|\lambda|^{-(N+1)})$
as $\lambda\to-\infty$ uniformly w. r. t. $x$ belonging to a small vicinity of
$P$.
Now, for each $0\leq k\leq N+1$ we have to address the following quantity
$R_{k}(x,P;\lambda):=\frac{u_{k}(x,y)}{(4\pi)^{3/2}}\int_{0}^{\infty}t^{k-\frac{3}{2}}e^{-\frac{d(x,P)^{2}}{4t}}e^{\lambda
t}dt.$
According to identity (36) below one has
$R_{0}(x,P;\lambda)=\frac{u_{0}(x,P)}{(4\pi)^{3/2}}\frac{2\sqrt{\pi}}{d(x,P)}e^{-d(x,P)\sqrt{-\lambda}}=$
(28) $\frac{1}{4\pi d(x,P)}-\frac{1}{4\pi}\sqrt{-\lambda}+o(1),$
as $d(x,P)\to 0$. For $k\geq 1$ one has
$R_{k}(x,P;\lambda)=\frac{u_{k}(x,P)}{(4\pi)^{3/2}}2^{3/2-k}\left(\frac{d(x,P)}{\sqrt{-\lambda}}\right)^{k-1/2}K_{k-\frac{1}{2}}(d(x,P)\sqrt{-\lambda})=$
(29) $-c_{k}(P)\frac{1}{(\sqrt{-\lambda})^{2k-1}}+o(1)$
as $d(x,P)\to 0$ (see [2], p. 146, f-la 29). Now (25) follows from (27), (28)
and (29). $\square$
Now from Lemma 5 it follows that
(30) $2\pi
i\tilde{\xi}^{\prime}(\lambda)=-\frac{1}{2\lambda}+O(|\lambda|^{-3/2})$
as $\lambda\to-\infty$, therefore, one can rewrite (11) as
(31) $\frac{\sin(\pi s)}{\pi}\left\\{\int_{-\infty}^{-C}|\lambda|^{-s}(2\pi
i\tilde{\xi}^{\prime}(\lambda)+\frac{1}{2\lambda})d\lambda+\frac{C^{-s}}{2s}\right\\}$
which is obviously analytic in $\Re s>-\frac{1}{2}$. Thus, it follows from (8)
that the function $\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})$ is regular at
$s=0$ and one can introduce the usual zeta-regularization
${\rm
det}(\Delta_{\alpha,P}-\tilde{\lambda})=\exp\\{-\zeta^{\prime}(0,\Delta_{\alpha,P}-\tilde{\lambda})\\}$
of ${\rm det}(\Delta_{\alpha,P}-\tilde{\lambda})$.
Moreover, differentiating (8) with respect to $s$ at $s=0$ similarly to (19)
we get
$\frac{d}{ds}\left[\zeta(s,\Delta_{\alpha,P}-\tilde{\lambda})-\zeta(s,\Delta-\tilde{\lambda})\right]\Big{|}_{s=0}=$
$2\pi i(\tilde{\xi}(\tilde{\lambda})-\tilde{\xi}(-C))+\int_{-\infty}^{-C}(2\pi
i\tilde{\xi}^{\prime}(\lambda)+\frac{1}{2\lambda})d\lambda-\frac{1}{2}\log C=$
which reduces after sending $-C\to-\infty$ to
$2\pi
i\tilde{\xi}(\tilde{\lambda})+\log\sin\alpha-\log(4\pi)+i\pi=-\log(\cot\alpha-F(\lambda;P))-\log(4\pi)+i\pi\,$
which implies the following theorem.
###### Theorem 2.
Let $\Delta_{\alpha,P}$ be the pseudo-laplacian on $X$ and
$\tilde{\lambda}\in{\mathbb{C}}\setminus({\rm Spectrum}(\Delta)\cup{\rm
Spectrum}(\Delta_{\alpha,P}))$. Then
(32) ${\rm
det}(\Delta_{\alpha,P}-\tilde{\lambda})=-4\pi(\cot\alpha-F(\tilde{\lambda};P)){\rm
det}(\Delta-\tilde{\lambda})\,.$
Sending $\tilde{\lambda}\to 0$ and noticing that relation (23) holds also in
case $d=3$ we get the following corollary.
###### Corollary 2.
(33) ${\rm det}\Delta_{\alpha,P}=-\frac{4\pi}{{\rm Vol}(X)}{\rm
det}^{*}\Delta\,.$
In what follows we consider two examples of three-dimensional compact
Riemannian manifolds for which there exist explicit expressions for the
resolvent kernels: a flat torus and the round (unit) $3d$-sphere. These
manifolds are homogeneous, so, as it is shown in [3], the scattering
coefficient $F(\lambda,P)$ is $P$-independent.
Example 1: Round $3d$-sphere.
###### Lemma 6.
Let $X=S^{3}$ with usual round metric. Then there is the following explicit
expression for scattering coefficient
(34)
$F(\lambda)=\frac{1}{4\pi}\coth\left(\pi\sqrt{-\lambda-1}\,\right)\cdot\sqrt{-\lambda-1}$
and, therefore, one has the following asymptotics as $\lambda\to-\infty$
(35) $F(\lambda)=\frac{1}{4\pi}\sqrt{|\lambda|-1}+O(|\lambda|^{-\infty})\,.$
###### Remark 2.
The possibility of finding an explicit expression for $F(\lambda)$ for $S^{3}$
was mentioned in [3]. However we failed to find (34) in the literature.
Proof. We will make use the well-known identity (see, e. g., [2], p. 146, f-la
28):
(36) $\int_{0}^{+\infty}e^{\lambda
t}t^{-3/2}e^{-\frac{d^{2}}{4t}}\,dt=2\frac{\sqrt{\pi}}{|d|}e^{-|d|\sqrt{-\lambda}};$
for $\lambda<0$ and $d\in{\mathbb{R}}$ and the following explicit formula for
the operator kernel $e^{-t}H(x,y;t)$ of the operator $e^{-t(\Delta+1)}$, where
$\Delta$ is the (positive) Laplacian on $S^{3}$ (see [4], (2.29)):
(37) $e^{-t}H(x,y;t)=-\frac{1}{2\pi}\frac{1}{\sin
d(x,y)}\frac{\partial}{\partial z}\Big{|}_{z=d(x,y)}\Theta(z,t)\,.$
Here $d(x,y)$ is the geodesic distance between $x,y\in S^{3}$ and
$\Theta(z,t)=\frac{1}{\sqrt{4\pi
t}}\sum_{k=-\infty}^{+\infty}e^{-(z+2k\pi)^{2}/4t}$
is the theta-function.
Denoting $d(x,y)$ by $\theta$ and using (37) and (36), one gets
$R(x,y;\lambda-1)=\int_{0}^{+\infty}e^{\lambda t}e^{-t}H(x,y;t)\,dt=$
$\frac{1}{4\pi}\frac{1}{\sin\theta}\left(-\sum_{k<0}e^{(\theta+2k\pi)\sqrt{-\lambda}}+\sum_{k\geq
0}e^{-(\theta+2k\pi)\sqrt{-\lambda}}\right)=$
$\frac{1}{4\pi}\frac{1}{\sin\theta}\frac{1}{1-e^{-2\pi\sqrt{-\lambda}}}\left[-e^{-2\pi\sqrt{-\lambda}}e^{\theta\sqrt{-\lambda}}+e^{-\theta\sqrt{-\lambda}}\right]=$
(38)
$\frac{1}{4\pi\theta}-\frac{1}{4\pi}\frac{1+e^{-2\pi\sqrt{-\lambda}}}{1-e^{-2\pi\sqrt{-\lambda}}}\sqrt{-\lambda}+o(1)$
as $\theta\to 0$, which implies the Lemma. $\square$
Example 2: Flat $3d$-tori.Let $\\{{\bf A,B,C}\\}$ be a basis of
${\mathbb{R}}^{3}$ and let $T^{3}$ be the quotient of ${\mathbb{R}}^{3}$ by
the lattice $\\{m{\bf A}+n{\bf B}+l{\bf C}:(m,n,l)\in{\mathbb{Z}}^{3}\\}$
provided with the usual flat metric.
Notice that the free resolvent kernel in $R^{3}$ is
$\frac{e^{-\sqrt{-\lambda}||x-y||}}{4\pi||x-y||}$
and, therefore,
(39)
$R(x,y;\lambda)=\frac{e^{-\sqrt{-\lambda}||x-y||}}{4\pi||x-y||}+\frac{1}{4\pi}\sum_{(m,n,l)\in{\mathbb{Z}}^{3}\setminus(0,0,0)}\frac{e^{-\sqrt{-\lambda}||x-y+m{\bf
A}+n{\bf B}+l{\bf C}||}}{||x-y+m{\bf A}+n{\bf B}+l{\bf C}||}\,.$
From (39) it follows that
$F(\lambda)=\frac{1}{4\pi}\sqrt{-\lambda}-\frac{1}{4\pi}\sum_{(m,n,l)\in{\mathbb{Z}}^{3}\setminus(0,0,0)}\frac{e^{-\sqrt{-\lambda}||m{\bf
A}+n{\bf B}+l{\bf C}||}}{||m{\bf A}+n{\bf B}+l{\bf C}||}=$
$\frac{1}{4\pi}\sqrt{-\lambda}+O(|\lambda|^{-\infty})$
as $\lambda\to-\infty$.
###### Remark 3.
It should be noted that explicit expressions for ${\rm det}^{*}\Delta$ in case
$X=S^{3}$ and $X=T^{3}$ are given in [13] and [6].
## References
* [1] Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H., Solvable models in quantum mechanics, AMS 2005
* [2] Erdélyi, A. and Bateman, H. Tables of integral transforms, volume 2, McGraw-Hill, New York, 1954
* [3] Yves Colin De Verdiere, Pseudo-laplaciens. I, Annales de l’institut Fourier, tome 32, N3 (1982), 275–286
* [4] J. Cheeger, M. Taylor, On the diffraction of waves by conical singularities. I, Communications on Pure and Applied Mathematics, Volume 35 (1982), Issue 3, 275 -331
* [5] Fay, John D., Kernel functions, analytic torsion, and moduli spaces, Memoirs of the AMS 464 (1992)
* [6] Furutani K., de Gosson S., Determinant of Laplacians on Heisenberg manifolds, J. Geom. Phys. 48 (2003), pp. 438 -479
* [7] L. Hillairet, A. Kokotov, Krein formula and S-matrix for euclidean surfaces with conical singularities, J. of Geom. Anal, 2012, to appear, arXiv:1011.5034v1
* [8] Klaus Kirsten, Paul Loya, Jinsung Park, Exotic expansions and pathological properties of $\zeta$-functions on conic manifolds, J. Geom. Anal., 18(2009), 835–888
* [9] Klaus Kirsten, Paul Loya, Jinsung Park, The very unusual properties of the resolvent, heat kernel, and zeta-function for the operator $-\frac{d^{2}}{dr^{2}}-1/(4r^{2})$, Journal of mathematical physics, 47(2006)
* [10] Loya P., McDonald P., Park J., Zeta regularized determinants for conic manifolds, Journal of Functional Analysis (2007), 242, N1, 195–229
* [11] Ray, D. B.; Singer, I. M., Analytic torsion for complex manifolds., Ann. of Math., 98 (1973), 154 177
* [12] Minakshisundaram, S.; Pleijel, A., Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canadian Journal of Mathematics 1 (1949): 242 -256
* [13] Kumagai H., The determinant of the laplacian on the $n$-sphere, Acta Arithmetica, XCL.3 (1999)
* [14] Ray D. B., Singer I. M., Analytic torsion for complex manifolds. Ann. of Math., Vol 98 (1973), N1, 154-177
* [15] Ueberschaer H., The trace formula for a point scatterer on a compact hyperbolic surface, arXiv:1109.4329v2
|
arxiv-papers
| 2012-02-17T21:54:12 |
2024-09-04T02:49:27.538682
|
{
"license": "Public Domain",
"authors": "Tayeb Aissiou, Luc Hillairet and Alexey Kokotov",
"submitter": "Alexey Kokotov Yu",
"url": "https://arxiv.org/abs/1202.4027"
}
|
1202.4219
|
# Turbulent convection model in the overshooting region: II. Theoretical
analysis
Q.S. Zhang11affiliation: National Astronomical Observatories/Yunnan
Observatory, Chinese Academy of Sciences, P.O. Box 110, Kunming 650011, China.
22affiliation: Laboratory for the Structure and Evolution of Celestial
Objects, CAS. 33affiliation: Graduate School of Chinese Academy of Sciences,
Beijing 100039, China. and Y. Li11affiliation: National Astronomical
Observatories/Yunnan Observatory, Chinese Academy of Sciences, P.O. Box 110,
Kunming 650011, China. 22affiliation: Laboratory for the Structure and
Evolution of Celestial Objects, CAS. zqs@ynao.ac.cn(QSZ); ly@ynao.ac.cn(YL)
###### Abstract
Turbulent convection models are thought to be good tools to deal with the
convective overshooting in the stellar interior. However, they are too complex
to be applied in calculations of stellar structure and evolution. In order to
understand the physical processes of the convective overshooting and to
simplify the application of turbulent convection models, a semi-analytic
solution is necessary. We obtain the approximate solution and asymptotic
solution of the turbulent convection model in the overshooting region, and
find some important properties of the convective overshooting: I. The
overshooting region can be partitioned into three parts: a thin region just
outside the convective boundary with high efficiency of turbulent heat
transfer, a power law dissipation region of turbulent kinetic energy in the
middle, and a thermal dissipation area with rapidly decreasing turbulent
kinetic energy. The decaying indices of the turbulent correlations $k$,
$\overline{u_{r}^{\prime}T^{\prime}}$, and $\overline{T^{\prime}T^{\prime}}$
are only determined by the parameters of the TCM, and there is an equilibrium
value of the anisotropic degree $\omega$. II. The overshooting length of the
turbulent heat flux $\overline{u_{r}^{\prime}T^{\prime}}$ is about
$1H_{k}$($H_{k}=|\frac{dr}{dlnk}|$). III. The value of the turbulent kinetic
energy at the convective boundary $k_{C}$ can be estimated by a method called
the maximum of diffusion. Turbulent correlations in the overshooting region
can be estimated by using $k_{C}$ and exponentially decreasing functions with
the decaying indices.
convection — diffusion — turbulence
## 1 Introduction
Convective overshooting is an important physical process in the stellar
structure and evolution. Phenomenologically, the acceleration of a fluid
element is zero at the convective boundary, but its speed is not zero. It is
able to go across the convective boundary into the dynamically stable zone.
This phenomenon is called the convective overshooting. The convective
overshooting transports heat and matter, and affects the structure and
evolution of stars. A phenomenological theory of the overshooting was
developed by Zahn(1991), which predicts an adiabatic overshooting region.
However, Xiong & Deng(2001) pointed out that the turbulent velocity and the
temperature are strongly correlated in Zahn’s theory. Recently, Christensen-
Dalsgaard et al.(2011) found that the convective overshooting only described
by the turbulent convection models could be in agreement with the helioseismic
data.
The turbulent convection models (TCMs) are based on fully hydrodynamic moment
equations, and applied on investigating the convective overshooting(Xiong,
1981, 1985, 1989; Xiong & Deng, 2001; Canuto, 1997; Canuto & Dubovikov, 1998;
Canuto, 1998, 1999; Marik & Petrovay, 2002; Deng & Xiong, 2006; Li & Yang,
2007; Deng & Xiong, 2008; Zhang & Li, 2009). There are two main difficulties
restricting the applications of the TCMs. One is to solve the equations of the
TCMs, which are highly non-linear and unstable in numerical calculations. The
other is to incorporate the TCMs into a stellar evolution code. In general,
solving the TCMs needs the parameters of the stellar structure(e.g.
temperature $T$, density $\rho$, pressure $P$, radius $r$, luminosity $L$, and
elements abundance vector), and solving the equations of stellar structure
requires the temperature gradient $\nabla$ which is determined by the TCMs.
Thus, in order to apply the TCMs, one must solve both the TCMs and the
equations of stellar structure, which shows enormous difficulty. Although
developing numerical technique is very important, getting an approximate
solution of the TCMs is more interesting because an approximate solution helps
to understand the physical processes and may significantly simplify the
application of the TCMs. Xiong(1989) found the asymptotic solution of his TCM
in the overshooting region, the turbulent correlations being exponentially
decreasing in the overshooting region. However, his solution of the heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is not suitable near the convective
boundary, and the initial turbulent kinetic energy $k_{0}$ is unknown so that
the value of the turbulent correlations in the overshooting region actually
can not be determined without numerical calculations.
In this paper, we investigate the properties of the convective overshooting by
analyzing Li & Yang’s TCM(Li & Yang, 2007), which was tested in the solar
convection zone(Li & Yang, 2007; Yang & Li, 2007). We try to get a semi-
analytical solution of the TCM in the overshooting region. We introduce the
TCM in Section 2, investigate the properties of the overshooting in Section 3,
and summarize the conclusions in Section 4.
## 2 Turbulent Convection Model
The closure assumptions of Li & Yang’s TCM are(Li & Yang, 2001, 2007): the
three-order moment terms are modeled with a gradient-type scheme; the
dissipation rate $\varepsilon$ of the turbulent kinetic energy $k$ is assumed
to be local; the dissipation rates of the turbulent heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ and the turbulent fluctuation of
temperature $\overline{T^{\prime}T^{\prime}}$ are assumed to be determined by
both the reciprocal timescale of the turbulent dissipation
$\tau_{1}^{-1}=\frac{\varepsilon}{k}$ and the thermal dissipation one
$\tau_{2}^{-1}=\frac{\lambda}{\rho c_{P}}\frac{\varepsilon^{2}}{k^{3}}$.
According to those closure assumptions, fully hydrodynamic moment equations on
the quasi-steady approximation result in the complete equations of two-order
moment terms(Li & Yang, 2007):
$\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial r}\left(C_{s}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{u_{r}^{\prime}u_{r}^{\prime}}}{\partial
r}\right)=\frac{2}{3}\varepsilon+\frac{2\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}+C_{k}\frac{\varepsilon}{k}\left(\overline{u_{r}^{\prime}u_{r}^{\prime}}-\frac{2}{3}k\right)$
(1) $\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial
r}\left(C_{s}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial
k}{\partial r}\right)=\varepsilon+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (2) $\displaystyle\frac{2}{\rho
r^{2}}\frac{\partial}{\partial r}\left(C_{t1}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}+C_{t}\left(\frac{\varepsilon}{k}+\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}\right)\overline{u_{r}^{\prime}T^{\prime}}$
(3) $\displaystyle\frac{1}{\rho r^{2}}\frac{\partial}{\partial
r}\left(C_{e1}\rho
r^{2}\frac{k}{\varepsilon}\overline{u_{r}^{\prime}u_{r}^{\prime}}\frac{\partial\overline{T^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\left(\frac{\varepsilon}{k}+\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}\right)\overline{T^{\prime}T^{\prime}}$ (4)
The temperature gradient is calculated as:
$\displaystyle\nabla=\nabla_{R}-\frac{H_{P}}{T}\frac{\rho
c_{P}}{\lambda}\overline{u_{r}^{\prime}T^{\prime}}$ (5)
The meaning of those equations and each term in them were described in
previous works(Li & Yang, 2007; Zhang & Li, 2009) in detail. We simply
introduce them here:
Equations (1-4) describe the equilibrium(time-independent) structure of the
radial kinetic energy $\overline{u_{r}^{\prime}u_{r}^{\prime}}$, the turbulent
kinetic energy $k$, the turbulent heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ and the turbulent fluctuation of
temperature $\overline{T^{\prime}T^{\prime}}$, respectively. On the left side
of those equations, there is the non-local term(i.e. the diffusion term) of
each turbulent correlation. On the right side, there are the local terms which
describe the generation and the dissipation of each turbulent correlation.
In Eq.(1) and (2), $\varepsilon$ is the turbulent dissipation rate of $k$ and
$\varepsilon=\frac{k^{\frac{3}{2}}}{l}$ where $l=\alpha H_{P}$, and the second
term on the right side is the generation rate of the kinetic energy due to the
contribution of the buoyancy. The last term in Eq.(1) is the return to
isotropy term which attempts to make the turbulent motion be isotropic. In
Eq.(3), the first two terms on the right side is the generation rate of the
turbulent heat flux $\overline{u_{r}^{\prime}T^{\prime}}$, and the last one is
the dissipation rate that comprises the turbulent dissipation and the thermal
dissipation. In Eq.(4), the first term on the right side is the generation
rate of the turbulent fluctuation of temperature
$\overline{T^{\prime}T^{\prime}}$, and the last one is the dissipation rate.
Meanings of other symbols are: $H_{P}=-\frac{dr}{dlnP}$ is the local pressure
scale height, $\beta=-(\frac{\partial ln\rho}{\partial lnT})_{P}$ the
expansion coefficient, $g_{r}=-\frac{GM_{r}}{r^{2}}$ the radial component of
gravity acceleration, $\nabla=\frac{dlnT}{dlnP}$ the temperature gradient in
the stellar interior, $\nabla_{ad}=(\frac{\partial lnT}{\partial lnP})_{S}$
the adiabatic temperature gradient, $\lambda=\frac{4acT^{3}}{3\kappa\rho}$ the
thermal conduction coefficient, $c_{P}=(\frac{\partial H}{\partial T})_{P}$
the specific heat, $C_{k}$ the parameter of the return to isotropy term,
($C_{s},C_{t1},C_{e1}$) the diffusion parameters and ($\alpha,C_{t},C_{e}$)
the dissipation parameters of turbulent
variations($k,\overline{u_{r}^{\prime}T^{\prime}},\overline{T^{\prime}T^{\prime}}$).
In Eqs.(1-4), overbars are only used in three turbulent correlations
$\overline{u_{r}^{\prime}u_{r}^{\prime}}$,
$\overline{u_{r}^{\prime}T^{\prime}}$ and $\overline{T^{\prime}T^{\prime}}$.
The other variations(density $\rho$ and the temperature $T$, etc.) are all
mean state quantities which should use overbars but we ignore them for
convenience.
Equation (5) describes the energy transport in the stellar interior by both
turbulent motions(i.e. convection and overshooting) and radiation.
$\nabla_{R}$ is the radiative temperature gradient.
## 3 Theoretical analysis of TCM in the overshooting region
In the previous work (Zhang & Li, 2009), we applied the TCM in the solar
overshooting region and found some properties of the overshooting region:
$\overline{u_{r}^{\prime}T^{\prime}}<0$, $\nabla_{R}<\nabla<\nabla_{ad}$, and
the peak of $\overline{T^{\prime}T^{\prime}}$, which are similar to
Xiong’s(1985) and Xiong & Deng’s(2001) works. In this section, we attempt to
get semi-analytical solutions of the TCM.
Some approximations are adopted to simplify Eqs.(1-5) in the overshooting
region:
Approximation I. Péclet number $P_{e}\gg 1$, where $P_{e}=\frac{\rho
C_{P}l\sqrt{k}}{\lambda}$. That is
$\frac{\varepsilon}{k}\gg\frac{\lambda}{\rho
c_{P}}\frac{\varepsilon^{2}}{k^{3}}$ which means the turbulent dissipation is
much stronger than the thermal dissipation. This assumption is reasonable in
most cases except for the region near the surface of a star or with very small
$k$.
Approximation II. All variations, except the turbulent fluctuations, are
thought to be constant because the turbulent fluctuations change much faster
than others in the overshooting region.
Approximation III. Far away from the convective boundary,
$\nabla\approx\nabla_{R}$. This assumption is acceptable if the heat flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is small.
### 3.1 Turbulent heat transport in the overshooting region
Defining the anisotropic degree
$\omega=\frac{\overline{u_{r}^{\prime}u_{r}^{\prime}}}{2k}$ which is the ratio
of radial kinetic energy to total kinetic energy, and applying Approximation
II and Eq.(5), we can rewrite Eq.(3) to:
$\displaystyle\frac{\partial}{\partial r}\left(4C_{t1}\omega
l\sqrt{k}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)=-\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}+[2\omega
P_{e}+C_{t}(1+P_{e}^{-1})]\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$
(6)
In the last bracket in Eq.(6), Approximation I($P_{e}\gg 1$) makes the
dissipation term
$C_{t}(1+P_{e}^{-1})\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$ be
ignorable. And, by using Eq.(5) and Approximation II, it is easy to find that
the diffusion term is on the same order of the ignorable dissipation term:
$\displaystyle\frac{\partial}{\partial r}\left(4C_{t1}\omega
l\sqrt{k}\frac{\partial\overline{u_{r}^{\prime}T^{\prime}}}{\partial
r}\right)\approx
2C_{t1}\alpha^{2}\omega\frac{dlnk}{dlnP}\cdot\frac{dln(\nabla_{R}-\nabla)}{dlnP}(\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}})\sim
Pe^{0}(\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}})$ (7)
Therefore the diffusion term is also ignorable. Equation (3) is in local
equilibrium:
$\displaystyle-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}u_{r}^{\prime}}+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}\approx 0$ (8)
In the overshooting region, the most important process is the diffusion of the
kinetic energy. Thus, we ignore the diffusion of
$\overline{T^{\prime}T^{\prime}}$(i.e., setting $C_{e1}=0$). The solution of
the TCM with $C_{e1}=0$ can be thought as the zero-order solution of the TCM.
Ignoring the diffusion of $\overline{T^{\prime}T^{\prime}}$ and the diffusion
and dissipation terms of $\overline{u_{r}^{\prime}T^{\prime}}$, using
Approximations I & II, one can rewrite Eqs.(1-4) as:
$\displaystyle\frac{2C_{s}l}{k}\frac{\partial}{\partial r}(\omega
k^{\frac{5}{2}}\frac{\partial\omega}{\partial
r})=(C_{k}-1)(\omega-\frac{1}{3})\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}(1-\omega)$ (9) $\displaystyle
2C_{s}l\frac{\partial}{\partial r}(\omega k^{\frac{1}{2}}\frac{\partial
k}{\partial r})=\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (10) $\displaystyle
0=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\omega k+\frac{\beta
g_{r}}{T}\overline{T^{\prime}T^{\prime}}$ (11) $\displaystyle
0=-\frac{2T}{H_{P}}(\nabla-\nabla_{ad})\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\frac{\varepsilon}{k}\overline{T^{\prime}T^{\prime}}$
(12)
Equation (9) results from Eq.(1) and (2), describing the equilibrium structure
of the anisotropic degree $\omega$. The left side is the diffusion of
$\omega$. The first term in the right side is the dissipation rate due to
return to isotropy term in Eq.(1). The last term is the generation rate of
$\omega$ due to the buoyancy.
Equations (11) and (12) show:
$\displaystyle
0=(\nabla-\nabla_{ad})(\overline{u_{r}^{\prime}T^{\prime}}+2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l})$ (13)
The solution is
$\overline{u_{r}^{\prime}T^{\prime}}=-2C_{e}\varepsilon\omega\frac{T}{\beta
g}$ or $\nabla=\nabla_{ad}$. The latter is equivalent to
$\overline{u_{r}^{\prime}T^{\prime}}=-\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})$. Because $\overline{u_{r}^{\prime}T^{\prime}}$
is close to zero near the convective boundary and gradually decreases far away
from the convective boundary(Xiong, 1989; Xiong & Deng, 2001; Zhang & Li,
2009), the physically acceptable result is:
$\displaystyle\overline{u_{r}^{\prime}T^{\prime}}=Max\\{-\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R}),-2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}\\}$ (14)
Equation (14) shows that there is an adiabatic stratification zone in the
overshooting region in the case of $C_{e1}=0$. In order to investigate the
property of heat transport in the overshooting region, we must know the length
of the adiabatic stratification zone. It is found in Eq.(14) that the boundary
of the adiabatic stratification is the location where
$\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})=2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$. Solving the equation of $\omega$ is not easy
because it is nonlinear. However, this problem is avoidable. Turbulent motions
are isotropic when $\omega=\frac{1}{3}$. In the convection zone,
$\omega>\frac{1}{3}$ because the buoyancy boosts radial turbulent motion. In
most part of overshooting region, $\omega$ should be less than $\frac{1}{3}$
because the buoyancy prevents radial turbulent motion. Therefore $\omega$
should be not far away from $\frac{1}{3}$ near the convective boundary.
Further more, taking $\omega$ as a constant, one can rewrite Eq.(10) as:
$\displaystyle 2C_{s}l\omega\frac{\partial}{\partial
r}(k^{\frac{1}{2}}\frac{\partial k}{\partial
r})=\frac{k^{\frac{3}{2}}}{l}+\frac{\beta
g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ (15)
Substituting Eq.(14) into the above equation, one can get the approximate
solution:
$\displaystyle k^{\frac{3}{2}}\approx
k_{C}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r-r_{C}}{l}|)$
(16)
if $\frac{T}{H_{P}}\frac{\lambda}{\rho c_{P}}(\nabla_{ad}-\nabla_{R})\leq
2C_{e}\omega\frac{T}{\beta g}\frac{k^{\frac{3}{2}}}{l}$, and:
$\displaystyle
k^{\frac{3}{2}}=k_{A}^{\frac{3}{2}}exp(-\sqrt{\frac{3(1+2C_{e}\omega)}{4C_{s}\omega}}|\frac{r-r_{A}}{l}|)$
(17)
if $\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})>2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$.
In Eq.(16), point C, which is the convective boundary where
$\nabla_{ad}=\nabla_{R}$, is set to be the initial point, $k_{C}$ and $r_{C}$
being $k$ and $r$ here. The contribution of the buoyancy term(i.e. the last
term in Eq.(15)) is ignored in obtaining the solution Eq.(16). In the deep
convection zone, turbulent motions are almost in local equilibrium, thus the
ratio of $-\frac{\beta g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ to
$\frac{k^{\frac{3}{2}}}{l}$ is about 1. However, near the convective boundary,
buoyancy is about zero, meanwhile the diffusion of $k$ dominates. Those make
the ratio be much less than 1. Therefore the buoyancy term is ignorable.
In Eq.(17), point A, where $k=k_{A}$ and $r=r_{A}$, is the boundary of the
adiabatic overshooting region. In the region beyond point A, the ratio of
$-\frac{\beta g_{r}}{T}\overline{u_{r}^{\prime}T^{\prime}}$ to
$\frac{k^{\frac{3}{2}}}{l}$ is $2C_{e}\omega$ which is on the order of $1$,
thus the buoyancy term remains.
The exponentially decreasing function of $k$ is due to the fact that there is
no generation in the overshooting region. Contrary to the situation in the
convection zone, the buoyancy dissipates $k$ because it prevents the radial
motion of fluid elements in the overshooting region. The distribution of $k$
results from the equilibrium between the diffusion and the dissipation. $k$
should decrease faster if the buoyancy is as effective as the turbulent
dissipation, which is found by comparing the exponential indices of Eq.(16)
and (17).
The location of point A is determined by $\frac{T}{H_{P}}\frac{\lambda}{\rho
c_{P}}(\nabla_{ad}-\nabla_{R})=2C_{e}\omega\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$. Using Eq.(16), we get a property of point A:
$\displaystyle
k_{C}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{A}-r_{C}}{l}|)=\frac{1}{2C_{e}\omega}\frac{\alpha\beta
g\lambda}{\rho c_{P}}(\nabla_{ad}-\nabla_{R,A})$ (18)
The relation between $r_{A}$ and $\nabla_{R,A}$ is needed in order to solve
this equation and to locate point A. Near the convective boundary, there is:
$\displaystyle|\nabla_{ad}-\nabla_{R,A}|\approx\nabla_{ad}|\chi(lnP_{A}-lnP_{C})|=\nabla_{ad}|\chi|\frac{l_{ad}}{H_{P}}$
(19)
where $l_{ad}=|r_{A}-r_{C}|$ is the length of the adiabatic overshooting
region, $P_{A}$ and $P_{C}$ the pressure at point A and C, and
$\chi=\frac{dln\nabla_{R}}{dlnP}$ which is approximately a constant.
Substituting Eq.(19) into Eq.(18), one finds:
$\displaystyle
k_{C}^{\frac{3}{2}}exp(-\frac{1}{\alpha}\sqrt{\frac{3}{4C_{s}\omega}}\frac{l_{ad}}{H_{P}})=\frac{1}{2C_{e}\omega}\frac{\alpha\beta
g\lambda}{\rho c_{P}}\nabla_{ad}|\chi|\frac{l_{ad}}{H_{P}}$ (20)
$l_{ad}$ can be worked out if $k_{C}$ is known. In the deep adiabatic
convection zone, turbulent diffusion is ignorable, and the localized TCM shows
$k^{\frac{3}{2}}_{Local}=\frac{\alpha\beta
g\lambda(\nabla_{R}-\nabla_{ad})}{\rho c_{P}}$ (see Appendix A). However,
$k_{C}$ can not be estimated as that because $\nabla_{R}=\nabla_{ad}$ thus
$k_{Local}=0$ at the convective boundary. Actually, the turbulent diffusion of
$k$ is effective near the convective boundary, and $k_{C}$ is determined by
the diffusion. We can estimate $k_{C}$ by a simple approach which will be
referred to as the maximum of diffusion hereafter. Setting point B at where
the diffusion becomes dominative in the convection zone, we get the relation
between $k_{C}$ and $k_{B}$ by solving Eq.(15):
$\displaystyle
k_{C}^{\frac{3}{2}}=k_{B}^{\frac{3}{2}}exp(-\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{C}-r_{B}}{l}|)$
(21)
where $k_{B}^{\frac{3}{2}}\approx\frac{\alpha\beta
g\lambda(\nabla_{R,B}-\nabla_{ad})}{\rho c_{P}}$. Equation (21) shows that
$k_{C}$ is a function of $r_{B}$. In reality, the diffusion leads to the
maximum of $k_{C}$. Therefore $r_{B}$ makes the derivation of the right side
of Equation (21) be zero. Noting that $\nabla_{R,B}-\nabla_{ad}$ is
approximately proportional to $r_{B}-r_{C}$, one can easily work out the
location of point B:
$\displaystyle\sqrt{\frac{3}{4C_{s}\omega}}|\frac{r_{C}-r_{B}}{l}|\approx 1$
(22)
It is found in Fig.1 that $k\approx k_{Local}$ in the deep convection zone
because the turbulent diffusion can be ignored here, and the turbulent
diffusion dominates in the layer beyond point B.
Using above results, we obtain:
$\displaystyle k_{C}^{\frac{3}{2}}=\frac{1}{e}\frac{\alpha\beta
g\lambda(\nabla_{R,B}-\nabla_{ad})}{\rho
c_{P}}\approx\frac{1}{e}\sqrt{\frac{4C_{s}\omega}{3}}\frac{\alpha^{2}\beta
g\lambda\nabla_{ad}|\chi|}{\rho c_{P}}$ (23)
Generally, $\frac{l_{ad}}{H_{P}}$ is very small. According to Eq.(20), the
length of the adiabatic overshooting region is:
$\displaystyle
l_{ad}\approx\frac{\sqrt{\frac{4C_{s}\omega}{3}}}{\frac{e}{2C_{e}\omega}+1}l$
(24)
In the area $|r-r_{C}|\leq|r_{A}-r_{C}|$ in the overshooting region, the
temperature gradient $\nabla$ is almost equal to the adiabatic one. In the
area $|r-r_{C}|>|r_{A}-r_{C}|$, however, according to Eq.(14), Eq.(17), and
Eq.(5), the temperature gradient $\nabla$ is gradually close to $\nabla_{R}$:
$\displaystyle\nabla-\nabla_{R}=(\nabla_{ad}-\nabla_{R,A})\cdot
exp[-\sqrt{\frac{3(1+2C_{e}\omega)}{4C_{s}\omega}}|\frac{r-r_{A}}{l}|]$ (25)
Although $\omega$ in Eq.(24) and Eq.(25) is still unknown, we can estimate it
roughly. Equation (24) and (25) describe the turbulent motion near the
convective boundary, thus we can use $\omega\approx\omega_{C}$ where
$\omega_{C}$ is $\omega$ at the convective boundary. In the deep convection
zone, $\omega$ is almost equal to the equilibrium value
$\omega_{cz}=\frac{2}{3C_{k}}+\frac{1}{3}$ which is derived from the localized
TCM (see Appendix A). $\omega_{C}<\omega_{cz}$ because the buoyancy is zero at
the boundary, and $\omega_{C}>\frac{1}{3}$ because the diffusion of $\omega$.
Therefore the typical value of $\omega_{C}$ can be taken as the average, i.e.
$\omega_{C}\approx\frac{1}{2}(\omega_{cz}+\frac{1}{3})$. If Eq.(25) is used in
the region far away from the convective boundary(beyond the peak of
$\overline{T^{\prime}T^{\prime}}$), $\omega\approx\omega_{C}$ is not
appropriate. One can use $\omega=\omega_{o}$, where $\omega_{o}$ is the
equilibrium value of $\omega$ in the overshooting region which is introduced
in the next subsection.
Another turbulent correlation is $\overline{T^{\prime}T^{\prime}}$, which can
be worked out by using Eq.(11):
$\displaystyle\overline{T^{\prime}T^{\prime}}\approx
0,(|r-r_{C}|\leq|r_{A}-r_{C}|)$ (26)
And:
$\displaystyle\overline{T^{\prime}T^{\prime}}=\frac{2T}{H_{P}}\frac{T}{\beta
g}(\nabla_{ad}-\nabla)\omega k,(|r-r_{C}|>|r_{A}-r_{C}|)$ (27)
Equation (26) seems to against Cauchy’s theorem
$\overline{u_{r}^{\prime}u_{r}^{\prime}}\overline{T^{\prime}T^{\prime}}\geq\overline{u_{r}^{\prime}T^{\prime}}^{2}$.
Actually, ${\overline{T^{\prime}T^{\prime}}\approx 0}$ is only an approximate
solution on the order of
($Pe^{1}\frac{\sqrt{k}}{l}\overline{u_{r}^{\prime}T^{\prime}}$), because
Eq.(8) is an approximation on that order. Numerical calculations show no
confliction.
Results obtained above are based on $C_{e1}=0$. Numerical results of $\nabla$
with both $C_{e1}=0$ and $C_{e1}\neq 0$ are shown in Fig.2. It is found that
the effects of the diffusion of $\overline{T^{\prime}T^{\prime}}$ are only
making $\nabla$ be smoother. However, there is no adiabatic overshooting
region when the diffusion of $\overline{T^{\prime}T^{\prime}}$ is present,
because $\overline{T^{\prime}T^{\prime}}$ increases near the convective
boundary due to the turbulent diffusion thus $\nabla$ decreases according to
Eq.(8). Numerical results of the turbulent correlations in both $C_{e1}=0$ and
$C_{e1}\neq 0$ with different TCM parameters and for different stellar models
are shown in Figs.3-5. It is found that the theoretical solutions well fit the
numerical solutions in the case of $C_{e1}=0$. This also validates that the
boundary value $k_{C}$ derived from the maximum of diffusion is a good
approximation. The diffusion of $\overline{T^{\prime}T^{\prime}}$ modifies and
smoothes the profile of $\overline{T^{\prime}T^{\prime}}$ and
$\overline{u_{r}^{\prime}T^{\prime}}$. However, $k$ is insensitive to the
diffusion of $\overline{T^{\prime}T^{\prime}}$ because that $k$ is mainly
dominated by the diffusion of itself. The diffusion of
$\overline{T^{\prime}T^{\prime}}$ doesn’t significantly change the integral
value of $\overline{T^{\prime}T^{\prime}}$. According to Eq.(8), the integral
value of $\nabla$ or $\overline{u_{r}^{\prime}T^{\prime}}$ is also insensitive
to the diffusion of $\overline{T^{\prime}T^{\prime}}$, which is found in
Figs.(2-5).
The distribution of $\overline{T^{\prime}T^{\prime}}$ reveals an important
property of the overshooting. In the nonadiabatic overshooting region, using
$\nabla\approx\nabla_{R}$, one finds that
$\overline{T^{\prime}T^{\prime}}\propto T(\nabla_{ad}-\nabla_{R})k$ according
to Eq.(27). This result indicates a maximum of
$\overline{T^{\prime}T^{\prime}}$(Xiong, 1985; Zhang & Li, 2009) which is
shown in Figs.3-5. Beyond the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$, the temperature of a turbulent element is
gradually close to the temperature of the environment, and the efficiency of
heat transport significantly decreases. Therefore the area between the
convective boundary and the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$ can be thought as the overshooting region of
$\overline{u_{r}^{\prime}T^{\prime}}$. It is found in Figs.3-5 that the width
of the valley of $\overline{u_{r}^{\prime}T^{\prime}}$ is approximately equal
to the distance from the convective boundary to the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$. In order to get the overshooting length of
heat transport, we need to locate the maximum of
$\overline{T^{\prime}T^{\prime}}$.
Using Eq.(17), defining
$\theta_{0}=\frac{dlnk}{dlnP}=\pm\frac{1}{\alpha}\sqrt{\frac{(1+2C_{e}\omega)}{3C_{s}\omega}}$
as the decaying index of $k$ (in the case of $C_{e1}=0$), we get:
$\displaystyle\overline{T^{\prime}T^{\prime}}\propto
T(\nabla_{ad}-\nabla_{R})P^{\theta_{0}}$ (28)
The derivative of $\overline{T^{\prime}T^{\prime}}$ is zero at the peak of
$\overline{T^{\prime}T^{\prime}}$. We get $\nabla_{R}$ there(denoted as
$\nabla_{R}^{*}$):
$\displaystyle(\nabla_{R}^{*}+\theta_{0})(\nabla_{ad}-\nabla_{R}^{*})-\chi\nabla_{R}^{*}\approx
0$ (29)
$\nabla_{R}^{*}$ is determined by only one turbulent parameter $\theta_{0}$.
The typical overshooting length of $\overline{u_{r}^{\prime}T^{\prime}}$ (or
$\nabla$) can be estimated with $\nabla_{R}^{*}$:
$\displaystyle|\chi|=|\frac{dln\nabla_{R}}{dlnP}|\approx\
|\frac{ln\nabla_{R,C}-ln\nabla_{R}^{*}}{lnP_{C}-lnP^{*}}|=|\frac{ln\nabla_{ad}-ln\nabla_{R}^{*}}{lnP_{C}-lnP^{*}}|=\frac{ln\frac{\nabla_{ad}}{\nabla_{R}^{*}}}{\frac{l_{\nabla}}{H_{P}}}$
(30)
where $\nabla_{R,C}$ is $\nabla_{R}$ at the convective boundary, $l_{\nabla}$
is the distance from the convective boundary to the location of the maximum of
$\overline{T^{\prime}T^{\prime}}$ and also the typical overshooting length of
$\nabla$.
$l_{\nabla}$ is worked out as:
$\displaystyle
l_{\nabla}\approx\frac{1}{|\chi|}ln\frac{\nabla_{ad}}{\nabla_{R}^{*}}H_{P}$
(31)
Usually, $|\theta_{0}|$ is much larger than $|\chi|$ and $\nabla_{ad}$, and
$\nabla_{R}^{*}$ can be approximately solved from Eq.(29):
$\displaystyle\nabla_{R}^{*}\approx(1-\frac{\chi}{\theta_{0}})\nabla_{ad}$
(32)
Finally, we find:
$\displaystyle l_{\nabla}\approx\frac{H_{P}}{|\theta_{0}|}=H_{k}$ (33)
where $H_{k}$ is the scale height of turbulent kinetic energy $k$ defined by
$H_{k}=|\frac{dr}{dlnk}|$. The result indicates that $\nabla$ is remarkably
modified by the overshooting only in about $1H_{k}$. It is found in Fig.3 that
$l_{\nabla}=ln\frac{k_{C}}{k_{*}}H_{k}\approx 0.8H_{k}$, which is in agreement
with Eq.(33). It is shown in Fig.2 that $\nabla$ is remarkably modified only
in $1H_{k}$.
### 3.2 Asymptotic analysis
In above subsection, we have discussed the turbulent heat transport and the
solution of turbulent correlations in the overshooting region near the
convective boundary based on the assumption $C_{e1}=0$. The diffusion of
$\overline{T^{\prime}T^{\prime}}$ only modifies turbulent correlations to be
smoother near the convective boundary. However, it makes more effects on
turbulent motions in the overshooting region further than $1H_{k}$ away from
the convective boundary. In this subsection, we investigate the turbulence
properties in the outer overshooting region(beyond $1H_{k}$).
In the numerical calculations of the TCM, we found that the anisotropic degree
$\omega$ always showed an equilibrium value in the overshooting region. A
typical numerical result is shown in Fig.6. In order to understand it, we
discuss the behave of the anisotropic degree $\omega$ in both convection zone
and overshooting region. $\omega$ should be larger than $\frac{1}{3}$ in the
convection zone because the buoyancy boosts radial movement of turbulent
elements. Actually, $\omega$ is almost equal to the equilibrium value in the
convection zone $\omega_{cz}=\frac{2}{3C_{k}}+\frac{1}{3}$(see Appendix A) due
to the equilibrium between the buoyancy and the return to isotropy term. When
turbulent elements go across the convective boundary into the overshooting
region, the buoyancy prevents convective elements moving, thus $\omega$
decreases to less than $\frac{1}{3}$ near the convective boundary. However, as
$\overline{u_{r}^{\prime}T^{\prime}}$ exponentially decreasing, the
equilibrium of $\omega$ is established again in the overshooting region. This
results in an asymptotic property of the overshooting region: there is an
equilibrium value of $\omega$ in the overshooting region,
$\omega\approx\omega_{o}$.
By using the asymptotic property $\omega\approx\omega_{o}$ and Approximations
I, II & III, it is easy to get the asymptotic solution of TCM in the
overshooting region(see Appendix B):
$\displaystyle\overline{u_{r}^{\prime}T^{\prime}}=\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}\frac{T}{\beta
g}\frac{k^{\frac{3}{2}}}{l}$ (34)
$\displaystyle\overline{T^{\prime}T^{\prime}}=2\omega_{o}(\nabla_{ad}-\nabla_{R})\frac{T^{2}}{\beta
gH_{P}}k$ (35) $\displaystyle k=k_{0}(\frac{P}{P_{0}})^{\theta}$ (36)
where $\theta$ is the asymptotic solution of $\frac{dlnk}{dlnP}$:
$\displaystyle\theta=\pm\frac{1}{\alpha}\sqrt{\frac{1}{3C_{s}\omega_{o}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]}$
(37)
$k$ takes the decreasing expression in the overshooting region, which means:
${}^{\prime}+^{\prime}$ is adopted in the upward overshooting region and
${}^{\prime}-^{\prime}$ in the downward one.
The equilibrium value $\omega_{o}$ is determined by:
$\displaystyle(2C_{s}C_{e}-C_{e1}C_{k}){\omega_{o}}^{2}+[\frac{1}{3}C_{e1}(C_{k}+2)-C_{s}(C_{k}+2C_{e}-1)]\omega_{o}+\frac{1}{3}C_{s}(C_{k}-1)=0$
(38)
The equilibrium value $\omega_{o}$ is only a function of turbulent parameters
$(C_{e},C_{e1},C_{s},C_{k})$. The fact that the buoyancy prevents the radial
movement of turbulent elements in the overshooting region restricts the
turbulent parameters to ensure $\omega_{o}<\frac{1}{3}$.
An important thing is where $\omega$ reaches its equilibrium value
$\omega_{o}$. According to Eq.(9), the equilibrium of $\omega$ can be realized
only if the buoyancy term synchronically decreases with $k$ decreasing.
Therefore $\omega$ starts to reach its equilibrium value $\omega_{o}$ beyond
the peak of $\overline{T^{\prime}T^{\prime}}$ due to
$|\overline{u_{r}^{\prime}T^{\prime}}|$ being decreasing.
Setting $C_{e1}=0$ in Eq.(38), we find that the asymptotic solution is the
same as the results in the overshooting region with
$|r-r_{C}|\geq|r_{A}-r_{C}|$ by setting $\omega=\omega_{o}$ in Eq(14),(17) &
(27). Because Eq.(8) is correct whether $C_{e1}=0$ or not, the conclusion that
the maximum of $\overline{T^{\prime}T^{\prime}}$ is located at about $1H_{k}$
is also correct in both cases.
It must be mentioned that we have used Approximation I(i.e. $P_{e}\gg 1$),
which means that the turbulent dissipation is much larger than the thermal
dissipation. If $k$ decreases enough to satisfy $P_{e}\ll 1$, the thermal
dissipation should become significant thus $\overline{T^{\prime}T^{\prime}}$
and the turbulent kinetic energy $k$ should rapidly decrease to zero. Then
$\omega$ also rapidly decreases as shown in Fig.6. In another word, turbulent
movement can hardly overshoot into the thermal dissipation zone where
$P_{e}\ll 1$.
According to discussions above, we can separate the overshooting region into
three parts as shown in Fig.7: the overshooting region of
$\overline{u_{r}^{\prime}T^{\prime}}$ or $\nabla$ with the length of about
$1H_{k}$, the turbulent dissipation region in which the asymptotic solution
holds, and the thermal dissipation region in which the turbulent movement
quickly vanishes. The boundaries among those parts are the peak of
$\overline{T^{\prime}T^{\prime}}$ and the location of $P_{e}=1$.
## 4 Conclusions and discussions
Turbulent convection models are better tools in dealing with the convective
overshooting than non-local mixing length theories. However, they are often
too complex to be applied in the calculations of stellar structure and
evolution. In order to investigate the property of the convective overshooting
and to make it easy to apply turbulent convection models, we have analyzed the
TCM developed by Li & Yang (Li & Yang, 2007) and obtained approximate and
asymptotic solutions of the TCM in the overshooting region with $P_{e}\gg 1$.
The main conclusions and corresponding discussions are listed as follows:
1\. The overshooting region can be partitioned into three parts: a thin
turbulent heat flux overshooting region, a power law dissipation region of
turbulent kinetic energy, and a thermal dissipation area with rapidly
decreasing $k$. The turbulent fluctuations $k$,
$\overline{u_{r}^{\prime}T^{\prime}}$, and $\overline{T^{\prime}T^{\prime}}$
exponentially decrease in the overshooting region as Eqs.(34-36). The
equilibrium value of the anisotropic degree $\omega_{o}$ and the exponential
indices of the turbulent fluctuations are only determined by the parameters of
the TCM. The decaying behaviors of the turbulent fluctuations are similar to
Xiong & Deng’s results(Xiong, 1989; Xiong & Deng, 2001).
2\. The peak of $\overline{T^{\prime}T^{\prime}}$ in the overshooting region
is located at about $1H_{k}$ away from the convective boundary. In this
distance, the modification of $\nabla$ caused by the overshooting is
remarkable. An approximate profile of $\nabla$ comprises an adiabatic
overshooting region with the length of $l_{ad}$ and an exponentially
decreasing function, as described in Eq.(24) and (25). Beyond $1H_{k}$, the
modification of $\nabla$ is ignorable and $\nabla\approx\nabla_{R}$. It should
be noted that the result of $1H_{k}$ overshooting distance of turbulent heat
transfer is independent of the parameters of TCM, so it may be a general
property of the overshooting. Our result is similar to Marik & Petrovay(2002)
whose result shows that the length between the peak of
$\overline{T^{\prime}T^{\prime}}$ and the convective boundary is about
$1.2H_{k}$. Meakin & Arnett(2010) simulated the turbulent convection of a
$23M_{\odot}$ star, the data of the turbulent kinetic energy and the
convective flux in the overshooting region being shown in Fig.8. It is found
that the overshooting length of the convective flux
$\overline{u_{r}^{\prime}T^{\prime}}$ is about $0.5\sim 2H_{k}$ which is in
agreement with our result.
3\. The value of the turbulent kinetic energy at the convective boundary
$k_{C}$ can be estimated by a method called the maximum of diffusion. The
value of turbulent fluctuations in the overshooting region can be estimated by
using the exponentially decreasing functions and the initial value $k_{C}$.
This may significantly simplify the application of the TCM in calculations of
the stellar structure and evolution.
There is a distinction between the non-local model of Zahn(1991) and our
results, i.e. the temperature gradient jumps from nearly adiabatic to
radiative in Zahn’s model but continuously changes in our results (see Fig.2).
This is caused by the assumption in Zahn’s model that the turbulent velocity
and temperature fluctuation are strongly correlated(Xiong & Deng, 2001). In
our results, the correlativity of turbulent velocity and temperature
fluctuation $R_{VT}=\frac{\overline{u_{r}^{\prime}T^{\prime}}}{\sqrt{2\omega
k\overline{T^{\prime}T^{\prime}}}}$ quickly decreases to zero then turns to be
negative near the convective boundary(see Fig.9), and the asymptotic solution
shows that $R_{VT}\propto\sqrt{k}$ and exponentially decreases in the
turbulent dissipation overshooting region. Our result is in agreement with
three-dimension simulations such as Fig.6 in Singh et al.(1995) and Fig.15 in
Meakin & Arnett(2007).
We thank the anonymous referee for valuable comments which help to improve the
paper. And we thank C. A. Meakin for providing the numerical data of Fig.8.
Fruitful discussions with J. Su, X. J. Lai and C. Y. Ding are highly
appreciated. This work is co-sponsored by the National Natural Science
Foundation of China through grant No.10673030 and No.10973035 and Science
Foundation of Yunnan Observatory No.Y0ZX011009.
## Appendix A The localized TCM in convection zone.
The localized TCM results from Eqs.(1-4) by ignoring the diffusion terms. It
is a good approximate of the TCM in the convection zone(Li & Yang, 2001). We
attempt to work out the solution in this appendix.
Some symbols are defined for conveniences:
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $V=\overline{T^{\prime}T^{\prime}}$,
$W=\sqrt{k}$, $A=\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})$, $B=-\frac{\beta
g_{r}}{T}$, $D=\frac{\lambda}{\rho C_{P}}$,
$f=\frac{\nabla-\nabla_{ad}}{\nabla_{R}-\nabla_{ad}}$.
Ignoring the diffusion terms of Eqs.(1-4), we get the localized TCM:
$\displaystyle
0=\frac{2}{3}\frac{W^{3}}{l}-2BU+2C_{k}(\omega-\frac{1}{3})\frac{W^{3}}{l}$
(A1) $\displaystyle 0=\frac{W^{3}}{l}-BU$ (A2) $\displaystyle 0=-2\omega
fAW^{2}-BV+C_{t}(1+P_{e}^{-1})\frac{WU}{l}$ (A3) $\displaystyle
0=-2fAU+2C_{e}(1+P_{e}^{-1})\frac{WV}{l}$ (A4) $\displaystyle U=AD(1-f)$ (A5)
Equation (A1) and (A2) show:
$\displaystyle\omega=\frac{2}{3C_{k}}+\frac{1}{3}$ (A6)
This is the equilibrium value $\omega_{cz}$ in convection zone.
Describing $W$, $V$, $U$ by $f$ and $P_{e}$($=\frac{lW}{D}$), we find:
$\displaystyle
f=\frac{C_{t}C_{e}P_{e}^{-1}(1+P_{e}^{-1})^{2}}{C_{t}C_{e}P_{e}^{-1}(1+P_{e}^{-1})^{2}+2C_{e}\omega(1+P_{e}^{-1})+1}$
(A7)
$W$, $V$ can be worked out as:
$\displaystyle W^{3}=ABDl(1-f)$ (A8) $\displaystyle
V=\frac{AfW^{2}}{C_{e}B(1+P_{e}^{-1})}$ (A9)
According to $P_{e}=\frac{lW}{D}$, Eq.(A8) and Eq.(A7), we get the equation of
$P_{e}$:
$\displaystyle aP_{e}^{4}+(b+1)P_{e}^{3}+2bP_{e}^{2}+(b-at)P_{e}-t=0$ (A10)
where $a=1+\frac{1}{2\omega C_{e}}$, $b=\frac{C_{t}}{2\omega}$,
$t=\frac{ABl^{4}}{D^{2}}$. $f$ is determined by $f=1-\frac{P_{e}^{3}}{t}$
according to Eq.(A8).
Solving Eq.(A10), we can obtain all turbulent fluctuations of the localized
TCM by using Eq.(A5), (A8), (A9) and (A11).
An important case is $t\gg 1$, thus $P_{e}\gg 1$ according to Eq.(A10). In
that case, Eq.(A7) shows:
$\displaystyle f=\frac{C_{e}C_{t}P_{e}^{-1}}{2C_{e}\omega+1}\approx 0$ (A11)
which corresponds to the adiabatic convection.
Finally, we obtain the turbulent fluctuations according to Eq.(A8), (A5) &
(A9):
$\displaystyle W^{3}\approx ABDl$ (A12) $\displaystyle
V\approx\frac{C_{t}}{2C_{e}\omega+1}\frac{AD}{Bl}W$ (A13) $\displaystyle
U\approx AD$ (A14)
and the correlativity of turbulent velocity and temperature $R_{VT}$:
$\displaystyle R_{VT}=\frac{U}{\sqrt{2\omega
W^{2}V}}\approx\sqrt{\frac{2C_{e}\omega+1}{2C_{t}\omega}}$ (A15)
## Appendix B Details of deriving the asymptotic solution of the TCM in
overshooting region.
There are the details of obtaining the asymptotic solution of the TCM in
overshooting region.
Some symbols are defined for conveniences:
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $V=\overline{T^{\prime}T^{\prime}}$,
$W=\sqrt{k}$,
$A=-\frac{T}{H_{P}}(\nabla-\nabla_{ad})\approx-\frac{T}{H_{P}}(\nabla_{R}-\nabla_{ad})$
(Approximation III is used), $B=-\frac{\beta g_{r}}{T}$.
Applying the asymptotic property $\omega=\omega_{o}$ and Approximations I, II
& III, one can rewrite TCM as:
$\displaystyle
0=(C_{k}-1)(\omega_{o}-\frac{1}{3})\frac{W^{3}}{l}-BU(1-\omega_{o})$ (B1)
$\displaystyle lC_{s}\omega_{o}\frac{\partial}{\partial r}(W\frac{\partial
W^{2}}{\partial r})=\frac{W^{3}}{l}-BU$ (B2) $\displaystyle
0=-BV+2A\omega_{o}W^{2}$ (B3) $\displaystyle
lC_{e1}\omega_{o}\frac{\partial}{\partial r}(W\frac{\partial V}{\partial
r})=AU+\frac{C_{e}}{l}WV$ (B4)
Equation (B1) is equivalent to:
$\displaystyle
U=\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}\frac{W^{3}}{Bl}$
(B5)
Taking it into Eq.(B2), one gets the equation of $W$:
$\displaystyle\frac{\partial^{2}W^{3}}{\partial
r^{2}}=\frac{3}{4C_{s}\omega_{o}l^{2}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]W^{3}$
(B6)
Equation (B3) is equivalent to:
$\displaystyle V=\frac{2A\omega_{o}}{B}W^{2}$ (B7)
According to Eq.(B4), (B5) and (B7), one gets another equation of $W$:
$\displaystyle\frac{\partial^{2}W^{3}}{\partial
r^{2}}=\frac{3}{4C_{e1}{\omega_{o}}^{2}l^{2}}[\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}+2C_{e}\omega_{o}]W^{3}$
(B8)
Comparing Eq.(B6) with Eq.(B8), one finds:
$\displaystyle\frac{3}{4C_{e1}{\omega_{o}}^{2}l^{2}}[\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}+2C_{e}\omega_{o}]=\frac{3}{4C_{s}\omega_{o}l^{2}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]$
(B9)
Therefore the equation of $\omega_{o}$ is:
$\displaystyle(2C_{s}C_{e}-C_{e1}C_{k}){\omega_{o}}^{2}+[\frac{1}{3}C_{e1}(C_{k}+2)-C_{s}(C_{k}+2C_{e}-1)]\omega_{o}+\frac{1}{3}C_{s}(C_{k}-1)=0$
(B10)
The asymptotic solution of $W$ is derived from Eq.(B6):
$\displaystyle
W=W_{0}exp{\\{\pm\frac{1}{2\alpha}\sqrt{\frac{1}{3C_{s}\omega_{o}}[1-\frac{(C_{k}-1)(\omega_{o}-\frac{1}{3})}{(1-\omega_{o})}]}ln(\frac{P}{P_{0}})\\}}$
(B11)
$W$ takes the decreasing expression in the overshooting region:
${}^{\prime}+^{\prime}$ is adopted in the upward overshooting region and
${}^{\prime}-^{\prime}$ in the downward one.
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Figure 1: Numerical results of $W=\sqrt{k}$, and
$W_{Local}\approx\sqrt[3]{\frac{\alpha\beta
g\lambda(\nabla_{R}-\nabla_{ad})}{\rho c_{P}}}$ which is the solution of
localized TCM (See Appendix A), for the solar model at present age. TCM
parameters are: $\alpha=0.84$, $C_{k}=2.5$, $C_{s}=0.1$, $C_{e1}=0$,
$C_{e}=0.2$, $C_{t}=7.0$, and $C_{t1}=0.01$, but $C_{t}$, $C_{t1}$ and
$C_{e1}$ are insensitive to the results. Point C is the boundary of the
convection zone, the location of point B is calculated by using Eq.(22).
Figure 2: Numerical results of temperature gradient near the convective
boundary in both $C_{e1}=0$ and $C_{e1}\neq 0$, $\nabla_{0}$ being the
temperature gradient of the model with $C_{e1}=0$, and $\nabla_{1}$
corresponding to $C_{e1}=0.01$. Dotted line $\nabla_{T}$, which is almost
identical to $\nabla_{0}$, is theoretical solution of the temperature gradient
with $C_{e1}=0$. The stellar model and other TCM parameters are the same as
Fig.1. Point A is the boundary of the adiabatic overshooting region. Our
theoretical result shows $l_{ad}\approx 0.013H_{P}$ in those TCM parameters
set, the numerical calculation being $0.015H_{P}$. Figure 3: Numerical results
of $\overline{T^{\prime}T^{\prime}}$, $\overline{u_{r}^{\prime}T^{\prime}}$,
$k$ near the convective boundary in both $C_{e1}=0$ and $C_{e1}\neq 0$, where
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model and other TCM parameters are the same as Fig.1.
Figure 4: Numerical results of $\overline{T^{\prime}T^{\prime}}$,
$\overline{u_{r}^{\prime}T^{\prime}}$, $k$ near the convective boundary in
both $C_{e1}=0$ and $C_{e1}\neq 0$, where
$U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model is a $7M_{\odot}$ star model at the top of RGB
phase. Others TCM parameters are: $\alpha=1.0$, $C_{k}=2.2$, $C_{s}=0.1$,
$C_{e}=1.0$, and $C_{t}=4.0$, $C_{t1}=0.01$. Figure 5: Numerical results of
$\overline{T^{\prime}T^{\prime}}$, $\overline{u_{r}^{\prime}T^{\prime}}$, $k$
near the boundary of the convective core in both $C_{e1}=0$ and $C_{e1}\neq
0$, where $U=\overline{u_{r}^{\prime}T^{\prime}}$, $W=\sqrt{k}$,
$V=\overline{T^{\prime}T^{\prime}}$. Dashed lines correspond to $C_{e1}=0$,
solid lines to $C_{e1}=0.01$. Dotted lines are the theoretical solutions with
$C_{e1}=0$. The stellar model is an early main sequence model of a
$3M_{\odot}$ star. Others TCM parameters are: $\alpha=1.0$, $C_{k}=2.1$,
$C_{s}=0.2$, $C_{e}=0.5$, and $C_{t}=3.0$, $C_{t1}=0.01$. Figure 6: Numerical
result of the structure of $\omega$ in overshooting region. The stellar model
is the solar model at present age. $C_{e1}=0.01$. The others TCM parameters
are the same as Fig.1, except $\alpha=0.2$ in order to enlarge $\theta$ to
show the thermal dissipation region in which $P_{e}\ll 1$. With those
parameters, the equilibrium value in convection zone is $\omega_{cz}=0.6$, and
the equilibrium value in overshooting region is $\omega_{o}=0.293$ which
denoted as the dotted line. Figure 7: The structure of the overshooting
region. $K=k$, $U=\overline{u_{r}^{\prime}T^{\prime}}$,
$V=\overline{T^{\prime}T^{\prime}}$. The stellar model is the solar model at
present age. $C_{e1}=0.01$, the others TCM parameters are the same as Fig.1,
except $\alpha=0.2$. With those parameters, in the turbulent dissipation
region with $P_{e}\gg 1$, theoretical result shows $\theta=17.5$ vs the
numerical result $17.6$, theoretical result of exponential decreasing index of
$U^{2}$ being $26.3$ vs the numerical result about $25.6$. $K$ is almost
parallel to $V$, which is in consistent with the asymptotic solution. In the
thermal dissipation region with $P_{e}\ll 1$, turbulent motion vanishes.
Figure 8: Numerical data of Meakin & Arnett (2010)’s results. The data of
model ’h1’ in their paper are plotted, where $U=F_{C}=\rho
C_{P}\overline{u_{r}^{\prime}T^{\prime}}$. Only the downward overshooting
region is shown. The distance from the convective boundary (where
$\overline{u_{r}^{\prime}T^{\prime}}=0$, about $R=0.62\times 10^{9}cm$) to the
right part of the valley of $\overline{u_{r}^{\prime}T^{\prime}}$ is about
$0.5\sim 2H_{k}$. Figure 9: Numerical results of the correlativity of
turbulent velocity and temperature $R_{VT}$. The stellar model is the solar
model at present age. Other TCM parameters are the same as Fig.1. $R_{VT}$
rapidly decreases to zero in the overshooting region. In the convection zone
near the convective boundary, the diffusion significantly enlarges
$\overline{T^{\prime}T^{\prime}}$ when $C_{e1}\neq 0$ (see Fig.3), and then
$R_{VT}$ is very small. In the interior of convection zone, localized TCM
shows the equilibrium value of $R_{VT}$ is
$R_{VT,cz}=\sqrt{\frac{2\omega_{cz}C_{e}+1}{2\omega_{cz}C_{t}}}$ (see Appendix
A). The TCM parameters show $R_{VT,cz}=0.384$.
|
arxiv-papers
| 2012-02-20T04:39:06 |
2024-09-04T02:49:27.550199
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Q. S. Zhang and Y. Li",
"submitter": "Qian-Sheng Zhang",
"url": "https://arxiv.org/abs/1202.4219"
}
|
1202.4311
|
# Comparative statistics of Garman-Klass, Parkinson, Roger-Satchell and bridge
estimators
S. Lapinova, A. Saichev National research University “Higher school of
economics”, RussiaETH Zurich – Department of Management, Technology and
Economics, Switzerland
###### Abstract
Comparative statistical properties of Parkinson, Garman-Klass, Roger-Satchell
and bridge oscillation estimators are discussed. Point and interval
estimations, related with mentioned estimators are considered
## 1 Examples of volatility estimators
Consider dependence on time $t$ of the price $P(t)$ of some financial
instrument. As a rule, at discussing of volatility, one consider its logarithm
$X(t):=\ln P(t).$
Let point out one of the conventional volatility $V(T)$ definition, which we
are using in this paper: It is the variance
$V(T):=\mathbb{Var}\left[Y(t,T)\right]=\mathbb{E}\left[Y^{2}(t,T)\right]-\mathbb{E}^{2}\left[Y(t,T)\right].$
(1)
of the log-price increment $Y(t,T):=X(t+T)-X(t)$ within given time interval
duration $T$.
Recall, Garman-Klass (G&K) [1], Parkinson (PARK) [2] and Roger-Satchell (R&S)
[3] volatility estimators are resting on the high and low values:
$H:=\sup_{t^{\prime}\in(0,T)}Y(t,t^{\prime}),\qquad
L:=\inf_{t^{\prime}\in(0,T)}Y(t,t^{\prime}).$ (2)
Accordingly, PARK estimator is equal to
$\hat{V}_{p}:=\frac{(H-L)^{2}}{\ln 16},$ (3)
while G&K estimator given by expression
$\begin{array}[]{c}\displaystyle\hat{V}_{g}:=k_{1}(H-L)^{2}-k_{2}(C(H-L)-2HL)-k_{3}C^{2},\\\
k_{1}=0.511,\qquad k_{2}=0.0109,\qquad k_{3}=0.383.\end{array}$ (4)
Here $C:=Y(t,T)$ is the close value of the log-price increment. Recall else
R&S estimator, equal to
$\hat{V}_{r}:=H(H-C)+L(L-C).$ (5)
Besides of mentioned well-known estimators, we discuss _bridge oscillation
estimator_. Below we call it shortly by _bridge estimator_. Before to define
it, recall bridge $Z(t,t^{\prime})$ stochastic process definition. It is equal
to
$Z(t,t^{\prime}):=Y(t,t^{\prime})-\frac{t^{\prime}}{T}~{}Y(t,T),\qquad
t^{\prime}\in(0,T).$ (6)
Let introduce high and low of the bridge:
$\mathcal{H}:=\max_{t^{\prime}\in(0,T)}Z(t,t^{\prime}),\qquad\mathcal{L}:=\min_{t^{\prime}\in(0,T)}Z(t,t^{\prime}).$
(7)
Accordingly, mentioned above bridge volatility estimator given by
$\hat{V}_{b}:=\kappa\left(\mathcal{H}-\mathcal{L}\right)^{2}.$ (8)
The value of the factor $\kappa$ will be calculated later.
## 2 Geometric Brownian motion
One of conventional models of price stochastic behavior is geometric Brownian
motion (see [4, 5, 6]). In particular, it is used in theoretical justification
of G&K, PARK and R&S estimators. Below we discuss statistics of mentioned
volatility estimators in frame of geometric Brownian motion model. Namely, we
assume that increment of the log-price is of the form
$Y(t,T)=\mu T+\sigma B(T).$ (9)
Here $\mu$ is the drift of the price, while $B(t)$ is the standard Brownian
motion $B(t)\sim\mathcal{N}(0,t)$. Factor $\sigma^{2}$ is the intensity of the
Brownian motion.
Recall, Brownian motion posses by self-similar property
$B(t)\sim\sqrt{T}\,B\left(\frac{t}{T}\right),\qquad\forall~{}T>0,$ (10)
where and below sign $\sim$ means identity in law.
Using pointed out self-similar property, one can ensure that
$\begin{array}[]{c}\displaystyle
Y(t,t^{\prime})\sim\sigma\sqrt{T}~{}x(\tau,\gamma),\\\\[11.38109pt]
\displaystyle
x(\tau,\gamma):=\gamma\tau+B(\tau),\qquad\gamma:=\frac{\mu}{\sigma}\sqrt{T},\qquad\tau:=\frac{t^{\prime}}{T}\in(0,1).\end{array}$
(11)
Henceforth we call process $x(\tau,\gamma)$ by _canonical Brownian motion_ ,
while factor $\gamma$ by _canonical drift_. Using relations (3), (4), (8) and
(11), one find that
$\begin{array}[]{c}\hat{V}_{p}\sim
V(T)\cdot\hat{v}_{p}(\gamma),\qquad\hat{V}_{g}\sim
V(T)\cdot\hat{v}_{g}(\gamma),\qquad\hat{V}_{b}\sim
V(T)\cdot\hat{v}_{b},\\\\[5.69054pt] \hat{V}_{r}\sim
V(T)\cdot\hat{v}_{r}(\gamma),\qquad V(T)=\sigma^{2}T.\end{array}$
We have used above _canonical estimators_ :
$\begin{array}[]{c}\displaystyle\hat{v}_{p}(\gamma):=\frac{d^{2}}{\ln
16},\qquad\hat{v}_{b}:=\kappa s^{2},\qquad d:=h-l,\qquad
s:=\xi-\zeta,\\\\[11.38109pt]
\hat{v}_{g}(\gamma):=k_{1}d^{2}-k_{2}(cd-2hc)-k_{3}c^{2},\qquad\hat{v}_{r}=h(h-c)+l(l-c),\end{array}$
(12)
containing high, low and close values
$h:=\sup_{\tau\in(0,1)}x(\tau,\gamma),\qquad
l:=\inf_{\tau\in(0,1)}x(\tau,\gamma),\qquad c:=x(1,\gamma),$ (13)
of canonical Brownian motion, and high and low values
$\xi:=\sup_{\tau\in(0,1)}z(\tau),\qquad\zeta:=\inf_{\tau\in(0,1)}z(\tau),$
(14)
of the canonical bridge
$z(\tau):=x(\tau,\gamma)-\tau x(1,\gamma)=B(\tau)-\tau\cdot
B(1),\qquad\tau\in(0,1).$ (15)
Plots of the typical paths of the canonical Brownian motion $x(\tau,\gamma)$
(11) for $\gamma=1$ and corresponding canonical bridge $z(\tau)$ (15) are
given in figure 1.
Figure 1: Typical paths of canonical Brownian motion $x(\tau,\gamma)$ (11) for
$\gamma=1$ and corresponding canonical bridge $z(\tau)$ (15)
It is worthwhile to note that the closer expected values of canonical
estimators $\hat{v}_{p}(\gamma)$, $\hat{v}_{g}(\gamma)$, $\hat{v}_{r}$ and
$\hat{v}_{b}$ to unity, the less biased corresponding original volatility
estimators. Analogously, the smaller variances of canonical estimators the
more efficient original volatility estimators $\hat{V}_{p}$, $\hat{V}_{g}$,
$\hat{V}_{r}$ and $\hat{V}_{b}$.
Notice additionally that canonical drift $\gamma$ of the canonical Brownian
motion $x(\tau,\gamma)$ (11) is, as a rule, unknown. Nevertheless, to get some
idea about dependence on drift $\mu$ of bias and efficiency of volatility
estimators, we will discuss below in details dependence of canonical
estimators statistical properties on possible values of the factor $\gamma$.
## 3 Comparative efficiency of PARK and bridge estimators
Resting on, given at Appendix, analytical formulas for probability density
functions (pdfs) of random variables (13) and (14), we explore in this section
some atatistical properties of canonical PARK estimator $\hat{v}_{p}(\gamma)$
and bridge one $\hat{v}_{b}$ (12).
Let check, first of all, unbiasedness of canonical PARK estimator. To make it,
let calculate, with help of pdf $q_{x}(\delta)$ (A.7), mean square of
oscillation $d=h-l$ of the canonical Brownian motion $x(\tau,\gamma)$ at the
zero canonical drift ($\gamma=0$). After simple calculations obtain
$\mathbb{E}[d^{2}]=2+\sum_{m=1}^{\infty}\frac{2}{m(4m^{2}-1)}=\ln 16.$ (16)
From here and from expression (12) of canonical PARK estimator
$\hat{v}_{p}(\gamma)$ one can see that the following expression is true
$\mathbb{E}[\hat{v}_{p}(\gamma=0)]=1.$
Let find now the factor $\kappa$ at expressions (8) and (12). To make it,
calculate first of all the mean square of the bridge oscillation. Due to
expression (A.9) for the bridge oscillation $s$ (12) pdf, one have
$\mathbb{E}[s^{2}]=\sum_{m=1}^{\infty}\frac{1}{m^{2}}=\frac{\pi^{2}}{6}.$
Accordingly, unbiased canonical bridge estimator has the form
$\mathbb{E}[\hat{v}_{b}]=1\quad\Rightarrow\quad\kappa=\frac{1}{\mathbb{E}[s^{2}]}\quad\Rightarrow\quad\hat{v}_{b}=\frac{6\,s^{2}}{\pi^{2}}.$
(17)
The great advantage of the bridge estimator is its unbiasedness for any drift.
This remarkable property of the pointed out estimator is the consequence of
the fact that bridge $Z(t,t^{\prime})$ (6) and its canonical counterpart
$z(\tau)$ don’t depend on the drift $\mu$ (canonical drift $\gamma$) at all.
On the contrary, PARK estimator becomes essentially biased at nonzero drift.
In figure 2 depicted dependence on $\gamma$ of canonical PARK estimator
expected value, illustrating bias of PARK estimator at nonzero drift.
Corresponding curve obtained with help of analytical expression (A.6) for
canonical bridge oscillation pdf.
Figure 2: Plot of canonical PARK estimator $\hat{v}_{p}(\gamma)$ mean value,
as function of canonical drift $\gamma$. It is seen that with growth of
$\gamma$ PARK estimator becomes more and more biased. Straight line is the
plot of canonical bridge $\hat{v}_{b}$, mean value
Let calculate variances of canonical PARK and bridge estimators. After
substitution into the rhs of expression
$\mathbb{E}[\hat{v}^{2}_{p}(\gamma=0)]:=\frac{1}{\ln^{2}16}\int_{0}^{\infty}\delta^{4}q_{x}(\delta)d\delta$
the sum (A.7) for the canonical Brownian motion oscillation pdf
$q_{x}(\delta)$, and after summation obtain for $\gamma=0$:
$\mathbb{E}[\hat{v}^{2}_{p}(\gamma=0)]=\frac{9\,\zeta(3)}{\ln^{2}16}\simeq
1.40733.$
Accordingly, variance of canonical PARK estimator $\hat{v}_{p}$ is
$\mathbb{Var}[\hat{v}_{p}(0)]=\frac{9\,\zeta(3)}{\ln^{2}16}-1\simeq 0.407.$
(18)
As the next step, we calculate variance of canonical bridge estimator
$\hat{v}_{b}$ (17). Sought variance is equal to
$\mathbb{Var}[\hat{v}_{b}]:=\frac{36}{\pi^{4}}~{}\mathbb{E}[s^{4}]-1.$
After substitution here, following from (A.9), relation
$\mathbb{E}[s^{4}]:=\int_{0}^{2}\delta^{4}q_{b}(\delta)d\delta=3\sum_{m=1}^{\infty}\frac{1}{m^{4}}=\frac{\pi^{4}}{30},$
obtain
$\mathbb{Var}[\hat{v}_{b}]=\frac{6}{5}-1=0.2.$ (19)
Comparing equalities (18) and (19), one can see that variance of bridge
estimator approximately twice smaller than variance of PARK estimator.
Recall, variance of bridge estimator does not depend on drift. On the
contrary, variance of PARK estimator essentially depends on the drift. One can
see it in figure 3, where depicted plot of dependence, on canonical drift
$\gamma$, of canonical PARK estimator variance.
Figure 3: Plots of dependence on $\gamma$ of canonical PARK estimator
variance. Straight line is the variance of canonical bridge estimator Figure
4: Plot of relative bias (20) of canonical PARK estimator as function of
canonical drift $\gamma$
Notice else that bias of some estimator is insignificant only if it is much
smaller than rms of corresponding estimator, i.e. is small the relative bias:
$\varrho:=\frac{\mathbb{E}[\hat{v}(\gamma)]-1}{\sqrt{\mathbb{Var}[\hat{v}(\gamma)]}}.$
(20)
Plot of canonical PARK estimator relative bias, as function of canonical drift
$\gamma$ depicted in figure 4.
## 4 Interval estimations on the basis of PARK and bridge estimators
Given at Appendix analytical expressions (A.6), (A.7) and (A.9) for canonical
Brownian motion and canonical bridge random oscillations pdfs allow us to
explore in details probabilistic properties of PARK and bridge canonical
estimators. Let find, at first, pdfs of mentioned canonical estimators random
values. It is well known from Probabilistic Theory that pdf $W_{p}(x;\gamma)$
of canonical PARK estimator is expressed through pdf $q_{x}(\delta;\gamma)$
(A.6) of canonical Brownian motion oscillation by the relation
$W_{p}(x;\gamma)=\sqrt{\frac{\alpha}{4x}}~{}q_{x}\left(\sqrt{\alpha
x};\gamma\right),\qquad\alpha=\ln 16.$ (21)
Similarly, pdf of canonical bridge estimator is equal to
$W_{b}(x)=\sqrt{\frac{\alpha}{4x}}~{}q_{b}\left(\sqrt{\alpha
x}\right),\qquad\alpha=\frac{\pi^{2}}{6}.$ (22)
Here $q_{b}(\delta)$ (A.9) is the pdf of canonical bridge oscillation. Plots
of canonical PARK estimator pdf, for $\gamma=0$, and pdf of canonical bridge
estimator are depicted in figure 5. In figure 6 are comparing pdfs of
canonical PARK estimator, for $\gamma=1$, and pdf of canonical bridge
estimator. It is seen in both figures that pdf of canonical bridge estimator
is better concentrated around its expected value $\mathbb{E}[\hat{v}_{b}]=1$
than canonical PARK estimator pdf.
Knowing estimators pdfs, one can produce interval estimations of possible
volatility values. Consider typical interval estimation: Let $\hat{V}$ is some
volatility estimator, equal to
$\hat{V}=V(T)\cdot\hat{v}.$ (23)
Here $\hat{v}$ is corresponding canonical estimator, while $V(T)$ is the
measured volatility. One needs to find probability
$F(N):=\mathbb{Pr}\left\\{V(T)<N\cdot\hat{V}\right\\}$
that unknown (random) volatility $V(T)$ is not more than $N$ times exceeds
known (measured) volatility estimated value $\hat{V}$. It follows from (23)
that following inequalities are equivalent:
$V(T)<N\cdot\hat{V}\qquad\Leftrightarrow\qquad\hat{v}>1\big{/}N.$
Last means in turn that sought probability $F(N)$ is expressed through pdf of
canonical estimator $\hat{v}$ by the following way:
$F(N)=\mathbb{Pr}\left\\{\hat{v}>1\big{/}N\right\\}=\int_{1/N}^{\infty}W(x)dx.$
(24)
Here $W(x)$ is the pdf of canonical estimator $\hat{v}$.
Figure 5: Plots of canonical PARK and bridge estimators pdfs, clearly
demonstrating “probabilistic preference” of bridge estimator in compare with
PARK one Figure 6: Plots of PARK and bridge canonical estimators pdfs for
$\gamma=1$
Calculations, resting on relations (21), (22), (24) give probability
$F_{b}(2)\simeq 0.918$ that true volatility is less than twice of given bridge
volatility estimator value $\hat{V}_{b}$. It is substantially larger than
analogous probability in the case of PARK estimator: $F_{p}(2,\gamma=0)\simeq
0.813$.
Plots of probabilities $F(N)$ (24) dependence on the level $N$, for PARK
estimator (in the case of zero drift $\mu=0$) and for bridge volatility
estimator are given in figure 7.
Figure 7: Plots of probabilities $F_{p}(N)$ and $F_{b}(N)$ that true
volatility is less than $N$ times exceeds values of PARK and bridge estimators
## 5 Comparative statistics of canonical estimators
Above, we explored in detail statistical properties of two, PARK and bridge
estimators. Here we compare their statistics and statistics of another well-
known volatility estimators: G&K and R&S one. Despite to previous chapters,
where we have used known analytical expressions for pdfs of canonical PARK and
the bridge estimators, below we use predominantly results of numerical
simulations.
Namely, we produce $M\gg 1$ numerical simulations of random sequences
$x_{n}(\gamma):=\gamma\frac{n}{N}+\frac{1}{\sqrt{N}}\sum_{n=1}^{N}\epsilon_{n},\qquad
n=0,1,\dots,N,\qquad x_{0}(\gamma)=0,$ (25)
where $\\{\epsilon_{n}\\}$ are iid Gaussian variables $\sim\mathcal{N}(0,1)$.
Notice that stochastic process $x_{n}(\gamma)$ of discrete argument $n$ rather
accurately approximates, for large $N\gg 1$, paths of canonical Brownian
motion $x(\tau,\gamma)$ (11).
Figure 8: Upper panel: Histogram of $M$ samples of canonical bridge estimator
$\hat{v}_{b}$. Solid line is the plot of canonical bridge estimator’s pdf,
given by analytical expression (22), (A.9). Dashed line is the pdf of
canonical PARK estimator for $\gamma=0$. Lower panel: Histogram of $M$ samples
of canonical G&K estimator $\hat{v}_{g}$ for $\gamma=0$. Solid line is the
plot of the canonical bridge estimator pdf. Dashed line is the canonical PARK
estimator pdf for $\gamma=0$
Knowing $M$ iid sequences $\\{x_{n}(\gamma)\\}$ one can find corresponding iid
samples of pointed out above canonical estimators. Everywhere below we take
number of iid samples $M$ and discretization number $N$ equal to
$N=5\cdot 10^{3},\qquad M=5\cdot 10^{5}.$
Plots in figure 8 demonstrate rather convincingly accuracy of numerical
simulations. In figure 9 are given two hundred samples of canonical G&K and
bridge estimators, ensuring “by naked eye” that canonical bridge estimator is
more efficient than G&K one.
In figure 10 are given, obtained by numerical simulations, plots of canonical
G&K, PARK, R&S and bridge estimators mean values, illustrating bias of G&K and
PARK estimators for nonzero canonical drift $\gamma\neq 0$, and actual absence
of bias for bridge and R&S estimators.
Eventually, in figure 12 are given plots of probabilities that true volatility
$V(T)$ is larger than half of corresponding estimator value and less than
twice of it:
$P_{\Delta}:=\mathbb{Pr}\left\\{\frac{\hat{V}}{2}<V(T)<2\hat{V}\right\\}=\int_{1/2}^{2}W(x)dx.$
(26)
It is seen that for any $\gamma$ mentioned probability is essentially larger
for bridge estimator, than for G&K, R&S and PARK estimators.
## 6 Acknowledgements
We are grateful for scientific and financial help of Higher school of
economics (Russia, Nizhny Novgorod) and Nizhny Novgorod State University
(Russia).
Figure 9: Plots of two hundreds samples of canonical estimators. Up to down
are samples of G&K, R&S, bridge and PARK estimators. It is seen even by “naked
eye” that bridge estimator estimates volatility more accurately than another
mentioned estimators Figure 10: Mean values $\bar{\hat{v}}$ of canonical PARK
($\blacksquare$), G&K ($\blacklozenge$), R&S ($\bigstar$) and bridge
($\blacktriangle$) estimators. Solid lines are theoretical expectations,
borrowing from figure 2 Figure 11: Estimations $\bar{D}$ of variance of PARK
($\blacksquare$), R&S ($\bigstar$), G&K ($\blacklozenge$) and bridge
($\blacktriangle$) canonical estimators. Solid lines are plots of theoretical
variances, borroved from the figure 3. It is seen that for any $\gamma$ bridge
estimator’s variance significantly smaller than variances of another mentioned
estimators
Figure 12: Estimations of probability $P_{\Delta}$ (26) at different $\gamma$
values, for PARK ($\blacksquare$), R&S ($\bigstar$), G&K ($\blacklozenge$) and
bridge ($\blacktriangle$) estimators. Solid lines are results of theoretical
calculations, resting on formula (26)
## References
* [1] Garman, M., and M. J. Klass. 1980. On the Estimation of Security Price Volatilities From Historical Data. Journal of Business 53: 67-78.
* [2] PARK, M. 1980. The extreme value method for estimating the variance of the rate of return. The Journal of Business 53: 61-65.
* [3] Rogers L. C. G., S. E. Satchell. 1991. Estimating variance from high, low and closing prices. The annals of Applied Probability 1: 504-512
* [4] Jeanblanc, M., M. Yor, M. Chesney. 2009. Mathematical Methods for Financial Markets. London: Springer Verlag.
* [5] Cont, R., P. Tankov. 2004. Financial Modelling With Jump Processes. London: CRC Press.
* [6] Saichev A., Ya. Malevergne, D. Sornette. 2010. Theory of Zipf’s Law and Beyond. Heidelberg: Springer Verlag.
* [7] Borodin, A. N., P. Salminen. 2002. Handbook of Brownian Motion – Facts and Formulae (Second Edition). Basel: Birkhäuser Verlag.
* [8] Saichev, A., D. Sornette. 2011. Time-Bridge Estimators of Integrated Variance. arXiv:1108.2611v1 [q-fin.ST] 12 Aug 2011.
## Appendix A Probabilistic properties of high, low and close values
Here are given pdfs of random variables $(h,l,c)$ (13) and variables
$(\xi,\zeta)$ (14), which one need for canonical estimators (12) statistical
analysis. Let begin with random variable $c=x(1,\gamma)$. Obviously, its pdf
is
$f(\chi;\gamma):=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{(\chi-\gamma)^{2}}{2}\right),\qquad\chi\in(-\infty,\infty).$
It is easy to show, additionally, that joint pdf $q_{x}(\eta,\chi;\gamma)$ of
high value $h$ (13) of canonical Brownian motion $x(\tau,\gamma)$ and the
close value $c=x(1,\gamma)$ is equal to
$\begin{array}[]{c}\displaystyle
q_{x}(\eta,\chi;\gamma)=\sqrt{\frac{2}{\pi}}\,(2\eta-\chi)\,e^{2\gamma\eta}\exp\left(-\frac{1}{2}(2\eta-x+\gamma)^{2}\right),\\\\[11.38109pt]
\displaystyle\chi<\eta,\qquad\eta>0.\end{array}$ (A.1)
In turn, pdf of high value $h$ (13)
$q_{x}(\eta;\gamma):=\int_{-\infty}^{h}q_{x}(\eta,\chi;\gamma)d\chi$
given by expression
$q_{x}(\eta;\gamma)=\sqrt{\frac{2}{\pi}}\exp\left(-\frac{(\eta-\gamma)^{2}}{2}\right)-\gamma
e^{2\gamma\eta}\,\text{erfc}\left(\frac{\eta+\gamma}{2}\right),\qquad\eta>0.$
(A.2)
Let write here explicit expression for joint pdf
$q_{x}(\eta,\ell,\chi;\gamma)$ of random variables $(h,l,c)$ (13). Using
formulas, given at the monograph [7] and in the article [8], one might show
that pointed out joint pdf given by:
$\begin{array}[]{c}\displaystyle
q_{x}(\eta,\ell,\chi;\gamma)=f(\chi;\gamma)\,\mathcal{S}(\eta,\ell|\chi),\\\\[8.53581pt]
\displaystyle\chi\in(\ell,\eta),\qquad
h>\chi\mathbb{1}(\chi),\qquad\ell<\chi\mathbb{1}(-\chi).\end{array}$ (A.3)
Here $\mathbb{1}(\chi)$ is the unit step function, equal to unity for $\chi>0$
and zero otherwise. Besides, above there is function
$\begin{array}[]{c}\mathcal{S}(\eta,\ell|\chi):=\\\\[2.84526pt]
\displaystyle\sum_{m=-\infty}^{\infty}m\left[m\mathcal{F}(m(\eta-\ell),\chi)+(1-m)\mathcal{F}(m(\eta-\ell)+\ell,\chi)\right],\\\\[8.53581pt]
\displaystyle\mathcal{F}(\eta,\chi):=\left[(\chi-2\eta)^{2}-1\right]e^{2\eta(\chi-\eta)}.\end{array}$
(A.4)
We need, at exploring statistical properties of canonical G&K estimator, in
joint pdf $q_{x}(\delta,\chi;\gamma)$ of canonical Brownian motion
$x(\tau,\gamma)$ (11) oscillation $d=h-l$ and the close value $c=x(1,\gamma)$.
As it follows from (A.3), (A.4), mentioned pdf is equal to
$\begin{array}[]{c}\displaystyle
q_{x}(\delta,\chi;\gamma)=4f(\chi;\gamma)\sum_{m=-\infty}^{\infty}m\times\\\\[11.38109pt]
\displaystyle\left[m(\delta-|\chi|)[(|\chi|+2m\delta)^{2}-1]-(m+1)(|\chi|+2m\delta)\right]e^{-2m\delta(|\chi|+m\delta)},\\\\[11.38109pt]
\displaystyle\delta>|\chi|,\qquad\chi\in(-\delta,\delta).\end{array}$ (A.5)
After integration above joint pdf over all $\chi$ values obtain pdf
$q_{x}(\delta;\gamma)$ of oscillation $d$:
$\begin{array}[]{c}\displaystyle
q_{x}(\delta;\gamma)=\sum_{m=-\infty}^{\infty}m\bigg{[}\sqrt{\frac{8}{\pi}}\exp\left(-\frac{(1+2m)^{2}\delta^{2}+2\delta\gamma+\gamma^{2}}{2}\right)\times\\\\[11.38109pt]
\displaystyle\bigg{(}2\exp\left(\frac{\delta(\delta+4m\delta+2\gamma)}{2}\right)(2m^{2}\delta^{2}-1-m(2+\gamma^{2}))+\\\\[8.53581pt]
\displaystyle(1+e^{2\delta\gamma})(1+m(2+\gamma^{2}))\bigg{)}-2\gamma\big{(}a(\delta,\gamma,m)-a(\delta,-\gamma,m)\big{)}\bigg{]},\\\\[5.69054pt]
\delta>0.\end{array}$ (A.6)
Here have used auxiliary function
$\begin{array}[]{c}\displaystyle
a(\delta,\gamma,m):=e^{2m\delta\gamma}\left[1+m(3+\gamma(\delta+2m\delta+\gamma))\right]\times\\\\[11.38109pt]
\displaystyle\left[\text{erf}\left(\frac{2m\delta+\gamma}{\sqrt{2}}\right)-\text{erf}\left(\frac{\delta+2m\delta+\gamma}{\sqrt{2}}\right)\right],\qquad\delta>0.\end{array}$
In particular case of zero drift ($\gamma=0$), one get from (A.6) following
expression
$\begin{array}[]{c}\displaystyle q_{x}(\delta)=\\\\[8.53581pt]
\displaystyle\sqrt{\frac{8}{\pi}}\sum_{m=-\infty}^{\infty}\left[(2m-1)^{2}\exp\left(-\frac{(2m-1)^{2}\delta^{2}}{2}\right)-4m^{2}e^{-2m^{2}\delta^{2}}\right],\\\\[11.38109pt]
\delta>0.\end{array}$ (A.7)
All statistical properties of high and low values (14) of canonical bridge
(15) are defined by their two-fold joint pdf $q_{b}(\eta,\ell)$, given by
relation
$\begin{array}[]{c}\displaystyle
q_{b}(\eta,\ell)=\sum_{m=-\infty}^{\infty}m\left[m\mathcal{F}(m(\eta-\ell))+(1-m)\mathcal{F}(m(\eta-\ell)+\ell)\right],\\\\[11.38109pt]
\displaystyle\mathcal{F}(\eta):=4(4\eta^{2}-1)e^{-2\eta^{2}}.\end{array}$
(A.8)
Following from here pdf $q_{b}(\delta)$ of canonical bridge oscillation
$s=\xi-\zeta$ given by equality
$q_{b}(\delta)=8\delta\sum_{m=1}^{\infty}m^{2}(4m^{2}\delta^{2}-3)e^{-2m^{2}\delta^{2}},\qquad\delta>0.$
(A.9)
|
arxiv-papers
| 2012-02-20T13:00:28 |
2024-09-04T02:49:27.562061
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexander Saichev and Svetlana Lapinova",
"submitter": "Alexander Saichev Prof",
"url": "https://arxiv.org/abs/1202.4311"
}
|
1202.4374
|
11institutetext: Dip. di Fisica and ICRA, Sapienza Università di Roma,
Piazzale Aldo Moro 5, I-00185 Rome, Italy, 22institutetext: ICRANet, Piazza
della Repubblica 10, I-65122 Pescara, Italy, 33institutetext: S. N. Bose
National Center for Basic Sciences, Salt Lake, Kolkata - 700098, India,
44institutetext: Indian Center for Space Physics, Garia, Kolkata - 700084,
India, 55institutetext: Universite de Nice Sophia Antipolis, CEDEX 2, Grand
Chateau Parc Valrose, Nice, France.
# A double component in GRB 090618: a proto-black hole and a genuinely long
GRB
L. Izzo 1122 R. Ruffini 112255 A. V. Penacchioni 1155 C. L. Bianco 1122 L.
Caito 1122 S. K. Chakrabarti 3344 Jorge A. Rueda 1122 A. Nandi 44 B.
Patricelli 1122
_Context:_ The joint X-ray and gamma-ray observations of GRB 090618 by a large
number of satellites offer an unprecedented possibility of testing crucial
aspects of theoretical models. In particular, it allows us to test (a) in the
process of gravitational collapse, the formation of an optically thick
$e^{+}e^{-}$ -baryon plasma self-accelerating to Lorentz factors in the range
200 $<\Gamma<$ 3000; (b) its transparency condition with the emission of a
component of $10^{53-54}$ baryons in the TeV region and (c) the collision of
these baryons with the circumburst medium (CBM) clouds, characterized by
dimensions of $10^{15-16}$ cm. In addition, these observations offer the
possibility of testing a new understanding of the thermal and power-law
components in the early phase of this GRB.
_Aims:_ We test the fireshell model of GRBs in one of the closest ($z=0.54$)
and most energetic ($E_{iso}=2.90\times 10^{53}$ ergs) GRBs, namely GRB
090618. It was observed at ideal conditions by several satellites, namely
Fermi, Swift, Konus-WIND, AGILE, RT-2 and Suzaku, as well as from on-ground
optical observatories.
_Methods:_ We analyze the emission from GRB 090618 using several spectral
models, with special attention to the thermal and power-law components. We
determine the fundamental parameters of a canonical GRB within the context of
the fireshell model, including the identification of the total energy of the
$e^{+}e^{-}$ plasma, $E_{tot}^{e^{+}e^{-}}$, the Proper GRB (P-GRB), the
baryon load, the density and structure of the CBM.
_Results:_ We find evidences of the existence of two different episodes in GRB
090618. The first episode lasts $50$ s and is characterized by a spectrum
consisting of thermal component, which evolves between $kT=54$ keV and $kT=12$
keV, and a power law with an average index $\gamma=1.75\pm 0.04$. The second
episode, which lasts for $\sim$ 100s, behaves as a canonical long GRB with a
Lorentz gamma factor at transparency of $\Gamma=495$, a temperature at
transparency of $29.22$ keV and with characteristic size of the surrounding
clouds of $R_{cl}\sim 10^{15-16}$ cm and masses of $\sim 10^{22-24}$ g.
_Conclusions:_ We support the recently proposed two-component nature of GRB
090618, namely, Episode 1 and Episode 2, by using specific theoretical
analysis. We further illustrate that the episode 1 cannot be considered to be
either a GRB or a part of a GRB event, but it appears to be related to the
progenitor of the collapsing bare core leading to the formation of the black
hole which we call a “proto-black hole”. Thus, for the first time, we are
witnessing the process of formation of a black hole from the phases just
preceding the gravitational collapse all the way up to the GRB emission.
###### Key Words.:
Gamma-ray burst: general — Gamma-ray burst: individual: GRB 090618 — Black
hole physics
## 1 Introduction
After the discovery of the GRBs by the Vela satellites (Klebesadel et al.,
1973; Strong & Klebesadel, 1974; Strong et al., 1974; Strong, 1975), the first
systematic analysis on a large sample of GRBs was possible thanks to the
observations of the BATSE instrument on board the Compton Gamma-Ray Observer
(CGRO) satellite (Meegan et al., 1992). The 4BATSE catalog (Meegan, 1997;
Paciesas et al., 1999; Kaneko et al., 2006) consists of 2704 confirmed GRBs,
and it is widely used by the science community as a reference for spectral and
timing analysis on GRBs. One of the outcomes of this early analysis of GRBs
led to the classification of GRBs as a function of their observed time
duration. $T_{90}$ was defined as the time interval over which the 90$\%$ of
the total BATSE background-subtracted counts are observed. The distribution of
the $T_{90}$ duration was bi-modal: the GRBs with $T_{90}$ less than $2$s were
classified as “short” while the ones with $T_{90}$ larger than $2$s were
classified as “long” (Klebesadel, 1992; Dezalay et al., 1992; Kouveliotou et
al., 1993; Tavani, 1998).
After the success of BATSE, a large number of space missions dedicated to the
GRB observations were launched. Particularly significant was the discovery of
an additional prolonged soft X-ray emission by Beppo-SAX (Costa et al., 1997),
following the usual hard X-ray emission observed by BATSE. The Beppo-SAX
observed emission was named as the “afterglow”, while the BATSE one as the
“prompt” radiation. The afterglow allowed to pinpoint more accurately the GRB
position in the sky and permitted the identification of their optical
counterpart by space and ground based telescopes. The measurement of the
cosmological redshift for GRBs became possible and their cosmological nature
was firmly established (van Paradijs et al., 1997).
The Beppo-SAX and related results led to rule out literally hundreds of
theoretical models of GRBs, (see for a review Ruffini, 2001). Among the
handful of surviving models, there was the one by Damour & Ruffini (1975)
based on the mass-energy formula of Black Holes. This model can naturally
explain the energetics up to 1054-55 erg, as requested by the cosmological
nature of GRBs, through the creation of an $e^{+}e^{-}$-plasma by vacuum
polarization processes in the Kerr-Newman geometry (for a recent review see
Ruffini et al., 2010b). This model was proposed a few months after the
presentation of the discovery of GRBs by Strong (Strong, 1975) at the AAAS
meeting in San Francisco.
It soon became clear that, as suggested by Goodman and Paczynski (Goodman,
1986; Paczynski, 1986), the presence of a Lorentz gamma factor larger than 100
could overcome the problem of opacity of the $e^{+}e^{-}$-plasma and justify
the $\gamma$-ray emission of GRBs at cosmological distances (see e.g. Piran,
2005). That the dynamics of an $e^{+}e^{-}$-plasma with a baryon load with
mass $M_{B}$ would naturally lead to Lorentz gamma factor in the range (102\-
103) was demonstrated by Shemi & Piran (1990); Piran et al. (1993); Meszaros
et al. (1993). The general solution for a baryon load
$B=M_{B}c^{2}/E_{tot}^{e^{+}e^{-}}$ between $0$ and $10^{-2}$ was obtained in
Ruffini et al. (2000). The interaction between the accelerated baryons with
the CBM, indicated by Meszaros & Rees (1993), was advocated to explain the
nature of the afterglow (see e.g. Piran, 1999, and references therein).
The unprecedented existence of such large Lorentz gamma factors led to
formulate the Relativistic Space-Time Transformations paradigm for GRBs
(Ruffini et al., 2001b). Such a paradigm made it a necessity to have a global,
instead of a piecewise, description of a GRB phenomenon (Ruffini et al.,
2001b). This global description led to the conclusion that the emission by the
accelerated baryons interacting with the CBM indeed occurs already in the
prompt emission phase in a fully radiative regime. A new interpretation of the
burst structure paradigm was then introduced (Ruffini et al., 2001a): the
existence of a characteristic emission at the transparency of an
$e^{+}e^{-}$-plasma, the Proper-GRB, followed by an extended-afterglow
emission. The relative intensity of these two components is a function of the
baryon load. It was proposed that the case of $B<10^{-5}$ corresponds to the
short GRBs, while the case of $B>3\times 10^{-4}$ corresponds to the long
GRBs.
This different parametrization of the prompt – afterglow versus the one of the
P-GRB – extended-afterglow could have originated years of academic
discussions. However a clear cut observational evidence came from the Swift
satellite, in favour of the second parametrization. The Norris-Bonnell
sources, characterized by an initial short spike-like emission in the hard
X-rays followed by a softer extended emission, had been indicated in the
literature as short bursts. There is a clear evidence that they belong to a
new class of “disguised” short GRBs, (Bernardini et al., 2007; Caito et al.,
2009, 2010; de Barros et al., 2011), where the initial spike is identified as
the P-GRB while the prolonged soft emission occurring from the extended-
afterglow emission in a CBM typically of the galatic halo. These sources have
a baryon load $10^{-4}<B<7\times 10^{-4}$: they are just long GRBs exploding
in a particularly low density CBM of the order of $10^{-3}$ particles/cm3.
This class of sources has given the first evidence of GRBs originating from
binary mergers, strongly supported also from direct optical observations
(Bloom et al., 2006; Fong et al., 2010).
It is interesting that, independent of the development of new missions, the
BATSE data continue to attract full scientific interests, even after the end
of the mission in the 2000. Important inferences, based on the BATSE data, on
the spectra of the early emission of the GRB have been made by Ryde (2004) and
Ryde et al. (2006). They have convincingly demonstrated that the spectral
feature composed by a blackbody and a power-law plays an important role in
selected episodes in the early part of the GRB emission. They have also shown,
in some cases, a power-law variation of the thermal component as a function of
time, following a broken power-law behavior, see Fig. 17.
The arrival of the Fermi and other satellites allowed further progresses in
the understanding of the GRB phenomenon in a much wider energy range. Thanks
to the Gamma-Ray Burst Monitor (GBM) (Meegan et al., 2009) and the Large Area
Telescope (LAT) (Atwood et al., 2009), additional data are obtained in the 8
keV - 40 MeV and 100 MeV - 300 GeV energy range. It has allowed, among others,
this first evidence of a GRB originating from the collapse of a core in the
late evolution of a massive star, what we have called the Proto Black Hole
(Ruffini, 2011; Penacchioni et al., 2012).
In the specific case of GRB 090618, it has been possible to obtain a complete
temporal coverage of the emission in gamma and X-rays, due to the joint
observations by Swift, Fermi, AGILE, RT-2/Coronas-PHOTON, Konus-WIND and
Suzaku-WAM telescopes. A full coverage in the optical bands, up to 100 days
from the burst trigger, has been obtained. This has allowed the determination
of the redshift, $z=0.54$, of the source from spectroscopical identification
of absorption lines (Cenko et al., 2009) and a recent claim of a possible
supernova emission $\sim$ 10 days after the GRB trigger. This GRB lasts for
$\sim$ 150 s in hard X-rays, and it is characterized by four prominent pulses.
In the soft X-rays there are observations up to 30 days from the burst
trigger.
We have pointed out in Ruffini et al. (2010a) that two different episodes are
present in GRB 090618. We have also showed that while the second episode may
fit a canonical GRB, the first episode is not expected to be either a part of
a GRB or an independent GRB (Ruffini et al., 2011).
In the present paper we enter in the merit of the nature of these two
episodes. In particular:
* •
in Section 2, we describe the observations and data reduction and analysis. We
obtain the Fermi GBM (8 keV - 1 MeV and 260 keV - 40 MeV) flux light curves,
shown in Fig. 2, following the standard data reduction procedure, and make a
detailed spectral analysis of the main emission features, using a Band and a
power-law with an exponential high energy cut-off spectral models.
* •
in Section 3, after a discussion about the most quoted GRB model, the
fireball, we recall the main features of the fireshell scenario, focusing on
the reaching of transparency at the end of the initial optically thick phase,
with the emission of the Proper-GRB (P-GRB). In Fig. 3 we give the theoretical
evolution of the Lorentz $\Gamma$ factor as a function of the radius, for
selected values of the baryon load, corresponding to fixed values of the total
energy $E_{tot}^{e^{+}e^{-}}$. The identification of the P-GRB is crucial in
determining the main fireshell parameters, which describe the canonical GRB
emission. The P-GRB emission is indeed characterized by the temperature, the
radius and the Lorentz $\Gamma$ factor at the transparency, which are related
with the $E_{tot}^{e^{+}e^{-}}$ energy and the baryon load, see Fig. 4. We
then recall the theoretical treatment, the simulation of the light curve and
spectrum of the extended-afterglow and, in particular, the determination of
the equations of motion, the role of the EQuiTemporal Surfaces (EQTS) (Bianco
& Ruffini, 2004, 2005a), as well as the ansatz of the spectral energy
distribution in the fireshell comoving frame, (see Patricelli et al., 2011,
and references therein).
The temporal variability of a GRB light curve has been interpreted in some
current models as due to internal shock (Rees & Meszaros, 1994). In the
fireshell model instead such temporal variability is produced by the
interaction of the ultra-relativistic baryons colliding with the
inhomogeneities of the CircumBurst Medium (CBM). This allows to perform a
tomography of the CBM medium around the location of the black hole formation,
see Fig. 10, gaining important information on its structure. These collisions
are described by three parameters: the $n_{CBM}$ average density, the filling
factor $\mathcal{R}$, the clumpiness on scales of 1015-16 cm and average
density contrast $10^{-1}\lesssim\langle\delta n/n\rangle\lesssim 10$. We then
refer also to the explanation of the observed hard-to-soft behavior due to the
drop of the Lorentz $\Gamma$ factor and the curvature effect of the EQTS. We
then recall the determination of the instantaneous spectra and the simulations
of the observed multi-band light curves in the chosen time interval, taking
into account all the thousands of convolutions of comoving spectra over each
EQTS leading to the observed spectrum. We also emphasize how these simulations
have to be performed together and optimized.
* •
in Section 4, we perform a spectral analysis of GRB 090618. We have divided
the total GRB emission in 6 time intervals, see Table 1, each one identifying
a significant feature in the emission process, see also Rao et al. (2011). We
have considered two different spectral models in the data fitting procedure: a
Band model (Band et al., 1993) and one by a blackbody plus a power-law
component, following e.g. Ryde (2004). We find that the first $50$s of
emission are well-fitted by both models, equally the following $9$s, from $50$
to $59$s. The remaining part, from $59$ to $151$ s, is fitted satisfactorily
only by the Band model, see Table 1.
* •
in Section 5, we proceed to the analysis of GRB 090618 in the fireshell
scenario. In Section 5.1, we attempt our first interpretation of GRB 090618
assuming it to be a single GRB. We recall that the blackbody is an expected
feature in the theory of P-GRB. From the spectral analysis of the first $50$
s, we find a spectral distribution consistent with a blackbody plus a power-
law component. We have first attempted a fit of the source identifying these
first $50$s as the P-GRB, see Fig. 6. We confirm the conclusion reached in
Ruffini et al. (2010a) that this interpretation is not sustainable for three
different reasons, based on: 1) the energetics of the source, 2) the time
duration and 3) the theoretical expected temperature for the P-GRB. We then
proceed, in the sub-section 5.2, to an interpretation of GRB 090618 as a multi
component system, following the procedure outlined in Ruffini et al. (2011),
in which we outline the possibility of the second episode between $50$ and
$151$s to be an independent GRB.
We identify the P-GRB of this second episode, as the first $4$s of emission.
We find that the spectrum in this initial emission can be fitted by a
blackbody plus a power-law component, see Fig. 8. Since this extra power-law
component can be due to the early onset of the extended-afterglow, we take it
into account to perform a fireshell simulation which is shown in Fig. 8, with
an energy $E_{tot}^{e^{+}e^{-}}=2.49\times 10^{53}$ erg and a baryon load
$B=1.98\pm 0.15\times 10^{-3}$. In Figs. 10,11,12, we report the results of
our simulations, summarized in Table 3. We notice, in particular, the presence
of a strong time lag in this GRB. A detailed analysis, see Rao et al. (2011),
about the time lags in the mean energy ranges of $35$ keV, $68$ keV and $125$
keV, reports a quite large lag , $\sim$ 7 s, in the first $50$s of the
emission which is unusual for GRBs, while in the following emission, from $51$
to $151$ s, the observed lags are quite normal, $\sim 1$ s.
* •
in Section 6, we perform a spectral analysis of the first $50$s, where we find
a strong spectral variation with time, as reported in Table 5 and in Figs.
16,17, with a chacteristic power-law time variation similar to the ones
identified by Ryde & Pe’er (2009) in a sample of 49 BATSE GRBs.
* •
in Section 7, we estimate the variability of the radius emitter, Fig. 18, and
proceed to an estimate of the early expansion velocity. We interpret this data
as originating in the expansion process occurring previous to the collapse of
the core of a massive star to a black hole, see e.g. Arnett & Meakin (2011):
this early $50$s of the emission are then defined as the proto-black hole
phenomenon.
* •
In Section 8, we proceed to the conclusions.
## 2 Observations
On 18th of June 2009, the Burst Alert Telescope (BAT) on board the Swift
satellite (Gehrels et al., 2009) triggered on GRB 090618 (Schady et al.,
2009). After 120 s the X-Ray Telescope (XRT) (Burrows et al., 2005) and the
UltraViolet Optical Telescope (UVOT) (Roming et al., 2005) on board the same
satellite, started the observations of the afterglow of GRB 090618. UVOT found
a very bright optical counterpart, with a white filter magnitude of 14.27
$\pm$ 0.01 (Schady, 2009) not corrected for the extinction, at the coordinates
RA(J2000) = 19:35:58.69 = 293.99456, DEC(J2000) = +78:21:24.3 = 78.35676. The
BAT light curve shows a multi peak structure, whose total estimated duration
is of $\sim$ 320 s, with the T90 duration in the (15-350) keV range was of 113
s (Baumgartner et al., 2009). The first 50 s of the light curve presents a
smooth decay trend, followed by a spiky emission, with three prominent peaks
at 62, 80 and 112 seconds after the trigger time, respectively, and each have
the typical appearance of the FRED pulse (see e.g. Fishman et al. (1994)), see
Fig. 2. The time integrated spectrum, (t0 \- 4.4, t0 \+ 213.6) s in the
(15-150)keV range, was found to be in agreement with a power-law spectral
model with an exponential cutoff, whose photon index was $\gamma$ = 1.42 $\pm$
0.08 and a cut-off energy $E_{peak}$ = 134 $\pm$ 19 keV (Sakamoto et al.,
2009). The XRT observations started 125 s after the BAT trigger time and
lasted $\sim$ 25.6 ks (Beardmore & Schady, 2009) and reported an initially
bright uncatalogued source, identified as the afterglow of GRB 090618. Its
early decay was very steep, ending at 310 s after the trigger time, when it
starts a shallower phase, the plateau. Then the light curve breaks to a more
steep last phase.
GRB 090618 was observed also by the Gamma-ray Burst Monitor (GBM) on board the
Fermi satellite (Meegan et al., 2009). From a first analysis, the time-
integrated spectrum, ($t_{0}$, $t_{0}$ \+ 140) s in the (8-1000)keV range, was
fitted by a Band (Band et al., 1993) spectral model, with a peak energy
$E_{peak}$ = 155.5 keV, $\alpha$ = $-1.26$ and $\beta$ = $-2.50$ (McBreen,
2009), but with strong spectral variations within the considered time
interval.
It is appropriate to compare and contrast the considerations of the time-
integrated spectral analysis, often adopted in the current literature of GRBs,
with the information from the time-resolved spectral analysis, as presented
e.g. in this article. For a traditional astrophysical source, steady during
the observation time, the time-integrated and time-resolved spectral analysis
usually coincide. In the case of GRBs, although the duration is only a few
seconds, each instantaneous observation corresponds to a very different
physical process and the two approaches have an extremely different physical
and astrophysical content.
The redshift of the source is $z=0.54$ and it was determined thanks to the
identification of the MgII, Mg I and FeII absorption lines, using the KAST
spectrograph mounted at the 3-m Shane telescope at the Lick observatory (Cenko
et al., 2009). Given the redshift, and the distance of the source, we computed
the emitted isotropic energy in the 8 - 10000 keV energy range, using the
Schaefer formula (Schaefer, 2007): using the fluence in the (8-1000 keV) as
observed by Fermi-GBM, Sobs = 2.7 $\times$ 10-4 (McBreen, 2009), and the
$\Lambda$CDM cosmological standard model $H_{0}$ = 70 km/s/Mpc, $\Omega_{m}$ =
0.27, $\Omega_{\Lambda}$ = 0.73, we obtain for the isotropic energy emitted
the value of Eiso = 2.90 $\times$ 1053 erg.
This GRB was observed also by Konus-WIND (Golenetskii et al., 2009), Suzaku-
WAM (Kono et al., 2009) and by the AGILE satellite (Longo et al., 2009), which
detected emission in the (18-60) keV and in the MCAL instrument, operating at
energies greater than 350 keV, but it did not observe high energy photons
above 30 MeV. GRB 090618 was the first GRB observed by the Indian payloads
RT-2 on board Russian Satellite CORONAS-PHOTON (Kotov et al., 2008; Nandi et
al., 2009; Rao et al., 2011). Two detectors, namely, RT-2/S and RT-2/G consist
of NaI(Tl)/CsI(Na) scintillators in phoswich assembly viewed by a
photomultiplier tube (PMT). RT-2/S has a viewing angle of $4^{\circ}\times
4^{\circ}$ and covers an energy range of 15 keV to 1 MeV, whereas RT-2/G has
an Al filter which sets the lower energy to $\sim 20$keV. The Mission was
launched from Plesetsk Cosmodrom, Russia on January 30, 2009. During the event
RT-2 payload was in the SHADOW mode (away from the Sun) during 08:16:10.207 UT
and ended at 08:37:35.465 UT and the GRB 090618 was detected at $77^{\circ}$
off-axis angle. During this period, the spectra was accumulated in every 100s
while the eight channel count rates for each detector are accumulated every
second. The entire episode was observed for a duration of more than 200
seconds. A closer examination of the data in the accumulated Channels 1:15-102
keV, 2:95-250 keV and 3:250-1000 keV indicates that the most significant
counts is in Channel 2 with a clear evidence of the followings: (a) The
emission in the first 50 s is prominent and broader in the lower channels, see
Fig. 1, (b) After the first 50 s, there is an evidence of a precursor of about
6 seconds duration before the main pulse (c) a break up into two peaks of the
main pulse at intermediate energies (35-200 keV) while at higher energies
(250-1000 keV) only the first peak of the main pulse survives, see Rao et al.
(2011) and also Fig. 2 in this manuscript.
Thanks to the complete data coverage of the optical afterglow of GRB 090618,
the possible presence of a supernova underlying the emission of the GRB 090618
optical afterglow (Cano et al., 2011) was reported. The evidence of a
supernova emission came from the presence of several bumps in the light curve
and by the change in $R_{c}$ \- $i$ color index over time: in the early
phases, the blue color is dominant, typical of the GRB afterglow, but then the
color index increases, suggesting a presence of a core-collapse SN. At late
times, the contribution from the host galaxy was dominant.
Figure 1: RT2 light curves of GRB 090618.
### 2.1 Data Analysis
We consider the BAT and XRT data of the Swift satellite, together with the
Fermi-GBM and RT2 data of the Coronas-PHOTON satellite. The data reduction was
done using the Heasoft v6.10 packages111http://heasarc.gsfc.nasa.gov/lheasoft/
for BAT and XRT, and the Fermi-Science tools for GBM.
We obtained the BAT light curve and spectra using the standard headas
procedure. After the data download from the gsfc
website222ftp://legacy.gsfc.nasa.gov/swift/data/obs/, we made a detector
quality map and corrected the event data for the known errors of the detector
and the hot pixels. We subtracted the background from the data, corrected for
the improved position, using the tool batmaskwtevt and obtained the 1-s binned
light curves and spectra in the main BAT energy band 15 - 150 keV and its
subranges, using the tool batbinevt. After the systematic corrections to the
spectrum, we created the response matrices and obtained the final spectra.
For the XRT data, we obtained a total dataset using the standard pipeline,
while for a time-resolved analysis we considered the on-line recipe, which is
well described in literature, see Evans et al. (2007, 2009). The GBM
data333ftp://legacy.gsfc.nasa.gov/fermi/data/gbm/, in particular the fourth
NaI detector in the (8 - 440 keV) and the b0 BGO detector (260 keV - 40 MeV),
were analyzed using the gtbindef tool to obtain a GTI file for the energy
distribution and the gtbin for the light curves and final spectra. In order to
obtain an energy flux lightcurve, we made a time resolved spectral analysis
dividing the count lightcurve in six time intervals, each of them
corresponding to a particular pulse, as described in the work of Rao et al.
(2011). All the time resolved spectra were fitted using the XSPEC data
analysis software (Arnaud, 1996) version 12.6.0q, included in the Heasoft data
package, and considering for each spectrum a classical Band spectral model
(Band et al., 1993) and a power-law model with an exponential energy cut-off,
folded through the detector response matrix. After the subtraction of the
background, we fit the spectrum by minimizing the $\chi^{2}$ between the
spectral models described above and the observed data, obtaining the best-fit
spectral parameters and the respective model normalization. In Table 1 we give
the results of our spectral analysis. The time reported in the first column
corresponds to the time after the GBM trigger time ttrig = 267006508 s, where
the $\beta$ parameter was not constrained, we used its averaged value, as
delineated in Guetta et al. (2011) $\beta$ = -2.3 $\pm$ 0.10. We have
considered the chi-square statistic for testing our data fitting procedure.
The reduced chi-square $\tilde{\chi}^{2}=\chi^{2}/N$, where $N$ is the number
of degrees of freedom (dof) which is $N=82$ for the NaI dataset and $N=121$
for the BGO one.
For the last pulse of the second episode, the Band model is not very precise
($\tilde{\chi}^{2}$ = 2.24), but a slightly better approximation is given by
the power-law with an exponential cut-off, whose fit results are shown for the
same intervals in the last two columns. From these values, we build the flux
light curves for both the detectors, which are shown in fig. 2.
---
Figure 2: Fermi-GBM flux light curve of GRB 090618 referring to the NaI (8-440 keV, _upper panel_) and BGO (260 keV - 40 MeV, _lower panel_) detectors. Table 1: Time-resolved spectral analysis of GRB 090618. We have considered six time intervals, each one corresponding to a particular emission feature in the light curve. We fit the GBM (8 keV - 10 MeV) observed emission with a Band model (Band et al., 1993) and a power-law function with an exponential cut-off. In the columns 2-4 are listed the Band low energy index $\alpha$, the high-energy $\beta$ and the break energy $E_{0}^{BAND}$, with the reduced chi-square value in the 6th column. In the last three columns are listed the power-law index $\gamma$, the cut-off energy $E_{0}^{cut}$ and the reduced chi-square value respectively, as obtained from the spectral fit with the cut-off power-law spectral function. Time Interval | $\alpha$ | $\beta$ | $E_{0}^{BAND}$ (keV) | $\tilde{\chi}^{2}_{BAND}$ | $\gamma$ | $E_{0}^{cut}$ (keV) | $\tilde{\chi}^{2}_{cut}$
---|---|---|---|---|---|---|---
0 - 50 | -0.77${}^{+0.38}_{-0.28}$ | -2.33${}^{+0.33}_{-0.28}$ | 128.12${}^{+109.4}_{-56.2}$ | 1.11 | 0.91${}^{+0.18}_{-0.21}$ | 180.9${}^{+93.1}_{-54.2}$ | 1.13
50 - 57 | -0.93${}^{+0.48}_{-0.37}$ | -2.30 $\pm$ 0.10 | 104.98${}^{+142.3}_{-51.7}$ | 1.22 | 1.11${}^{+0.25}_{-0.30}$ | 168.3${}^{+158.6}_{-70.2}$ | 1.22
57 - 68 | -0.93${}^{+0.09}_{-0.08}$ | -2.43${}^{+0.21}_{-0.67}$ | 264.0${}^{+75.8}_{-54.4}$ | 1.85 | 1.01${}^{+0.06}_{-0.06}$ | 340.5${}^{+56.0}_{-45.4}$ | 1.93
68 - 76 | -1.05${}^{+0.08}_{-0.07}$ | -2.49${}^{+0.21}_{-0.49}$ | 243.9${}^{+57.1}_{-53.0}$ | 1.88 | 1.12${}^{+0.04}_{-0.04}$ | 311.0${}^{+38.6}_{-32.9}$ | 1.90
76 - 103 | -1.06${}^{+0.08}_{-0.08}$ | -2.65${}^{+0.19}_{-0.34}$ | 125.7${}^{+23.27}_{-19.26}$ | 1.23 | 1.15${}^{+0.06}_{-0.06}$ | 157.7${}^{+22.2}_{-18.6}$ | 1.39
103 - 150 | -1.50${}^{+0.20}_{-0.18}$ | -2.30 $\pm$ 0.10 | 101.1${}^{+58.3}_{-30.5}$ | 1.07 | 1.50${}^{+0.18}_{-0.20}$ | 102.8${}^{+56.8}_{-30.4}$ | 1.06
We turn now to the XRT which started to observe GRB 090618 $\sim$ 120 s after
the BAT trigger. Its early data show a continued activity of the prompt
emission, fading away $\sim$ 200 s after the BAT trigger time. Then the light
curve is well approximated with a power-law decay. In view of the lack of soft
X-ray data before the onset of the XRT, we cannot exclude a previous pulse in
the X-ray light curve emission of GRB 090618. The following shallow and late
decay phases, well-known in literature (Sari et al., 1999; Nousek et al.,
2006), will not be analyzed in this paper since we focus in the first 200 s of
the GRB emission.
## 3 A brief review of the fireshell and alternative models
### 3.1 The GRB prompt emission in the fireball scenario
A variety of models have been developed to theoretically explain the
observational properties of GRBs. One of the most quoted is the fireball model
(see for a review Piran (2005)). The model was first proposed by Cavallo &
Rees (1978), Goodman (1986) and Paczynski (1986), who have shown that the
sudden release of a large quantity of energy in a compact region can lead to
an optically thick photon-lepton plasma and to the production of $e^{+}e^{-}$
pairs. The total annihilation of the $e^{+}e^{-}$ plasma was assumed, leading
to a vast release of energy pushing on the CBM: the “fireball”.
An alternative approach, originating in the gravitational collapse to a black
hole, is the fireshell model (see for a review Ruffini et al. (2010b) and
(Ruffini, 2011)). There the GRBs originate from an optically thick
electron–positron plasma in thermal equilibrium, having a total energy of
$E_{tot}^{e^{\pm}}$. Such plasma is initially confined between the radius of a
black hole $r_{h}$ and the dyadosphere radius
$r_{ds}=r_{h}\left[2\alpha\frac{E_{tot}^{e^{+}e^{-}}}{m_{e}c^{2}}\left(\frac{\hbar/m_{e}c}{r_{h}}\right)^{3}\right]^{1/4},$
(1)
where, $\alpha$ is the usual fine structure constant, $\hbar$ and $c$ the
Planck constant and the speed of light, and $m_{e}$ the mass of the electron.
The lower limit of $E_{tot}^{e^{\pm}}$ coincides with $E_{iso}$. The condition
of thermal equilibrium assumed in this model as shown by Aksenov et al.
(2007), differentiates this approach from the alternative ones (e.g. the one
by Cavallo & Rees, 1978), see Section 3.2.
In the fireball model, the prompt emission, including the sharp luminosity
variations (Ramirez-Ruiz & Fenimore, 2000) are due to the prolonged and
variable activity of the “inner engine” (Rees & Meszaros, 1994; Piran, 2005).
The conversion of the fireball energy to radiation originates by shocks,
either internal (when faster moving matter takes over a slower moving shell,
see Rees & Meszaros (1994)) or external (when the moving matter is slowed down
by the external medium surrounding the burst, see Rees & Meszaros (1992)).
Much attention has been given to the Synchrotron emission from relativistic
electrons, possibly accompanied by SSC emission to explain the observed GRB
spectrum. These processes were found to be consistent with the observational
data of many GRBs (Tavani, 1996; Frontera et al., 2000). However, several
limitations have been evidenced in relation with the low energy spectral
slopes of time-integrated spectra (Crider et al., 1997; Preece et al., 2002;
Ghirlanda et al., 2002, 2003; Daigne et al., 2009) and time-resolved spectra
(Ghirlanda et al., 2003). Additional limitations on SSC have also been pointed
out by Kumar & McMahon (2008a) and Piran et al. (2009).
The latest phases of the afterglow are described in the fireball model by
assuming an equation of motion given by the Blandford-McKee self-similar
power-law solution (Blandford & McKee, 1976). The maximum Lorentz factor of
the fireball is estimated from the temporal occurrence of the peak of the
optical emission, which is identified with the peak of the forward external
shock emission (Molinari et al., 2007; Rykoff et al., 2009) in the thin shell
approximation (Sari & Piran, 1999). There have been developed partly
alternative and/or complementary scenarios to the fireball model, e.g. the
ones based on: quasi-thermal Comptonization (Ghisellini & Celotti, 1999),
Compton drag emission (Zdziarski et al., 1991; Shemi, 1994), Synchrotron
emission from a decaying magnetic field (Pe’er & Zhang, 2006), jitter
radiation (Medvedev, 2000), Compton scattering of synchrotron self absorbed
photons (Panaitescu & Mészáros, 2000; Stern & Poutanen, 2004), photospheric
emission (Eichler & Levinson, 2000; Mészáros & Rees, 2000; Mészáros, 2002;
Daigne & Mochkovitch, 2002; Giannios, 2006; Ryde & Pe’er, 2009; Lazzati &
Begelman, 2010). In particular, Ryde & Pe’er (2009) pointed out that the
photospheric emission overcomes some of the difficulties of pure non-thermal
emission models.
### 3.2 The fireshell scenario
In the fireshell model, the rate equation for the $e^{+}e^{-}$ pairs and its
dynamics have been given by Ruffini et al. (2000) (the pair-electromagnetic
pulse or PEM pulse for short). This plasma engulfs the baryonic material left
over in the process of gravitational collapse having mass $M_{B}$, still
keeping thermal equilibrium between electrons, positrons and baryons. The
baryon load is measured by the dimensionless parameter
$B=M_{B}c^{2}/E_{tot}^{e^{+}e^{-}}$. It was shown (Ruffini, 1999) that no
relativistic expansion of the plasma can be found for $B>10^{-2}$. The
fireshell is still optically thick and self-accelerates to ultrarelativistic
velocities (the pair-electromagnetic-baryonic pulse or PEMB pulse for short,
Ruffini, 1999). Then the fireshell becomes transparent and the Proper - GRB
(P-GRB) is emitted (Ruffini et al., 2001a). The final Lorentz gamma factor at
transparency can vary in a vast range between $10^{2}$ and $10^{3}$ as a
function of $E_{tot}^{e^{+}e^{-}}$ and $B$, see Fig. 3. For the final
determination it is necessary to integrate explicitly the rate equation of the
$e^{+}e^{-}$ annihilation process and evaluate, for a given black hole mass
and a given $e^{+}e^{-}$ plasma radius, the reaching of the transparency
condition Ruffini et al. (2000), see Fig. 4.
The fireshell scenario does not require any prolonged activity of the inner
engine. After transparency, the remaining accelerated baryonic matter still
expands ballistically and starts to slow down by the collisions with the CBM,
having average density $n_{cbm}$. In the standard fireball scenario (Meszaros,
2006), the spiky light curve is assumed to be caused by internal shocks. In
the fireshell model the entire extended-afterglow emission is assumed to
originate from an expanding thin shell enforcing energy and momentum
conservation in the collision with the CBM. The condition of a fully radiative
regime is assumed (Ruffini et al., 2001a). This, in turn, allows to estimate
the characteristic inhomogeneities of the CBM, as well as its average value.
It is appropriate to recall a further difference between our treatment and the
ones in the current literature. The complete analytic solution of the
equations of motion of the baryonic shell has been developed (Bianco &
Ruffini, 2004, 2005b), while in the current literature usually the Blandford -
McKee (Blandford & McKee, 1976) self-similar solution has been uncritically
adopted (e.g. Meszaros et al., 1993; Sari, 1997, 1998; Waxman, 1997; Rees &
Meszaros, 1998; Granot et al., 1999; Panaitescu & Meszaros, 1998; Gruzinov &
Waxman, 1999; van Paradijs et al., 2000; Mészáros, 2002). The analogies and
differences between the two approaches have been explicitly pointed out in
Bianco & Ruffini (2005a).
From this general approach, a canonical GRB bolometric light curve composed of
two different parts is defined: the P-GRB and the extended-afterglow. The
relative energetics of these two components, the observed temporal separation
between the corresponding peaks, is a function of the above three parameters
$E_{tot}^{e^{+}e^{-}}$, $B$, and the average value of the $n_{cbm}$; the first
two parameters are inherent to the accelerator characterizing the GRB, i.e.,
the optically thick phase, while the third one is inherent to the GRB
surrounding environment which gives rise to the extended-afterglow. If one
goes to the observational properties of this model of a relativistic expanding
shell, a crucial concept has been the introduction of the EQTS. In this topic,
also, our model differs from the ones in the literature for deriving the
analytic expression of the EQTS from the analytic solutions of the equations
of motion (Bianco & Ruffini, 2005a).
In this paper, we assume $E_{tot}^{e^{+}e^{-}}=E_{iso}$. This assumption is
based on the very accurate information we have on the luminosity and the
spectral properties of the source. In other GRBs, we have assumed
$E_{tot}^{e^{+}e^{-}}>E_{iso}$ to take into account the observational
limitations, due to detector thresholds, distance effects and lack of data.
### 3.3 The emission of the P-GRB
The lower limit of $E_{tot}^{e^{+}e^{-}}$ is given by the observed isotropic
energy emitted in the GRB, $E_{iso}$. The identification of the energy of the
afterglow and of the P-GRB determines the baryon load $B$ and, from these, it
is possible to determine, see Fig. 4: the value of the Lorentz $\Gamma$ factor
at transparency, the observed temperature as well as the temperature in the
comoving frame and the laboratory radius at transparency. We can determine
indeed from the spectral analysis of the P-GRB candidate, the temperature
$kT_{obs}$ and the energy emitted in the transparency $E_{PGRB}$. The relation
between these parameters can not be expressed by an analytical formulation:
they can be only obtained by a numerical integration of the entire fireshell
equations of motion. In practice we need to perform a trial and error
procedure to find the set of values which fit the observations.
Figure 3: The evolution of the Lorentz $\Gamma$ factor until the transparency
emission, for a GRB of a fixed $E_{tot}^{e^{+}e^{-}}$ = 1.22 $\times$ 1055
(upper panel),and $E_{tot}^{e^{+}e^{-}}$ = 1.44 $\times$ 1049, for different
values of the baryon load $B$. This computation refers to a mass of the black
hole of 10 M☉ and a $\tau$ =
$\int_{R}dr(n_{e^{\pm}}+n_{e^{-}}^{b})\sigma_{T}=0.67$, where $\sigma_{T}$ is
the Thomson cross-section and the integration is over the thickness of the
fireshell (Ruffini, 1999).
(a)
(b)
(c)
(d)
Figure 4: The fireshell temperature in the comoving and observer frame and the
laboratory radius at the transparency emission (panels (a) and (b)), the
Lorentz $\Gamma$ factor at the transparency (panel (c)) and the energy
radiated in the P-GRB and in the afterglow in units of $E_{tot}^{e^{+}e^{-}}$
(panel (d)) as a function of the baryon load $B$, for 4 different values of
$E_{tot}^{e^{+}e^{-}}$.
As we are going to see in the case of GRB 090618, the direct measure of the
temperature of the thermal component at the transparency offers a very
important new information in the determination of the GRB parameters. In the
emission of the P-GRB two different phases are present: one corresponding to
the emission of the photons when the transparency is reached, and the second
is the early interaction of the ultra-relativistic protons and electrons with
the CBM. A spectral energy distribution with a thermal component and a non-
thermal one should be expected to occur.
### 3.4 The extended-afterglow
The majority of works in the current literature has addressed the analysis of
the afterglow emission as due to various combinations of Synchrotron and
Inverse Compton processes, see e.g. Piran (2005). It appears, however, that
this description is not fully satisfactory (see e.g. Ghirlanda et al., 2003;
Kumar & McMahon, 2008b; Piran et al., 2009).
We have adopted in the fireshell model a pragmatic approach by making the full
use of the knowledge of the equations of motion, of the EQTS formulations
(Bianco & Ruffini, 2005b) as well as of the correct relativistic
transformations between the comoving frame of the fireshell and the observer
frame. These equations, that relate the four time variables, are necessary for
the interpretation of the GRB data. They are: a) the comoving time, b) the
laboratory time, c) the arrival time, and d) the arrival time at the detector
corrected by the cosmological effects. This is the content of the Relative
Space-Time Transformations paradigm, essential for the interpretation of GRBs
data (Ruffini et al., 2001b). Such a paradigm made it a necessity to have a
global, instead of a piecewise, description of a GRB phenomenon (Ruffini et
al., 2001b). This global description led to a new interpretation of the burst
structure paradigm (Ruffini et al., 2001a). As recalled in the Introduction, a
new conclusion, arising from the burst structure paradigm, has been that the
emission by the accelerated baryons interacting with the CBM is indeed
occurring already in the prompt emission phase, just after the P-GRB emission.
This is the extended-afterglow emission, which presents in its “light curve” a
rising part, a peak, and a decaying tail. Following this paradigm, the prompt
emission phase is therefore composed by both the P-GRB emission and the peak
of the extended-afterglow.
To evaluate the extended-afterglow spectral properties, we have adopted an
ansatz on the spectral properties of the emission in the collisions between
the baryons and the CBM in the comoving frame. We have then evaluate all the
observational properties in the observer frame by integrating on the EQTS. The
initial ansatz of thermal spectrum (Ruffini et al., 2001a), has been recently
modified to
$\frac{dN_{\gamma}}{dVd\epsilon}=\left(\frac{8\pi}{h^{3}c^{3}}\right)\left(\frac{\epsilon}{k_{B}T}\right)^{\alpha}\frac{\epsilon^{2}}{exp\left(\frac{\epsilon}{k_{B}T}\right)-1},$
(2)
where $\alpha$ is a phenomenological parameter defined in the comoving frame
of the fireshell (Patricelli et al., 2011), determined by the optimization of
the simulation of the observed data. It is well known that in the
ultrarelativistic collision of protons and electrons with the CBM, collective
processes of ultrarelativistic plasma physics are expected, not yet fully
explored and understood (e.g. Weibel instability, see Medvedev & Loeb (1999)).
Promising results along this line have been already obtained by Spitkovsky
(2008) and Medvedev & Spitkovsky (2009), and may lead to the understanding of
the physycal origin of the $\alpha$ parameter in Eq. 2.
In order to take into due account the filamentary, clumpy and porous structure
of the CBM, we have introduced the additional parameter $\mathcal{R}$, which
describes the fireshell surface filling factor. It is defined as the ratio
between the effective emitting area of the fireshell $A_{eff}$ and its total
visible area $A_{vis}$ (Ruffini et al., 2002, 2005).
One of the main features of the GRB afterglow has been the observation of hard
to soft spectral variation, which is generally absent in the first spike-like
emission, which we have identified as the P-GRB, Bernardini et al. (2007);
Caito et al. (2009, 2010); de Barros et al. (2011). An explanation of the
hard-to-soft spectral variation has been advanced on the ground of two
different contributions: the curvature effect and the intrinsic spectral
evolution. In particular, in the work of Peng et al. (2011) the authors use
the model developed in Qin (2002) for the spectral lag analysis, taking into
account an intrinsic Band model for the GRBs and a Gaussian profile for the
GRB pulses, in order to take into account the angular effects, and they find
that both causes provide a very good explanation for the observed time lags.
Within the fireshell model we can indeed explain a hard-to-soft spectral
variation very naturally, in the extended-afterglow emission. Since the
Lorentz $\Gamma$ factor decreases with time, the observed effective
temperature of the fireshell will drop as the emission goes on, so the peak of
the emission will occur at lower energies. This effect is amplified by the
presence of the curvature effect, which has origin in the EQTS concept. Both
these observed features are considered as the responsible for the time lag
observed in GRBs.
### 3.5 The simulation of a GRB light curve and spectra of the extended-
afterglow
The simulation of a GRB light curve and the respective spectrum requires also
the determination of the filling factor $\mathcal{R}$ and of the CBM density
$n_{CBM}$. These extra parameters are extrinsic and they are just functions of
the radial coordinate from the source. The parameter $\mathcal{R}$, in
particular, determines the effective temperature in the comoving frame and the
corresponding peak energy of the spectrum, while $n_{cbm}$ determines the
temporal behavior of the light curve. It is found that the CBM is typically
formed of “clumps” of width $\sim 10^{15-16}$ cm and average density contrast
$10^{-1}\lesssim\langle\delta n/n\rangle\lesssim 10$ centered on the value of
4 $particles/cm^{3}$, see Fig. 10, and clumps of masses $M_{clump}\approx
10^{22-24}$ g. Particularly important is the determination of the average
value of $n_{cbm}$. Values of the order of $0.1$-$10$ particles/cm3 have been
found for GRBs exploding inside star forming region galaxies, while values of
the order of $10^{-3}$ particles/cm3 have been found for GRBs exploding in
galactic halos (Bernardini et al., 2007; Caito et al., 2009; de Barros et al.,
2011). The presence of such a clumpy medium, already predicted in pioneering
works of Fermi in the theoretical study of interstellar matter in our galaxy
(Fermi, 1949, 1954), is by now well-established both from the GRB observations
and by additional astrophysical observations, see e.g. the circum-burst medium
observed in novae (Shara et al., 1997), or by theoretical considerations on
supergiant, massive stars, clumpy wind (Ducci et al., 2009). Interesting are
the considerations by Arnett and Meakin (Arnett & Meakin, 2011), who have
shown how realistic 2D simulations of the late evolution of a core collapse
show processes of violent emission of clouds: there the 2D simulations differ
from the one in 1D, which show a much more regular and wind behavior around
the collapsing core. Consequently, attention should be given also to
instabilities prior to the latest phases of the evolution of the core,
possibly giving origin to the cloud pattern observed in the CBM of GRB
phenomenon (D.Arnett private communication).
The determination of the $\mathcal{R}$ and $n_{CBM}$ parameters depends
essentially on the reproduction of the shape of the extended-afterglow and of
the respective spectral emission, in a fixed energy range. Clearly, the
simulation of a source within the fireshell model is much more complex than
simply fitting the $N(E)$ spectrum with phenomenological analytic formulas for
a finite temporal range of the data. It is a consistent picture, which has to
find the best value for the parameters of the source, the P-GRB (Ruffini et
al., 2001a), its spectrum, its temporal structure, as well as its energetics.
For each spike in the light curve are computed the parameters of the
corresponding CBM clumps, taking into account all the thousands of
convolutions of comoving spectra over each EQTS leading to the observed
spectrum (Bianco & Ruffini, 2005b, a). It is clear that, since the EQTS
encompass emission processes occurring at different comoving times weighted by
their Lorentz and Doppler factors, the “fitting” of a single spike is not only
a function of the properties of the specific CBM clump but of the entire
previous history of the source. Any mistake at any step of the simulation
process affects the entire evolution that follows and, conversely, at any step
a fit must be made consistently with all the previous history: due to the non-
linearity of the system and to the EQTS, any change in the simulation produces
observable effects up to a much later time. This brings to an extremely
complex procedure by trial and error in the data simulation, in which the
variation of the parameters defining the source are further and further
narrowed down, reaching very quickly the uniqueness. Of course, we cannot
expect the latest parts of the simulation to be very accurate, since some of
the basic hypothesis on the equations of motion, and possible fragmentation of
the shell, can affect the procedure.
In particular, the theoretical photon number spectrum to be compared with the
observational data is obtained by an averaging procedure of instantaneous
spectra. In turn, each instantaneous spectrum is linked to the simulation of
the observed multiband light curves in the chosen time interval. Therefore,
both the simulation of the spectrum and of the observed multiband light curves
have to be performed together and simultaneously optimized.
## 4 Spectral analysis of GRB 090618
We proceed now to the detailed spectral analysis of GRB 090618. We divide the
emission in six time intervals, shown in Table 1, each one identifying a
significant feature in the emission process. We then fit for each time
interval the spectra by a Band model as well as by a blackbody with an extra
power-law component, following Ryde (2004). In particular, we are interested
in the estimation of the temperature $kT$ and the observed energy flux
$\phi_{obs}$ of the blackbody component. The specific intensity of emission of
a thermal spectrum at energy $E$ in energy range $dE$ into solid angle
$\Delta\Omega$ is
$I(E)dE=\frac{2}{h^{3}c^{2}}\frac{E^{3}}{\exp(E/kT)-1}\Delta\Omega dE.$ (3)
The source of radius $R$ is seen within a solid angle $\Delta\Omega=\pi
R^{2}/D^{2}$, and its full luminosity is $L=4\pi R^{2}\sigma T^{4}$. What we
are fitting however is the background-subtracted photon spectra $A(E)$, which
is obtained by dividing the specific intensity $I(E)$ by the energy $E$:
$\displaystyle A(E)dE\equiv\frac{I(E)}{E}dE$ $\displaystyle=$
$\displaystyle\frac{k^{4}L}{2\sigma(kT)^{4}D^{2}h^{3}c^{2}}\frac{E^{2}dE}{\exp(E/kT)-1}$
(4) $\displaystyle=$
$\displaystyle\frac{15\phi_{obs}}{\pi^{4}(kT)^{4}}\frac{E^{2}dE}{\exp(E/kT)-1},$
where $h$, $k$ and $\sigma$ are respectively the Planck, the Boltzmann and the
Stefan-Boltzmann constants, $c$ is the speed of light and $\phi_{obs}=L/(4\pi
D^{2})$ is the observed energy flux of the blackbody emitter. The great
advantage of Eq. (4) is that it is written in terms of the observables
$\phi_{obs}$ and $T$, so from a spectral fitting procedure we can obtain the
values of these quantities for each time interval considered. In order to
determine these parameters, we must perform an integration of the actual
photon spectrum $A(E)$ over the instrumental response $R(i,E)$ of the detector
which observe the source, where $i$ denotes the different instrument energy
channels. The result is a predicted count spectrum
$C_{p}(i)=\int_{E_{min}(i)}^{E_{max}(i)}A(E)R(i,E)dE,$ (5)
where $E_{min}(i)$ and $E_{max}(i)$ are the boundaries of the $i$-th energy
channel of the instrument. Eq. (5) must be compared with the observed data by
a fit statistic.
The main parameters obtained from the fitting procedure are shown in Table 2.
We divide the entire GRB in two main episodes, as advanced in Ruffini et al.
(2011): one lasting the first 50 s and the other from 50 to 151 s after the
GRB trigger time, see Fig. 5. It is easy to see that the first 50 s of
emission, corresponding to the first episode, are well fitted by a Band model
as well as a black-body with an extra power-law model, Fig. 6. The same
happens for the first 9 s of the second episode (from 50 to 59 s after the
trigger time), Fig. 7. For the subsequent three intervals corresponding to the
main peaks in the light curve, the black-body plus a power-law model does not
provide a satisfactory fit. Only the Band model fits the spectrum with good
accuracy, with the exception of the first main spike (compare the values of
$\chi^{2}$ in the table). We find also that the last peak can be fitted by a
simple power-law model with a photon index $\gamma$ = 2.20 $\pm$ 0.03, better
than by a Band model.
The result of this analysis points to a different emission mechanism in the
first 50 s of GRB 090618 and in the following 9 s. A sequence of very large
pulses follow, which spectral energy distribution is not attributable either
to a blackbody or a blackbody and an extra power-law component. The evidence
for the transition is well represented by the test of the data fitting, whose
indicator is given by the changing of the $\tilde{\chi^{2}}$ ($N_{dof}=169$)
for the blackbody plus a power-law model for the different time intervals, see
table 2. Although the Band spectral model is an empirical model without a
clear physical origin, we do check its validity in all of the time-detailed
spectra with the sole exception of the first main pulse of the second episode.
The $\chi^{2}$ corresponding to the Band model for such a main pulse, although
better than the one corresponding to the blackbody and power-law case, is
unsatisfactory. We are now going to a direct application of the fireshell
model in order to make more stringent the above conclusions and reach a better
understanding of the source.
Table 2: Time-resolved spectral analysis (8 keV - 10 MeV) of the second episode in GRB 090618. | Time Interval (s) | $\alpha$ | $\beta$ | $E_{0}(keV)$ | $\tilde{\chi}^{2}_{BAND}$ | $kT(keV)$ | $\gamma$ | $\tilde{\chi}^{2}_{BB+po}$
---|---|---|---|---|---|---|---|---
A | 0 - 50 | -0.74 $\pm$ 0.10 | -2.32 $\pm$ 0.16 | 118.99 $\pm$ 21.71 | 1.12 | 32.07 $\pm$ 1.85 | 1.75 $\pm$ 0.04 | 1.21
B | 50 - 59 | -1.07 $\pm$ 0.06 | -3.18 $\pm$ 0.97 | 195.01 $\pm$ 30.94 | 1.23 | 31.22 $\pm$ 1.49 | 1.78 $\pm$ 0.03 | 1.52
C | 59 - 69 | -0.99 $\pm$ 0.02 | -2.60 $\pm$ 0.09 | 321.74 $\pm$ 14.60 | 2.09 | 47.29 $\pm$ 0.68 | 1.67 $\pm$ 0.08 | 7.05
D | 69 - 78 | -1.04 $\pm$ 0.03 | -2.42 $\pm$ 0.06 | 161.53 $\pm$ 11.64 | 1.55 | 29.29 $\pm$ 0.57 | 1.78 $\pm$ 0.01 | 3.05
E | 78 - 105 | -1.06 $\pm$ 0.03 | -2.62 $\pm$ 0.09 | 124.51 $\pm$ 7.93 | 1.20 | 24.42 $\pm$ 0.43 | 1.86 $\pm$ 0.01 | 2.28
F | 105 - 151 | -2.63 $\pm$ -1 | -2.06 $\pm$ 0.02 | unconstrained | 1.74 | 16.24 $\pm$ 0.84 | 2.23 $\pm$ 0.05 | 1.15
Figure 5: The two episode nature of GRB 090618.
---
Figure 6: Time-integrated spectra for the first episode (from 0 to 50 s) of
GRB 090618 fitted with the Band, $\tilde{\chi}^{2}$ = 1.12 (left) and
blackbody + power-law (right) models, $\tilde{\chi}^{2}$ = 1.28. In the
following we will consider the case of a blackbody + power-law model and infer
some physical consequences. The corresponding considerations in the case of
the Band model are currently being considered and will be published elsewhere.
---
Figure 7: Time-integrated spectra for the first 9 s of the second episode
(from 50 to 59 s after the trigger time) of GRB 090618 fitted with the Band,
$\tilde{\chi}^{2}$ = 1.23 (left) and blackbody + power-law (right) models,
$\tilde{\chi}^{2}$ = 1.52.
## 5 Analysis of GRB 090618 in the fireshell scenario: from a single GRB to a
multi-component GRB
### 5.1 Attempt for a single GRB scenario: the role of the first episode
We first approach the analysis of GRB 090618 by assuming that we are in
presence of a single GRB and attempt to identify its components in a canonical
GRB scenario, based on the fireshell model. We first attempt the
identification of the P-GRB emission. We have already seen that the integrated
first 50 s can be well-fitted with a black-body at a temperature $kT$ = 32.07
$\pm$ 1.85 keV and an extra power-law component with the photon index $\gamma$
= -1.75 $\pm$ 0.04, see panel A in Fig. 7 and Table 2. Being the presence of a
blackbody component the distinctive feature of the P-GRB, we have first
attempted an interpretation of GRB 090618 as a single GRB with the first 50 s
as the P-GRB Ruffini et al. (2010a). We have first proceeded to evaluate if
the energetics of the emission in the first 50 s can be interpreted as due to
a P-GRB. The energy emitted by the sole blackbody is $E_{BB}$ =
8.35${}^{+0.27}_{-0.36}$ $\times$ 1051 ergs. Recalling that the isotropic
energy of the entire GRB 090618 is $E_{iso}$ = (2.90 $\pm$ 0.02) $\times$ 1053
ergs, we have that the blackbody component would be $\sim$ 2.9 $\%$ of the
total energy emitted in the burst. This would imply, see lower panel in fig.
4, a baryon load $B\sim 10^{-3}$ with a corresponding Lorentz $\Gamma$ factor
of $\sim$ 800 and a temperature of $\sim$ 52 keV. This value is in
disagreement with the observed temperature $kT_{obs}$ = 32.07 keV.
One may attempt to reconcile the value of the theoretically predicted GRB
temperature with the observed one by increasing $E_{tot}^{e^{+}e^{-}}$. This
would lead to an $E_{tot}^{e^{+}e^{-}}$ = 4 $\times$ 1054 ergs and a
corresponding baryon load of $B\approx 10^{-4}$. This would imply three major
discrepancies: a) there would be an unjustified complementary unobserved
energy; b) in view of the value of the baryon load, and the corresponding
Lorentz $\Gamma$ factor, the duration of the extended-afterglow emission would
be more than an order of magnitude smaller than the observed 100 s (Bianco et
al., 2008); c) the duration of this first 50 s is much longer than the one
typically expected for all P-GRBs identified in other GRBs (Ruffini et al.,
2007), which is at maximum of the order of $\sim$ 10 s. We have therefore
considered hopeless this approach and proceeded to a different one looking for
multiple components.
### 5.2 The multi-component scenario: the second episode as an independent
GRB
#### 5.2.1 The identification of the P-GRB of the second episode
We now proceed to the analysis of the data between 50 and 150 s after the
trigger time, as a canonical GRB in the fireshell scenario, namely the second
episode, see Fig. 5, (Ruffini et al., 2011). We proceed to identify the P-GRB
within the emission between 50 and 59 s, since we find a blackbody signature
in this early second-episode emission. Considerations based on the time
variability of the thermal component bring us to consider the first 4 s of
such time interval as due to the P-GRB emission. The corresponding spectrum
(8-440 keV) is well fitted ($\tilde{\chi}^{2}=1.15$) with a blackbody of a
temperature $kT=29.22\pm 2.21$ keV (norm = 3.51 $\pm$ 0.49), and an extra
power-law component with photon index $\gamma$ = 1.85 $\pm$ 0.06, (norm =
46.25 $\pm$ 10.21), see Fig. 8. The fit with the Band model is also acceptable
($\tilde{\chi}^{2}=1.25$). The fit gives a low energy power-law index
$\alpha=-1.22\pm 0.08$, a high energy index $\beta=-2.32\pm 0.21$ and a break
energy $E_{0}=193.2\pm 50.8$, see Fig. 8. In view of the theoretical
understanding of the thermal component in the P-GRB, see Section 3.2, we shall
focus in the following on the blackbody + power-law spectral model.
The isotropic energy of the second episode is $E_{iso}$ = (2.49 $\pm$ 0.02)
$\times$ 1053 ergs. The simulation within the fireshell scenario is done
assuming $E_{tot}^{e^{+}e^{-}}\equiv E_{iso}$. From the upper panel in Fig. 4
and the observed temperature, we can then derive the corresponding value of
the baryon load. The observed temperature of the blackbody component is
$kT=29.22\pm 2.21$, so that we can determine a value of the baryon load of
$B=1.98\pm 0.15\times$ 10-3, and deduce the energy of the P-GRB as a fraction
of the total $E_{tot}^{e^{+}e^{-}}$. We so obtain a value of the P-GRB energy
of 4.33${}^{+0.25}_{-0.28}$ $\times$ 1051 erg.
Now, from the second panel in Fig. 4 we can derive the radius of the
transparency condition, to occur at $r_{tr}$ = 1.46 $\times$ 1014 cm. From the
third panel we derive the bulk Lorentz factor of $\Gamma_{th}$ = 495. We
compare this value with the energy measured in the sole blackbody component of
$E_{BB}$ = 9.24${}^{+0.50}_{-0.58}$ $\times$ 1050 erg, and with the energy in
the blackbody plus the power-law component of $E_{BB+po}$ =
5.43${}^{+0.07}_{-0.11}$ $\times$ 1051 erg, and verify that the theoretical
value is in between these observed energies. We have found this result quite
satisfactory: it represents the first attempt to relate the GRB properties to
the details of the black hole responsible for the overall GRB energetics. The
above theoretical estimates have been based on a non rotating black hole of 10
M☉, a total energy of $E_{tot}^{e^{+}e^{-}}$ = 2.49 $\times$ 1053 erg and a
mean temperature of the initial plasma of $e^{+}e^{-}$ of 2.4 MeV, derived
from the expression of the dyadosphere radius, Eq. 1. Any refinement of the
direct comparison between theory and observations will have to address a
variety of fundamental issues such as, for example: 1) the possible effect of
rotation of the black hole, leading to a more complex dyadotorus structure; 2)
a more detailed analysis of the transparency condition of the $e^{+}e^{-}$
plasma, simply derived from the condition $\tau$ =
$\int_{R}dr(n_{e^{\pm}}+n_{e^{-}}^{b})\sigma_{T}=0.67$ (Ruffini, 1999); 3) an
analysis of the general relativistic, electrodynamical, strong interactions
descriptions of the gravitational collapse core leading to a black hole
formation, (Cherubini et al., 2009; Ruffini et al., 2003; Ruffini, 1999).
---
Figure 8: On the left panel it is shown the time-integrated power spectra
(8-440 keV) for the P-GRB emission episode (from 50 to 54 s after the trigger
time) of GRB 090618 fitted with the blackbody + power-law models,
$\tilde{\chi}^{2}$ = 1.15, while on the right it is shown the fit with a Band
model, $\tilde{\chi}^{2}$ = 1.25. Figure 9: The fireshell simulation, green
line, and the sole blackbody emission, red line, of the time integrated
(t0+50, t0+54 s) spectrum of the P-GRB emission. The sum of the two
components, the blue line, is the total simulated emission in the first 4 s of
the second episode.
#### 5.2.2 The analysis of the extended-afterglow of the second episode
The extended-afterglow starts at the above given radius of the transparency,
with an initial value of the Lorentz $\Gamma$ factor of $\Gamma_{0}$ = 495. In
order to simulate the extended-afterglow emission, we need to determine the
radial distribution of the CBM around the burst site, which we assume for
simplicity to be spherically symmetric, we infer characteristic size of
$\Delta R=10^{15-16}$ cm. We have already recalled how the simulate of the
spectra and of the observed multi band light curves have to be performed
together and jointly optimized, leading to the determination of the
fundamental parameters characterizing the CBM medium (Ruffini et al., 2007).
This radial distribution is shown in Fig. 10, and is characterized by a mean
value of $<n>$ = 0.6 part/cm3 and an average density contrast with a $<\delta
n/n>$ $\approx$ 2, see Fig. 10 and Table 4. The data up to 8.5 $\times$ 1016
cm are simulated with a value for the filling factor $\mathcal{R}=3\times
10^{-9}$, while the data from this value on with $\mathcal{R}=9\times
10^{-9}$. From the radial distribution of the CBM density, and considering the
$1/\Gamma$ effect on the fireshell visible area, we found that the CBM clumps
originating the spikes in the extended-afterglow emission have masses of the
order of $10^{22-24}$ g. The value of the $\alpha$ parameter has been found to
be -1.8 along the total duration of the GRB.
In Figs. 11 we show the simulated light curve (8-1000 keV) of GRB and the
corresponding spectrum, using the spectral model described in (Bianco &
Ruffini, 2004), (Patricelli et al., 2011).
We focus our attention, in particular, on the structure of the first spikes.
The comparison between the spectra of the first main spike (t0+59, t0+66 s) of
the extended-afterglow of GRB 090618, obtained with three different
assumptions is shown in Fig. 12: in the upper panel we show the fireshell
simulation of the integrated spectrum (t0+59, t0+66 s) of the first main
spike, in the middle panel we show the best fit with a blackbody and a power-
law component model and in the lower panel the best fit using a simple power-
law spectral model.
We can see that the fit with the last two models is not satisfactory: the
corresponding $\tilde{\chi}^{2}$ is 7 for the blackbody + power-law and $\sim$
15 for the simple power-law. We cannot give the $\tilde{\chi}^{2}$ of the
fireshell simulation, since it is not represented by an explicit analytic
fitting function, but it originates by a sequence of complex high non-linear
procedure, summarized in Sec. 3. It is clear by a direct scrutiny that it
correctly reproduces the low energy emission, thanks in particular to the role
of the $\alpha$ parameter, which was described previously. At higher energies,
the theoretically predicted spectrum is affected by the cut-off induced by the
thermal spectrum. The temporal variability of the first two spikes are well
simulated.
We are not able to accurately reproduce the last spikes of the light curve,
since the equations of motion of the accelerated baryons become very
complicated after the first interactions of the fireshell with the CBM
(Ruffini et al., 2007). This happens for different reasons. First, a possible
fragmentation of the fireshell can occur (Ruffini et al., 2007). Moreover, at
larger distances from the progenitor the fireshell visible area becomes larger
than the transverse dimension of a typical blob of matter, consequently a
modification of the code for a three-dimensional description of the
interstellar medium will be needed. This is unlike the early phases in the
prompt emission, which is the main topic we address at the moment, where a
spherically simmetric approximation applies. The fireshell visible area is
smaller than the typical size of the CBM clouds in the early phases of the
prompt radiation, (Izzo et al., 2010).
The second episode, lasting from 50 to 151 s, agrees with a canonical GRB in
the fireshell scenario. Particularly relevant is the problematic of the P-GRB.
It interfaces with the fundamental physics issues, related to the physics of
the gravitational collapse and the black hole formation. There is an interface
between the reaching of transparency of the P-GRB and the early part of the
extended-afterglow. This connection has already been introduced in literature
(Pe’er et al., 2010). We have studied this interface in the fireshell by
analyzing the thermal emission at the transparency with the early interaction
of the baryons with the CBM matter, see Fig. 9.
We turn now to reach a better understanding of the meaning of the first
episode, between 0 and 50 s of the GRB emission. To this end we examine the
two episodes in respect to: 1) the Amati relation, 2) the hardness variation
and 3) the observed time lag.
Figure 10: Radial CBM density distribution in the case of GRB 090618. The characteristic masses of each cloud are of the order of $\sim$ 1022-24 g and 1016 cm in radii. Table 3: Final results of the simulation of GRB 090618 in the fireshell scenario Parameter | Value
---|---
$E_{tot}^{e^{+}e^{-}}$ | 2.49 $\pm$ 0.02 $\times$ 1053 ergs
$B$ | 1.98 $\pm$ 0.15 $\times$ 10-3
$\Gamma_{0}$ | 495 $\pm$ 40
$kT_{th}$ | 29.22 $\pm$ 2.21 keV
$E_{P-GRB,th}$ | 4.33 $\pm$ 0.28 $\times$ 1051 ergs
$<n>$ | $0.6\,part/cm^{3}$
$<\delta n/n>$ | $2\,part/cm^{3}$
Table 4: Physical properties of the three clouds surrounding the burst site: the Distance from the burst site (2nd column, the radius $r$ of the cloud, 3rd column, the particle density $\rho$, 4th column and the mass $M$ in the last column Cloud | Distance (cm) | r (cm) | $\rho$ (#/cm3) | M (g)
---|---|---|---|---
First | 4.0 $\times$ 1016 | 1 $\times$ 1016 | 1 | 2.5 $\times$ 1024
Second | 7.4 $\times$ 1016 | 5 $\times$ 1015 | 1 | 3.1 $\times$ 1023
Third | 1.1 $\times$ 1017 | 2 $\times$ 1015 | 4 | 2.0 $\times$ 1022
---
Figure 11: Simulated light curve and time integrated (t0+58, t0+150 s)
spectrum (8-440 keV) of the extended-afterglow of GRB 090618.
---
Figure 12: Simulated time integrated (t0+58, t0+66 s) count spectrum (8-440
keV) of the extended-afterglow of GRB 090618 (upper panel), count spectrum (8
keV - 10 MeV) of the main pulse emission (t0+58, t0+66) and best fit with a
blackbody + power-law model (middle panel) and a simple power-law model (lower
panel).
## 6 The Amati relation, the HR and the time lag of the two episodes
### 6.1 The first episode as an independent GRB?
We first check if the two episodes fulfill separately the Amati relation,
(Amati et al., 2002). By using the Band spectrum we verify that the first
episode presents an intrinsic peak energy value of $E_{p,1st}$ = 223.01 $\pm$
24.15 keV, while the second episode presents an $E_{p,2nd}$ = 224.57 $\pm$
17.4 keV. The isotropic energies emitted in each single episode are
$E_{iso,1st}$ = 4.09 $\pm$ 0.07 $\times$ 1052 ergs and $E_{iso,2nd}$ = 2.49
$\pm$ 0.02 $\times$ 1053 ergs, so we have that both episodes satisfy the Amati
relation, see fig. 13. The fulfillment of the Amati relation of episode 2 was
expected, being the second episode a canonical GRB. What we find surprising is
the fulfillment of the Amati relation of the first episode.
Figure 13: Position of the first and second component of GRB 090618 in the
$E_{p,i}$ \- $E_{iso}$ plane respect the best fit of the Amati relation, as
derived following the procedure described in (Capozziello & Izzo, 2010). The
red circle corresponds to the first emission while the green circle
corresponds to the second one.
We first examine the episode 1 as a single GRB. We notice a sharp rise in the
luminosity in the first 6 s of emission. We therefore attempted a first
interpretation by assuming the first 6 s as the P-GRB component of this
independent GRB, as opposed to the remaining 44 s as the extended-afterglow of
this GRB. A value of the fit gives $E_{tot}^{e^{+}e^{-}}=3.87\times 10^{52}$
ergs and $B=1.5\times 10^{-4}$. This would imply a very high value for the
Lorentz factor at the transparency of $\sim$ 5000\. In turn, this value would
imply (Ruffini, 1999) a spectrum of the P-GRB peaking around $\sim$ 300 keV,
which is in contrast with the observed temperature of 58 keV. Alternatively,
we have attempted a second simulation by assuming all the observed data be
part of the extended-afterglow of a GRB, with a P-GRB below the detector
threshold. Assuming in this case $E_{iso}$ = $E_{tot}^{e^{+}e^{-}}$, $B$ =
10-2, and assuming for the P-GRB a duration smaller than 10 s, as confirmed
from the observations of all existing P-GRBs (Ruffini et al., 2007), we should
obtain an energy of the P-GRB greater than 10-8 ergs/cm2/s, which should have
been easily detectable from Fermi and Swift. Also this second possibility is
therefore not viable. We can then conclude generally that in no way we can
interpret this episode either as a P-GRB of the second episode, as proved in
paragraph 3.2 or, as proved here, as a separate GRB. We then conclude that the
fulfillment of the Amati relation does not imply for the source to be
necessarily a GRB.
### 6.2 The HR variation and the time lag of the two episodes
We finally address a further difference between the two episodes, related to
the Hardness-Ratio behavior (HR) and their observed time-lag. The first
evidence of an evolution of the GRBs power-law slope indexes with time was
observed in the BATSE GRB photon spectra (Crider et al., 1997). In the context
of the fireshell scenario, as recalled earlier, the spectral evolution comes
out naturally from the evolution of the comoving temperature, the decrease of
the bulk Lorentz $\Gamma$ factor and from the curvature effect (Bianco &
Ruffini, 2004), with theoretically predicted values, in excellent agreement
with observations in past GRBs.
In order to build the HR ratio, we considered the data from three different
instruments: Swift-BAT, Fermi-GBM and the CORONAS-PHOTON-RT-2. The plots
obtained with these instruments confirm the existence of a peculiar trend of
the hardness behavior: in the first 50 s it is evident a monotonic hard-to-
soft behavior, as due to the blackbody evolution of the first episode. For the
second episode, the following 50 to 151 s of the emission, there is a soft-to-
hard trend in the first 4 s of emission, and a hard-to-soft behavior modulated
by the spiky emission in the following 100 s. For the HR ratio we considered
the ratio of the count rate detected from a higher energy channel to that of a
lower energy channel: HR = ctg(HE)/ctg(LE). In particular, we considered the
count rate subtracted for the background, even when this choice provides bad
HR data in time region dominated by the background, where the count rate can
be zero or negative. For the Swift data, we consider the HR ratio for two
different energy subranges: the HR1 ratio shows the ratio of the (50-150 keV)
over the (15-50 keV) emission while the HR2 ratio shows the ratio of the
(25-50 keV) over the (15-25 keV) emission, see Fig. 14.
Figure 14: Hardness-Ratio ratios for the Swift BAT data in two different
energy channels: HR1 = cts(25-50 keV)/cts(15-25 keV), HR2 = cts(50-150
keV)/cts(15-50 keV). Figure 15: Hardness-Ratio ratio for the Fermi data. We
considered the cts observed in the (260 keV - 40 MeV) energy range over the (8
- 260 keV) energy range. The time reported on the x-axis is in terms of the
Mission Elapsed Time (MET). The presence of some negative data points is due
to the presence of noise, in other terms the non-presence of GRB emission, in
the background-subtracted BGO count light curve.
A similar trend is found for the Fermi-GBM NaI and RT-2 instruments, see
Fig.15. In particular, the HR from Fermi observations was done considering the
counts observed by the b0 BGO detector in the range (260 keV - 40 MeV) and the
ones observed by the n4 NaI detector in the range (8 - 260 keV). In Fig. 15 it
is shown the HR ratio for the Fermi observations, where we rebinned the counts
in time intervals of 3 seconds. From this analysis we see that the HR ratio
peaks at the beginning of each pulse, also for the second episode pulses, but
each peak of the second episode pulses is softer than the previous one,
suggesting that these pulses are consequential in the second episode and are
in general agreement with the advance of a fireshell in the CBM. Since RT-2
data clearly show both the episodes up to 1 MeV it complements the results
obtained by Swift (up to 200 keV) and FERMI (up to 440 keV) in the high and
the most interesting energy range. Hardness ratio plot of (250-1000
keV)/(8-250 keV) indicates that first phase of both episodes are the hardest.
Finally, the evident asymmetry of the first episode, supported by the
observations of a large time lag in the high and low energy channels, see fig.
2, suggests a different process at work. There is a very significant softening
of the first episode, as reported in Rao et al. (2011), where it is observed a
large time lag between the 15-25 keV energy range and the 100-150 keV one: the
high energy photons peak $\sim$ 7 seconds before the photons detected in the
15-25 keV energy range. This large time lag is not observed in the second
episode, where the lags are of the order of $\sim$ 1 s.
Motivated by these results, we proceed to a most accurate time-resolved
spectral analysis of the first episode to identify its physical and
astrophysical origin.
## 7 A different emission process in the first episode
### 7.1 The time resolved spectra and temperature variation
One of the most significant outcome of the multi-year work of Felix Ryde and
his collaborators, (see e.g. Ryde et al. (2010) and references therein), has
been the identification and the detailed analysis of the thermal plus power-
law features observed in a time limited intervals in selected BATSE GRBs.
Similar features have been also observed recently in the data acquired by the
Fermi satellite (Ryde et al., 2010; Guiriec et al., 2011). We propose to
divide these observations in two broad families. The first family presents a
thermal plus power-law(s) feature, with a temperature changing in time
following precise power-law behavior. The second family is also characterized
by a thermal plus power-law component, but with the blackbody emission
generally varying without specific power-law behavior and on shorter time
scales. It is our goal to study these features within the fireshell scenario,
in order to possibly identify the underlying physical processes. We have
already identified in Sec. 4 that the emission of the thermal plus power-law
component characterizes the P-GRB emission. We have also emphasized that the
P-GRB emission is the most relativistic regime occurring in GRBs, uniquely
linked to the process of the black hole formation, see Sec. 5. This process
appears to belong to the second family above considered. Our aim here is to
see if the first episode of GRB 090618 can lead to the identification of the
above first family of events: the ones with temperature changing with time
following a power-law behavior on time scales from 1 to 50 s. We have already
pointed out in the previous section that the hardness-ratio evolution and the
large time lag observed for the first episode (Rao et al., 2011) points to a
distinct origin for the first 50 s of emission, corresponding to the first
episode.
We have made a detailed time-resolved analysis of the first episode,
considering different time bin durations in order to have a good statistic in
the spectra and to take into account the sub-structures in the light curve. We
have then used two different spectral models to fit the observed data, a
classical Band spectrum (Band et al., 1993), and a blackbody with a power-law
component.
In order to have more accurate constraints on the spectral parameters, we made
a joint fit considering the observations from both the n4 NaI and the b0 BGO
detectors, covering in this way a wider energy range, from 8 keV to 40 MeV. To
avoid some bias due to low photon statistic, we considered an energy upper
limit of the value of 10 MeV. We report in the last three columns of the Table
5 also the spectral analysis performed in the energy range of the BATSE LAD
instrument (20-1900 keV), as analyzed in Ryde & Pe’er (2009), just as a
comparison tool with the results described in that paper. Our analysis has
been summarized in Figs. 16, 17 and in Table 5, where we report the residual
ratio diagram as well as the reduced-$\chi^{2}$ values for the spectral models
considered.
Figure 16: Evolution of the BB+powerlaw spectral model in the $\nu\,F(\nu)$ spectrum of the first emission of GRB 090618. It is evident the cooling of the black-body and of the associated non-thermal component with the time. In this picture we have preferred to plot just the fitting functions, in order to prevent some confusion. Table 5: Time-resolved spectral analysis of the first episode in GRB 090618. We have considered seven time intervals, as described in the text, and we used two spectral models, whose best-fit parameters are shown here. The last three columns, marked with a LAD subscript, report the same analysis but in the energy range $20-1900$ keV, which is the same energy range of the BATSE-LAD detector as used in the work of Ryde & Pe’er (2009). Time | $\alpha$ | $\beta$ | $E_{0}$ (keV) | $\tilde{\chi}^{2}_{BAND}$ | $kT$ (keV) | $\gamma$ | $\tilde{\chi}^{2}_{BB+po}$ | $kT_{LAD}$ (keV) | $\gamma_{LAD}$ | $\tilde{\chi}^{2}_{BB+po,LAD}$
---|---|---|---|---|---|---|---|---|---|---
A:0 - 5 | -0.45 $\pm$ 0.11 | -2.89 $\pm$ 0.78 | 208.9 $\pm$ 36.13 | 0.93 | 59.86 $\pm$ 2.72 | 1.62 $\pm$ 0.07 | 1.07 | 52.52 $\pm$ 23.63 | 1.42 $\pm$ 0.06 | 0.93
B:5 - 10 | -0.16 $\pm$ 0.17 | -2.34 $\pm$ 0.18 | 89.84 $\pm$ 17.69 | 1.14 | 37.57 $\pm$ 1.76 | 1.56 $\pm$ 0.05 | 1.36 | 37.39 $\pm$ 2.46 | 1.55 $\pm$ 0.06 | 1.27
C:10 - 17 | -0.74 $\pm$ 0.08 | -3.36 $\pm$ 1.34 | 149.7 $\pm$ 21.1 | 0.98 | 34.90 $\pm$ 1.63 | 1.72 $\pm$ 0.05 | 1.20 | 36.89 $\pm$ 2.40 | 1.75 $\pm$ 0.06 | 1.10
D:17 - 23 | -0.51 $\pm$ 0.17 | -2.56 $\pm$ 0.26 | 75.57 $\pm$ 16.35 | 1.11 | 25.47 $\pm$ 1.38 | 1.75 $\pm$ 0.06 | 1.19 | 25.70 $\pm$ 1.76 | 1.75 $\pm$ 0.08 | 1.19
E:23 - 31 | -0.93 $\pm$ 0.13 | unconstr. | 104.7 $\pm$ 21.29 | 1.08 | 23.75 $\pm$ 1.68 | 1.93 $\pm$ 0.10 | 1.13 | 24.45 $\pm$ 2.24 | 1.95 $\pm$ 0.12 | 1.31
F:31 - 39 | -1.27 $\pm$ 0.28 | -3.20 $\pm$ 1.00 | 113.28 $\pm$ 64.7 | 1.17 | 18.44 $\pm$ 1.46 | 2.77 $\pm$ 0.83 | 1.10 | 18.69 $\pm$ 1.89 | 4.69 $\pm$ 4.2 | 1.08
G:39 - 49 | -3.62 $\pm$ 1.00 | -2.19 $\pm$ 0.17 | 57.48 $\pm$ 50.0 | 1.15 | 14.03 $\pm$ 2.35 | 3.20 $\pm$ 1.38 | 1.10 | 14.71 $\pm$ 3.52 | 3.06 $\pm$ 3.50 | 1.09
We conclude that both the Band and the proposed blackbody + power-law spectral
models fit very well the observed data. Particularly interesting is the clear
evolution in the time-resolved spectra, corresponding to the blackbody and
power-law component, see Fig. 16. In particular the $kT$ parameter of the
blackbody presents a strong decay, with a temporal behavior well described by
a double broken power-law function, see upper panel in Fig. 17. From a fitting
procedure we obtain the best fit (R2-statistic = 0.992) for the two decay
indexes for the temperature variation are $a_{kT}$ = -0.33 $\pm$ 0.07 and
$b_{kT}$ = -0.57 $\pm$ 0.11. In Ryde & Pe’er (2009) an average value for these
parameters on a set of 49 GRBs is given: $\left\langle a_{kT}\right\rangle$ =
-0.07 $\pm$ 0.19 and $\left\langle b_{kT}\right\rangle$ = -0.68 $\pm$ 0.24. We
note however that in the sample considered in Ryde & Pe’er (2009) only few
bursts shows a break time around 10 s, as it is in our case, see Fig. 17.
There are two of these bursts, whose analysis presents many similarities with
our presented source GRB 090618: GRB 930214 and GRB 990102. These bursts are
characterized by a simple FRED pulse, whose total duration is $\sim$ 40 s,
quite close to the one corresponding to the first episode of GRB 090618. The
break time $t_{b}$ in these two burst are respectively at 12.9 and 8.1 s,
while the decay indexes are $a_{kT}$ = -0.25 $\pm$ 0.02 and $b_{kT}$ = -0.78
$\pm$ 0.04 for GRB 930214 and $a_{kT}$ = -0.36 $\pm$ 0.03 and $b_{kT}$ = -0.64
$\pm$ 0.04 for GRB 990102, see Table 1 in Ryde & Pe’er (2009), in very good
agreement with the values observed for the first episode of GRB 090618. We
conclude that the values we observe in GRB 090618 are very close to the values
of these two bursts. We shall return to compare and contrast our results with
the other sources considered in (Ryde & Pe’er, 2009), as well as GRB 970828
(Pe’er et al., 2007) in a forthcoming publication.
The results presented in Figs. 16,17, as well as in Table 5, point to a rapid
cooling of the thermal emission with time of the first episode. The evolution
of the corresponding power-law spectral component, also, appears to be
strictly related to the change of the temperature $kT$. The power-law $\gamma$
index falls, or softens, with the temperature, see Fig. 16. An interesting
feature appears to occur at the transition of the two power-law describing the
observed decrease of the temperature. The large time lag observed in the first
episode and reported in section 6.1 has a clear explanation in the power-law
behavior of the temperature and corresponding evolution of the photon index
$\gamma$, Figs. 16,17.
### 7.2 The radius of the emitting region
We turn now to estimate an additional crucial parameter for the identification
of the nature of the blackbody component: the radius of the emitter $r_{em}$.
We have proved that the first episode is not an independent GRB, not a part of
a GRB. We can therefore provide the estimate of the radius of the emitter from
non-relativistic considerations, just corrected for the cosmological redshift
$z$. We have, in fact, that the temperature of the emitter
$T_{em}=T_{obs}(1+z)$, and that the luminosity of the emitter, due to the
blackbody emission, is
$L=4\pi r_{em}^{2}\sigma T_{em}^{4}=4\pi r_{em}^{2}\sigma
T_{obs}^{4}(1+z)^{4},$ (6)
where $r_{em}$ is the radius of the emitter and $\sigma$ is the Stefan
constant. From the luminosity distance definition, we also have that the
observed flux $\phi_{obs}$ is given by:
$\phi_{obs}=\frac{L}{4\pi D^{2}}=\frac{r_{em}^{2}\sigma
T_{obs}^{4}(1+z)^{4}}{D^{2}}.$ (7)
We then obtain
$r_{em}=\left(\frac{\phi_{obs}}{\sigma
T_{ob}^{4}}\right)^{1/2}\frac{D}{(1+z)^{2}}.$ (8)
The above radius differs from the radius $r_{ph}$ given in Eq. (1) of Ryde &
Pe’er (2009) and clearly obtained by interpreting the early evolution of GRB
970828 as belonging to the photospheric emission of a GRB and assuming a
relativistic expansion with a Lorentz gamma factor $\Gamma$:
$r_{ph}=\hat{\mathcal{R}}D\left(\frac{\Gamma}{(1.06)(1+z)^{2}}\right),$ (9)
where $\hat{\mathcal{R}}=\left(\phi_{obs}/(\sigma T_{ob}^{4})\right)^{1/2}$
and the prefactor 1.06 arises from the dependence of $r_{ph}$ on the angle of
sight (Pe’er, 2008). Typical values of $r_{ph}$ are at least two orders of
magnitude larger than our radius $r_{em}$. We shall return on the analysis of
GRB 970828 in a forthcoming paper.
Assuming a standard cosmological model ($H_{0}=70$ km/s/Mpc, $\Omega_{m}=0.27$
and $\Omega_{\Lambda}=0.73$) for the estimate of the luminosity distance $D$,
and using the values for the observed flux $\phi_{obs}$ and the temperature
$kT_{obs}$, we have given in Fig. 18 the evolution of the radius of the
surface emitting the blackbody $r_{em}$ as a function of time.
Assuming an exponential evolution with time $t^{\delta}$ of the radius in the
comoving frame, we obtain from a fitting procedure the value $\delta=0.59\pm
0.11$, well compatible with $\delta=0.5$. We also notice a steeper behavior
for the variation of the radius with time corresponding to the first 10 s,
which corresponds to the emission before the break of the double power-law
behavior of the temperature. We estimate an average velocity of
$\bar{v}=4067\pm 918$ km/s, R2 = 0.91, in these first 10 s of emission. In
episode 1 the observations lead to a core of an initial radius of $\sim$ 12000
km expanding in the early phase with a sharper initial velocity of $\sim$ 4000
km/s. The effective Lorentz $\Gamma$ factor is very low, $\Gamma-1\sim$ 10-5.
---
Figure 17: Evolution of the $kT$ observed temperature of the black-body
component and the corresponding evolution of the photon index of the power-
law. The blue line in the upper panel corresponds to the fit of the time
evolution of the temperature with a broken power-law function. It is evident a
break time $t_{b}$ around 11 s after the trigger time, as obtained from the
fitting procedure. Figure 18: Evolution of the radius of the first episode
emitter, as given by Eq. (8).
## 8 Conclusions
GRB 090618 is one of the closest ($z=0.54$) and most energetic ($E_{iso}$ =
2.9 $\times$ 1053 ergs ) GRBs up to date. It has been observed simultaneously
by the largest number of X and $\gamma$ ray telescopes: Fermi, Swift, AGILE,
Konus-WIND, Suzaku-WAM and the CORONAS-PHOTON-RT2. These circumstances have
produced an unprecedented set of high quality data as well as the coverage of
the instantaneous spectral properties and of the time variability in
luminosity of selected bandwidth of the source, see e.g. Figs. 1,2. In
addition there is also the possibility of identifying an underlying supernova
event from the optical observations in the light curve of well-defined bumps,
as well as from the correspective change in colour after around 10 days from
the main event (Cano et al., 2011). Unfortunately a spectroscopic confirmation
of the presence of such supernova is lacking. We have restricted our attention
in this paper to the sole X and $\gamma$ ray emission of the GRB, without
addressing the possible supernova component.
By applying our analysis within the fireshell scenario, see section 4, we have
supported that GRB 090618 is actually composed of two different episodes
(Ruffini et al., 2010a): episode 1, lasting from 0 to 50 s and episode 2 from
50 s to 151 s after the trigger time. We have also illustrated the recent
conclusions presented in Ruffini et al. (2011), that episode 1 cannot be
either a GRB nor a part of a GRB, see section 5. By a time-resolved spectral
analysis we have fitted the instantaneous spectra by a blackbody plus an extra
power-law component. The temperature of the blackbody appears to have a
regular dependence with time, described by two power-law functions: a first
power-law with decay index akT = -0.33 $\pm$ 0.07 and the second one with bkT
= -0.57 $\pm$ 0.11, see Section 7 . All these features follow precisely some
of the results obtained by Felix Ryde and his collaborators (Ryde & Pe’er,
2009), where the authors analyzed selected temporal episodes in some GRBs
observed by BATSE.
We have also examined with particular attention, see section 6, the radius
$r_{em}$ of the blackbody emitter observed in the first episode, given by Eq.
(8). We interpret the nature of this episode 1 as originating from what we
have defined a proto-black hole, (Ruffini et al., 2010a): the collapsing bare
core leading to the black hole formation. Within this interpretation, the
radius $r_{em}$ depends only on the observed energy flux of the blackbody
component $\phi_{obs}$, the temperature $kT$ as well as on the luminosity
distance of the source $D$. We obtained a radius of the emitting region
smoothly varying between $\sim$ 12000 and 70000 km, see Fig. 18. Other
interpretations associating the origin of this early emission to the GRB main
event (Pe’er et al., 2007) lead to a different definition for the radius of
the blackbody emitter, which results to be larger than our radius by at least
two orders of magnitude. We are planning a systematic search for other systems
presenting these particular features.
Episode 2 is identified as a canonical long GRB which originates from the
black hole formation process and lasts in arrival time from 50 s to 151 s
after the trigger time. The good quality of data allowed us to search for the
P-GRB signature in the early emission of the episode 2. From a detailed
analysis we find that the first 4 s of episode 2 are in good agreement with
the theoretically predicted P-GRB emission, see section 5.2. The observed
spectrum integrated over these 4 s is well fitted by a blackbody with an extra
power-law component, where this latter component is mainly due to the early
emission of the extended-afterglow, see Fig. 8. From the temperature observed
in the P-GRB, $kT_{PGRB}$ = 29.22 $\pm$ 2.21, and the $E_{tot}^{e^{+}e^{-}}$
energy of the second episode, which we assumed equal to the isotropic
equivalent energy of this episode, $E_{tot}^{e^{+}e^{-}}$ = 2.49 $\times$ 1053
ergs, we obtained the value of the baryon load of the GRB, see also Fig. 4,
$B=(1.98\pm 0.15)\times 10^{-3}$, and a consequent Lorentz $\Gamma$ factor at
the transparency of $\Gamma_{\circ}=495\pm 40$. We have been able to simulate
the temporal and the spectral emission of the second episode, as seen by the
Fermi-GBM instrument (8 keV – 10 MeV). As we have shown in Fig. 12, our
simulation succeeds in fitting the light curves as well as the spectral energy
distribution emitted in the first main spike of the second episode. The
residual emission of the last spikes is reasonably fitted, taking into due
account the difficulties in integrating the equations of motion, which after
the first interactions of the fireshell with the CBM become hardly
predictable. The energetic of the simulation is fulfilled and we find that the
emission is due to blobs of matter in the CBM with typical dimensions of
$r_{bl}=10^{16}$ cm and average density contrast $\delta n/n$ $\simeq$ 2
particles/cm3 in an overall average density of 1 particle/cm3. We need to find
additional cases of such phenomena to augment our statistic and improve its
comprehension.
Particularly relevant are the first two-dimensional hydrodynamical simulations
of the progenitor evolution of a $23M_{\sun}$ star close to core-collapse,
leading to a naked core, as shown in the recent work of Arnett and Meakin
(Arnett & Meakin, 2011). In that work, pronounced asymmetries and strong
dynamical interactions between burning shells are seen: the dynamical behavior
proceeds to large amplitudes, enlarging deviations from the spherical symmetry
in the burning shells. It is of clear interest to find a possible connection
between the proto black hole concept, introduced in this work, with the Arnett
and Meakin results: to compare the radius, the temperature and the dynamics of
the core we have found in the present work with the naked core obtained by
Arnett and Meakin from the thermonuclear evolution of the progenitor star.
Particularly relevant is the presence, during this phase of collapse, of
strong waves, originated in the mixing of the different element’ shells. Such
waves should become compressional, as they propagate inward, but they should
also dissipate in non-convective regions, causing heating and slow mixing in
these regions of the star. Since the wave heating is faster than radiative
diffusion (which is very slow), an expansion phase of the boundary layers will
occur, while the iron (Fe) core will contract (Arnett & Meakin, 2011). There
is also the interesting possibility that the CBM clouds observed in GRBs be
related to the vigorous dynamics in violent activity of matter ejected in the
evolution of the original massive star, well before the formation of the naked
core (Arnett D., private communication).
It is appropriate to emphasize that these results have no relation with the
study of precursors in GRBs done in the current literature (see e.g. Burlon et
al., 2008, and references therein). Episode 1 and episode 2 are not temporally
separated by a quiescent time. The spectral feature of episode 1 and episode 2
are strikingly different and, moreover, the episode 1 is very energetic, which
is quite unusual for a typical precursor event. We finally conclude that for
the first time we witness the process of formation of the black hole from the
phases just preceding the gravitational collapse all the way up to the GRB
emission.
There is now evidence that the Proto Black Hole formation has been observed
also in other GRB sources. After the submission of this article a second
example has been found in GRB 101023, then and a paper about this source was
submitted on November 4th 2011 and then published on February 1st 2012
(Penacchioni et al., 2012). There, extremely novel considerations in the
structure of the late phase of the emission in X-ray at times larger than 200
s have been presented in favour of a standard signature in these sources (see
also the considerations made in Page et al., 2011). The possible use of this
new family of GRBs as distance indicators is being considered.
###### Acknowledgements.
We thank David Arnett for most fruitful discussions, the participants of the
Les Houches workshop “From Nuclei to White Dwarfs and Neutron Stars” held in
April 2011 (Eds. A. Mezzacappa and R. Ruffini, World Scientific 2011, in
press), as well as the members of the AlbaNova University High Energy
Astrophysics group. We are thankful to an anonymous referee for her/his
important remarks both on the content and the presentation of our work which
have improved the presentation of our paper. LI is especially grateful to
Marco Muccino for fruitful discussions about the work concerning this
manuscript. We are also greateful to the Swift and Fermi teams for their
assistance. One of us, AVP, acknowledges the support for the fellowship
awarded for the Erasmus Mundus IRAP PhD program. This work made use of data
supplied by the UK Swift Science Data Centre at the University of Leicester.
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|
arxiv-papers
| 2012-02-20T16:46:04 |
2024-09-04T02:49:27.571393
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Izzo, R. Ruffini, A. V. Penacchioni, C. L. Bianco, L. Caito, S. K.\n Chakrabarti, Jorge A. Rueda, A. Nandi, B. Patricelli",
"submitter": "Luca Izzo",
"url": "https://arxiv.org/abs/1202.4374"
}
|
1202.4525
|
# Numerically erasure-robust frames
Matthew Fickus Department of Mathematics and Statistics, Air Force Institute
of Technology, Wright-Patterson Air Force Base, OH 45433, USA;
matthew.fickus@afit.edu and Dustin G. Mixon Program in Applied and
Computational Mathematics, Princeton University, Princeton, New Jersey 08544,
USA; E-mail: dmixon@princeton.edu
###### Abstract.
Given a channel with additive noise and adversarial erasures, the task is to
design a frame that allows for stable signal reconstruction from transmitted
frame coefficients. To meet these specifications, we introduce numerically
erasure-robust frames. We first consider a variety of constructions, including
random frames, equiangular tight frames and group frames. Later, we show that
arbitrarily large erasure rates necessarily induce numerical instability in
signal reconstruction. We conclude with a few observations, including some
implications for maximal equiangular tight frames and sparse frames.
###### Key words and phrases:
frames, erasures, well-conditioned
###### 2000 Mathematics Subject Classification:
42C15, 15A12
The authors thank the anonymous referee for very helpful comments and
suggestions. MF was supported by NSF Grant No. DMS-1042701 and AFOSR Grant
Nos. F1ATA01103J001 and F1ATA00183G003, and DGM was supported by the A.B.
Krongard Fellowship. The views expressed in this article are those of the
authors and do not reflect the official policy or position of the United
States Air Force, Department of Defense, or the U.S. Government.
## 1\. Introduction
Modern communication networks are rooted in both information theory and
algebraic coding theory. In these contexts, after deciding on a probabilistic
noise model for a given communication channel, one chooses an appropriate
error-correcting code to achieve reliable communication with a maximal
information rate. For linear codes in particular, encoding and decoding reduce
to problems in linear algebra over finite fields. Beginning with Goyal et al.
[16], finite frame theorists have studied the generalizations of these
problems to real and complex vector spaces. This generalization allows one to
use certain mathematical tools, such as matrix norms and condition numbers,
which are not well-defined in the finite-field setting.
This paper is concerned with a channel characterized by additive noise and
adversarial erasures. We encode a signal $x\in\mathbb{C}^{M}$ using inner
products $\langle x,f_{n}\rangle$ with members of a spanning sequence of
vectors $\\{f_{n}\\}_{n=1}^{N}\subseteq\mathbb{C}^{M}$; such a sequence is
called a frame. In transmitting these inner products, we expect additive noise
due to various phenomena such as atmospheric interactions or round-off error.
If these were the only sources of noise, then it would be reasonable to
reconstruct the original signal by applying the Moore-Penrose pseudoinverse.
To be precise, letting $F$ denote the $M\times N$ matrix whose columns are the
$f_{n}$’s, we transmit $y=F^{*}x$. At the receiver, an estimate of $x$ is then
found by computing
$\hat{x}=\big{(}(FF^{*})^{-1}F\big{)}(y+e)=x+(FF^{*})^{-1}Fe,$
where $e$ is additive noise. Assuming the channel has a “signal-to-noise
ratio” of $R=\|y\|/\|e\|$, we can estimate how the size of the estimate error
$(FF^{*})^{-1}Fe$ compares with the size of the original signal $x$. Indeed,
$\|(FF^{*})^{-1}Fe\|\leq\frac{C}{R}\|x\|$, where
$C:=\\!\\!\\!\sup_{\begin{subarray}{c}x\in\mathbb{C}^{M}\setminus\\{0\\}\\\
e\in\mathbb{C}^{N}\setminus\\{0\\}\end{subarray}}\\!\\!\\!R\cdot\frac{\|(FF^{*})^{-1}Fe\|}{\|x\|}=\\!\\!\\!\sup_{\begin{subarray}{c}x\in\mathbb{C}^{M}\setminus\\{0\\}\\\
e\in\mathbb{C}^{N}\setminus\\{0\\}\end{subarray}}\\!\\!\\!\frac{\|F^{*}x\|}{\|x\|}\cdot\frac{\|(FF^{*})^{-1}Fe\|}{\|e\|}=\|F\|_{2}\|(FF^{*})^{-1}F\|_{2}.$
Here, $C$ is the condition number of $F$, denoted $\mathrm{Cond}(F)$, which is
equal to the ratio of the greatest singular value of $F$ to its smallest one.
From this perspective, the best possible frames are those with
$\mathrm{Cond}(F)=1$, a fact which occurs precisely when
$FF^{*}=A\mathrm{I}_{M}$ for some $A>0$; such $F$’s are called tight frames.
We consider channels that, in addition to additive noise, suffer from
erasures. To be precise, the transmitted signal is a sequence of inner
products: $F^{*}x=\\{\langle x,f_{n}\rangle\\}_{n=1}^{N}$. Like [16], we
consider channels which completely delete some of these inner products and add
noise to the remaining ones. However, whereas [16] focuses on average
reconstruction performance, we instead follow [11] and [17] by focusing on
worst-case reconstruction performance. In particular, by considering worst-
case performance, we design frames which are robust against the erasure of any
fixed number of inner products. Such frames could be particularly useful in
situations where an adversary is actively deleting our most useful frame
coefficients, i.e., active jamming. We say that such frames are robust against
adversarial erasures.
To design such frames, we first acknowledge that we cannot reconstruct the
$M$-dimensional signal $x$ without at least $M$ inner products. As such, we
must impose some constraint on the adversary. For the highly constrained
adversary, Casazza and Kovačević [11] show that tight frames of unit-norm
vectors, called unit norm tight frames, are optimally robust against one
erasure. Soon thereafter, Holmes and Paulsen [17] showed that equiangular
tight frames—explicitly defined in the next section—are optimal for two
erasures. To combat the highly destructive adversary, Püschel and Kovačević
[20] propose frames which are maximally robust to erasures in the sense that
the original signal can be recovered from any $M$ of the $N$ inner products.
Other constructions of such maximally robust frames are given in [1], where
they are dubbed full spark frames. It remains unclear whether the deletion of
any $N-M$ frame coefficients will allow for numerically stable reconstruction;
this is an important distinction between invertible submatrices—the subject of
[1, 20]—and well-conditioned submatrices, which is our focus here.
To be clear, in this paper we consider the case where the adversary is only
capable of removing a proportion $p$ of the $N$ transmitted inner products.
Then the remaining $(1-p)N$ inner products correspond to a subcollection of
$(1-p)N$ columns of $F$, which we require to be well-conditioned for our
reconstruction to properly combat the additive noise. Since erasures occur
according to the will of an adversary, as opposed to a random process, we must
ensure that every subcollection of $(1-p)N$ columns of $F$ is well-
conditioned. This leads to the following definition:
###### Definition 1.
Given $p\in[0,1]$ and $C\geq 1$, an $M\times N$ frame $F$ is a
$(p,C)$-numerically erasure-robust frame (NERF) if for every
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$ of size $K:=(1-p)N$, the corresponding
$M\times K$ submatrix $F_{\mathcal{K}}$ has condition number
$\mathrm{Cond}(F_{\mathcal{K}})\leq C$.
The purpose of this paper is to make the first strides in studying NERFs. In
the following section, we use a variety of techniques to form different NERF
constructions. Taking inspiration from matrix design problems in compressed
sensing, we first investigate frames whose entries are independent Gaussian
random variables. Next, we consider equiangular tight frames, with which we
get stronger results at the price of higher redundancy in the frame. Later, we
show how the symmetry of group frames makes them naturally amenable to NERF
analysis. In Section 3, we report a result on the fundamental limits of NERFs:
that NERFs cannot stably support erasure rates $p$ which are arbitrarily close
to $1$. Finally, we conclude with a few interesting observations in Section 4.
## 2\. Constructions
### 2.1. Random frames
The reader may have noticed some similarity between the definition of
numerically erasure-robust frames and a matrix property which comes from the
compressed sensing literature: the restricted isometry property (RIP). To be
clear, an $M\times N$ matrix $F$ is RIP if it acts as a near-isometry on
sufficiently sparse vectors, that is, $\|Fx\|\approx\|x\|$ for all vectors $x$
with sufficiently few nonzero entries [12]. In other words, submatrices
$F_{\mathcal{K}}$ composed of sufficiently few columns from $F$ have
$F_{\mathcal{K}}^{*}F_{\mathcal{K}}$ particularly close to the identity
matrix, meaning $F_{\mathcal{K}}^{*}F_{\mathcal{K}}$ is particularly well-
conditioned. The key difference between NERFs and RIP matrices is that well-
conditioned NERF submatrices $F_{\mathcal{K}}$ have $K:=|\mathcal{K}|\geq M$
columns, whereas $F_{\mathcal{K}}$ has fewer than $M$ columns in the RIP case.
Regardless, in constructing NERFs, we can exploit some intuition from the
construction of RIP matrices. In particular, the RIP matrices which support
the largest sparsity levels to date arise from random processes. As an
example, one may draw the entries independently from a Gaussian distribution
of mean zero and variance $\frac{1}{M}$; this was originally established in
Lemma 3.1 of [13]. What follows is the analogous NERF result:
###### Theorem 2.
Fix $\varepsilon>0$ and pick an $M\times N$ frame $F$ by drawing each entry
independently from a standard normal distribution. Then $F$ is a
$(p,C)$-numerically erasure-robust frame with overwhelming probability
provided
$\sqrt{\frac{M}{N}}\leq\frac{C-1}{C+1}\sqrt{1-p}-\sqrt{\varepsilon+2p(1-\log
p)}.$ (1)
Note that (1) requires its right-hand side to be positive, which in turn
implies
$\sqrt{1-p}-\sqrt{2p(1-\log p)}>0.$
This occurs whenever $p\leq 0.1460$. That is, the random construction in
Theorem 2 is numerically robust to erasure rates of up to almost 15%. However,
approaching a 15% erasure rate while satisfying (1) will admittedly cost a
large worst-case condition number $C$ along with high redundnacy $\frac{N}{M}$
in the frame. Still, Theorem 2 provides a useful guarantee. For example, a
Gaussian matrix of redundancy $\frac{N}{M}=5$ will, with overwhelming
probability, be robust to 1% erasures with a worst-case condition number of
10. We proceed with the proof:
###### Proof of Theorem 2.
Pick $\mathcal{K}\subseteq\\{1,\ldots,N\\}$ of size $K=(1-p)N$. Note the
assumption (1) implies that $\frac{M}{N}\leq 1-p$ and so $K=(1-p)N\geq M$. As
such, Theorem II.13 of [14] gives bounds on the singular values of the random
“tall” $K\times M$ matrix $F_{\mathcal{K}}^{*}$:
$\mathrm{Pr}\big{[}\sqrt{K}-\sqrt{M}-t\leq\sigma_{\mathrm{min}}(F_{\mathcal{K}}^{*})\leq\sigma_{\mathrm{max}}(F_{\mathcal{K}}^{*})\leq\sqrt{K}+\sqrt{M}+t\big{]}\geq
1-2\mathrm{e}^{-t^{2}/2}\qquad\forall t\geq 0.$
This probabilistic bound on the extreme singular values implies
$\mathrm{Pr}\bigg{[}\mathrm{Cond}(F_{\mathcal{K}})\leq\frac{\sqrt{K}+\sqrt{M}+t}{\sqrt{K}-\sqrt{M}-t}\bigg{]}\geq
1-2\mathrm{e}^{-t^{2}/2}\qquad\forall t\geq 0.$
Taking a union bound over all
$\binom{N}{K}=\binom{N}{N-K}\leq(\frac{\mathrm{e}N}{N-K})^{N-K}$ choices for
$\mathcal{K}$ gives
$\displaystyle\mathrm{Pr}\bigg{[}\exists\mathcal{K}\mbox{ s.t.
}\mathrm{Cond}(F_{\mathcal{K}})>\frac{\sqrt{K}+\sqrt{M}+t}{\sqrt{K}-\sqrt{M}-t}\bigg{]}$
$\displaystyle\leq\binom{N}{N-K}2e^{-t^{2}/2}$ $\displaystyle\leq
2\exp\bigg{(}-\frac{t^{2}}{2}+(N-K)\log\frac{\mathrm{e}N}{N-K}\bigg{)}$
$\displaystyle=2\exp\bigg{(}-\frac{t^{2}}{2}+Np\log\frac{\mathrm{e}}{p}\bigg{)}\qquad\forall
t\geq 0.$ (2)
Now pick $t$ such that $C=\frac{\sqrt{K}+\sqrt{M}+t}{\sqrt{K}-\sqrt{M}-t}$,
namely $t=\sqrt{N}(\frac{C-1}{C+1}\sqrt{1-p}-\sqrt{\frac{M}{N}})$. Note that
(1) implies $t\geq 0$, and so we may substitute it into (2) and simplify the
result:
$\displaystyle\mathrm{Pr}\big{[}\exists\mathcal{K}\mbox{ s.t.
}\mathrm{Cond}(F_{\mathcal{K}})>C\big{]}$ $\displaystyle\leq
2\exp\Bigg{[}-\frac{N}{2}\Bigg{(}\bigg{(}\frac{C-1}{C+1}\sqrt{1-p}-\sqrt{\frac{M}{N}}\bigg{)}^{2}-2p(1-\log
p)\Bigg{)}\Bigg{]}$ $\displaystyle\leq 2\mathrm{e}^{-N\varepsilon/2}.$
Thus, the probability of $F$ not being a $(p,C)$-NERF is
$\mathrm{O}(N^{-\alpha})$ for every fixed $\alpha$, meaning $F$ is a
$(p,C)$-NERF with overwhelming probability. ∎
### 2.2. Equiangular tight frames
The previous subsection constructed a random family of numerically erasure-
robust frames by following intuition from known constructions of matrices with
the restricted isometry property. Indeed, state-of-the-art RIP matrices are
built according to random processes, while deterministic constructions have
found less success [5]. In this subsection, the analogy between RIP matrices
and NERFs will break down, as we will construct deterministic NERFs which
outperform the random counterparts with much larger erasure rates, albeit at
the price of high redundancy.
In [17], Holmes and Paulsen show that frames of pairwise dissimilar unit-norm
vectors are robust to two erasures. This dissimilarity is measured in terms of
worst-case coherence, which is known to satisfy the Welch bound:
###### Theorem 3 (Welch bound [25]).
Every $M\times N$ frame $\\{f_{n}\\}_{n=1}^{N}$ of unit-norm vectors has
worst-case coherence
$\max_{\begin{subarray}{c}n,n^{\prime}\in\\{1,\ldots,N\\}\\\ n\neq
n^{\prime}\end{subarray}}|\langle
f_{n},f_{n^{\prime}}\rangle|\geq\sqrt{\frac{N-M}{M(N-1)}}.$
Specifically, Proposition 2.2 of [17] gives that minimizers of worst-case
coherence are optimally robust to two erasures. For many values of $M$ and
$N$, there exist frames which achieve equality in the Welch bound. In fact, a
sequence of unit-norm vectors $F=\\{f_{n}\\}_{n=1}^{N}$ achieves the Welch
bound if and only if it is an equiangular tight frame (ETF), meaning that it
is a tight frame (i.e., $FF^{*}=A\mathrm{I}_{M}$) which also satisfies the
equiangularity condition that $|\langle f_{n},f_{n^{\prime}}\rangle|$ is
constant over all choices of $n\neq n^{\prime}$ [22]. Not only are ETFs
minimizers of worst-case coherence, they also have combinatorial symmetries
related to strongly regular graphs, difference sets and Steiner systems; these
combinatorial structures have each been used to build the only general ETF
constructions to date [15, 24, 26].
In this subsection, we consider an ETF construction based on a particular
difference set. Let $q$ be a prime power, take $M=q+1$ and $N=q^{2}+q+1$, and
consider the trace map
$\mathrm{Tr}:\mathbb{F}_{q^{3}}\rightarrow\mathbb{F}_{q}$ defined by
$\mathrm{Tr}(\beta)=\beta+\beta^{q}+\beta^{q^{2}}$. Given a generator $\alpha$
of the multiplicative group of $\mathbb{F}_{q^{3}}$, define the $M$-element
subset $\mathcal{M}\subseteq\mathbb{Z}_{N}$ by
$\mathcal{M}=\\{t:\mathrm{Tr}(\alpha^{t})=0\\}$. By construction,
$\mathcal{M}$ has the property that every nonzero member of $\mathbb{Z}_{N}$
can be uniquely expressed as the difference of two elements of $\mathcal{M}$;
this set is called the $(N,M,1)$-Singer difference set [18]. As shown in [26],
any difference set $\mathcal{M}\subseteq\mathbb{Z}_{N}$ can be used to build
an ETF by taking rows from the $N\times N$ discrete Fourier transform matrix
which are indexed by members of $\mathcal{M}$ and then normalizing the
resulting columns. This construction has the following guarantee:
###### Theorem 4.
Take $M=q+1$ and $N=q^{2}+q+1$ for some prime power $q$, and let $F$ be the
$M\times N$ equiangular tight frame $F$ constructed from the $(N,M,1)$-Singer
difference set, as in [26]. Then $F$ is a $(p,C)$-numerically erasure-robust
frame for every $p\leq\frac{1}{2}-\frac{C^{2}}{C^{4}+1}$.
This result essentially states that such ETFs are numerically robust to
erasure rates of up to 50%. Compared to the random construction of the
previous section, which required less than 15% erasures, this is quite an
improvement. Certainly, the frame redundancy $\frac{N}{M}$ is unbounded in
this case since $N$ scales as $M^{2}$, but the reward is significant. For
example, such ETFs are robust to 49% erasures with a worst-case condition
number of 10. Meanwhile, for $N\gg M$, Theorem 2 only guarantees—with
overwhelming probability—a worst-case condition number of 10 when less than 9%
of the frame is erased.
###### Proof of Theorem 4.
Pick some $\mathcal{K}\subseteq\\{1,\ldots,N\\}$ of size $K=(1-p)N$, and let
$\\{\lambda_{\mathcal{K};m}\\}_{m=1}^{M}$ denote the eigenvalues of
$F_{\mathcal{K}}F_{\mathcal{K}}^{*}$. Taking
$\delta_{\mathcal{K}}:=\max_{m}|\frac{M}{K}\lambda_{\mathcal{K};m}-1|$, we
have
$\big{(}\mathrm{Cond}(F_{\mathcal{K}})\big{)}^{2}=\mathrm{Cond}(F_{\mathcal{K}}F_{\mathcal{K}}^{*})=\frac{\lambda_{\mathrm{max}}(F_{\mathcal{K}}F_{\mathcal{K}}^{*})}{\lambda_{\mathrm{min}}(F_{\mathcal{K}}F_{\mathcal{K}}^{*})}\leq\frac{1+\delta_{\mathcal{K}}}{1-\delta_{\mathcal{K}}}$
(3)
provided $\delta_{\mathcal{K}}<1$; if $\delta_{\mathcal{K}}\geq 1$, then
$F_{\mathcal{K}}$ could be rank deficient. Moreover, the fact that
$F_{\mathcal{K}}F_{\mathcal{K}}^{*}$ and $\mathrm{I}_{M}$ are simultaneously
diagonalizable implies
$\delta_{\mathcal{K}}^{2}=\tfrac{M^{2}}{K^{2}}\max_{m\in\\{1,\ldots,M\\}}|\lambda_{\mathcal{K};m}-\tfrac{K}{M}|^{2}\leq\tfrac{M^{2}}{K^{2}}\sum_{m=1}^{M}|\lambda_{\mathcal{K};m}-\tfrac{K}{M}|^{2}=\tfrac{M^{2}}{K^{2}}\mathrm{Tr}[(F_{\mathcal{K}}F_{\mathcal{K}}^{*}-\tfrac{K}{M}\mathrm{I}_{M})^{2}].$
(4)
From here, the cyclic property of the trace and the fact that $F$ has unit-
norm columns give
$\displaystyle\mathrm{Tr}[(F_{\mathcal{K}}F_{\mathcal{K}}^{*}-\tfrac{K}{M}\mathrm{I}_{M})^{2}]$
$\displaystyle=\mathrm{Tr}[(F_{\mathcal{K}}F_{\mathcal{K}}^{*})^{2}]-\tfrac{2K}{M}\mathrm{Tr}[F_{\mathcal{K}}F_{\mathcal{K}}^{*}]+\tfrac{K^{2}}{M^{2}}\mathrm{Tr}[\mathrm{I}_{M}]$
$\displaystyle=\mathrm{Tr}[(F_{\mathcal{K}}^{*}F_{\mathcal{K}})^{2}]-\tfrac{2K}{M}\mathrm{Tr}[F_{\mathcal{K}}^{*}F_{\mathcal{K}}]+\tfrac{K^{2}}{M}$
$\displaystyle=\sum_{k\in\mathcal{K}}\sum_{k^{\prime}\in\mathcal{K}}|\langle
f_{k},f_{k^{\prime}}\rangle|^{2}-\tfrac{K^{2}}{M}.$ (5)
Since $F$ is an ETF, the inner products between distinct frame elements
achieve equality in the Welch bound: $|\langle
f_{k},f_{k^{\prime}}\rangle|^{2}=\frac{N-M}{M(N-1)}$ for every $k\neq
k^{\prime}$. Applying this to (5) and substituting into (4) then gives
$\delta_{\mathcal{K}}^{2}\leq\frac{M^{2}}{K^{2}}\bigg{(}K+K(K-1)\frac{N-M}{M(N-1)}-\frac{K^{2}}{M}\bigg{)}=\frac{M(M-1)(N-K)}{K(N-1)}=\frac{pM(M-1)}{(1-p)(N-1)}.$
(6)
According to the theorem statement, $N=M^{2}-M+1$ and
$p\leq\frac{1}{2}-\frac{C^{2}}{C^{4}+1}$, and so
$\delta_{\mathcal{K}}^{2}\leq\frac{p}{1-p}\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}}.$
Substituting this into (3) therefore gives $\mathrm{Cond}(F_{\mathcal{K}})\leq
C$. ∎
We note that (6) together with the necessary condition
$\delta_{\mathcal{K}}^{2}<1$ indicate that of all $M\times N$ ETFs, the above
proof technique will only work for those with $N=\Omega(M^{2})$ frame
elements. However, as noted in Proposition 2.3 of [4], $M\times N$ ETFs
necessarily have $N\leq M^{2}$, and so the ETFs for which the above proof can
demonstrate NERF are asymptotically maximal. A long-standing open problem in
frame theory concerns the existence of $M\times N$ ETFs with $N=M^{2}$, or
maximal ETFs, and it is easy to verify that Theorem 4 also holds for this
conjectured family; to date, these are only known to exist for finitely many
$M$’s [3]. As for asymptotically maximal ETFs, the difference set construction
of Theorem 4 is the only such infinite family known to the authors.
Regardless, a version of Theorem 4 holds for every family of asymptotically
maximal ETFs, which follows directly from (6):
###### Theorem 5.
Every $M\times N$ equiangular tight frame with $\frac{N-1}{M(M-1)}\geq\alpha$
is a $(p,C)$-numerically erasure-robust frame for every
$p\leq\frac{\alpha(C^{2}-1)^{2}}{\alpha(C^{2}-1)^{2}+(C^{2}+1)^{2}}$.
Since maximal ETFs are particularly difficult to construct, different fields
have turned to mutually unbiased bases (MUBs) to fill their need for large
frames with low coherence [19, 22]. There are several $M\times M^{2}$ MUB
constructions, all of which have the property that the inner product between
any two columns is of size $0$ or $1/\sqrt{M}$ [2, 9, 19]. As the Welch bound
in this case is $1/\sqrt{M+1}$, MUBs are “almost” ETFs. It is therefore
surprising that the above proof techniques fail to show that MUBs are NERFs.
To illustrate this fact, we consider the MUB version of (6):
$\delta_{\mathcal{K}}^{2}\leq\frac{M^{2}}{K^{2}}\bigg{(}K+K(K-1)\frac{1}{M}-\frac{K^{2}}{M}\bigg{)}=\frac{M(M-1)}{K}=\frac{M-1}{(1-p)M}.$
(7)
Due to the necessity of $\delta_{\mathcal{K}}<1$, this bound will not be
useful unless $p<\frac{1}{M}$. However, even in this case, substituting (7)
into (3) gives
$\big{(}\mathrm{Cond}(F_{\mathcal{K}})\big{)}^{2}\leq\frac{1+\delta_{\mathcal{K}}}{1-\delta_{\mathcal{K}}}\leq\frac{\sqrt{(1-p)M}+\sqrt{M-1}}{\sqrt{(1-p)M}-\sqrt{M-1}}.$
(8)
Further since $0\leq p\leq\frac{1}{M}$, separately bounding the numerator and
denominator gives that the right-hand side of (8) is always at least
$2\sqrt{M-1}$, meaning (8) says very little about the worst-case condition
number, regardless of the erasure rate.
It remains to be seen whether this is a true distinction between ETFs and MUBs
or is instead an artifact of our proof techniques. One way to improve this
analysis is to find a better bound on the frame potential (5), see [7]. To be
clear, we can certainly bound it in general using worst-case coherence, and
such a bound is tight whenever the frame is equiangular. However, when the
frame is not equiangular, this bound is less than optimal. For a better bound
in the general case, suppose that for every $n\in\\{1,\ldots,N\\}$, the
distribution of the squares of inner products $\\{|\langle
f_{n},f_{n^{\prime}}\rangle|^{2}\\}_{n^{\prime}=1}^{N}$ is identical. In this
case, let $d_{F}\in\mathbb{R}^{N}$ denote the common sequence of squared inner
products, sorted in nonincreasing order. We can then bound the sum in (5) by
exploiting this structure:
$\sum_{k\in\mathcal{K}}\sum_{k^{\prime}\in\mathcal{K}}|\langle
f_{k},f_{k^{\prime}}\rangle|^{2}\leq K\sum_{k=1}^{K}d_{F}[k].$ (9)
Combining bounds (4), (5) and (9) then yields
$\delta_{\mathcal{K}}^{2}\leq\frac{M^{2}}{K^{2}}\bigg{(}\sum_{k\in\mathcal{K}}\sum_{k^{\prime}\in\mathcal{K}}|\langle
f_{k},f_{k^{\prime}}\rangle|^{2}-\frac{K^{2}}{M}\bigg{)}\leq
M^{2}\bigg{(}\frac{1}{K}\sum_{k=1}^{K}d_{F}[k]-\frac{1}{M}\bigg{)}.$ (10)
In particular, in order to use (10) to guarantee $\delta_{\mathcal{K}}<1$, we
want the average of the $K$ largest values of $d_{F}[k]$ to be close to
$\frac{1}{M}$. Further note that if $F$ is a unit norm tight frame, which
necessarily has tight frame constant $A=\frac{N}{M}$, then the average of all
values of $d_{F}[k]$ is $\frac{1}{M}$:
$\frac{1}{N}\sum_{k=1}^{N}d_{F}[k]=\frac{1}{N}\sum_{n=1}^{N}|\langle
f_{n},f_{n^{\prime}}\rangle|^{2}=\frac{1}{N}\frac{N}{M}\|f_{n^{\prime}}\|^{2}=\frac{1}{M}.$
In such cases, using (10) to estimate the NERF properties of a given frame
reduces to finding how quickly (as a function of $K$) the average of the $K$
largest values of $d_{F}[k]$ converges to the average of all of its values.
With this refined analysis, we can prove that MUBs are actually NERFs. We note
that the bound (9) is identical to the worst-case coherence bound unless $K$
is large, since $d_{F}$ in this case has one copy of $1$, $M(M-1)$ copies of
$\frac{1}{M}$, and $M-1$ copies of $0$ [2, 9, 19]. Indeed, analysis with (9)
can only show that MUBs are NERFs when the erasure rate is small:
###### Theorem 6.
An $M\times M^{2}$ frame of mutually unbiased bases is a $(p,C)$-numerically
erasure-robust frame for every
$p\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}(M+1)}$.
Note that the above guarantee is not nearly as good as the one we got for
ETFs, or even for random frames. However, the result is still of some use; for
example, when $M$ is sufficiently large, removing any $0.96M$ of the $M^{2}$
frame vectors will leave a submatrix of condition number smaller than $10$.
###### Proof of Theorem 6.
Applying (10) to the distribution $d_{F}$ of the $M\times M^{2}$ MUB yields
$\delta_{\mathcal{K}}^{2}\leq\frac{M^{2}}{K}\bigg{(}\sum_{k=1}^{K}d_{F}[k]-\frac{K}{M}\bigg{)}\\\
=\frac{M^{2}}{K}\bigg{(}1+M(M-1)\frac{1}{M}-\frac{K}{M}\bigg{)}\\\
=\frac{M(M^{2}-K)}{K}.$
Since $K=(1-p)N$ and $N=M^{2}$, we can simplify and apply
$p\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}(M+1)}$ to get
$\displaystyle\delta_{\mathcal{K}}^{2}\leq\frac{pM}{1-p}$
$\displaystyle\leq\frac{(C^{2}-1)^{2}M}{(C^{2}+1)^{2}(M+1)-(C^{2}-1)^{2}}$
$\displaystyle\leq\frac{(C^{2}-1)^{2}M}{(C^{2}+1)^{2}(M+1)-(C^{2}+1)^{2}}=\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}}.$
(11)
Substituting this into (3) therefore gives $\mathrm{Cond}(F_{\mathcal{K}})\leq
C$. ∎
### 2.3. Group frames
In the previous subsection, we demonstrated that mutually unbiased bases are
NERFs by exploiting an important property: the distribution of the squares of
inner products $\\{|\langle
f_{n},f_{n^{\prime}}\rangle|^{2}\\}_{n^{\prime}=1}^{N}$ is identical for every
$f_{n}$. In this subsection, we will consider a much larger class of unit norm
tight frames that enjoy this identical distribution property: group frames.
Given a seed vector $f\in\mathbb{C}^{M}$ and a finite subgroup $G$ of the
group of all $M\times M$ unitary matrices, the corresponding group frame is
the orbit $\\{Uf\\}_{U\in G}$ of $f$ under the action of this group, though
$\\{Uf\\}_{U\in G}$ should only be called a frame if the $Uf$’s span. In fact,
if $\|f\|=1$, then $\\{Uf\\}_{U\in G}$ will be a unit norm tight frame
provided the group $G$ is irreducible, meaning that for any nonzero
$x\in\mathbb{C}^{M}$ the vectors $\\{Ux\\}_{U\in G}$ necessarily span
$\mathbb{C}^{M}$; for this and other interesting facts about group frames, see
[23]. Note that for any $U,U^{\prime}\in G$,
$\langle Uf,U^{\prime}f\rangle=\langle f,U^{*}U^{\prime}f\rangle=\langle
f,U^{-1}U^{\prime}f\rangle.$
Since each $U^{-1}$ acts as a permutation on $G$, we conclude that $\\{\langle
Uf,U^{\prime}f\rangle\\}_{U^{\prime}\in G}$ is a permutation of $\\{\langle
f,U^{\prime}f\rangle\\}_{U^{\prime}\in G}$, thereby confirming our above claim
that each row of the Gram matrix $F^{*}F$ is identically distributed.
To illustrate the usefulness of group frame ideas in estimating
$\delta_{\mathcal{K}}$ with (10), we will apply it to group frames generated
by the symmetric group of the simplex. First, we define a (regular) simplex to
be any $M\times(M+1)$ matrix $\Psi$ whose $(M+1)\times(M+1)$ Gram matrix is
$\Psi^{*}\Psi=\frac{M+1}{M}\mathrm{I}_{M+1}-\frac{1}{M}\mathrm{J}_{M+1}$,
where $\mathrm{J}_{M+1}$ denotes an $(M+1)\times(M+1)$ matrix of ones. Notice
that the spectrum of $\Psi^{*}\Psi$ consists of $M$ copies of $\frac{M+1}{M}$
and one value of $0$; since this is a zero-padded version of the spectrum of
the $M\times M$ frame operator $\Psi\Psi^{*}$, we conclude that
$\Psi\Psi^{*}=\frac{M+1}{M}\mathrm{I}_{M}$, meaning $\Psi$ is a tight frame.
In fact, since the off-diagonal entries of $\Psi^{*}\Psi$ are all equal in
size (to the Welch bound), $\Psi$ is an equiangular tight frame.
The simplex plays an important role in finite frame theory. Indeed, the
Mercedes-Benz frame and the vertices of the tetrahedron, being 2- and
3-dimensional realizations of the simplex, serve as fundamental examples of
frames [7, 23]. Simplices can also be easily expressed in higher dimensions by
removing the row of 1’s from an $(M+1)\times(M+1)$ discrete Fourier transform
matrix or Hadamard matrix and then normalizing the resulting columns. This
representation of simplices plays a key role in the construction of Steiner
ETFs [15]. In this paper, we are specifically interested in the symmetries of
the simplex. In general, the symmetry group of a frame is the set of all
matrices which, when acting on frame elements, permute them. The following
result gives a particularly nice description of the symmetry group of the
simplex:
###### Lemma 7.
The symmetry group of an $M\times(M+1)$ regular simplex $\Psi$ is the set of
all matrices of the form $U=\frac{M}{M+1}\Psi P\Psi^{*}$, where $P$ is an
$(M+1)\times(M+1)$ permutation matrix.
###### Proof.
The symmetry group of $\Psi$ is the set of all matrices $U$ for which there
exists a permutation matrix $P$ such that $U\Psi=\Psi P$. Note this implies
$U\Psi\Psi^{*}=\Psi P\Psi^{*}$ which, since
$\Psi\Psi^{*}=\frac{M+1}{M}\mathrm{I}_{M}$, further implies
$U=\frac{M}{M+1}\Psi P\Psi^{*}$. In other words, for each member $U$ of the
symmetry group of $\Psi$, there is a unique permutation matrix $P$ such that
$U\Psi=\Psi P$. Thus, all that remains to be shown is that for each
permutation matrix $P$, the matrix $U=\frac{M}{M+1}\Psi P\Psi^{*}$ satisfies
$U\Psi=\Psi P$. To this end, note
$U\Psi=\tfrac{M}{M+1}\Psi P\Psi^{*}\Psi=\tfrac{M}{M+1}\Psi
P(\tfrac{M+1}{M}\mathrm{I}_{M+1}-\tfrac{1}{M}\mathrm{J}_{M+1})=\Psi
P-\tfrac{1}{M+1}\Psi P\mathrm{J}_{M+1}.$
It therefore suffices to show that $\Psi P\mathrm{J}_{M+1}=0$. To do this,
factor $J_{M+1}$ as an outer product of an all-ones vector with itself, a
vector which happens to be preserved by permutations: $\Psi
P\mathrm{J}_{M+1}=\Psi P1_{M+1}1_{M+1}^{*}=\Psi 1_{M+1}1_{M+1}^{*}$. Then note
that $\Psi 1_{M+1}=0$:
$\|\Psi 1_{M+1}\|^{2}=1_{M+1}^{*}\Psi^{*}\Psi
1_{M+1}=1_{M+1}^{*}(\tfrac{M+1}{M}\mathrm{I}_{M+1}-\tfrac{1}{M}1_{M+1}1_{M+1}^{*})1_{M+1}=0.\qed$
From Lemma 7, we can deduce that the symmetry group of an $M\times(M+1)$
simplex $\Psi$ is the symmetric group on $M+1$ letters, and so we denote it by
$S_{M+1}$. We are interested in the frames formed by applying the $(M+1)!$
members of $S_{M+1}$ to unit vectors. We claim that such frames are
automatically unit norm tight frames. Moreover, motivated by (10), we further
seek the distribution $d_{F}$ of the squared-moduli of the inner products of
the frame elements with each other.
Here, it is helpful to note that $\Phi^{*}:=\sqrt{M/(M+1)}\Psi^{*}$ is a
unitary transformation between $\mathbb{C}^{M}$ and the $M$-dimensional
orthogonal complement $1_{M+1}^{\perp}$ of the $(M+1)$-dimensional all-ones
vector; the proof of this fact is straightforward and is not included here.
Indeed, writing any unit-norm vector $f\in\mathbb{C}^{M}$ as $f=\Phi g$ where
$g\in 1_{M+1}^{\perp}$ has $\|g\|=1$, we have inner products of the form:
$\langle f,Uf\rangle=\langle f,\tfrac{M}{M+1}\Psi
P\Psi^{*}f\rangle=\langle\Phi^{*}f,P\Phi^{*}f\rangle=\langle g,Pg\rangle.$
(12)
Moreover, as noted above, our group frame will be tight provided that for any
$x\neq 0$ the following vectors span $\mathbb{C}^{M}$:
$\\{Ux\\}_{U\in G}=\\{\tfrac{M}{M+1}\Psi P\Psi^{*}x\\}_{P\in S_{M+1}}=\\{\Phi
P\Phi^{*}x\\}_{P\in S_{M+1}},$
which is equivalent to having that $\\{Py\\}_{P\in S_{M+1}}$ spans
$1_{M+1}^{\perp}$ for any nonzero $y\in 1_{M+1}^{\perp}$. This in turn is
equivalent to showing that $z=0$ is the only choice of $z\in 1_{M+1}^{\perp}$
for which $\langle z,Py\rangle=0$ for all permutations $P$. To do this, fix
any indices $n_{1}\neq n_{2},n_{3}\neq n_{4}$ from $\\{1,\dotsc,M+1\\}$, and
consider the zero inner product $\langle z,P_{1}y\rangle$ that arises from any
permutation $P_{1}$ which takes $n_{3}$ to $n_{1}$ and $n_{4}$ to $n_{2}$.
From this, now subtract the zero inner product from a permutation $P_{2}$
which is identical to $P_{1}$, except that it takes $n_{3}$ to $n_{2}$ and
$n_{4}$ to $n_{1}$:
$\displaystyle 0$ $\displaystyle=\langle z,P_{1}y\rangle-\langle
z,P_{2}y\rangle$
$\displaystyle=z[n_{1}]\overline{y[n_{3}]}+z[n_{2}]\overline{y[n_{4}]}-z[n_{1}]\overline{y[n_{4}]}-z[n_{2}]\overline{y[n_{3}]}$
$\displaystyle=(z[n_{1}]-z[n_{2}])\overline{(y[n_{3}]-y[n_{4}])}.$ (13)
Now, since $0\neq y\in 1_{M+1}^{\perp}$ we have that $y$ is a nonzero vector
whose entries sum to zero, and so in particular there exists indices $n_{3}$
and $n_{4}$ such that $y[n_{3}]-y[n_{4}]\neq 0$. As such, (13) implies that
$z[n_{1}]=z[n_{2}]$ for every choice of $n_{1}\neq n_{2}$, namely that the
entries of $z$ are all equal. Since $z\in 1_{M+1}^{\perp}$, this means $z=0$
as claimed. We summarize these facts below:
###### Theorem 8.
Let $\Psi$ be an $M\times(M+1)$ matrix whose unit columns form a regular
simplex in $\mathbb{C}^{M}$. Let $f=\sqrt{M/(M+1)}\Psi g$, where $g$ is any
unit-norm vector $g\in\mathbb{C}^{M+1}$ whose entries sum to zero. Then the
group frame
$\\{Uf\\}_{U\in G}:=\\{\tfrac{M}{M+1}\Psi P\Psi^{*}f\\}_{P\in S_{M+1}}$
is a unit norm tight frame of $(M+1)!$ elements for $\mathbb{C}^{M}$.
Moreover, each row of the Gram matrix of this frame has entries of the form
$\\{\langle f,Uf\rangle\\}_{U\in G}=\\{\langle g,Pg\rangle\\}_{P\in S_{M+1}}$.
Here, $P$ ranges over all $(M+1)\times(M+1)$ permutation matrices.
We now use these ideas to construct a frame to be used in conjunction with the
bound (10), where $d_{F}[k]$ denotes the $k$th largest value of the form
$|\langle f,Uf\rangle|^{2}=|\langle g,Pg\rangle|^{2}$. In particular, our goal
is to find a unit norm vector $g\in 1_{M+1}^{\perp}$ for which the average of
the $K$ largest values of $d_{F}[k]$ is very close to the average of all of
its values: $\frac{1}{M}$.
Moreover, considering the underlying application of NERFs, we prefer not to
transmit as many as $(M+1)!$ frame coefficients to convey an $M$-dimensional
signal. For this reason, we seek vectors $g$ which are fixed by a large
subgroup of permutation matrices, namely, vectors with large level sets; this
way, we can get away with only using representatives of distinct cosets of
this large subgroup. In this paper, we only consider vectors of two level
sets, say
$g=(\underbrace{a,a,\ldots,a}_{L\mbox{\tiny{
times}}},\\!\\!\underbrace{b,b,\ldots,b}_{M+1-L\mbox{\tiny{ times}}}\\!\\!).$
(14)
Choosing $g$ in this way guarantees that the corresponding group frame only
has $\binom{M+1}{L}$ distinct elements. Moreover, since each of these unique
elements appears the same number of times, namely $L!(M+1-L)!$ times, the
$\binom{M+1}{L}$-element subframe is still tight.
To estimate the NERF properties of such frames using (10), we first need to
find explicit expressions for $a$ and $b$. Here, the condition $\langle
g,1_{M+1}\rangle=0$ implies $La+(M+1-L)b=0$. Combining this with the fact that
$g$ has unit norm then gives
$a=\sqrt{\frac{M+1-L}{(M+1)L}},\qquad b=-\sqrt{\frac{L}{(M+1)(M+1-L)}},$ (15)
where we take $a>0$ without loss of generality. Next, note that $\langle
g,Pg\rangle$ is completely determined by the number $J$ of indices $n$ for
which $g[n]=(Pg)[n]=a$. This leads to the following calculation:
$\langle f,Uf\rangle=\langle
g,Pg\rangle=Ja^{2}+2(L-J)ab+(M+1+J-2L)b^{2}=\frac{J(M+1)-L^{2}}{L(M+1-L)}.$
Moreover, of the $\binom{M+1}{L}$ distinct $Uf$’s in this construction, there
are $\binom{L}{J}\binom{M+1-L}{L-J}$ which produce the above inner product,
since $J$ of the $a$’s in $Pg$ must align with $a$’s in $g$, while the other
$L-J$ $a$’s in $Pg$ align with $b$’s in $g$. In the special case where $g$ has
$L=2$ $a$’s, we have a total of $\binom{M+1}{2}$ distinct $Uf$’s, and the
distribution of inner products is given by
$\\{\langle f,Uf\rangle\\}=\left\\{\begin{array}[]{cl}1&\mbox{with
multiplicity }1,\\\ \frac{M-3}{2(M-1)}&\mbox{with multiplicity }2(M-1),\\\
-\frac{2}{M-1}&\mbox{with multiplicity
}\frac{1}{2}(M-1)(M-2).\end{array}\right.$ (16)
As verified below, substituting this fact into (10) yields the following
result:
###### Theorem 9.
Pick $M\geq 7$ and consider the $M\times\binom{M+1}{2}$ frame $F$ with columns
of the form $\sqrt{M/(M+1)}\Psi Pg$, where $\Psi$ is an $M\times(M+1)$ regular
simplex and the $Pg$’s are distinct permutations of $g$, which is defined by
(14) and (15) with $L=2$. Then $F$ is a $(p,C)$-numerically erasure-robust
frame for every $p\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}(M+1)}$.
The above guarantee bears a striking resemblance to Theorem 6, despite the
distribution $d_{F}$ being significantly different. Again, while this result
is not nearly as good as the ones we got for ETFs or random frames, it still
gives something; for example, removing any $0.48M$ of the $\binom{M+1}{2}$
frame vectors will leave a submatrix of condition number smaller than 10. As
one would expect, there are similar NERF results for the frames that
correspond to larger values of $L$, but we do not report them here.
###### Proof of Theorem 9.
Since $M\geq 7$, the sizes of the inner products in (16) are nonincreasing,
and so $d_{F}$ is defined accordingly. Also, taking $K=(1-p)N$ with
$p\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}(M+1)}\leq\frac{1}{M+1}$, we claim
that $K\geq 2(M-1)+1$. Indeed,
$K\geq\Big{(}1-\frac{1}{M+1}\Big{)}N=\frac{M^{2}}{2}\geq 2(M-1)+1,$
where the last inequality follows from $M\geq 7\geq 2+\sqrt{2}$. Since $K\geq
2(M-1)+1$, then applying (10) to (16) yields
$\displaystyle\delta_{\mathcal{K}}^{2}$
$\displaystyle\leq\frac{M^{2}}{K}\bigg{(}\sum_{k=1}^{K}d_{F}[k]-\frac{K}{M}\bigg{)}$
$\displaystyle=\frac{M^{2}}{K}\bigg{(}1+2(M-1)\Big{(}\frac{M-3}{2(M-1)}\Big{)}^{2}+\big{(}K-(2M-1)\big{)}\Big{(}\frac{2}{M-1}\Big{)}^{2}-\frac{K}{M}\bigg{)}$
$\displaystyle=\bigg{(}\frac{M(M+1)-2K}{2K}\bigg{)}\bigg{(}\frac{M(M^{2}-6M+1)}{(M-1)^{2}}\bigg{)}.$
Since $K=(1-p)N$ and $N=\binom{M+1}{2}$, we can simplify to get
$\delta_{\mathcal{K}}^{2}\leq\frac{pM(M^{2}-6M+1)}{(1-p)(M-1)^{2}}\leq\frac{pM}{1-p}.$
From here, $p\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}(M+1)}$ and (11) together
imply $\delta_{\mathcal{K}}^{2}\leq\frac{(C^{2}-1)^{2}}{(C^{2}+1)^{2}}$, which
we substitute into (3) to conclude that $\mathrm{Cond}(F_{\mathcal{K}})\leq
C$. ∎
## 3\. Limiting our expectations
The previous section gave four different constructions of numerically erasure-
robust frames. The last three constructions were deterministic, and their
proofs hinged on how coherent a subcollection of frame vectors can be. In this
section, we shed some light on the fundamental limits of NERFs by again
considering the coherence of frame subcollections. We start with the following
lemma, which says that a matrix with similar columns will have a large
condition number:
###### Lemma 10.
Take an $M\times N$ matrix $F$ with unit-norm columns. Then for every unit
vector $x\in\mathbb{R}^{M}$,
$\big{(}\mathrm{Cond}(F)\big{)}^{2}\geq\frac{(M-1)\|F^{*}x\|^{2}}{N-\|F^{*}x\|^{2}}.$
###### Proof.
First, we have
$\lambda_{\mathrm{max}}(FF^{*})=\|F^{*}\|_{2}^{2}\geq\|F^{*}x\|^{2}$. Next,
take $\\{x_{m}\\}_{m=1}^{M}$ to be some orthonormal basis with $x_{1}=x$. Then
$\lambda_{\mathrm{min}}(FF^{*})\leq\|F^{*}x_{m}\|^{2}$ for every $m$, and so
averaging over $m=2,\ldots,M$ gives
$\lambda_{\mathrm{min}}(FF^{*})\leq\frac{1}{M-1}\sum_{m=2}^{M}\|F^{*}x_{m}\|^{2}=\frac{1}{M-1}\sum_{n=1}^{N}\sum_{m=2}^{M}|\langle
x_{m},f_{n}\rangle|^{2}.$
Since each $f_{n}$ has unit norm and $\\{x_{m}\\}_{m=1}^{M}$ is an orthonormal
basis with $x_{1}=x$, we continue:
$\lambda_{\mathrm{min}}(FF^{*})\leq\frac{1}{M-1}\sum_{n=1}^{N}\Big{(}1-|\langle
x,f_{n}\rangle|^{2}\Big{)}=\frac{N-\|F^{*}x\|^{2}}{M-1}.$
Combining this with our lower bound on $\lambda_{\mathrm{max}}(FF^{*})$ gives
the result. ∎
To be explicit, the lower bound in Lemma 10 is exceedingly large when the
columns of $F$ each have a large inner product with $x$. We now use this lemma
to prove the following statement on the fundamental limits of NERFs:
###### Theorem 11.
Take a sequence of real $M\times N_{M}$ frames $\\{F_{M}\\}_{M=1}^{\infty}$,
pick $C>1$, and take a sequence of erasure rates $\\{p_{M}\\}_{M=1}^{\infty}$
such that
$\liminf_{M\rightarrow\infty}p_{M}>1-2Q(C),\qquad
Q(t):=\frac{1}{\sqrt{2\pi}}\int_{t}^{\infty}\mathrm{e}^{-u^{2}/2}\,\mathrm{d}u.$
(17)
Then for all sufficiently large $M$, $F_{M}$ is not a $(p_{M},C)$-numerically
erasure-robust frame.
###### Proof.
For notational simplicity, we write $F=F_{M}$, $N=N_{M}$ and $p=p_{M}$.
Further let $\mathbb{S}^{M-1}$ denote the unit sphere in $\mathbb{R}^{M}$. For
any $x\in\mathbb{S}^{M-1}$, consider the “polar caps” of the sphere about $\pm
x$, namely the set $B(x):=\\{y\in\mathbb{S}^{M-1}:|\langle
x,y\rangle|^{2}\geq\frac{C^{2}}{M}\\}$. For any such bi-cap, we may count the
number of frame elements that it contains, namely the cardinality of the set
$B(x)\cap\\{f_{n}\\}_{n=1}^{N}$. Let $x_{0}$ denote the point on the sphere
whose bi-cap contains the most frame elements. By the pigeonhole principle,
the fraction of frame elements contained in this bi-cap is at least the
fraction of its surface area to the surface area of the entire sphere:
$\Big{|}B(x_{0})\cap\\{f_{n}\\}_{n=1}^{N}\Big{|}\geq
N\cdot\frac{\mathrm{Area}(B(x))}{\mathrm{Area}(\mathbb{S}^{M-1})}.$
Assuming for the moment that
$1-p\leq\mathrm{Area}(B(x))/\mathrm{Area}(\mathbb{S}^{M-1})$, we may take
$\mathcal{K}$ to be the indices of any $K=(1-p)N$ of the $f_{n}$’s in
$B(x_{0})\cap\\{f_{n}\\}_{n=1}^{N}$. Then
$\|F_{\mathcal{K}}^{*}x_{0}\|^{2}=\sum_{k\in\mathcal{K}}|\langle
x_{0},f_{k}\rangle|^{2}\geq K\frac{C^{2}}{M},$
and so applying Lemma 10 to the $M\times K$ matrix $F_{\mathcal{K}}$ gives
$\big{(}\mathrm{Cond}(F_{\mathcal{K}})\big{)}^{2}\geq\frac{(M-1)\|F_{\mathcal{K}}^{*}x_{0}\|^{2}}{K-\|F_{\mathcal{K}}^{*}x_{0}\|^{2}}\geq\frac{(M-1)K\frac{C^{2}}{M}}{K-K\frac{C^{2}}{M}}=\frac{M-1}{M-C^{2}}C^{2}>C^{2},$
as claimed. Thus, it only remains to show that
$1-p\leq\frac{\mathrm{Area}(B(x))}{\mathrm{Area}(\mathbb{S}^{M-1})}$ (18)
for sufficiently large $M$.
To this end, pick $M$ large enough so that $\frac{C^{2}}{M}<1$ and take
$\theta\in(0,\frac{\pi}{2})$ such that $\cos^{2}\theta=\frac{C^{2}}{M}$. Then
$B(x)$ is the union of both polar caps of angular radius $\theta$ centered at
$\pm x$. Using hyperspherical coordinates, we find that
$\mathrm{Area}(B(x))=2~{}\mathrm{Area}(\mathbb{S}^{M-2})\int_{0}^{\theta}\sin^{M-2}\varphi\,\mathrm{d}\varphi.$
(19)
Next, we can substitute $t=\cos\varphi$ to get
$\int_{0}^{\theta}\sin^{M-2}\varphi\,\mathrm{d}\varphi=\int_{0}^{\theta}\sin^{M-3}\varphi\sin\varphi\,\mathrm{d}\varphi=\int_{\cos\theta}^{1}(1-t^{2})^{\frac{M-3}{2}}\,\mathrm{d}t.$
(20)
Note that the area of $\mathbb{S}^{M-1}$ is given by replacing $\theta$ with
$\frac{\pi}{2}$ in $\eqref{eq.bicap area}$ and (20), and so
$\frac{\mathrm{Area}(B(x))}{\mathrm{Area}(\mathbb{S}^{M-1})}=\frac{\int_{\cos\theta}^{1}(1-t^{2})^{\frac{M-3}{2}}\,\mathrm{d}t}{\int_{0}^{1}(1-t^{2})^{\frac{M-3}{2}}\,\mathrm{d}t}.$
Substituting $u=t\sqrt{M-3}$ and recalling that
$\cos^{2}\theta=\frac{C^{2}}{M}$ results in new integrals which converge as
$M$ grows large:
$\frac{\mathrm{Area}(B(x))}{\mathrm{Area}(\mathbb{S}^{M-1})}=\frac{\displaystyle\int_{C\sqrt{\frac{M-3}{M}}}^{\sqrt{M-3}}\Big{(}1-\frac{2}{M-3}\frac{u^{2}}{2}\Big{)}^{\frac{M-3}{2}}\,\mathrm{d}u}{\displaystyle\int_{0}^{\sqrt{M-3}}\Big{(}1-\frac{2}{M-3}\frac{u^{2}}{2}\Big{)}^{\frac{M-3}{2}}\,\mathrm{d}u}.$
Specifically, since $(1+\frac{x}{n})^{n}$ converges from below to
$\mathrm{e}^{x}$ for all $x\geq 0$, we can apply the Lebesgue dominated
convergence theorem to the Gaussian to obtain
$\frac{\mathrm{Area}(B(x))}{\mathrm{Area}(\mathbb{S}^{M-1})}\longrightarrow\frac{\int_{C}^{\infty}\mathrm{e}^{-u^{2}/2}du}{\int_{0}^{\infty}\mathrm{e}^{-u^{2}/2}du}=2Q(C).$
This implies that as $M$ grows large, our assumption (17) guarantees (18), as
needed. ∎
As a corollary to Theorem 11, note that if $p_{M}\rightarrow 1$ as $M$ gets
large, then the worst-case condition number diverges to infinity.
Specifically, this establishes that $M\times N$ full spark frames with
$M=\mathrm{o}(N)$ cannot be “maximally robust to erasures” in a numerical
sense; for sufficiently large $M$, the adversary can delete $N-M$ columns of
the frame in a way that leaves an arbitrarily ill-conditioned square
submatrix. This highlights the value of a theory of numerically erasure-robust
frames.
## 4\. Implications and remaining problems
Having constructed several numerically erasure-robust frames, and having
further proved certain fundamental limits, we conclude with a few interesting
observations. First, we consider an implication for maximal ETFs: no $M\times
N$ $(p,C)$-NERF can have $(1-p)N$ zeros in a common row, since otherwise the
adversary can delete the other $pN$ columns and leave a rank-deficient
submatrix. Since Theorem 4 also applies to maximal ETFs, this implies that
there is no basis over which half of a maximal ETF’s vectors share a common
zero coordinate. That is, if maximal ETFs exist, then they cannot be too
sparse in any basis.
Due to their computational benefits, frames which have a sparse representation
have recently become a subject of active research [8, 10]. In this vein, one
attractive feature of Steiner ETFs is their naturally sparse representation;
in fact, the proportion of nonzero entries in an $M\times N$ Steiner ETF is
$\mathrm{O}(M^{-1/2})$ [15]. However, no Steiner ETF can be maximal, for they
have at most $N=\mathrm{O}(M^{3/2})$. The work presented here reinforces this
fact: since no $M\times N$ $(p,C)$-NERF can be very sparse, and since ETFs
with $N=\Omega(M^{2})$ are NERFs by Theorem 5, we see that neither Steiner
ETFs—nor any generalization of the Steiner construction with similar levels of
sparsity—will ever be able to produce ETFs in which $N=\Omega(M^{2})$.
Recall that $M\times N$ full spark frames have the defining property that
every subcollection of $M$ columns spans; trivially, this implies that every
subcollection of size _at least_ $M$ also spans. By analogy, it is natural to
ask whether a $(p,C)$-NERF is also a $(p^{\prime},C)$-NERF for every
$p^{\prime}\in[0,p)$. However, it is not clear whether this is the case, since
deleting columns does not necessarily worsen a frame’s conditioning. As an
example, the union of an orthonormal basis with some unit vector is not as
well conditioned as the orthonormal basis which survives the deletion of the
last vector. While this open question is interesting, it is inconsequential in
practice: If the adversary deletes less than $pN$ of the frame vectors, we can
neglect more of them to guarantee a well-conditioned subframe.
Another remark: Reviewing the results of this paper, we know there exist NERFs
with $p<\frac{1}{2}$ by Theorem 4. Meanwhile, Theorem 11 states that for any
fixed $C$, there do not exist NERFs with values of $p$ that grow arbitrarily
close to $1$. Various questions remain: Do there exist NERFs with
$p\in[\frac{1}{2},1)$? If so, what is the largest $p$ for which $(p,C)$-NERFs
exist? Interestingly, this “one-half barrier” appears to be more than a mere
artifact of Theorem 4. To be clear, every matrix $F$ whose entries are $\pm
1$’s cannot be a NERF with $p\geq\frac{1}{2}$; for any two rows of $F$, the
corresponding entries are either equal or opposite, and so the adversary can
delete the columns corresponding to the less popular relationship and leave a
rank-deficient matrix. Moreover, random matrix methods [6, 21] apply to
matrices of $\pm 1$ entries without loss of effectiveness, and so breaking the
one-half barrier, if it is even possible, will likely require other methods.
## References
* [1] B. Alexeev, J. Cahill, D.G. Mixon, Full spark frames, submitted, Available online: arXiv:1110.3548
* [2] W.O. Alltop, Complex sequences with low periodic correlations, IEEE Trans. Inform. Theory, 26 (1980) 350–354.
* [3] D.M. Appleby, Symmetric informationally complete-positive operator valued measures and the extended Clifford group, J. Math. Phys. 46 (2005) 052107/1–29.
* [4] B. Balan, B.G. Bodmann, P.G. Casazza and D. Edidin, Painless Reconstruction from Magnitudes of Frame Vectors, J. Fourier Anal. Appl. 15 (2009) 488–501.
* [5] A.S. Bandeira, M. Fickus, D.G. Mixon, P. Wong, The road to deterministic matrices with the restricted isometry property, submitted, Available online: arXiv:1202.1234
* [6] R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrics, Constr. Approx. 28 (2008) 253–263.
* [7] J.J. Benedetto, M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003) 357–385.
* [8] R. Calderbank, P.G. Casazza, A. Heinecke, G. Kutyniok, A. Pezeshki, Sparse fusion frames: Existence and construction, Adv. Comput. Math. 35 (2011) 1–31.
* [9] P.G. Casazza, M. Fickus, Fourier transforms of finite chirps, EURASIP J. Appl. Signal Process. 2006 (2006) 70204/1–7.
* [10] P.G. Casazza, A. Heinecke, F. Krahmer, G. Kutyniok, Optimally sparse frames, IEEE Trans. Inform. Theory 57 (2011) 7279–7287.
* [11] P.G. Casazza, J. Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003) 387–430.
* [12] E.J. Candès, The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 589–592.
* [13] E.J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 44 (2005) 4203–4215.
* [14] K.R. Davidson and S.J. Szarek, Local operator theory, random matrices and Banach spaces, In: Handbook in Banach Spaces Vol I, ed. W.B. Johnson, J. Lindenstrauss, Elsevier (2001), 317–366.
* [15] M. Fickus, D.G. Mixon and J.C. Tremain, Steiner equiangular tight frames, Linear Algebra Appl. 436 (2012) 1014–1027.
* [16] V.K. Goyal, J. Kovačević, J.A. Kelner, Quantized frame expansions with erasures, Appl. Comp. Harmon. Anal. 10 (2001) 203–233.
* [17] R.B. Holmes, V.I. Paulsen, Optimal frames for erasures, Linear Algebra Appl. 377 (2004) 31–51.
* [18] D. Jungnickel, A. Pott, K.W. Smith, Difference sets, In: C.J. Colbourn, J.H. Dinitz (Eds.), Handbook of Combinatorial Designs (2007) 419–435.
* [19] M. Planat, H.C. Rosu, S. Perrine, A survey of finite algebraic geometrical structures underlying mutually unbiased quantum measurements, Found. Phys. 36 (2006) 1662–1680.
* [20] M. Püschel, J. Kovačević, Real, tight frames with maximal robustness to erasures, Proc. Data Compr. Conf. (2005) 63–72.
* [21] M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements, Comm. Pure Appl. Math. 61 (2008) 1025–1045.
* [22] T. Strohmer and R.W. Heath, Grassmannian frames with applications to coding and communication, Appl. Comp. Harmon. Anal. 14 (2003) 257–275.
* [23] R. Vale, S. Waldron, Tight frames and their symmetries, Constr. Approx. 21 (2005) 83–112.
* [24] S. Waldron, On the construction of equiangular frames from graphs, Linear Algebra Appl. 431 (2009) 2228–2242.
* [25] L.R. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory 20 (1974) 397–399.
* [26] P. Xia, S. Zhou and G.B Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory 51 (2005) 1900–1907.
|
arxiv-papers
| 2012-02-21T04:51:28 |
2024-09-04T02:49:27.591002
|
{
"license": "Public Domain",
"authors": "Matthew Fickus and Dustin G. Mixon",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1202.4525"
}
|
1202.4615
|
119–126
# Surface Brightness Variation of the Contact Binary SW Lac: Clues From
Doppler Imaging
Hakan Volkan Şenavcı1 1University of Ankara, Faculty of Science, Department of
Astronomy and Space Sciences, TR-06100 Tandoğan-Ankara, TURKEY
email: hvsenavci@ankara.edu.tr
(2011)
###### Abstract
In this study, we present the preliminary light curve analysis of the contact
binary SW Lac, using B, V light curves of the system spanning 2 years (2009 -
2010). During the spot modeling process, we used the information coming from
the Doppler maps of the system, which was performed using the high resolution
and phase dependent spectra obtained at the 2.1 m Otto Struve Telescope of the
McDonald Observatory, in 2009. The results showed that the spot modeling from
the light curve analysis are in accordance with the Doppler maps, while the
non-circular spot modeling technique is needed in order to obtain much better
and reliable spot models.
###### keywords:
techniques: photometric, (stars:) binaries: eclipsing, stars: spots
††volume: 282††journal: From Interacting Binaries to Exoplanets: Essential
Modeling Tools††editors: A.C. Editor, B.D. Editor & C.E. Editor, eds.
## 1 Introduction
The light variability of the short-period contact binary SW Lac (P 0.d32,
Vmax=8.m91) is very well known and studied by several investigators since its
discovery by [Miss Ashall (1918), Miss Ashall (1918)]. The first photoelectric
UBV light curves of the system were obtained by [Brownlee (1956), Brownlee
(1956)], who also pointed out the light curve asymmetries from cycle to cycle.
These asymmetries were confirmed and attributed to the existence of cool spot
regions by several authors (see [Albayrak et al. 2004, Albayrak et al. 2004],
[Alton & Terrell 2006, Alton & Terrell 2006] and references there in). The
spectral studies of the system including spectral classification, mass ratio
determination and UV/X-ray region spectral analysis were carried out by
several investigators, who revealed that the system is a W-type contact binary
showing chromospheric and coronal activity (see [Şenavcı et al. 2011, Şenavcı
et al. 2011] and references there in, for details).
The aim of this study is to perform the light curve analysis with the spot
modeling, using the 2009 and 2010 light curves of the system with the help of
the information coming from the Doppler maps obtained by [Şenavcı et al. 2011,
Şenavcı et al. (2011)].
## 2 Observations and Data Reduction
The 2009 and 2010 BV band light curves of the contact binary SW Lac were
obtained at the Ankara University Observatory, using an Apogee Alta U47 CCD
camera attached to a 40 cm Schmidt-Cassegrain telescope. BD+37∘ 4715 and
BD+37∘ 4711 were chosen as comparison and check stars, respectively. The
nightly extinction coefficients for each passband were determined by using the
observations of the comparison star. A total of 700 and 995 data points were
obtained in each passband, while the probable error of a single observation
point was estimated to be $\pm 0.003/0.004$ and $\pm 0.004/0.004$ for 2009 and
2010 BV bands, respectively.
## 3 The Light Curve Analysis
The 2009 and 2010 BV light curves were analysed simultaneously with the radial
velocity curves of the system obtained by [Rucinski et al. (2005), Rucinski et
al. (2005)] using the interface version of the Wilson-Devinney code ([Wilson &
Devinney 1971, Wilson & Devinney 1971]), PHOEBE ([Prsa & Zwitter 2005, Prsa &
Zwitter 2005]). Since the surface reconstructions of the system were performed
using the time series spectra obtained in 2009, we first adopted the spot
modeling, as three main circular spot regions, to 2009 light curves and
carried out the LC modeling (see Fig.1). The results from the LC and spot
modeling were represented in Fig.2.
Figure 1: The Doppler maps and the adopted spots for LC modeling of the
system.
Figure 2: Observational and theoretical light curves with O-C residuals for
2009 and 2010.
## 4 Conclusion
The analysis showed that the theoretical light curves are compatible with the
observed ones, though the circular spot modeling was performed. However, in
order to perform more reliable spot modeling, a code with a none circular
shaped spot approximation is needed as the Doppler maps clearly show us the
spots are not circular.
## References
* [Albayrak et al. 2004] Albayrak, B., Djurasevic, G., Erkapic, S., & Tanrıverdi, T. 2004, A&A, 420, 1039
* [Alton & Terrell 2006] Alton, K.B., & Terrell, D. 2006, JAVSO, 34, 188
* [Miss Ashall (1918)] Leavitt, H. 1918, Harvard Obs. Circ., No. 207
* [Brownlee (1956)] Brownlee, R.R. 1956, AJ, 61, 2
* [Prsa & Zwitter 2005] Prsa, A. & Zwitter, T. 2005, AJ, 628, 426
* [Rucinski et al. (2005)] Rucinski, S.M., Pych, W., Ogloza, W., DeBond, H., Thomson, J.R., Mochnacki, S.W., Capobianco, C.C., Conidis, G. & Rogoziecki, P. 2005, AJ, 130, 767
* [Şenavcı et al. 2011] Şenavcı, H.V., Hussain, G.A.J., O’Neal, D. & Barnes, J.R. 2011, A&A, 529, 11
* [Wilson & Devinney 1971] Wilson, R.E. & Devinney, E.J. 1971, AJ, 166, 605
|
arxiv-papers
| 2012-02-21T12:09:00 |
2024-09-04T02:49:27.604995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H.V. \\c{S}enavc{\\i}",
"submitter": "Hakan Volkan Senavci",
"url": "https://arxiv.org/abs/1202.4615"
}
|
1202.4629
|
# MIPS 24-160 $\mu$m photometry for the Herschel-SPIRE Local Galaxies
Guaranteed Time Programs
G. J. Bendo1,2, F. Galliano3, S. C. Madden3
1 UK ALMA Regional Centre Node, Jodrell Bank Centre for Astrophysics, School
of Physics and Astronomy, University of Manchester, Oxford Road,
Manchester M13 9PL, United Kingdom
2 Astrophysics Group, Imperial College, Blackett Laboratory, Prince Consort
Road, London SW7 2AZ, United Kingdom
3 Laboratoire AIM, CEA, Université Paris Diderot, IRFU/Service
d’Astrophysique, Bat. 709, 91191 Gif-sur-Yvette, France
###### Abstract
We provide an overview of ancillary 24, 70, and 160 $\mu$m data from the
Multiband Imaging Photometer for Spitzer (MIPS) that are intended to
complement the 70-500 $\mu$m Herschel Space Observatory photometry data for
nearby galaxies obtained by the Herschel-SPIRE Local Galaxies Guaranteed Time
Programs and the Herschel Virgo Cluster Survey. The MIPS data can be used to
extend the photometry to wave bands that are not observed in these Herschel
surveys and to check the photometry in cases where Herschel performs
observations at the same wavelengths. Additionally, we measured globally-
integrated 24-160 $\mu$m flux densities for the galaxies in the sample that
can be used for the construction of spectral energy distributions. Using MIPS
photometry published by other references, we have confirmed that we are
obtaining accurate photometry for these galaxies.
###### keywords:
infrared: galaxies, galaxies: photometry, catalogues
††pagerange: MIPS 24-160 $\mu$m photometry for the Herschel-SPIRE Local
Galaxies Guaranteed Time Programs–References
## 1 Introduction
The Herschel-SPIRE Local Galaxies Guaranteed Time Programs (SAG2) comprise
several Herschel Space Observatory (Pilbratt et al., 2010) programs that used
primarily the Photodetector Array Camera and Spectrometer (PACS; Poglitsch et
al., 2010) and Spectral and Photometric Imaging Receiver (SPIRE; Griffin et
al., 2010) to perform far-infrared and submillimetre observations of galaxies
in the nearby universe. Three of the programs include photometric surveys of
galaxies. The Very Nearby Galaxies Survey (VNGS; PI: C. D. Wilson) has
performed 70-500 $\mu$m photometric and spectroscopic observations of 13
archetypal nearby galaxies that includes Arp 220, M51, and M81. The Dwarf
Galaxy Survey (DGS; PI: S. C. Madden) is a 70-500 $\mu$m photometric and
spectroscopic survey of 48 dwarf galaxies selected to span a range of
metallicities (with 12+log(O/H) values ranging from 7.2 to 8.5). The Herschel
Reference Survey (HRS; Boselli et al., 2010) is a 250-500 $\mu$m photometric
survey of a volume-limited sample of 323 nearby galaxies designed to include
both field and Virgo Cluster galaxies. The HRS also significantly overlaps
with the Herschel Virgo Cluster Survey (HeViCS Davies et al., 2010a), a
100-500 $\mu$m survey that will image 60 square degrees of the Virgo Cluster,
and both collaborations will be sharing their data.
The far-infrared and submillimetre photometric data from these surveys can be
used to construct spectral energy distributions (SEDs) of the dust emission
and to map the distribution of cold dust within these galaxies. However, the
surveys benefit greatly from the inclusion of 24, 70, and 160 $\mu$m data from
the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al., 2004), the
far-infrared photometric imager on board the Spitzer Space Telescope (Werner
et al., 2004). The 24 $\mu$m MIPS data are particularly important either when
attempting to model the complete dust emission from individual galaxies, as it
provides constraints on the hot dust emission, or when attempting to measure
accurate star formation rates, as 24 $\mu$m emission has been shown to be
correlated with other star formation tracers (Calzetti et al., 2005, 2007;
Prescott et al., 2007; Kennicutt et al., 2007, 2009; Zhu et al., 2008). The 70
$\mu$m MIPS data are less critical for the VNGS and DGS galaxies, which have
been mapped with PACS at 70 $\mu$m, but the data are more important for the
HRS galaxies, most of which will not be mapped with PACS at 70 $\mu$m. None
the less, the MIPS 70 $\mu$m data can be used to check the PACS photometry,
and the data may be useful as a substitute for PACS photometry in situations
where the MIPS data are able to detect emission at higher signal-to-noise
levels but where the higher resolution of PACS is not needed. For galaxies
without 70 $\mu$m PACS observations, the MIPS data will provide an important
additional data point that is useful for constraining the part of the far-
infrared SED that represents the transition between the $\sim 20$ K dust
emission from the diffuse interstellar medium and the hot dust emission from
large grains in star forming regions and very small grains. The 160 $\mu$m
MIPS data are less important, as 160 $\mu$m PACS observations with equivalent
sensitivities and smaller PSFs have been performed on the VNGS and DGS samples
as well as the fraction of the HRS sample that falls within the HeViCS fields.
For these galaxies, the MIPS 160 $\mu$m data can primarily be used to check
PACS 160 $\mu$m photometry. An additional follow-up program (Completing the
PACS coverage of the Herschel Reference Survey, P.I.: L. Cortese) has been
submitted to perform PACS 160 $\mu$m observations on the HRS galaxies outside
the HeViCS field. However, those observations have not yet been performed at
the time of this writing, so the MIPS 160 $\mu$m data can serve as a
substitute for the missing PACS data.
The pipeline processing from the MIPS archive is not optimized for
observations of individual galaxies. The final 24 $\mu$m images may include
gradients from zodiacal light emission, incomplete flatfielding, and
foreground asteroids, while the 70 and 160 $\mu$m images may include short-
term variations in the background signal (“drift”). Moreover, many galaxies
are often observed multiple times in multiple Astronomical Observation
Requests (AORs), and optimal images can often be produced by combining the
data from these multiple AORs, which is something that the MIPS pipeline is
not designed to do. Hence, to get the best MIPS images for analysis, it is
necessary to reprocess the archival data.
Work on reprocessing the archival MIPS data for the SAG2 and HeViCS programs
has been ongoing since before the launch of Herschel. Either these reprocessed
MIPS data or earlier versions of the data have already been used in multiple
papers from the SAG2 collaboration (Cortese et al., 2010a; Eales et al., 2010;
Galametz et al., 2010; Gomez et al., 2010; O’Halloran et al., 2010; Pohlen et
al., 2010; Sauvage et al., 2010; Auld et al., 2011; Bendo et al., 2012; Smith
et al., 2011; Foyle et al., 2012) and the HeViCS collaboration (de Looze et
al., 2010; Smith et al., 2010; Davies et al., 2012), and the data have also
been used in other publications outside of these collaboration (Young et al.,
2009; Wilson et al., 2009; Whaley et al., 2009; Galametz et al., 2010; Cortese
et al., 2010b; Bendo et al., 2010; de Looze et al., 2011). The data processing
has been described with some details in some of these papers but not in
others. Global photometry measurements (printed numerical values, not just
data points shown in figures) have only been published for 11 galaxies, and
some of the measurements are based either on older versions of the data
processing or on images created before all of the MIPS data for the targets
were available.
The goal of this paper is to describe the MIPS data processing for SAG2 in
detail and to present photometry for all of the SAG2 galaxies as well as the
500 $\mu$m flux-limited sample of HeViCS galaxies published by Davies et al.
(2012). While the MIPS data is incomplete for the DGS, HRS, and HeViCS samples
and hence cannot be used to create statistically complete datasets, the data
are still useful for constructing SEDs for individual galaxies and subsets of
galaxies in the SAG2 and HeViCS samples. The paper is divided into two primary
sections. Section 2 describes the data processing in detail. Section 3
describes the globally-integrated photometry for these galaxies, which can be
used as a reference for other papers, and also discusses how the photometry
compares to the MIPS photometry from other surveys.
## 2 Data processing
### 2.1 Overview of MIPS
This section gives a brief overview of the MIPS instrument and the type of
data produced by the instrument. Additional information on the instrument and
the arrays can be found in the MIPS Instrument Handbook (MIPS Instrument and
MIPS Instrument Support Teams, 2011)181818The MIPS Instrument Handbooks is
available at http://irsa.ipac.caltech.edu/data/SPITZER/docs/mips/
mipsinstrumenthandbook/MIPS_Instrument_Handbook.pdf ..
MIPS has four basic observing modes, but most observations were performed in
one of the two imaging modes. The photometry map mode produced maps of
multiple dithered frames that were usually $\sim 5$ arcmin in size. The
observing mode could also be used to produce raster maps or could be used in
cluster mode to produce maps of multiple objects that are close to each other.
Although intended to be used for observing sources smaller than 5 arcmin, the
mode was sometimes used to image larger objects. Because the 24, 70, and 160
$\mu$m arrays are offset from each other in the imaging plane, observations in
each wave band need to be performed in separate pointings. The scan map
observing mode was designed to be used primarily for observing objects larger
than 5 arcmin. The telescope scans in a zig-zag pattern where each of the
arrays in the instrument pass over the target region in each scan leg. In
typical observations, the telescope scans a region that is 1 degree in length,
although longer scan maps were also produced with the instrument. In both
observing modes, a series of individual data frames are taken in a cycle with
the telescope pointing at different offsets from the target. These cycles
include stimflash observations, which are frames in which the arrays are
illuminated with an internal calibration source. Between 6 and 32 frames may
be taken during a photometry map observation. In scan map observations, the
number of frames per cycle may vary, but the data are always bracketed by
stimflash frames. In typical 1 degree long scan map legs taken with the medium
scan rate, each scan leg contains 4 cycles of data, and each cycle contains 25
frames.
The other two observing modes were a 65-97 $\mu$m low resolution spectroscopy
mode using the 70 $\mu$m array and a total power mode that could be used to
measure the total emission from the sky. However, since our interest is in
working with photometric images of individual galaxies, we did not use the
data from either of these observing modes.
Details on the three arrays are given in Table 1. The 70 $\mu$m array is
actually a $32\times 32$ array, but half of the array was effectively
unusable, so the array effectively functions as a $32\times 16$ array. Details
on the effective viewing area are given in the table. Also, the 70 $\mu$m
array can be used in wide field-of-view and super-resolution modes for
producing photometry maps, but virtually no super-resolution data was taken
for our target galaxies, so we only list data for the wide field-of-view mode.
Table 1: Data on the three MIPS arraysa
Wave Band | Pixel Size | Array Size | PSF FWHMb | Flux Conversion Factors | Calibration
---|---|---|---|---|---
($\mu$m) | (arcsec) | (pixels) | (arcmin) | (arcsec) | (MJy/sr) [MIPS unit]-1 | Uncertainty
24 | $2.5\times 2.6$ | $128\times 128$ | $5.4\times 5.4$ | 6c | $4.54\times 10^{-2}$c | 4%c
70 | $9.9\times 10.1$ | $32\times 16$ | $5.2\times 2.6$ | 18d | $702^{d}$ | 10%d
160 | $16\times 18$ | $2\times 20$ | $2.1\times 5.3$ | 38e | $41.7^{e}$ | 12%e
a Except where noted, these data come from the MIPS Instrument Handbook (MIPS
Instrument and MIPS Instrument Support Teams, 2011).
b This is the full-width and half-maximum (FWHM) of the point spread function
(PSF).
c Data are from Engelbracht et al. (2007).
d Data are from Gordon et al. (2007).
e Data are from Stansberry et al. (2007).
### 2.2 Overview of data
#### 2.2.1 Archival data
Spitzer observations of multiple galaxies within the SAG2 samples were
performed in other survey programs before SAG2 began working on the MIPS
analysis and data reduction. The only Spitzer observing program devoted to
SAG2 photometry that was awarded observing time was a program that included
MIPS 24 $\mu$m observations for 10 of the DGS galaxies, which is described in
the next subsection. All other MIPS data originate from an assortment of
programs. Some galaxies were observed as specific targets in surveys of nearby
galaxies. Others were observed in surveys of wide fields, such as the wide
field surveys of the Virgo Cluster. Still others were serendipitously observed
in observations with other targets, such as scan map observations of zodiacal
light. Both photometry maps and scan map data are available for these
galaxies. Consequently, the observed areas vary significantly among the
galaxies. The coverage (the number of data frames covering each pixel in the
final mosaics) and on-source integration times also vary among the galaxies.
Given the inhomogeneity of the data as well as the incomplete coverage of the
galaxies in the sample, we opted to use all data available for every galaxy to
produce the best images for each galaxy. This means that the data set will not
be uniform and that the noise levels in the data will vary among the galaxies
in the sample, but the resulting images will be the best on hand for analysis.
While we generally attempted to use all available, we made some judgments on
selecting data for final images. When both scan map and photometry map data
were available for individual galaxies, we used only the scan map data to
create final images if the optical discs of the galaxies were larger than the
areas covered in the photometry maps or if the background area in the
photometry map was too small to allow us to apply data processing steps that
rely on measurements from the background in on-target frames. We also did not
use observations that covered less than half of the optical discs of
individual objects. When multiple objects were covered in regions covered in
multiple overlapping or adjacent AORs, we made larger mosaics using all of the
data whenever technically feasible. Also, for photometry map data, we often
used the serendipitous data taken when individual arrays were in off-target
positions if those fields covered galaxies in our samples, and when multiple
fields were observed using the cluster option in the photometry map data (see
the MIPS Instrument handbook by the MIPS Instrument and MIPS Instrument
Support Teams, 2011), we combined the data from all pointings that covered
SAG2 or HeViCS galaxies.
#### 2.2.2 SAG2 observations of dwarf galaxies
Ten of the dwarf galaxies were observed by DGS with MIPS in cycle 5 as part of
the program Dust Evolution in Low-Metallicity Environments: Bridging the Gap
Between Local Universe and Primordial Galaxies (PI: F. Galliano; ID: 50550).
Since these were objects smaller than 5 arcmin in diameter and since SAG2
intended to rely upon Herschel for 70 and 160 $\mu$m photometry, these
galaxies were mapped only at 24 $\mu$m using the photometry map mode. One AOR
was performed per object. Each observation uses a dither pattern to cover a
$\sim 6$ arcmin square region around the targets, and the integration times
were set to 3 s per frame, giving a total time of 328 s per AOR.
### 2.3 Data processing for individual data frames
The raw data from the Spitzer archive were reprocessed using the MIPS Data
Analysis Tools (Gordon et al., 2005) along with additional processing steps,
some of which are performed by software from the MIPS Instrument Team and some
of which were developed independently. The scan map data processing is a
variant of the data processing pipeline used in the fourth data delivery of
MIPS data from the Spitzer Infrared Nearby Galaxies Survey (SINGS; Kennicutt
et al., 2003), although changes have been made to the background subtraction,
and an asteroid removal step has been added to the 24 $\mu$m data processing.
Although other data processing software for MIPS is available from the Spitzer
Science Center, we have continued to use the MIPS DAT because of our
familiarity with the software and because we have developed an extensive range
of tools to work with the intermediate and final data products produced by the
MIPS DAT.
Separate sections are used to describe the processing steps applied to the 24
$\mu$m data frames and the steps applied to the 70 and 160 $\mu$m data frames,
as the data from the 24 $\mu$m silicon-based detectors differs somewhat from
the data from the 70 and 160 $\mu$m germanium-gallium detectors. The tools for
processing photometry map data frames differ slightly from the tools for the
scan map data frames. However, the differences are small enough that it is
possible to describe the data processing for both observing modes in the same
sections. The mosaicking and post-processing steps applied to all data are
very similar, and so these steps are described in the last subsection.
#### 2.3.1 MIPS 24 $\mu$m data frame processing
The raw 24 $\mu$m data consist of slopes to the ramps measured by the
detectors (the counts accumulated in each pixel during non-destructive
readouts). The following data processing steps were applied to MIPS 24 $\mu$m
data frames:
* $1$.
The MIPS DAT program mips_sloper was applied to the frames. This applies a
droop correction, which removes an excess signal in each detector that is
proportional to the signal in the entire array, a dark current subtraction,
and an electronic nonlinearity correction.
* $2$.
The MIPS DAT program mips_caler was applied to the data frames. This corrects
the detector responsivity using a mirror-position dependent flatfield that
removes spots from the images caused by material on the scan mirror. This data
processing step also included a correction for variations in the readout
offsets between different columns in the data frames.
* $3$.
To remove latent images from bright sources, pixels with signals above 2500
MIPS units in individual frames were masked out in the following three frames.
In a few cases, this threshold was lowered to remove additional latent image
effects.
* $4$.
When some 24 $\mu$m data frames were made, the array was hit by strong cosmic
rays that also caused severe “jailbar” effects or background offsets in the
data. When we have identified data frames with these problems or other severe
artefacts, we masked out those data frames manually at this stage in the data
processing.
* $5$.
A mirror-position independent flatfield was created from on-target frames
falling outside “exclusion” regions that included the optical disc of target
galaxies and bright foreground or background sources. These flatfields correct
for responsivity variations in the array that are specific to each
observation. This flatfield was then applied to the data frames. In the case
of some photometry map data, not enough background area was available for
properly making flatfields. In these cases, data from the off-target pointings
were used to build the mirror-position independent flatfields that were then
applied to the data.
* $6$.
Gradients in the background signal, primarily from zodiacal light, were then
subtracted from the data frames. This step differs between the photometry and
scan map modes. For photometry map data, the background signal outside the
exclusion regions in each frame was fit with a plane, and then this plane was
subtracted from the data (although this step was skipped if not enough area
was available in the data frames to measure the background). In the scan map
data, two different approaches were used. Before applying either of these
methods, we typically discarded the first five frames of data from each scan
leg because the background signal was often ramping up to a stable background
level; these frames usually did not cover any targets. In the standard
approach, the background was subtracted in two steps. First, the median signal
for data outside the exclusion regions in each data frame were fit with a
second-order polynomial that was a function of time, and then this function
was subtracted from the data. Second, we measured the mean residual background
signal as a function of the frame position within a stimflash cycle and
subtracted these background variations from the data. The alternate background
subtraction approach relies upon using data from multiple scan legs; it was
generally applied when the standard approach did not properly subtract the
background. It was also sometimes used in place of the standard approach on
data that did not scan 1 degree with the medium scan rate (6.5 arcsec s-1), as
the code was simply more flexible to use. For all forward scan leg data or all
reverse scan leg data, we measured the median background level as a position
of location within the scan leg. This gives the background signal as a
function of position in a scan leg and scan direction that is then applied to
each scan leg. Note that these steps will also remove large scale structure
outside of the exclusion regions from the data but do not significantly affect
signal from compact and unresolved sources.
* $7$.
In cases where we had data from multiple AORs that overlapped similar regions,
we compared the data from pairs of AORs to perform asteroids removal in a
three step process that involved. In the first step, we used the mips_enhancer
in the MIPS DAT to make preliminary mosaics of the data from each AOR. In the
second step, we subtracted the data from each AOR to produce difference maps
in which asteroids and other transient sources will appear as either bright or
dark sources but where stationary objects will appear as noise. To identify
locations that contained signal from asteroids, we looked for data where
signal in either of the AORs was above a set S/N threshold, where the signal
in the difference maps was above a set S/N threshold, and where the coverage
was above a set threshold; these thresholds needed to be manually adjusted for
each comparison. When performing this step, we visually confirmed that the
software was identifying transient sources and not stationary sources or
background noise. In the final step, we went through the data frames from each
AOR and masked out data within 5 pixels ($\sim 12.5$ arcsec) of pixels
identified as containing signal from asteroids. In cases with bright
asteroids, we may identify multiple pixels containing signal from asteroids,
and so we often masked out regions signficantly larger than 11 pixels.
#### 2.3.2 MIPS 70-160 $\mu$m data frame processing
The raw 70 and 160 $\mu$m data consist of the counts accumulated in each pixel
during non-destructive readouts, which are referred to as ramps. We applied
the following processing steps to the 70 and 160 $\mu$m data frames:
* $1$.
The MIPS DAT program mips_sloper was applied to the individual data frames to
convert the ramps into slopes. This step also removes cosmic rays and readout
jumps, and it includes a nonlinearity corrections.
* $2$.
The MIPS DAT program mips_caler was applied to adjust the detector
responsivity relative to the stim flashes observed during the observations and
to apply illumination corrections. This step also includes electronic
nonlinearity and dark current corrections.
* $3$.
Short term drift in the signal was removed from the data on a pixel-by-pixel
basis. The background signal was measured in data falling outside the optical
disc of the galaxy and other sources that we identified in exclusion regions
similar to those described in the 24 $\mu$m data processing. In the 70 $\mu$m
photometry map data, the background was measured as a function of time and
then subtracted from the data. The 160 $\mu$m photometry map observations
often did not include enough background data to perform this step properly,
and the background variations in the 160 $\mu$m data was not problematic.
However, when the 160 $\mu$m photometry map data were to be combined with scan
map data, we did measure median background signals in the areas outside the
exclusion regions on a frame-by-frame basis and subtract these backgrounds
from the data. In the case of the scan map data, the median background signal
was measured for each pixel during each stim flash cycle, a spline procedure
was used to describe the background signal as a function of time during the
entire AOR, and then this background was subtracted from the data. This
procedure also removes gradients and large-scale structure from regions
outside the exclusion regions but will generally not affect compact and
unresolved sources.
* $4$.
In scan map data, residual variations in the background signal as a function
of time since the last stim flash were measured in data outside the exclusion
regions and then subtracted from the data.
* $5$.
Any problematic data that we have identified, such as individual 160 $\mu$m
detector pixels with very poor drift correction over a subset of the data
frames or cosmic ray hits on 160 $\mu$m detectors that were not filtered out
in the previous data processing steps, were masked out manually.
### 2.4 Mosaicking data and post-processing
Final images for the galaxies were created using all suitable AORs using the
mips_enhancer in a two step process. In the first step, the mips_enhancer is
used to identify pixels from individual frames that are statistical outliers
compared to co-spatial pixels from other frames. These pixels are then masked
out in enhanced versions of the data frames. In the second step, the
mips_enhancer is used to create the final maps. In these images, north is up,
east is left, and the pixel scales are set to 1.5, 4.5, and 9.0 arcsec
pixel-1. The pixel scales are based on a convention originally adopted by
SINGS, as it allows for fine sampling of PSF substructure and as the pixel
scales are integer multiples of each other, which allows for easier
comparisons among the images.
The CRPIX keywords in the final FITS images correspond to the centres of the
optical discs of the individual target galaxies as given by the NASA/IPAC
Extragalactic Database. In cases where two or more galaxies fell in contiguous
areas, we sometimes produced separate final mosaics for each galaxy in which
the final maps were constructed using different CRPIX values. We also
attempted to do this for a large amount of contiguous data for the Virgo
Cluster covering a $\sim 5^{\circ}$ region centered on a point near NGC 4486
and an overlapping $\sim 2.5^{\circ}$ region approximately centered on
RA=12:28:10 Dec=+80:31:35. While we succeeded at doing this with the 70 and
160 $\mu$m data, mips_enhancer failed to execute properly when we attempted
this with the 24 $\mu$m data, probably because of the relatively large angular
area compared to the pixel size. We therefore produced final 24 $\mu$m mosaics
of each galaxy in this region based on subsets of the contiguous data. In
doing this, we ensured that, when producing a 24 $\mu$m image of an individual
galaxy, we mosaicked all AORs that covered each galaxy that was being mapped.
NGC 4380 is an exception, as it lies near the ends of a $\sim 5^{\circ}$ scan
to the north and a $\sim 2.5^{\circ}$ scan to the south. We therefore measured
the 24 $\mu$m flux density for this galaxy in the map produced for NGC 4390,
which is nearby and which falls in almost all of the scan maps centered on or
to the north of NGC 4380. We also had problems with producing 24 $\mu$m maps
of NGC 4522 with the CRPIX values set to the central coordinates of the
galaxy, so we measured the flux density in the map centered on NGC 4519. In
the cases of NGC 3226/NGC 3227 and NGC 4567/NGC4568, where the galaxies appear
close enough that their optical discs overlap, we only made one map with the
central position set to the centre of the galaxy that is brighter at optical
wavelengths.
We performed a few post-processing steps to the final mosaics. We applied the
flux calibration factors given in Table 1 to produce maps in units of MJy
sr-1. Next, we applied a non-linearity correction to 70 $\mu$m pixels that
exceeded 66 MJy sr-1. This correction, given by Dale et al. (2007) as
$f_{70\mu m}(\mbox{true})=0.581(f_{70\mu m}(\mbox{measured}))^{1.13}$ (1)
is based on data from Gordon et al. (2007). When applying this correction, we
adjusted the calculations to include the median background signal measured in
the individual data frames before the drift removal steps. We then measured
and subtracted residual background surface brightnesses outside the optical
discs of the galaxies in regions that did not contain any nearby, resolved
galaxies (regardless of whether they were detected in the MIPS bands) or
point-like sources. In the case of the 24 $\mu$m data, we used multiple small
circular regions around the centres of targets. For the 70 and 160 $\mu$m
images, we used whenever possible two or more regions that were as large as or
larger than the optical discs of the target galaxies and that straddled the
optical disc of the galaxy. In some of the smaller photometry maps, however,
we could not often do this, so we made our best effort to measure the
background levels within whatever background regions were observed. In cases
where multiple galaxies fall within the final mosaics, we only performed this
background subtraction for the central galaxy, although when performing
photometry on the other galaxies in these fields, we measured the backgrounds
in the same way around the individual targets.
The final images have a few features and artefacts that need to be taken into
consideration when using the data. First of all, the large scale structure
outside of the target galaxies in the images has been mostly removed. Although
the images, particularly the 160 $\mu$m images, may contain some cirrus
structure, most of the large scale features in the cirrus have been removed.
Second, all scan map data may contain some residual striping. Additionally,
the 70 $\mu$m images for bright sources are frequently affected by latent
image effects that manifest themselves as positive or negative streaks aligned
with the scan direction. Finally, many objects falling within the Virgo
Cluster as well as a few objects in other fields were observed in fields
covered only with MIPS scan map data taken using the fast scan rate. The
resulting 160 $\mu$m data contain large gaps in the coverage, and the data
appear more noisy than most other 160 $\mu$m data because of the poor
sampling.
## 3 Photometry
### 3.1 Description of measurements
For most galaxies, we performed aperture photometry within elliptical
apertures with major and minor axes that were the greater of either 1.5 times
the axis sizes of the D25 isophotes given by (de Vaucouleurs et al., 1991) or
3 arcmin. The same apertures were used in all three bands for consistency. The
lower limit of 3 arcmin on the measurement aperture dimensions ensures that we
can measure the total flux densities of 160 $\mu$m sources without needing to
apply aperture corrections. We performed tests with measuring some unresolved
sources in the DGS with different aperture sizes and found that the fraction
of the total flux not included within a 3 arcmin aperture for these sources is
below the 12% calibration uncertainty of the 160 $\mu$m band. In galaxies much
larger than 3 arcmin, we found that apertures that were 1.5 times the D25
isophote contained all of the measurable signal from the target galaxies. The
measured flux densities in apertures larger than this did not change
significantly, but the measured flux densities decreased if we used smaller
apertures.
For the elliptical galaxies NGC 3640, NGC 4125, NGC 4365, NGC 4374, NGC 4406,
NGC 4472, NGC 4486, NGC 4552, NGC 4649, NGC 4660, and NGC 5128, however, we
used measurement apertures that were the same size as the D25 isophotes.
Additionally, for the nearby dwarf elliptical galaxy NGC 205, we used a
measurement aperture that was 0.5 times the size of the D25 isophote. These
were all cases where the 70 and 160 $\mu$m emission across most of the optical
disc is within $5\sigma$ of the background noise, and in many cases, the
emission from the galaxies is not detected. Using smaller apertures in these
specific cases allows us to avoid including background sources and artefacts
from the data processing, thus allowing us to place better constraints on the
flux densities. We also treated NGC 4636 as a special case in which, at 160
$\mu$m, we only measured the flux density for the central source because of
issues with possible background sources falling within the optical disc of the
galaxy (although the background sources are not as problematic at 24 $\mu$m,
and so the 24 $\mu$m measurement is still for the entire optical disc).
Additional details on NGC 4636 are given in Section 3.1.1.
A few galaxies in the various samples are so close to each other or so close
to other galaxies at equivalent distances that attempting to separate the
infrared emission from the different sources would be very difficult. Objects
where this is the case are Mrk 1089 (within NGC 1741), NGC 3395/3396, NGC
4038/4039, NGC 4567/4568, NGC 5194/5195, and UM 311 (within NGC 450). In these
cases, we used measurement apertures that were large enough to encompass the
emission from the target galaxy and all other nearby sources. Details on the
other apertures are given in Table 2.
Table 2: Special measurement apertures
Galaxy | R.A. | Dec. | Axis sizes | Position
---|---|---|---|---
| (J2000) | (J2000) | (arcmin) | Anglea
Mrk 1089 | 05:01:37.8 | -04:15:28 | $3.0\times 3.0$ | $0^{\circ}$
NGC 891 | 02:22:33.4 | +42:20:57 | $20.3\times 10.0$ | $22^{\circ}$
NGC 3395/3396 | 10:49:50.1 | +32:58:58 | $6.0\times 6.0$ | $0^{\circ}$
NGC 4038/4039 | 12:01:53.0 | -18:52:10 | $10.4\times 10.4$ | $0^{\circ}$
NGC 4567/4568 | 12:36:34.3 | +11:14:20 | $8.5\times 8.5$ | $0^{\circ}$
NGC 5194/5195 | 13:29:52.7 | +47:11:43 | $19.6\times 19.6$ | $0^{\circ}$
NGC 6822 | 19:44:56.6 | -14:47:21 | $30.0\times 30.0$ | $0^{\circ}$
UM 311 | 01:15:30.4 | -00:51:39 | $4.7\times 3.5$ | $72^{\circ}$
a Position angle is defined as degrees from north through east.
Many of the galaxies in the DGS do not have optical discs defined by de
Vaucouleurs et al. (1991), and some do not have optical discs defined anywhere
in the literature. These are generally galaxies smaller than the minimum 3
arcmin diameter aperture that we normally use, so we used measurement
apertures of that size in many cases. However, for sources fainter than 100
mJy in the 24 $\mu$m data, we found that background noise could become an
issue when measuring 24 $\mu$m flux densities over such large apertures;
although the galaxy would clearly be detected at a level much higher than
$5\sigma$ in the centre of the aperture, the integral of the aperture would
make the detection appear weaker. Hence, for 24 $\mu$m DGS sources that were
fainter than 10 mJy and did not appear extended in the 24 $\mu$m data, we used
apertures with 1 arcmin diameters and divided the data by 0.93, which is an
aperture correction that we derived empirically from bright point-like sources
in the DGS.
NGC 891 and NGC 6822 were treated as special cases for selecting the
measurement apertures. Details are given in the photometry notes below, and
the parameters describing the measurement apertures are given in Table 2.
Before performing the photometry on individual galaxies, we identified and
masked out emission that appeared to be unrelated to the target galaxies. We
visually identified and masked out artefacts from the data processing in the
final mosaics, such as bright or dark pixels near the edges of mapped field
and streaking in the 70 $\mu$m images related to latent image effects. We also
statistically checked for pixels that were $5\sigma$ below the background,
which are almost certainly associated with artefacts except when this becomes
statistically probable in apertures containing large numbers of pixels. In
cases where we determined that the $<-5\sigma$ pixels were data processing
artefacts or excessively noisy pixels, we masked them out. When other galaxies
appeared close to individual galaxies in which we were measauring flux
densities but when the optical discs did not overlap significantly, we masked
out the adjacent galaxies. We also masked out emission from unresolved
sources, particularly unresolved 24 $\mu$m sources, that did not appear to be
associated with the target galaxies and that appeared signficantly brighter
than the emission in the regions where we measured the background. Most of
these sources appeared between the D25 isophote and the measurement aperture.
In cases where the galaxies contained very compact 24 $\mu$m emission (as is
the case for many elliptical and S0 galaxies), we also masked out unresolved
sources within but near the D25 isophote. A few unresolved sources within the
D25 isophote appeared as bright, unresolved sources in Digitized Sky Survey or
2MASS data, indicating that they were foreground stars, and we masked them out
as well. In many 24 $\mu$m images, the measured flux densities changed by less
than 4% (the calibration uncertainty) when the unresolved sources were
removed.
As stated above, in cases where the MIPS 160 $\mu$m data for individual
galaxies consists of only scan map data taken at the fast scan rate, our final
160 $\mu$m maps include gaps in the coverage. To make 160 $\mu$m measurements,
we have interpolated the signal across these gaps using nearest neighbor
sampling techniques. We also applied this interpolation technique to 160
$\mu$m data for the regions in the optical discs (but not in the whole
measurement aperture, which may fall outside the scan region) of IC 1048, NGC
4192, NGC 4535, and NGC 5692. In many other cases, the observed regions did
not completely cover the optical discs of the target galaxies. We normally
measured the flux densities for the regions covered in the observed regions.
Cases where the observed regions did not cover $~{}\hbox
to0.0pt{$>$\hss}{\lower 4.30554pt\hbox{$\sim$}}90$% of the optical discs are
noted in the photometry tables. Although we believe that these data are
reliable (especially since the observations appear to cover most of the
emission that is seen in the other bands), people using these data should
still be aware of the limitations of these data.
As a quality check on the photometry, we examined the 24/70, 24/160, and
70/160 $\mu$m flux density ratios to identify any galaxies that may have
discrepant colours (for example, abnormally high 24/70 and low 70/160 $\mu$m
colours, which would be indicative of problems with unmasked negative pixels
in the final images). In such discrepant cases, we examined the images for
unmasked artefacts, masked out the artefacts when identified, and repeated the
photometry.
The globally-integrated flux densities for the galaxies in the four different
samples are listed in Tables 3-6. No colour corrections have been applied to
these data. We include three sources of uncertainty. The first source is the
calibration uncertainty. The second source is the uncertainty based on the
error map. Each pixel in the error map is based on the standard deviation of
the overlapping pixels from the individual data frames; the uncertainties will
include both instrumental background noise and shot noise from the
astronomical sources. To calculate the total uncertainty traced by the data in
the error map, we used the square root of the sum of the square of the error
map pixels in the measurement region. The third source of uncertainty is from
background noise (which includes both instrumental and astronomical sources of
noise) measured in the background regions. The total uncertainties are
calculated by adding these three sources of uncertainty in quadrature. Sources
that are less than $5\sigma$ detections within the measurement apertures
compared to the combination of the error map and background noise are reported
as $5\sigma$ upper limits. Sources in which the surface brightness within the
measurement aperture is not detected at the $5\sigma$ level for regions
unaffected by foreground/background sources or artefacts are reported as upper
limits; in these cases, the integrated flux densities within the apertures are
used as upper limits. This second case occurs when the target aperture
includes emission from diffuse, extended emission (as described for NGC 4552
below) or large scale artefacts that are impossible to mask out for the
photometry.
Table 3: Photometry for the Very Nearby Galaxies Survey
Galaxy | Optical Disc | Wavelength | Flux Density | Flux Density Uncertainty (Jy)d
---|---|---|---|---
| R. A. | Declination | Axes | Position | ($\mu$m) | Measurement | Calibration | Error | Background | Total
| (J2000)a | (J2000)a | (arcmin)b | Anglebc | | (Jy) | | Map | |
NGC 205 | 00:40:22.0 | +41:41:07 | $21.9\times 11.0$ | $170^{\circ}$ | 24 | 0.1089 | 0.0044 | 0.0005 | 0.0008 | 0.0044
| | | | | 70 | 1.302 | 0.130 | 0.019 | 0.023 | 0.134
| | | | | 160 | 8.98 | 1.08 | 0.03 | 0.05 | 1.08
NGC 891e | 02:22:33.4 | +42:20:57 | $13.5\times 2.5$ | $22^{\circ}$ | 24 | 6.4531 | 0.2581 | 0.0005 | 0.0007 | 0.2581
| | | | | 70 | 97.122 | 9.712 | 0.045 | 0.018 | 9.712
| | | | | 160 | 287.27 | 34.47 | 8.72 | 0.04 | 35.56
NGC 1068 | 02:42:40.7 | -00:00:48 | $7.1\times 6.0$ | $70^{\circ}$ | 24 | | | | |
| | | | | 70 | 189.407 | 18.941 | 0.491 | 0.058 | 18.947
| | | | | 160 | 237.39 | 28.49 | 5.53 | 0.06 | 29.02
NGC 2403 | 07:36:51.4 | +65:36:09 | $21.9\times 12.3$ | $127^{\circ}$ | 24 | 6.0161 | 0.2406 | 0.0022 | 0.0019 | 0.2407
| | | | | 70 | 81.710 | 8.171 | 0.057 | 0.052 | 8.171
| | | | | 160 | 221.04 | 26.53 | 0.24 | 0.11 | 26.53
NGC 3031 | 09:55:33.1 | +69:03:55 | $26.9\times 14.1$ | $157^{\circ}$ | 24 | 5.2748 | 0.2110 | 0.0017 | 0.0024 | 0.2110
| | | | | 70 | 81.049 | 8.105 | 0.063 | 0.080 | 8.106
| | | | | 160 | 316.30 | 37.96 | 0.97 | 0.40 | 37.97
NGC 4038f | | | | | 24 | 5.8226 | 0.2329 | 0.0073 | 0.0012 | 0.2330
| | | | | 70 | 45.949 | 4.595 | 0.148 | 0.035 | 4.597
| | | | | 160 | 80.28 | 9.63 | 3.62 | 0.06 | 10.29
NGC 4125 | 12:08:06.0 | +65:10:27 | $5.8\times 3.2$ | $95^{\circ}$ | 24 | 0.0790 | 0.0032 | 0.0002 | 0.0003 | 0.0032
| | | | | 70 | 1.014 | 0.101 | 0.008 | 0.008 | 0.102
| | | | | 160 | 1.37 | 0.16 | 0.01 | 0.01 | 0.17
NGC 4151 | 12:10:32.5 | +39:24:21 | $6.3\times 4.5$ | $50^{\circ}$ | 24 | 4.5925 | 0.1837 | 0.0104 | 0.0005 | 0.1840
| | | | | 70 | 5.415 | 0.541 | 0.027 | 0.013 | 0.542
| | | | | 160 | 9.38 | 1.13 | 0.02 | 0.02 | 1.13
NGC 5128 | 13:25:27.6 | -43:01:09 | $25.7\times 20.0$ | $35^{\circ}$ | 24 | 24.0374 | 0.9615 | 0.0135 | 0.0028 | 0.9616
| | | | | 70 | 263.165 | 26.316 | 0.226 | 0.068 | 26.318
| | | | | 160 | 582.51 | 69.90 | 22.50 | 0.14 | 73.43
NGC 5194f | | | | | 24 | 14.2309 | 0.5692 | 0.0037 | 0.0015 | 0.5693
| | | | | 70 | 151.000 | 15.100 | 0.123 | 0.045 | 15.101
| | | | | 160 | 458.44 | 55.01 | 7.80 | 0.11 | 55.56
NGC 5236 | 13:37:00.9 | -29:51:57 | 12.9 | | 24 | 40.4266 | 1.6171 | 0.0263 | 0.0017 | 1.6173
| | | | | 70 | 312.808 | 31.281 | 0.290 | 0.051 | 31.282
| | | | | 160 | 798.23 | 95.78 | 9.95 | 0.13 | 96.30
Arp 220 | 15:34:57.1 | +23:30:11 | 1.5 | | 24 | | | | |
| | | | | 70 | 74.976 | 7.498 | 0.309 | 0.023 | 7.504
| | | | | 160 | 54.88 | 6.59 | 1.38 | 0.02 | 6.73
a Data are from NED.
b Data are from de Vaucouleurs et al. (1991) unless otherwise specified. If de
Vaucouleurs et al. (1991) specify both the minor/major axis ratio and the
position angle, then both axes and the position angle are listed. If de
Vaucouleurs et al. (1991) did not specify either of these data, then we
performed photometry on circular regions, and so only the major axis is
specified.
c The position angle is defined as degrees from north through east.
d Details on the sources of these uncertainties are given in Section 3.1.
e A special measurement aperture was used for NGC 891. See Table 2.
f These objects consist of two galaxies with optical discs that overlap. See
Table 2 for the dimensions of the measurement apertures for these objects.
Table 4: Photometry for the Dwarf Galaxies Survey Galaxy | Optical Disc | Wavelength | Flux Density | Flux Density Uncertainty (Jy)d
---|---|---|---|---
| R. A. | Declination | Axes | Position | ($\mu$m) | Measurement | Calibration | Error | Background | Total
| (J2000)a | (J2000)a | (arcmin)b | Angle${}^{b}c$ | | (Jy) | | Map | |
IC 10 | 00:20:17.3 | +59:18:14 | 6.3 | | 24 | 9.8188 | 0.3928 | 0.0136 | 0.0013 | 0.3930
| | | | | 70 | | | | |
| | | | | 160 | | | | |
HS 0017+1055 | 00:20:21.4 | +11:12:21 | | | 24 | 0.0237 | 0.0009 | 0.0005 | 0.0009 | 0.0014
| | | | | 70 | | | | |
| | | | | 160 | | | | |
Haro 11 | 00:36:52.4 | -33:33:19 | | | 24 | 2.3046 | 0.0922 | 0.0123 | 0.0005 | 0.0930
| | | | | 70 | 4.912 | 0.491 | 0.038 | 0.007 | 0.493
| | | | | 160 | 2.01 | 0.24 | 0.01 | 0.02 | 0.24
HS 0052+2536 | 00:54:56.3 | +25:53:08 | | | 24 | 0.0207 | 0.0008 | 0.0004 | 0.0008 | 0.0012
| | | | | 70 | | | | |
| | | | | 160 | | | | |
UM 311e | | | | | 24 | 0.3289 | 0.0132 | 0.0008 | 0.0009 | 0.0132
| | | | | 70 | 3.075 | 0.308 | 0.008 | 0.008 | 0.308
| | | | | 160 | 6.62 | 0.79 | 0.02 | 0.01 | 0.79
NGC 625 | 01:35:04.6 | -41:26:10 | $5.8\times 1.9$ | $92^{\circ}$ | 24 | 0.8631 | 0.0345 | 0.0016 | 0.0003 | 0.0346
| | | | | 70 | 6.252 | 0.625 | 0.036 | 0.012 | 0.626
| | | | | 160 | 7.87 | 0.94 | 0.03 | 0.02 | 0.95
UGCA 20 | 01:43:14.7 | +19:58:32 | $3.1\times 0.8$ | $153^{\circ}$ | 24 | $<0.0085$ | | | |
| | | | | 70 | | | | |
| | | | | 160 | | | | |
UM 133 | 01:44:41.2 | +40:53:26 | | | 24 | 0.0094 | 0.0004 | 0.0002 | 0.0003 | 0.0005
| | | | | 70 | | | | |
| | | | | 160 | | | | |
UM 382 | 01:58:09.3 | -00:06:38 | | | 24 | | | | |
| | | | | 70 | $<0.070$ | | | |
| | | | | 160 | | | | |
NGC 1140 | 02:54:33.5 | -10:01:40 | $1.7\times 0.9$ | $10^{\circ}$ | 24 | 0.3764 | 0.0151 | 0.0009 | 0.0006 | 0.0151
| | | | | 70 | 3.507 | 0.351 | 0.020 | 0.008 | 0.351
| | | | | 160 | 3.67 | 0.44 | 0.01 | 0.01 | 0.44
SBS 0335-052 | 03:37:44.0 | -05:02:40 | | | 24 | 0.0768 | 0.0031 | 0.0005 | 0.0005 | 0.0032
| | | | | 70 | 0.051 | 0.005 | 0.005 | 0.006 | 0.009
| | | | | 160 | $<0.07$ | | | |
NGC 1569 | 04:30:49.0 | -64:50:53 | $3.6\times 1.8$ | $120^{\circ}$ | 24 | 7.7189 | 0.3088 | 0.0091 | 0.0010 | 0.3089
| | | | | 70 | 46.120 | 4.612 | 0.068 | 0.029 | 4.613
| | | | | 160 | 33.49 | 4.02 | 0.11 | 0.02 | 4.02
NGC 1705 | 04:54:13.5 | -53:21:40 | $1.9\times 1.4$ | $50^{\circ}$ | 24 | 0.0532 | 0.0021 | 0.0000 | 0.0001 | 0.0021
| | | | | 70 | 1.315 | 0.132 | 0.002 | 0.004 | 0.132
| | | | | 160 | 1.29 | 0.16 | 0.01 | 0.01 | 0.16
Mrk 1089e | | | | | 24 | 0.5252 | 0.0210 | 0.0008 | 0.0003 | 0.0210
| | | | | 70 | 1.123 | 0.112 | 0.004 | 0.004 | 0.112
| | | | | 160 | | | | |
II Zw 40 | 05:55:42.6 | +03:23:32 | | | 24 | 1.6545 | 0.0662 | 0.0063 | 0.0006 | 0.0665
| | | | | 70 | 5.438 | 0.544 | 0.031 | 0.011 | 0.545
| | | | | 160 | | | | |
Tol 0618-402 | 06:20:02.5 | -40:18:09 | | | 24 | $<0.0015$ | | | |
| | | | | 70 | $<0.037$ | | | |
| | | | | 160 | $<0.42$ | | | |
NGC 2366 | 07:28:54.6 | +69:12:57 | $8.1\times 3.3$ | $25^{\circ}$ | 24 | 0.6919 | 0.0277 | 0.0013 | 0.0007 | 0.0277
| | | | | 70 | 5.230 | 0.523 | 0.021 | 0.019 | 0.524
| | | | | 160 | 5.50 | 0.66 | 0.21 | 0.03 | 0.69
HS 0822+3542 | 08:25:55.5 | +35:32:32 | | | 24 | 0.0032 | 0.0001 | 0.0001 | 0.0002 | 0.0003
| | | | | 70 | 0.043 | 0.004 | 0.004 | 0.006 | 0.008
| | | | | 160 | $<0.04$ | | | |
He 2-10 | 08:36:15.1 | -26:24:34 | | | 24 | 5.7368 | 0.2295 | 0.0262 | 0.0007 | 0.2310
| | | | | 70 | 17.969 | 1.797 | 0.102 | 0.009 | 1.800
| | | | | 160 | 13.41 | 1.61 | 0.05 | 0.01 | 1.61
UGC 04483 | 08:37:03.0 | +69:46:31 | | | 24 | 0.0101 | 0.0004 | 0.0001 | 0.0003 | 0.0005
| | | | | 70 | 0.142 | 0.014 | 0.003 | 0.006 | 0.016
| | | | | 160 | 0.27 | 0.03 | 0.01 | 0.00 | 0.03
Table 4: Photometry for the Dwarf Galaxies Survey (continued) Galaxy | Optical Disc | Wavelength | Flux Density | Flux Density Uncertainty (Jy)d
---|---|---|---|---
| R. A. | Declination | Axes | Position | ($\mu$m) | Measurement | Calibration | Error | Background | Total
| (J2000)a | (J2000)a | (arcmin)b | Angle${}^{b}c$ | | (Jy) | | Map | |
I Zw 18 | 09:34:02.0 | +55:14:28 | | | 24 | 0.0061 | 0.0002 | 0.0001 | 0.0002 | 0.0003
| | | | | 70 | 0.042 | 0.004 | 0.002 | 0.004 | 0.006
| | | | | 160 | $<0.12$ | | | |
Haro 2 | 10:32:31.9 | +54:24:03 | | | 24 | 0.8621 | 0.0345 | 0.0015 | 0.0001 | 0.0345
| | | | | 70 | 3.988 | 0.399 | 0.019 | 0.005 | 0.399
| | | | | 160 | 3.09 | 0.37 | 0.01 | 0.01 | 0.37
Haro 3 | 10:45:22.4 | +55:57:37 | | | 24 | 0.8514 | 0.0341 | 0.0027 | 0.0004 | 0.0342
| | | | | 70 | 4.898 | 0.490 | 0.018 | 0.007 | 0.490
| | | | | 160 | 3.93 | 0.47 | 0.01 | 0.01 | 0.47
Mrk 153 | 10:49:05.0 | +52:20:08 | | | 24 | 0.0358 | 0.0014 | 0.0003 | 0.0005 | 0.0015
| | | | | 70 | 0.260 | 0.026 | 0.004 | 0.007 | 0.027
| | | | | 160 | | | | |
VII Zw 403 | 11:27:59.8 | +78:59:39 | | | 24 | 0.0329 | 0.0013 | 0.0002 | 0.0005 | 0.0014
| | | | | 70 | 0.425 | 0.043 | 0.005 | 0.007 | 0.043
| | | | | 160 | 0.31 | 0.04 | 0.00 | 0.01 | 0.04
Mrk 1450 | 11:38:35.6 | +57:52:27 | | | 24 | 0.0570 | 0.0023 | 0.0003 | 0.0004 | 0.0023
| | | | | 70 | 0.264 | 0.026 | 0.004 | 0.005 | 0.027
| | | | | 160 | 0.15 | 0.02 | 0.00 | 0.01 | 0.02
UM 448 | 11:42:12.4 | +00:20:03 | | | 24 | 0.6425 | 0.0257 | 0.0018 | 0.0007 | 0.0258
| | | | | 70 | 3.703 | 0.370 | 0.021 | 0.015 | 0.371
| | | | | 160 | 2.67 | 0.32 | 0.01 | 0.01 | 0.32
UM 461 | 11:51:33.3 | -02:22:22 | | | 24 | 0.0344 | 0.0014 | 0.0002 | 0.0029 | 0.0032
| | | | | 70 | 0.090 | 0.009 | 0.003 | 0.011 | 0.014
| | | | | 160 | 0.10 | 0.01 | 0.00 | 0.01 | 0.01
SBS 1159+545 | 12:02:02.3 | +54:15:50 | | | 24 | 0.0062 | 0.0002 | 0.0001 | 0.0002 | 0.0004
| | | | | 70 | | | | |
| | | | | 160 | | | | |
SBS 1211+540 | 12:14:02.4 | +53:45:17 | | | 24 | 0.0033 | 0.0001 | 0.0001 | 0.0002 | 0.0003
| | | | | 70 | | | | |
| | | | | 160 | | | | |
NGC 4214 | 12:15:39.1 | +36:19:37 | 8.5 | | 24 | 2.1044 | 0.0842 | 0.0015 | 0.0012 | 0.0842
| | | | | 70 | 24.049 | 2.405 | 0.043 | 0.032 | 2.406
| | | | | 160 | 38.18 | 4.58 | 0.34 | 0.05 | 4.59
Tol 1214-277 | 12:17:17.0 | -28:02:33 | | | 24 | 0.0068 | 0.0003 | 0.0001 | 0.0002 | 0.0003
| | | | | 70 | 0.073 | 0.007 | 0.004 | 0.005 | 0.010
| | | | | 160 | | | | |
HS 1222+3741 | 12:24:36.7 | +37:24:37 | | | 24 | | | | |
| | | | | 70 | 0.062 | 0.006 | 0.004 | 0.007 | 0.010
| | | | | 160 | | | | |
Mrk 209 | 12:26:16.0 | +48:29:37 | | | 24 | 0.0587 | 0.0023 | 0.0003 | 0.0005 | 0.0024
| | | | | 70 | 0.466 | 0.047 | 0.004 | 0.004 | 0.047
| | | | | 160 | 0.18 | 0.02 | 0.00 | 0.01 | 0.02
NGC 4449 | 12:28:11.8 | +44:05:40 | $6.2\times 4.4$ | $45^{\circ}$ | 24 | 3.2863 | 0.1315 | 0.0010 | 0.0008 | 0.1315
| | | | | 70 | 43.802 | 4.380 | 0.053 | 0.019 | 4.381
| | | | | 160 | 78.09 | 9.37 | 0.70 | 0.03 | 9.40
SBS 1249+493 | 12:51:52.4 | +49:03:28 | | | 24 | 0.0043 | 0.0002 | 0.0001 | 0.0002 | 0.0003
| | | | | 70 | | | | |
| | | | | 160 | | | | |
NGC 4861 | 12:59:02.3 | +34:51:34 | $4.0\times 1.5$ | $15^{\circ}$ | 24 | 0.3657 | 0.0146 | 0.0012 | 0.0008 | 0.0147
| | | | | 70 | 1.971 | 0.197 | 0.012 | 0.010 | 0.198
| | | | | 160 | 2.00 | 0.24 | 0.01 | 0.02 | 0.24
HS 1304+3529 | 13:06:24.1 | +35:13:43 | | | 24 | 0.0122 | 0.0005 | 0.0004 | 0.0007 | 0.0009
| | | | | 70 | | | | |
| | | | | 160 | | | | |
Pox 186 | 13:25:48.6 | -11:36:38 | | | 24 | 0.0108 | 0.0004 | 0.0005 | 0.0009 | 0.0011
| | | | | 70 | | | | |
| | | | | 160 | | | | |
NGC 5253 | 13:39:55.9 | -31:38:24 | $5.0\times 1.9$ | $45^{\circ}$ | 24 | | | | |
| | | | | 70 | 23.626 | 2.363 | 0.074 | 0.015 | 2.364
| | | | | 160 | 17.35 | 2.08 | 0.05 | 0.03 | 2.08
Table 4: Photometry for the Dwarf Galaxies Survey (continued)
Galaxy | Optical Disc | Wavelength | Flux Density | Flux Density Uncertainty (Jy)d
---|---|---|---|---
| R. A. | Declination | Axes | Position | ($\mu$m) | Measurement | Calibration | Error | Background | Total
| (J2000)a | (J2000)a | (arcmin)b | Angle${}^{b}c$ | | (Jy) | | Map | |
SBS 1415+437 | 14:17:01.3 | +43:30:05 | | | 24 | 0.0187 | 0.0007 | 0.0003 | 0.0005 | 0.0009
| | | | | 70 | 0.177 | 0.018 | 0.004 | 0.006 | 0.019
| | | | | 160 | $<0.06$ | | | |
HS 1424+3836 | 14:26:28.1 | +38:22:59 | | | 24 | | | | |
| | | | | 70 | $<0.024$ | | | |
| | | | | 160 | | | | |
HS 1442+4250 | 14:44:12.8 | +42:37:44 | | | 24 | 0.0066 | 0.0003 | 0.0001 | 0.0001 | 0.0003
| | | | | 70 | 0.079 | 0.008 | 0.004 | 0.006 | 0.010
| | | | | 160 | $<0.10$ | | | |
SBS 1533+574 | 15:34:13.8 | +57:17:06 | | | 24 | | | | |
| | | | | 70 | 0.270 | 0.027 | 0.004 | 0.005 | 0.028
| | | | | 160 | | | | |
NGC 6822f | 19:44:56.6 | -14:47:21 | 15.5 | | 24 | 4.5230 | 0.1809 | 0.0027 | 0.0032 | 0.1810
| | | | | 70 | 52.413 | 5.241 | 0.082 | 0.096 | 5.243
| | | | | 160 | 109.44 | 13.13 | 0.61 | 0.20 | 13.15
Mrk 930 | 23:31:58.2 | +28:56:50 | | | 24 | 0.1985 | 0.0079 | 0.0005 | 0.0006 | 0.0080
| | | | | 70 | 1.159 | 0.116 | 0.007 | 0.006 | 0.116
| | | | | 160 | 0.96 | 0.12 | 0.01 | 0.02 | 0.12
HS 2352+2733 | 23:54:56.7 | +27:49:59 | | | 24 | 0.0026 | 0.0001 | 0.0001 | 0.0003 | 0.0003
| | | | | 70 | | | | |
| | | | | 160 | | | | |
a Data are from NED.
b Data are from de Vaucouleurs et al. (1991) unless otherwise specified. If de
Vaucouleurs et al. (1991) specify both the minor/major axis ratio and the
position angle, then both axes and the position angle are listed. If de
Vaucouleurs et al. (1991) did not specify either of these data, then we
performed photometry on circular regions, and so only the major axis is
specified. If no optical dimensions are specified, then we performed
photometry on a 3 arcmin diameter circular region centered on the source
c The position angle is defined as degrees from north through east.
d Details on the sources of these uncertainties are given in Section 3.1.
e Special measurement apertures were used for these targets because of the
presence of nearby associated sources. See Table 2.
f A special measurement aperture was used for NGC 6822. See Table 2.
Table 5: Photometry for the Herscher Reference Survey Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 3226 | 3 | 10:23:27.4 | +19:53:55 | $3.2\times 2.8$ | $15^{\circ}$ | 24 | 0.0250 | 0.0010 | 0.0003 | 0.0006 | 0.0012
| | | | | | 70 | 0.459 | 0.046 | 0.009 | 0.011 | 0.048
| | | | | | 160 | | | | |
NGC 3227 | 4 | 10:23:30.5 | +19:51:54 | $5.4\times 3.6$ | $155^{\circ}$ | 24 | 1.7173 | 0.0687 | 0.0067 | 0.0010 | 0.0690
| | | | | | 70 | 9.033 | 0.903 | 0.044 | 0.018 | 0.905
| | | | | | 160 | 18.19f | 2.18 | 0.05 | 0.02 | 2.18
NGC 3254 | 8 | 10:29:19.9 | +29:29:31 | $5.0\times 1.6$ | $46^{\circ}$ | 24 | 0.0927 | 0.0037 | 0.0005 | 0.0007 | 0.0038
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3338 | 15 | 10:42:07.5 | +13:44:49 | $5.9\times 3.6$ | $100^{\circ}$ | 24 | 0.4578 | 0.0183 | 0.0003 | 0.0005 | 0.0183
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3370 | 17 | 10:47:04.0 | +17:16:25 | $3.2\times 1.8$ | $148^{\circ}$ | 24 | 0.3836 | 0.0153 | 0.0005 | 0.0009 | 0.0154
| | | | | | 70 | 5.194 | 0.519 | 0.018 | 0.012 | 0.520
| | | | | | 160 | 10.30 | 1.24 | 0.02 | 0.02 | 1.24
NGC 3395 | 20 | | | | | 24 | 1.1400 | 0.0456 | 0.0013 | 0.0010 | 0.0456
/3396h | /(N/A) | | | | | 70 | 11.927 | 1.193 | 0.025 | 0.027 | 1.193
| | | | | | 160 | 17.26 | 2.07 | 0.03 | 0.04 | 2.07
NGC 3414 | 22 | 10:51:16.2 | +27:58:30 | 3.5 | | 24 | 0.0430 | 0.0017 | 0.0004 | 0.0007 | 0.0019
| | | | | | 70 | 0.428 | 0.043 | 0.011 | 0.016 | 0.047
| | | | | | 160 | | | | |
NGC 3424 | 23 | 10:51:46.3 | +32:54:03 | $2.8\times 0.8$ | $112^{\circ}$ | 24 | 0.7181 | 0.0287 | 0.0012 | 0.0005 | 0.0288
| | | | | | 70 | 9.398 | 0.940 | 0.035 | 0.012 | 0.941
| | | | | | 160 | 15.93 | 1.91 | 0.04 | 0.03 | 1.91
NGC 3430 | 24 | 10:52:11.4 | +32:57:02 | $4.0\times 2.2$ | $30^{\circ}$ | 24 | 0.4101 | 0.0164 | 0.0004 | 0.0006 | 0.0164
| | | | | | 70 | 5.683 | 0.568 | 0.015 | 0.021 | 0.569
| | | | | | 160 | 14.36 | 1.72 | 0.03 | 0.02 | 1.72
NGC 3448 | 31 | 10:54:39.2 | +54:18:19 | $5.6\times 1.8$ | $65^{\circ}$ | 24 | 0.5782 | 0.0231 | 0.0009 | 0.0005 | 0.0232
| | | | | | 70 | 6.730 | 0.673 | 0.024 | 0.012 | 0.673
| | | | | | 160 | 9.43f | 1.13 | 0.20 | 0.47 | 1.24
NGC 3485 | 33 | 11:00:02.3 | +14:50:30 | 2.3 | | 24 | 0.1853 | 0.0074 | 0.0002 | 0.0003 | 0.0074
| | | | | | 70 | 2.279 | 0.228 | 0.008 | 0.012 | 0.228
| | | | | | 160 | | | | |
NGC 3499 | 35 | 11:03:11.0 | +56:13:18 | 0.8 | | 24 | 0.0124 | 0.0005 | 0.0001 | 0.0002 | 0.0005
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3504 | 36 | 11:03:11.2 | +27:58:21 | 2.7 | | 24 | 3.0895 | 0.1236 | 0.0131 | 0.0003 | 0.1243
| | | | | | 70 | 19.268 | 1.927 | 0.089 | 0.017 | 1.929
| | | | | | 160 | 21.44 | 2.57 | 0.04 | 0.04 | 2.57
NGC 3512 | 37 | 11:04:02.9 | +28:02:13 | 1.6 | | 24 | 0.1365 | 0.0055 | 0.0001 | 0.0001 | 0.0055
| | | | | | 70 | 1.982 | 0.198 | 0.007 | 0.010 | 0.199
| | | | | | 160 | | | | |
NGC 3608 | 43 | 11:16:58.9 | +18:08:55 | $3.2\times 2.6$ | $75^{\circ}$ | 24 | 0.0223 | 0.0009 | 0.0002 | 0.0004 | 0.0010
| | | | | | 70 | $<0.110$ | | | |
| | | | | | 160 | $<0.48$ | | | |
NGC 3640 | 49 | 11:21:06.8 | +03:14:05 | $4.0\times 3.2$ | $100^{\circ}$ | 24 | 0.0236 | 0.0009 | 0.0003 | 0.0006 | 0.0011
| | | | | | 70 | $<0.137$ | | | |
| | | | | | 160 | $<0.72$ | | | |
NGC 3655 | 50 | 11:22:54.6 | +16:35:25 | $1.5\times 1.0$ | $30^{\circ}$ | 24 | 0.7768 | 0.0311 | 0.0002 | 0.0001 | 0.0311
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3659 | 51 | 11:23:45.5 | +17:49:07 | $2.1\times 1.1$ | $60^{\circ}$ | 24 | 0.1419 | 0.0057 | 0.0002 | 0.0003 | 0.0057
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3666 | 53 | 11:24:26.0 | +11:20:32 | $4.4\times 1.2$ | $100^{\circ}$ | 24 | 0.2577 | 0.0103 | 0.0003 | 0.0004 | 0.0103
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3681 | 54 | 11:26:29.8 | +16:51:47 | 2.5 | | 24 | 0.0772 | 0.0031 | 0.0002 | 0.0003 | 0.0031
| | | | | | 70 | 1.374 | 0.137 | 0.008 | 0.012 | 0.138
| | | | | | 160 | | | | |
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 3683 | 56 | 11:27:31.8 | +56:52:37 | $1.9\times 0.7$ | $128^{\circ}$ | 24 | 1.1755 | 0.0470 | 0.0003 | 0.0001 | 0.0470
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3686 | 57 | 11:27:43.9 | +17:13:27 | $3.2\times 2.5$ | $15^{\circ}$ | 24 | 0.5463 | 0.0219 | 0.0004 | 0.0004 | 0.0219
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3729 | 60 | 11:33:49.3 | +53:07:32 | $2.8\times 1.9$ | $15^{\circ}$ | 24 | 0.4591 | 0.0184 | 0.0012 | 0.0002 | 0.0184
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 3945 | 71 | 11:53:13.7 | +60:40:32 | $5.2\times 3.5$ | $165^{\circ}$ | 24 | 0.0914 | 0.0037 | 0.0003 | 0.0005 | 0.0037
| | | | | | 70 | 0.456 | 0.046 | 0.012 | 0.020 | 0.051
| | | | | | 160 | 3.30f | 0.40 | 0.01 | 0.01 | 0.40
NGC 3953 | 73 | 11:53:48.9 | +52:19:36 | $6.9\times 3.5$ | $13^{\circ}$ | 24 | 1.0606 | 0.0424 | 0.0006 | 0.0009 | 0.0424
| | | | | | 70 | 12.034 | 1.203 | 0.025 | 0.036 | 1.204
| | | | | | 160 | 47.64 | 5.72 | 0.05 | 0.06 | 5.72
NGC 3982 | 74 | 11:56:28.1 | +55:07:31 | 2.3 | | 24 | 0.7506 | 0.0300 | 0.0008 | 0.0006 | 0.0300
| | | | | | 70 | 9.222 | 0.922 | 0.024 | 0.012 | 0.923
| | | | | | 160 | 14.39 | 1.73 | 0.03 | 0.02 | 1.73
NGC 4030 | 77 | 12:00:23.6 | -01:06:00 | $4.2\times 3.0$ | $27^{\circ}$ | 24 | 1.9186 | 0.0767 | 0.0005 | 0.0004 | 0.0767
| | | | | | 70 | 18.994 | 1.899 | 0.046 | 0.011 | 1.900
| | | | | | 160 | 57.33 | 6.88 | 1.19 | 0.03 | 6.98
KUG 1201 | 82 | 12:03:35.9 | +16:03:20 | $1.0\times 1.0$ | $0^{\circ}$ | 24 | 0.0596 | 0.0024 | 0.0002 | 0.0003 | 0.0024
+163 | | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4116 | 86 | 12:07:37.1 | +02:41:26 | $3.8\times 2.2$ | $155^{\circ}$ | 24 | 0.2172 | 0.0087 | 0.0003 | 0.0004 | 0.0087
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4178 | 89 | 12:12:46.4 | +10:51:57 | $5.1\times 1.8$ | $30^{\circ}$ | 24 | 0.3898 | 0.0156 | 0.0003 | 0.0004 | 0.0156
| | | | | | 70 | 5.138 | 0.514 | 0.011 | 0.011 | 0.514
| | | | | | 160 | 14.16 | 1.70 | 0.03 | 0.02 | 1.70
NGC 4192 | 91 | 12:13:48.2 | +14:54:01 | $9.8\times 2.8$ | $155^{\circ}$ | 24 | 1.0139 | 0.0406 | 0.0006 | 0.0007 | 0.0406
| | | | | | 70 | 11.914 | 1.191 | 0.046 | 0.023 | 1.193
| | | | | | 160 | 42.78g | 5.13 | 0.04 | 0.06 | 5.13
NGC 4203 | 93 | 12:15:05.0 | +33:11:50 | $3.4\times 3.2$ | $10^{\circ}$ | 24 | 0.0759 | 0.0030 | 0.0004 | 0.0006 | 0.0031
| | | | | | 70 | 0.895 | 0.090 | 0.013 | 0.017 | 0.092
| | | | | | 160 | 4.11 | 0.49 | 0.01 | 0.02 | 0.49
NGC 4207 | 95 | 12:15:30.4 | +09:35:06 | $1.6\times 0.8$ | $124^{\circ}$ | 24 | 0.2359 | 0.0094 | 0.0003 | 0.0003 | 0.0094
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4208 | 96 | 12:15:39.3 | +13:54:05 | $3.2\times 1.9$ | $75^{\circ}$ | 24 | 0.7779 | 0.0311 | 0.0009 | 0.0006 | 0.0311
| | | | | | 70 | 9.041 | 0.904 | 0.040 | 0.015 | 0.905
| | | | | | 160 | 20.97 | 2.52 | 0.03 | 0.02 | 2.52
NGC 4237 | 100 | 12:17:11.4 | +15:19:26 | $2.1\times 1.3$ | $108^{\circ}$ | 24 | 0.3020 | 0.0121 | 0.0002 | 0.0003 | 0.0121
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4251 | 101 | 12:18:08.3 | +28:10:31 | $3.6\times 1.5$ | $100^{\circ}$ | 24 | 0.0259 | 0.0010 | 0.0003 | 0.0005 | 0.0012
| | | | | | 70 | $<0.082$ | | | |
| | | | | | 160 | $<0.13$ | | | |
NGC 4254 | 102 | 12:18:49.6 | +14:24:59 | 5.4 | | 24 | 4.2582 | 0.1703 | 0.0008 | 0.0008 | 0.1703
| | | | | | 70 | 44.920 | 4.492 | 0.051 | 0.023 | 4.492
| | | | | | 160 | 123.29 | 14.80 | 0.72 | 0.05 | 14.81
NGC 4260 | 103 | 12:19:22.2 | +06:05:55 | $2.7\times 1.3$ | $58^{\circ}$ | 24 | 0.0290 | 0.0012 | 0.0002 | 0.0003 | 0.0012
| | | | | | 70 | 0.375 | 0.037 | 0.010 | 0.009 | 0.040
| | | | | | 160 | 1.38 | 0.17 | 0.01 | 0.02 | 0.17
NGC 4262 | 105 | 12:19:30.5 | +14:52:40 | 1.9 | | 24 | 0.0182 | 0.0007 | 0.0002 | 0.0003 | 0.0008
| | | | | | 70 | $<0.152$ | | | |
| | | | | | 160 | $<0.35$ | | | |
NGC 4294 | 110 | 12:21:17.8 | +11:30:38 | $3.2\times 1.2$ | $155^{\circ}$ | 24 | 0.2259 | 0.0090 | 0.0002 | 0.0003 | 0.0090
| | | | | | 70 | 3.860 | 0.386 | 0.009 | 0.008 | 0.386
| | | | | | 160 | 6.69 | 0.80 | 0.02 | 0.02 | 0.80
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 4298 | 111 | 12:21:32.7 | +14:36:22 | $3.2\times 1.8$ | $140^{\circ}$ | 24 | 0.5129 | 0.0205 | 0.0002 | 0.0003 | 0.0205
| | | | | | 70 | 5.672 | 0.567 | 0.010 | 0.008 | 0.567
| | | | | | 160 | 19.18 | 2.30 | 0.03 | 0.01 | 2.30
NGC 4302 | 113 | 12:21:42.4 | +14:35:54 | $5.5\times 1.0$ | $178^{\circ}$ | 24 | 0.4855 | 0.0194 | 0.0002 | 0.0004 | 0.0194
| | | | | | 70 | 6.286 | 0.629 | 0.014 | 0.010 | 0.629
| | | | | | 160 | 26.88 | 3.23 | 0.03 | 0.02 | 3.23
NGC 4303 | 114 | 12:21:54.8 | +04:28:25 | 6.5 | | 24 | 3.9380 | 0.1575 | 0.0021 | 0.0029 | 0.1576
| | | | | | 70 | | | | |
| | | | | | 160 | 99.78f | 11.97 | 0.20 | 0.04 | 11.98
NGC 4305 | 116 | 12:22:03.6 | +12:44:27 | $2.2\times 1.2$ | $32^{\circ}$ | 24 | $<0.0043$ | | | |
| | | | | | 70 | $<0.128$ | | | |
| | | | | | 160 | $<0.21^{g}$ | | | |
NGC 4312 | 119 | 12:22:31.3 | +15:32:17 | $4.6\times 1.1$ | $170^{\circ}$ | 24 | 0.2225 | 0.0089 | 0.0004 | 0.0004 | 0.0089
| | | | | | 70 | 3.070 | 0.307 | 0.016 | 0.009 | 0.308
| | | | | | 160 | | | | |
NGC 4313 | 120 | 12:22:38.5 | +11:48:03 | $4.0\times 1.0$ | $143^{\circ}$ | 24 | 0.1512 | 0.0060 | 0.0004 | 0.0006 | 0.0061
| | | | | | 70 | 1.666 | 0.167 | 0.010 | 0.010 | 0.167
| | | | | | 160 | 5.42g | 0.65 | 0.02 | 0.04 | 0.65
NGC 4321 | 122 | 12:22:54.9 | +15:49:21 | $7.4\times 6.3$ | $30^{\circ}$ | 24 | 3.4082 | 0.1363 | 0.0009 | 0.0009 | 0.1363
| | | | | | 70 | 36.015 | 3.602 | 0.066 | 0.028 | 3.602
| | | | | | 160 | 123.21 | 14.79 | 0.43 | 0.04 | 14.79
NGC 4330 | 124 | 12:23:17.2 | +11:22:05 | $4.5\times 0.9$ | $59^{\circ}$ | 24 | 0.1086 | 0.0043 | 0.0002 | 0.0003 | 0.0044
| | | | | | 70 | 1.382 | 0.138 | 0.007 | 0.009 | 0.139
| | | | | | 160 | 4.77 | 0.57 | 0.02 | 0.01 | 0.57
IC 3259 | 128 | 12:23:48.5 | +07:11:13 | $1.7\times 0.9$ | $15^{\circ}$ | 24 | | | | |
| | | | | | 70 | $<0.107$ | | | |
| | | | | | 160 | | | | |
NGC 4350 | 129 | 12:23:57.8 | +16:41:36 | $3.0\times 1.4$ | $28^{\circ}$ | 24 | 0.0370 | 0.0015 | 0.0001 | 0.0002 | 0.0015
| | | | | | 70 | 0.641 | 0.064 | 0.006 | 0.008 | 0.065
| | | | | | 160 | 1.06 | 0.13 | 0.01 | 0.02 | 0.13
NGC 4351 | 130 | 12:24:01.5 | +12:12:17 | $2.0\times 1.3$ | $80^{\circ}$ | 24 | 0.0635 | 0.0025 | 0.0002 | 0.0003 | 0.0026
| | | | | | 70 | 0.951 | 0.095 | 0.005 | 0.006 | 0.095
| | | | | | 160 | 2.41 | 0.29 | 0.01 | 0.01 | 0.29
NGC 4356 | 134 | 12:24:14.5 | +08:32:09 | $2.8\times 0.5$ | $40^{\circ}$ | 24 | 0.0890 | 0.0036 | 0.0007 | 0.0008 | 0.0037
| | | | | | 70 | 0.650 | 0.065 | 0.018 | 0.023 | 0.071
| | | | | | 160 | 1.93 | 0.23 | 0.01 | 0.02 | 0.23
NGC 4365 | 135 | 12:24:28.2 | +07:19:03 | $6.9\times 5.0$ | $40^{\circ}$ | 24 | 0.0571 | 0.0023 | 0.0010 | 0.0015 | 0.0029
| | | | | | 70 | $<0.352$ | | | |
| | | | | | 160 | $<1.03^{g}$ | | | |
NGC 4370 | 136 | 12:24:54.9 | +07:26:42 | $1.4\times 0.7$ | $83^{\circ}$ | 24 | 0.0483 | 0.0019 | 0.0005 | 0.0007 | 0.0021
| | | | | | 70 | 1.284 | 0.128 | 0.023 | 0.019 | 0.132
| | | | | | 160 | 2.88g | 0.35 | 0.01 | 0.02 | 0.35
NGC 4371 | 137 | 12:24:55.4 | +11:42:15 | $4.0\times 2.2$ | $95^{\circ}$ | 24 | 0.0251 | 0.0010 | 0.0007 | 0.0010 | 0.0016
| | | | | | 70 | 0.123 | 0.012 | 0.005 | 0.006 | 0.014
| | | | | | 160 | $<0.27^{g}$ | | | |
NGC 4374 | 138 | 12:25:03.7 | +12:53:13 | $6.5\times 5.6$ | $135^{\circ}$ | 24 | 0.1299 | 0.0052 | 0.0005 | 0.0011 | 0.0053
| | | | | | 70 | 0.584 | 0.058 | 0.027 | 0.033 | 0.072
| | | | | | 160 | 0.99g | 0.12 | 0.02 | 0.04 | 0.13
NGC 4376 | 139 | 12:25:18.0 | +05:44:28 | $1.4\times 0.9$ | $157^{\circ}$ | 24 | 0.0467 | 0.0019 | 0.0002 | 0.0003 | 0.0019
| | | | | | 70 | 0.790 | 0.079 | 0.008 | 0.008 | 0.080
| | | | | | 160 | 2.04 | 0.24 | 0.01 | 0.02 | 0.25
NGC 4378 | 140 | 12:25:18.1 | +04:55:31 | $2.9\times 2.7$ | $167^{\circ}$ | 24 | 0.0820 | 0.0033 | 0.0005 | 0.0007 | 0.0034
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4380 | 141 | 12:25:22.2 | +10:00:57 | $3.5\times 1.9$ | $153^{\circ}$ | 24 | 0.1301 | 0.0052 | 0.0002 | 0.0003 | 0.0052
| | | | | | 70 | 1.237 | 0.124 | 0.006 | 0.009 | 0.124
| | | | | | 160 | 6.25 | 0.75 | 0.02 | 0.01 | 0.75
NGC 4383 | 142 | 12:25:25.5 | +16:28:12 | $1.9\times 1.0$ | $28^{\circ}$ | 24 | 0.9641 | 0.0386 | 0.0010 | 0.0002 | 0.0386
| | | | | | 70 | 1.237 | 0.124 | 0.006 | 0.009 | 0.124
| | | | | | 160 | 9.10 | 1.09 | 0.02 | 0.01 | 1.09
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
IC 3322A | 143 | 12:25:42.5 | +07:13:00 | $3.5\times 0.4$ | $157^{\circ}$ | 24 | 0.1816 | 0.0073 | 0.0007 | 0.0009 | 0.0074
| | | | | | 70 | 2.872 | 0.287 | 0.023 | 0.024 | 0.289
| | | | | | 160 | 7.97g | 0.96 | 0.04 | 0.05 | 0.96
NGC 4388 | 144 | 12:25:46.7 | +12:39:44 | $5.6\times 1.3$ | $92^{\circ}$ | 24 | 2.5714 | 0.1029 | 0.0041 | 0.0004 | 0.1029
| | | | | | 70 | 10.730 | 1.073 | 0.029 | 0.010 | 1.073
| | | | | | 160 | 16.08 | 1.93 | 0.20 | 0.02 | 1.94
NGC 4390 | 145 | 12:25:50.6 | +10:27:33 | $1.7\times 1.3$ | $95^{\circ}$ | 24 | 0.0775 | 0.0031 | 0.0004 | 0.0006 | 0.0032
| | | | | | 70 | 0.980 | 0.098 | 0.005 | 0.007 | 0.098
| | | | | | 160 | 2.12g | 0.25 | 0.01 | 0.03 | 0.26
IC 3322 | 146 | 12:25:54.1 | +07:33:17 | $2.3\times 0.5$ | $156^{\circ}$ | 24 | 0.0603 | 0.0024 | 0.0005 | 0.0007 | 0.0026
| | | | | | 70 | 1.000 | 0.100 | 0.017 | 0.020 | 0.103
| | | | | | 160 | 2.23g | 0.27 | 0.02 | 0.02 | 0.27
NGC 4396 | 148 | 12:25:58.8 | +15:40:17 | $3.3\times 1.0$ | $125^{\circ}$ | 24 | 0.1290 | 0.0052 | 0.0002 | 0.0003 | 0.0052
| | | | | | 70 | 2.090 | 0.209 | 0.006 | 0.008 | 0.209
| | | | | | 160 | 4.95 | 0.59 | 0.02 | 0.01 | 0.59
NGC 4402 | 149 | 12:26:07.5 | +13:06:46 | $3.9\times 1.1$ | $90^{\circ}$ | 24 | 0.6473 | 0.0259 | 0.0002 | 0.0003 | 0.0259
| | | | | | 70 | 8.281 | 0.828 | 0.017 | 0.010 | 0.828
| | | | | | 160 | 22.05 | 2.65 | 0.04 | 0.02 | 2.65
NGC 4406 | 150 | 12:26:11.7 | +12:56:46 | $8.9\times 5.8$ | $130^{\circ}$ | 24 | 0.1221 | 0.0049 | 0.0003 | 0.0005 | 0.0049
| | | | | | 70 | $<0.204$ | | | |
| | | | | | 160 | 0.97g | 0.12 | 0.03 | 0.02 | 0.12
NGC 4407 | 151 | 12:26:32.2 | +12:36:40 | $2.3\times 1.5$ | $60^{\circ}$ | 24 | 0.1445 | 0.0058 | 0.0002 | 0.0003 | 0.0058
| | | | | | 70 | 1.381 | 0.138 | 0.005 | 0.008 | 0.138
| | | | | | 160 | 3.78 | 0.45 | 0.01 | 0.01 | 0.45
NGC 4412 | 152 | 12:26:36.0 | +03:57:53 | 1.4 | | 24 | 0.4029 | 0.0161 | 0.0011 | 0.0003 | 0.0162
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4416 | 153 | 12:26:46.7 | +07:55:08 | 1.7 | | 24 | 0.1114 | 0.0045 | 0.0005 | 0.0007 | 0.0045
| | | | | | 70 | 1.506 | 0.151 | 0.016 | 0.019 | 0.153
| | | | | | 160 | 3.44g | 0.41 | 0.02 | 0.02 | 0.41
NGC 4411B | 154 | 12:26:47.2 | +08:53:05 | 2.5 | | 24 | 0.0601 | 0.0024 | 0.0006 | 0.0008 | 0.0026
| | | | | | 70 | 1.188 | 0.119 | 0.017 | 0.022 | 0.122
| | | | | | 160 | 2.79g | 0.34 | 0.02 | 0.02 | 0.34
NGC 4417 | 155 | 12:26:50.6 | +09:35:03 | $3.4\times 1.3$ | $49^{\circ}$ | 24 | 0.0213 | 0.0009 | 0.0006 | 0.0009 | 0.0014
| | | | | | 70 | $<0.115$ | | | |
| | | | | | 160 | $<0.26^{g}$ | | | |
NGC 4419 | 156 | 12:26:56.4 | +15:02:51 | $3.3\times 1.1$ | $133^{\circ}$ | 24 | 1.2483 | 0.0499 | 0.0022 | 0.0003 | 0.0500
| | | | | | 70 | 8.091 | 0.809 | 0.032 | 0.008 | 0.810
| | | | | | 160 | 13.71 | 1.65 | 0.03 | 0.02 | 1.65
NGC 4409 | 157 | 12:26:58.5 | +02:29:40 | $2.0\times 1.0$ | $8^{\circ}$ | 24 | 0.2485 | 0.0099 | 0.0002 | 0.0003 | 0.0099
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4424 | 159 | 12:27:11.5 | +09:25:14 | $3.6\times 1.8$ | $95^{\circ}$ | 24 | 0.3235 | 0.0129 | 0.0004 | 0.0003 | 0.0130
| | | | | | 70 | 3.636 | 0.364 | 0.014 | 0.008 | 0.364
| | | | | | 160 | 5.19 | 0.62 | 0.02 | 0.01 | 0.62
NGC 4429 | 161 | 12:27:26.5 | +11:06:28 | $5.6\times 2.6$ | $99^{\circ}$ | 24 | 0.1452 | 0.0058 | 0.0010 | 0.0014 | 0.0060
| | | | | | 70 | 2.856 | 0.286 | 0.039 | 0.037 | 0.291
| | | | | | 160 | 4.45g | 0.53 | 0.26 | 0.05 | 0.60
NGC 4435 | 162 | 12:27:40.4 | +13:04:44 | $2.8\times 2.0$ | $13^{\circ}$ | 24 | 0.1342 | 0.0054 | 0.0002 | 0.0003 | 0.0054
| | | | | | 70 | 2.569 | 0.257 | 0.014 | 0.008 | 0.257
| | | | | | 160 | 4.20 | 0.50 | 0.02 | 0.01 | 0.50
NGC 4438 | 163 | 12:27:45.5 | +13:00:32 | $8.5\times 3.2$ | $27^{\circ}$ | 24 | 0.3026 | 0.0121 | 0.0004 | 0.0006 | 0.0121
| | | | | | 70 | 5.932 | 0.593 | 0.024 | 0.018 | 0.594
| | | | | | 160 | 15.01 | 1.80 | 0.04 | 0.02 | 1.80
NGC 4440 | 164 | 12:27:53.5 | +12:17:36 | 1.9 | | 24 | 0.0191 | 0.0008 | 0.0005 | 0.0007 | 0.0011
| | | | | | 70 | $<0.117$ | | | |
| | | | | | 160 | $<0.11^{g}$ | | | |
NGC 4442 | 166 | 12:28:03.8 | +09:48:13 | $4.6\times 1.8$ | $87^{\circ}$ | 24 | 0.0429 | 0.0017 | 0.0003 | 0.0006 | 0.0018
| | | | | | 70 | $<0.057$ | | | |
| | | | | | 160 | $<0.47$ | | | |
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 4445 | 167 | 12:28:15.9 | +09:26:10 | $2.6\times 0.5$ | $106^{\circ}$ | 24 | 0.0366 | 0.0015 | 0.0006 | 0.0008 | 0.0018
| | | | | | 70 | 0.538 | 0.054 | 0.012 | 0.017 | 0.058
| | | | | | 160 | 1.62g | 0.19 | 0.02 | 0.07 | 0.21
UGC 7590 | 168 | 12:28:18.7 | +08:43:46 | $1.3\times 0.4$ | $168^{\circ}$ | 24 | 0.0320 | 0.0013 | 0.0005 | 0.0007 | 0.0015
| | | | | | 70 | 0.528 | 0.053 | 0.015 | 0.019 | 0.058
| | | | | | 160 | 0.94g | 0.11 | 0.01 | 0.02 | 0.11
NGC 4450 | 170 | 12:28:29.6 | +17:05:06 | $5.2\times 3.9$ | $175^{\circ}$ | 24 | 0.2089 | 0.0084 | 0.0004 | 0.0006 | 0.0084
| | | | | | 70 | 2.890 | 0.289 | 0.013 | 0.015 | 0.290
| | | | | | 160 | 13.54 | 1.62 | 0.02 | 0.03 | 1.62
NGC 4451 | 171 | 12:28:40.5 | +09:15:32 | $1.5\times 1.0$ | $162^{\circ}$ | 24 | 0.1504 | 0.0060 | 0.0005 | 0.0007 | 0.0061
| | | | | | 70 | 2.497 | 0.250 | 0.023 | 0.018 | 0.251
| | | | | | 160 | 4.48g | 0.54 | 0.02 | 0.02 | 0.54
IC 3392 | 172 | 12:28:43.2 | +14:59:58 | $2.3\times 1.0$ | $40^{\circ}$ | 24 | 0.1184 | 0.0047 | 0.0002 | 0.0003 | 0.0047
| | | | | | 70 | 1.579 | 0.158 | 0.007 | 0.007 | 0.158
| | | | | | 160 | 3.64 | 0.44 | 0.09 | 0.02 | 0.45
NGC 4457 | 173 | 12:28:59.0 | +03:34:14 | 2.7 | | 24 | 0.4012 | 0.0160 | 0.0005 | 0.0003 | 0.0161
| | | | | | 70 | 5.478 | 0.548 | 0.021 | 0.010 | 0.548
| | | | | | 160 | 9.16 | 1.10 | 0.02 | 0.01 | 1.10
NGC 4459 | 174 | 12:29:00.0 | +13:58:43 | $3.5\times 2.7$ | $110^{\circ}$ | 24 | 0.1292 | 0.0052 | 0.0008 | 0.0010 | 0.0053
| | | | | | 70 | 2.364 | 0.236 | 0.033 | 0.028 | 0.240
| | | | | | 160 | 3.71g | 0.45 | 0.02 | 0.06 | 0.45
NGC 4461 | 175 | 12:29:03.0 | +13:11:02 | $3.5\times 1.4$ | $9^{\circ}$ | 24 | 0.0222 | 0.0009 | 0.0005 | 0.0005 | 0.0011
| | | | | | 70 | $<0.153$ | | | |
| | | | | | 160 | $<0.16^{g}$ | | | |
NGC 4469 | 176 | 12:29:28.0 | +08:44:60 | $3.8\times 1.3$ | $89^{\circ}$ | 24 | 0.0876 | 0.0035 | 0.0007 | 0.0010 | 0.0037
| | | | | | 70 | 1.367 | 0.137 | 0.024 | 0.026 | 0.141
| | | | | | 160 | 3.24g | 0.39 | 0.02 | 0.08 | 0.40
NGC 4470 | 177 | 12:29:37.7 | +07:49:27 | $1.3\times 0.9$ | $0^{\circ}$ | 24 | 0.1511 | 0.0060 | 0.0003 | 0.0005 | 0.0061
| | | | | | 70 | 2.352 | 0.235 | 0.022 | 0.018 | 0.237
| | | | | | 160 | 3.93 | 0.47 | 0.02 | 0.01 | 0.47
NGC 4472 | 178 | 12:29:46.7 | +08:00:02 | $10.2\times 8.3$ | $155^{\circ}$ | 24 | 0.2047 | 0.0082 | 0.0013 | 0.0019 | 0.0082
| | | | | | 70 | $<0.354$ | | | |
| | | | | | 160 | $<1.38^{g}$ | | | |
NGC 4473 | 179 | 12:29:48.8 | +13:25:46 | $4.5\times 2.5$ | $100^{\circ}$ | 24 | 0.0335 | 0.0013 | 0.0003 | 0.0005 | 0.0014
| | | | | | 70 | $<0.203$ | | | |
| | | | | | 160 | $<0.21^{g}$ | | | |
NGC 4477 | 180 | 12:30:02.1 | +13:38:12 | $3.8\times 3.5$ | $15^{\circ}$ | 24 | 0.0518 | 0.0021 | 0.0005 | 0.0007 | 0.0022
| | | | | | 70 | 0.682 | 0.068 | 0.025 | 0.032 | 0.080
| | | | | | 160 | 0.81g | 0.10 | 0.02 | 0.05 | 0.11
NGC 4478 | 181 | 12:30:17.4 | +12:19:43 | $1.9\times 1.6$ | $140^{\circ}$ | 24 | 0.0256 | 0.0010 | 0.0005 | 0.0007 | 0.0013
| | | | | | 70 | $<0.117$ | | | |
| | | | | | 160 | $<0.17$ | | | |
NGC 4486 | 183 | 12:30:49.4 | +12:23:28 | 8.3 | | 24 | 0.2511 | 0.0100 | 0.0014 | 0.0020 | 0.0105
| | | | | | 70 | 0.429 | 0.043 | 0.028 | 0.042 | 0.066
| | | | | | 160 | 0.30g | 0.04 | 0.03 | 0.05 | 0.07
NGC 4491 | 184 | 12:30:57.1 | +11:29:01 | $1.7\times 0.9$ | $148^{\circ}$ | 24 | 0.3183 | 0.0127 | 0.0021 | 0.0007 | 0.0129
| | | | | | 70 | 2.490 | 0.249 | 0.038 | 0.019 | 0.253
| | | | | | 160 | 1.71g | 0.21 | 0.02 | 0.03 | 0.21
NGC 4492 | 185 | 12:30:59.7 | +08:04:40 | 1.7 | | 24 | 0.0415 | 0.0017 | 0.0005 | 0.0007 | 0.0019
| | | | | | 70 | 0.288 | 0.029 | 0.012 | 0.020 | 0.037
| | | | | | 160 | 1.62g | 0.19 | 0.01 | 0.02 | 0.20
NGC 4494 | 186 | 12:31:24.0 | +25:46:30 | 4.8 | | 24 | 0.0600 | 0.0024 | 0.0005 | 0.0007 | 0.0025
| | | | | | 70 | 0.342 | 0.034 | 0.015 | 0.017 | 0.041
| | | | | | 160 | 0.43f | 0.05 | 0.01 | 0.02 | 0.06
NGC 4496 | 187 | 12:31:39.2 | +03:56:22 | $4.0\times 3.0$ | $70^{\circ}$ | 24 | 0.4920 | 0.0197 | 0.0006 | 0.0006 | 0.0197
| | | | | | 70 | 5.956 | 0.596 | 0.022 | 0.025 | 0.597
| | | | | | 160 | 13.84 | 1.66 | 0.02 | 0.02 | 1.66
NGC 4498 | 188 | 12:31:39.5 | +16:51:10 | $3.0\times 1.6$ | $133^{\circ}$ | 24 | 0.1343 | 0.0054 | 0.0002 | 0.0002 | 0.0054
| | | | | | 70 | 2.050 | 0.205 | 0.006 | 0.008 | 0.205
| | | | | | 160 | 5.48 | 0.66 | 0.02 | 0.01 | 0.66
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
IC 797 | 189 | 12:31:54.7 | +15:07:26 | $1.3\times 0.9$ | $108^{\circ}$ | 24 | 0.0731 | 0.0029 | 0.0002 | 0.0003 | 0.0029
| | | | | | 70 | 1.105 | 0.111 | 0.006 | 0.010 | 0.111
| | | | | | 160 | 2.31 | 0.28 | 0.01 | 0.02 | 0.28
NGC 4501 | 190 | 12:31:59.2 | +14:25:14 | $6.9\times 3.7$ | $140^{\circ}$ | 24 | 2.2216 | 0.0889 | 0.0004 | 0.0006 | 0.0889
| | | | | | 70 | 28.284 | 2.828 | 0.024 | 0.016 | 2.829
| | | | | | 160 | 98.41 | 11.81 | 0.08 | 0.04 | 11.81
NGC 4506 | 192 | 12:32:10.5 | +13:25:11 | $1.6\times 1.1$ | $110^{\circ}$ | 24 | 0.0126 | 0.0005 | 0.0005 | 0.0007 | 0.0010
| | | | | | 70 | 0.357 | 0.036 | 0.012 | 0.018 | 0.042
| | | | | | 160 | 0.34g | 0.04 | 0.01 | 0.02 | 0.05
NGC 4517 | 194 | 12:32:45.5 | +00:06:54 | $10.5\times 1.5$ | $83^{\circ}$ | 24 | 1.1476 | 0.0459 | 0.0007 | 0.0007 | 0.0459
| | | | | | 70 | 11.273 | 1.127 | 0.023 | 0.019 | 1.128
| | | | | | 160 | 45.06 | 5.41 | 0.11 | 0.03 | 5.41
NGC 4516 | 195 | 12:33:07.5 | +14:34:30 | $1.7\times 1.0$ | $0^{\circ}$ | 24 | 0.0077 | 0.0003 | 0.0005 | 0.0007 | 0.0009
| | | | | | 70 | | | | |
| | | | | | 160 | $<0.16^{g}$ | | | |
NGC 4519 | 196 | 12:33:30.2 | +08:39:17 | $3.2\times 2.5$ | $145^{\circ}$ | 24 | 0.5386 | 0.0215 | 0.0023 | 0.0011 | 0.0217
| | | | | | 70 | 5.006 | 0.501 | 0.029 | 0.020 | 0.502
| | | | | | 160 | 8.44g | 1.01 | 0.04 | 0.03 | 1.01
NGC 4522 | 197 | 12:33:40.0 | +09:10:30 | $3.7\times 1.0$ | $33^{\circ}$ | 24 | 0.1542 | 0.0062 | 0.0002 | 0.0003 | 0.0062
| | | | | | 70 | 2.011 | 0.201 | 0.009 | 0.008 | 0.201
| | | | | | 160 | 5.53 | 0.66 | 0.02 | 0.01 | 0.66
IC 800 | 199 | 12:33:56.6 | +15:21:17 | $1.5\times 1.1$ | $148^{\circ}$ | 24 | 0.0421 | 0.0017 | 0.0002 | 0.0003 | 0.0017
| | | | | | 70 | 0.636 | 0.064 | 0.005 | 0.010 | 0.065
| | | | | | 160 | 1.35 | 0.16 | 0.01 | 0.03 | 0.16
NGC 4526 | 200 | 12:34:03.0 | +07:41:57 | $7.2\times 2.4$ | $113^{\circ}$ | 24 | 0.3144 | 0.0126 | 0.0009 | 0.0010 | 0.0126
| | | | | | 70 | 8.098f | 0.810 | 0.047 | 0.017 | 0.811
| | | | | | 160 | 11.84f | 1.42 | 0.03 | 0.04 | 1.42
IC 3510 | 202 | 12:34:14.8 | +11:04:17 | $0.9\times 0.6$ | $0^{\circ}$ | 24 | $<0.0043$ | | | |
| | | | | | 70 | $<0.112$ | | | |
| | | | | | 160 | $<0.65^{g}$ | | | |
NGC 4532 | 203 | 12:34:19.3 | +06:28:04 | $2.8\times 1.1$ | $160^{\circ}$ | 24 | 0.8125 | 0.0325 | 0.0004 | 0.0003 | 0.0325
| | | | | | 70 | 9.742 | 0.974 | 0.022 | 0.008 | 0.974
| | | | | | 160 | 12.93 | 1.55 | 0.02 | 0.02 | 1.55
NGC 4535 | 204 | 12:34:20.3 | +08:11:52 | $7.1\times 5.0$ | $0^{\circ}$ | 24 | 1.7829 | 0.0713 | 0.0024 | 0.0015 | 0.0714
| | | | | | 70 | 16.427 | 1.643 | 0.052 | 0.031 | 1.644
| | | | | | 160 | 58.76g | 7.05 | 0.05 | 0.05 | 7.05
NGC 4536 | 205 | 12:34:27.1 | +02:11:16 | $7.6\times 3.2$ | $130^{\circ}$ | 24 | 3.5045 | 0.1402 | 0.0047 | 0.0008 | 0.1403
| | | | | | 70 | 26.991 | 2.699 | 0.122 | 0.019 | 2.702
| | | | | | 160 | 49.47 | 5.94 | 0.05 | 0.03 | 5.94
NGC 4548 | 208 | 12:35:26.4 | +14:29:47 | $5.4\times 4.3$ | $150^{\circ}$ | 24 | 0.4331 | 0.0173 | 0.0003 | 0.0006 | 0.0173
| | | | | | 70 | 4.350 | 0.435 | 0.013 | 0.014 | 0.435
| | | | | | 160 | 26.01 | 3.12 | 0.03 | 0.03 | 3.12
NGC 4546 | 209 | 12:35:29.5 | -03:47:35 | $3.3\times 1.4$ | $78^{\circ}$ | 24 | 0.0498 | 0.0020 | 0.0003 | 0.0005 | 0.0021
| | | | | | 70 | 0.206 | 0.021 | 0.010 | 0.015 | 0.028
| | | | | | 160 | | | | |
NGC 4550 | 210 | 12:35:30.6 | +12:13:15 | $3.3\times 0.9$ | $178^{\circ}$ | 24 | 0.0273 | 0.0011 | 0.0004 | 0.0005 | 0.0013
| | | | | | 70 | 0.406 | 0.041 | 0.018 | 0.022 | 0.050
| | | | | | 160 | $<0.11^{g}$ | | | |
NGC 4552 | 211 | 12:35:39.8 | +12:33:23 | 5.1 | | 24 | 0.0960 | 0.0038 | 0.0003 | 0.0006 | 0.0039
| | | | | | 70 | 0.119 | 0.012 | 0.009 | 0.011 | 0.019
| | | | | | 160 | $<0.87^{g}$ | | | |
NGC 4561 | 212 | 12:36:08.1 | +19:19:21 | $1.5\times 1.3$ | $30^{\circ}$ | 24 | 0.0805 | 0.0032 | 0.0001 | 0.0002 | 0.0032
| | | | | | 70 | 1.551 | 0.155 | 0.005 | 0.006 | 0.155
| | | | | | 160 | 2.50 | 0.30 | 0.01 | 0.01 | 0.30
NGC 4565 | 213 | 12:36:20.7 | +25:59:16 | $15.8\times 2.1$ | $136^{\circ}$ | 24 | 1.6495 | 0.0660 | 0.0005 | 0.0007 | 0.0660
| | | | | | 70 | 19.257 | 1.926 | 0.026 | 0.022 | 1.926
| | | | | | 160 | 86.49 | 10.38 | 0.07 | 0.03 | 10.38
NGC 4564 | 214 | 12:36:26.9 | +11:26:22 | $3.5\times 1.5$ | $47^{\circ}$ | 24 | 0.0138 | 0.0006 | 0.0006 | 0.0009 | 0.0012
| | | | | | 70 | $<0.168$ | | | |
| | | | | | 160 | $<0.14^{g}$ | | | |
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 4567 | 215 | | | | | 24 | 2.0960 | 0.0838 | 0.0019 | 0.0020 | 0.0839
/4568h | /216 | | | | | 70 | 27.005 | 2.700 | 0.109 | 0.054 | 2.703
| | | | | | 160 | 68.02g | 8.16 | 0.49 | 0.05 | 8.18
NGC 4569 | 217 | 12:36:49.7 | +13:09:46 | $9.5\times 4.4$ | $23^{\circ}$ | 24 | 1.4279 | 0.0571 | 0.0019 | 0.0009 | 0.0572
| | | | | | 70 | 11.451 | 1.145 | 0.041 | 0.021 | 1.146
| | | | | | 160 | 35.98 | 4.32 | 0.05 | 0.05 | 4.32
NGC 4570 | 218 | 12:36:53.4 | +07:14:48 | $3.8\times 1.1$ | $159^{\circ}$ | 24 | 0.0287 | 0.0011 | 0.0002 | 0.0003 | 0.0012
| | | | | | 70 | $<0.047$ | | | |
| | | | | | 160 | $<0.07$ | | | |
NGC 4578 | 219 | 12:37:30.5 | +09:33:18 | $3.3\times 2.5$ | $35^{\circ}$ | 24 | 0.0206 | 0.0008 | 0.0003 | 0.0006 | 0.0011
| | | | | | 70 | $<0.119$ | | | |
| | | | | | 160 | $<0.30$ | | | |
NGC 4579 | 220 | 12:37:43.6 | +11:49:05 | $5.9\times 4.7$ | $95^{\circ}$ | 24 | 0.8077 | 0.0323 | 0.0007 | 0.0008 | 0.0323
| | | | | | 70 | 9.585 | 0.958 | 0.020 | 0.019 | 0.959
| | | | | | 160 | 36.16 | 4.34 | 0.30 | 0.05 | 4.35
NGC 4580 | 221 | 12:37:48.3 | +05:22:07 | $2.1\times 1.6$ | $165^{\circ}$ | 24 | 0.1448 | 0.0058 | 0.0002 | 0.0003 | 0.0058
| | | | | | 70 | 1.937 | 0.194 | 0.007 | 0.007 | 0.194
| | | | | | 160 | 5.65 | 0.68 | 0.02 | 0.01 | 0.68
NGC 4584 | 222 | 12:38:17.8 | +13:06:36 | 5.1 | | 24 | 0.0751 | 0.0030 | 0.0022 | 0.0029 | 0.0047
| | | | | | 70 | | | | |
| | | | | | 160 | 0.60g | 0.07 | 0.01 | 0.03 | 0.08
NGC 4592 | 227 | 12:39:18.7 | -00:31:55 | $5.8\times 1.5$ | $97^{\circ}$ | 24 | 0.1504 | 0.0060 | 0.0004 | 0.0005 | 0.0061
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4596 | 231 | 12:39:55.9 | +10:10:34 | $4.0\times 3.0$ | $135^{\circ}$ | 24 | 0.0554 | 0.0022 | 0.0005 | 0.0007 | 0.0024
| | | | | | 70 | 0.650 | 0.065 | 0.025 | 0.033 | 0.077
| | | | | | 160 | | | | |
NGC 4606 | 232 | 12:40:57.5 | +11:54:44 | $3.2\times 1.6$ | $33^{\circ}$ | 24 | 0.0877 | 0.0035 | 0.0002 | 0.0003 | 0.0035
| | | | | | 70 | 1.335 | 0.133 | 0.008 | 0.008 | 0.134
| | | | | | 160 | 2.80 | 0.34 | 0.05 | 0.02 | 0.34
NGC 4607 | 233 | 12:41:12.4 | +11:53:12 | $2.9\times 0.7$ | $2^{\circ}$ | 24 | 0.2676 | 0.0107 | 0.0002 | 0.0002 | 0.0107
| | | | | | 70 | 4.108 | 0.411 | 0.013 | 0.007 | 0.411
| | | | | | 160 | 8.55 | 1.03 | 0.05 | 0.01 | 1.03
NGC 4612 | 235 | 12:41:32.7 | +07:18:53 | $2.5\times 1.9$ | $145^{\circ}$ | 24 | 0.0139 | 0.0006 | 0.0002 | 0.0005 | 0.0008
| | | | | | 70 | $<0.101$ | | | |
| | | | | | 160 | $<0.11$ | | | |
NGC 4621 | 236 | 12:42:02.3 | +11:38:49 | $5.4\times 3.7$ | $165^{\circ}$ | 24 | 0.1028 | 0.0041 | 0.0011 | 0.0015 | 0.0045
| | | | | | 70 | $<0.240$ | | | |
| | | | | | 160 | $<0.83^{g}$ | | | |
NGC 4630 | 237 | 12:42:31.1 | +03:57:37 | $1.8\times 1.3$ | $10^{\circ}$ | 24 | 0.2877 | 0.0115 | 0.0006 | 0.0003 | 0.0115
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4638 | 240 | 12:42:47.4 | +11:26:33 | $2.2\times 1.4$ | $125^{\circ}$ | 24 | 0.0159 | 0.0006 | 0.0003 | 0.0004 | 0.0008
| | | | | | 70 | $<0.110$ | | | |
| | | | | | 160 | $<0.25$ | | | |
NGC 4636 | 241 | 12:42:49.8 | +02:41:16 | $6.0\times 4.7$ | $150^{\circ}$ | 24 | 0.1134 | 0.0045 | 0.0009 | 0.0014 | 0.0048
| | | | | | 70 | | | | |
| | | | | | 160 | 0.40 | 0.05 | 0.01 | 0.01 | 0.05
NGC 4639 | 242 | 12:42:52.3 | +13:15:27 | $2.8\times 1.9$ | $123^{\circ}$ | 24 | 0.1554 | 0.0062 | 0.0003 | 0.0006 | 0.0062
| | | | | | 70 | | | | |
| | | | | | 160 | 3.55g | 0.43 | 0.12 | 0.13 | 0.46
NGC 4647 | 244 | 12:43:32.3 | +11:34:55 | $2.9\times 2.3$ | $125^{\circ}$ | 24 | 0.6396 | 0.0256 | 0.0006 | 0.0008 | 0.0256
| | | | | | 70 | 7.480 | 0.748 | 0.030 | 0.021 | 0.749
| | | | | | 160 | 16.30g | 1.96 | 0.14 | 0.15 | 1.97
NGC 4649 | 245 | 12:43:39.9 | +11:33:10 | $7.4\times 6.0$ | $105^{\circ}$ | 24 | 0.1758 | 0.0070 | 0.0010 | 0.0014 | 0.0075
| | | | | | 70 | $<0.570$ | | | |
| | | | | | 160 | $<1.95^{g}$ | | | |
NGC 4651 | 246 | 12:43:42.6 | +16:23:36 | $4.0\times 2.6$ | $80^{\circ}$ | 24 | 0.5691 | 0.0228 | 0.0002 | 0.0003 | 0.0228
| | | | | | 70 | 7.818 | 0.782 | 0.014 | 0.009 | 0.782
| | | | | | 160 | 20.72 | 2.49 | 0.03 | 0.02 | 2.49
Table 5: Photometry for the Herscher Reference Survey (continued) Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 4654 | 247 | 12:43:56.5 | +13:07:36 | $4.9\times 2.8$ | $128^{\circ}$ | 24 | 1.6726 | 0.0669 | 0.0005 | 0.0004 | 0.0669
| | | | | | 70 | 19.503 | 1.950 | 0.021 | 0.012 | 1.950
| | | | | | 160 | 48.60 | 5.83 | 0.07 | 0.02 | 5.83
NGC 4660 | 248 | 12:44:31.9 | +11:11:26 | $2.2\times 1.6$ | $100^{\circ}$ | 24 | 0.0173 | 0.0007 | 0.0001 | 0.0001 | 0.0007
| | | | | | 70 | $<0.072$ | | | |
| | | | | | 160 | $<0.05^{g}$ | | | |
IC 3718 | 249 | 12:44:45.9 | +12:21:05 | $2.7\times 1.0$ | $72^{\circ}$ | 24 | $<0.0104$ | | | |
| | | | | | 70 | $<0.232$ | | | |
| | | | | | 160 | $<0.85$ | | | |
NGC 4666 | 251 | 12:45:08.5 | -00:27:43 | $4.6\times 1.3$ | $42^{\circ}$ | 24 | 3.2683 | 0.1307 | 0.0018 | 0.0008 | 0.1307
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4688 | 252 | 12:47:46.5 | +04:20:10 | 3.2 | | 24 | 0.1753 | 0.0070 | 0.0005 | 0.0006 | 0.0071
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4689 | 254 | 12:47:45.5 | +13:45:46 | 4.3 | | 24 | 0.4682 | 0.0187 | 0.0003 | 0.0004 | 0.0187
| | | | | | 70 | 4.949 | 0.495 | 0.011 | 0.015 | 0.495
| | | | | | 160 | 17.45 | 2.09 | 0.04 | 0.02 | 2.09
NGC 4698 | 257 | 12:48:22.9 | +08:29:14 | $4.0\times 2.5$ | $170^{\circ}$ | 24 | 0.1173 | 0.0047 | 0.0002 | 0.0003 | 0.0047
| | | | | | 70 | 0.688 | 0.069 | 0.007 | 0.010 | 0.070
| | | | | | 160 | 5.41 | 0.65 | 0.02 | 0.02 | 0.65
NGC 4697 | 258 | 12:48:35.9 | -05:48:03 | $7.2\times 4.7$ | $70^{\circ}$ | 24 | 0.0786 | 0.0031 | 0.0009 | 0.0012 | 0.0035
| | | | | | 70 | 0.502 | 0.050 | 0.028 | 0.031 | 0.065
| | | | | | 160 | 1.32 | 0.16 | 0.09 | 0.04 | 0.19
NGC 4701 | 259 | 12:49:11.5 | +03:23:19 | $2.8\times 2.1$ | $45^{\circ}$ | 24 | 0.2414 | 0.0097 | 0.0003 | 0.0004 | 0.0097
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4725 | 263 | 12:50:26.6 | +25:30:03 | $10.7\times 7.6$ | $35^{\circ}$ | 24 | 0.8748 | 0.0350 | 0.0008 | 0.0012 | 0.0350
| | | | | | 70 | 7.469 | 0.747 | 0.023 | 0.028 | 0.748
| | | | | | 160 | 51.70 | 6.20 | 0.05 | 0.04 | 6.20
NGC 4754 | 269 | 12:52:17.5 | +11:18:49 | $4.6\times 2.5$ | $23^{\circ}$ | 24 | 0.0401 | 0.0016 | 0.0002 | 0.0004 | 0.0017
| | | | | | 70 | $<0.065$ | | | |
| | | | | | 160 | $<0.82^{f}$ | | | |
NGC 4762 | 272 | 12:52:56.0 | +11:13:51 | $8.7\times 1.7$ | $32^{\circ}$ | 24 | 0.0463 | 0.0019 | 0.0003 | 0.0004 | 0.0019
| | | | | | 70 | $<0.072$ | | | |
| | | | | | 160 | $<0.17^{f}$ | | | |
NGC 4772 | 274 | 12:53:29.1 | +02:10:06 | $3.4\times 1.7$ | $147^{\circ}$ | 24 | 0.0568 | 0.0023 | 0.0003 | 0.0005 | 0.0023
| | | | | | 70 | 0.748 | 0.075 | 0.011 | 0.015 | 0.077
| | | | | | 160 | | | | |
UGC 8041 | 279 | 12:55:12.6 | +00:06:60 | $3.1\times 1.9$ | $165^{\circ}$ | 24 | 0.0859 | 0.0034 | 0.0002 | 0.0004 | 0.0035
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 4808 | 283 | 12:55:48.9 | +04:18:15 | $2.8\times 1.1$ | $127^{\circ}$ | 24 | 0.6398 | 0.0256 | 0.0003 | 0.0002 | 0.0256
| | | | | | 70 | 8.688 | 0.869 | 0.018 | 0.008 | 0.869
| | | | | | 160 | 15.99 | 1.92 | 0.03 | 0.01 | 1.92
NGC 4941 | 288 | 13:04:13.1 | -05:33:06 | $3.6\times 1.9$ | $15^{\circ}$ | 24 | 0.4272 | 0.0171 | 0.0010 | 0.0007 | 0.0171
| | | | | | 70 | 1.845 | 0.184 | 0.021 | 0.034 | 0.189
| | | | | | 160 | | | | |
NGC 5147 | 293 | 13:26:19.7 | +02:06:03 | $1.9\times 1.5$ | $120^{\circ}$ | 24 | 0.2554 | 0.0102 | 0.0003 | 0.0005 | 0.0102
| | | | | | 70 | 4.023 | 0.402 | 0.010 | 0.012 | 0.403
| | | | | | 160 | 6.18 | 0.74 | 0.02 | 0.02 | 0.74
NGC 5248 | 295 | 13:37:32.0 | +08:53:06 | $6.2\times 4.5$ | $110^{\circ}$ | 24 | 2.4003 | 0.0960 | 0.0016 | 0.0013 | 0.0960
| | | | | | 70 | 28.384 | 2.838 | 0.081 | 0.028 | 2.840
| | | | | | 160 | 66.48 | 7.98 | 0.19 | 0.05 | 7.98
NGC 5273 | 296 | 13:42:08.3 | +35:39:15 | $2.8\times 2.5$ | $10^{\circ}$ | 24 | 0.1076 | 0.0043 | 0.0006 | 0.0004 | 0.0044
| | | | | | 70 | 0.817 | 0.082 | 0.011 | 0.015 | 0.084
| | | | | | 160 | 0.91 | 0.11 | 0.01 | 0.02 | 0.11
NGC 5303 | 298 | 13:47:44.9 | +38:18:17 | $0.9\times 0.4$ | $92^{\circ}$ | 24 | 0.2983 | 0.0119 | 0.0003 | 0.0002 | 0.0119
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
Table 5: Photometry for the Herscher Reference Survey (continued)
Galaxy | HRS | Optical Disc | Wave- | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---|---
| Numbera | R. A. | Declination | Axes | Position | length | Measurement | Calibration | Error | Back- | Total
| | (J2000)b | (J2000)b | (arcmin)c | Anglecd | ($\mu$m) | (Jy) | | Map | ground |
NGC 5363 | 306 | 13:56:07.2 | +05:15:17 | $4.1\times 2.6$ | $135^{\circ}$ | 24 | 0.1421 | 0.0057 | 0.0004 | 0.0007 | 0.0057
| | | | | | 70 | 2.167 | 0.217 | 0.030 | 0.018 | 0.220
| | | | | | 160 | 4.54 | 0.54 | 0.12 | 0.04 | 0.56
NGC 5576 | 312 | 14:21:03.6 | +03:16:16 | $3.5\times 2.2$ | $95^{\circ}$ | 24 | 0.0270 | 0.0011 | 0.0003 | 0.0006 | 0.0013
| | | | | | 70 | $<0.096$ | | | |
| | | | | | 160 | $<0.33^{f}$ | | | |
NGC 5577 | 313 | 14:21:13.1 | +03:26:09 | $3.4\times 1.0$ | $56^{\circ}$ | 24 | 0.0879 | 0.0035 | 0.0003 | 0.0003 | 0.0035
| | | | | | 70 | | | | |
| | | | | | 160 | | | | |
NGC 5669 | 319 | 14:32:43.4 | +09:53:26 | $4.0\times 2.8$ | $50^{\circ}$ | 24 | 0.1959 | 0.0078 | 0.0004 | 0.0007 | 0.0079
| | | | | | 70 | 3.006 | 0.301 | 0.024 | 0.036 | 0.304
| | | | | | 160 | 7.83 | 0.94 | 0.01 | 0.03 | 0.94
NGC 5668 | 320 | 14:33:24.3 | +04:27:02 | 3.3 | | 24 | 0.2561 | 0.0102 | 0.0004 | 0.0007 | 0.0103
| | | | | | 70 | 4.444 | 0.444 | 0.025 | 0.024 | 0.446
| | | | | | 160 | 11.49 | 1.38 | 0.02 | 0.02 | 1.38
NGC 5692 | 321 | 14:38:18.1 | +03:24:37 | $0.9\times 0.6$ | $35^{\circ}$ | 24 | 0.1163 | 0.0047 | 0.0005 | 0.0007 | 0.0047
| | | | | | 70 | 1.873 | 0.187 | 0.022 | 0.017 | 0.189
| | | | | | 160 | 2.07g | 0.25 | 0.18 | 0.09 | 0.32
IC 1048 | 323 | 14:42:58.0 | +04:53:22 | $2.2\times 0.7$ | $163^{\circ}$ | 24 | 0.1624 | 0.0065 | 0.0010 | 0.0013 | 0.0067
| | | | | | 70 | | | | |
| | | | | | 160 | 5.08g | 0.61 | 0.15 | 0.12 | 0.64
a The HRS number corresponds to the numbers given by Boselli et al. (2010).
b Data are from NED.
c Data are from de Vaucouleurs et al. (1991) unless otherwise specified. If de
Vaucouleurs et al. (1991) specify both the minor/major axis ratio and the
position angle, then both axes and the position angle are listed. If de
Vaucouleurs et al. (1991) did not specify either of these data, then we
performed photometry on circular regions, and so only the major axis is
specified.
d The position angle is defined as degrees from north through east.
e Details on the sources of these uncertainties are given in Section 3.1.
f These measurements are from data in which significant portions of the
optical discs ($>10$%) of the galaxies were not covered in this specific wave
band. The measurements here are for the region that was covered in the MIPS
data. We have applied no corrections for the missing flux density.
g These 160 $\mu$m measurements are for galaxies that were covered in scan map
observations in which the final 160 $\mu$m images for these galaxies contain
NaN values within the optical disc as a consequence of incomplete coverage.
This typically occurs when scan maps are performed using the fast scan rate,
although NaN values within the optical discs of galaxies occasionally appear
in other data. The 160 $\mu$m measurements for these galaxies is based upon
interpolating over these pixels; see the text for details.
h These objects consist of two galaxies with optical discs that overlap. See
Table 2 for the dimensions of the measurement apertures for these objects.
Table 6: Photometry for additional Herschel Virgo Cluster Survey galaxiesa
Galaxy | Optical Disc | Wavelength | Flux Density | Flux Density Uncertainty (Jy)e
---|---|---|---|---
| R. A. | Declination | Axes | Position | ($\mu$m) | Measurement | Calibration | Error | Background | Total
| (J2000)b | (J2000)b | (arcmin)c | Anglecd | | (Jy) | | Map | |
NGC 4165 | 12:12:11.7 | +13:14:47 | $1.3\times 0.9$ | $160^{\circ}$ | 24 | 0.0264 | 0.0011 | 0.0003 | 0.0004 | 0.0012
| | | | | 70 | | | | |
| | | | | 160 | | | | |
NGC 4234 | 12:17:09.1 | +03:40:59 | 1.3 | | 24 | 0.1547 | 0.0062 | 0.0002 | 0.0003 | 0.0062
| | | | | 70 | | | | |
| | | | | 160 | | | | |
NGC 4252 | 12:18:30.8 | +05:33:34 | $1.5\times 0.4$ | $48^{\circ}$ | 24 | 0.0098 | 0.0004 | 0.0002 | 0.0002 | 0.0005
| | | | | 70 | 0.183 | 0.018 | 0.005 | 0.005 | 0.020
| | | | | 160 | 0.46 | 0.06 | 0.00 | 0.02 | 0.06
NGC 4266 | 12:19:42.3 | +05:32:18 | $2.0\times 0.4$ | $76^{\circ}$ | 24 | 0.0329 | 0.0013 | 0.0004 | 0.0005 | 0.0015
| | | | | 70 | 0.494 | 0.049 | 0.007 | 0.011 | 0.051
| | | | | 160 | 2.04 | 0.24 | 0.01 | 0.02 | 0.25
NGC 4273 | 12:19:56.0 | +05:20:36 | $2.3\times 1.5$ | $10^{\circ}$ | 24 | 1.0295 | 0.0412 | 0.0011 | 0.0005 | 0.0412
| | | | | 70 | 12.387 | 1.239 | 0.030 | 0.013 | 1.239
| | | | | 160 | 18.50 | 2.22 | 0.03 | 0.02 | 2.22
NGC 4299 | 12:21:40.9 | +11:30:12 | $1.7\times 1.6$ | $26^{\circ}$ | 24 | 0.2350 | 0.0094 | 0.0002 | 0.0002 | 0.0094
| | | | | 70 | 3.346 | 0.335 | 0.008 | 0.006 | 0.335
| | | | | 160 | 4.32 | 0.52 | 0.02 | 0.01 | 0.52
NGC 4309 | 12:22:12.3 | +07:08:40 | $1.9\times 1.1$ | $85^{\circ}$ | 24 | 0.0620 | 0.0025 | 0.0003 | 0.0003 | 0.0025
| | | | | 70 | | | | |
| | | | | 160 | | | | |
IC 3258 | 12:23:44.4 | +12:28:42 | 1.6 | | 24 | 0.0764 | 0.0031 | 0.0005 | 0.0007 | 0.0032
| | | | | 70 | 0.776 | 0.078 | 0.014 | 0.019 | 0.081
| | | | | 160 | 0.87 | 0.10 | 0.01 | 0.02 | 0.11
NGC 4411 | 12:26:30.1 | +08:52:20 | 2.0 | | 24 | 0.0234 | 0.0009 | 0.0005 | 0.0006 | 0.0012
| | | | | 70 | 0.474 | 0.047 | 0.014 | 0.017 | 0.052
| | | | | 160 | 1.40 | 0.17 | 0.01 | 0.01 | 0.17
UGC 7557 | 12:27:11.0 | +07:15:47 | 3.0 | | 24 | 0.0326 | 0.0013 | 0.0007 | 0.0010 | 0.0018
| | | | | 70 | 0.659 | 0.066 | 0.020 | 0.029 | 0.075
| | | | | 160 | 1.55f | 0.19 | 0.03 | 0.04 | 0.19
NGC 4466 | 12:29:30.5 | +07:41:47 | $1.3\times 0.4$ | $101^{\circ}$ | 24 | 0.0243 | 0.0010 | 0.0005 | 0.0007 | 0.0013
| | | | | 70 | 0.602 | 0.060 | 0.014 | 0.020 | 0.065
| | | | | 160 | 1.13f | 0.14 | 0.01 | 0.02 | 0.14
IC 3476 | 12:32:41.8 | +14:03:02 | $2.1\times 1.8$ | $30^{\circ}$ | 24 | 0.1881 | 0.0075 | 0.0006 | 0.0007 | 0.0076
| | | | | 70 | 1.961 | 0.196 | 0.016 | 0.019 | 0.198
| | | | | 160 | 2.88f | 0.35 | 0.01 | 0.02 | 0.35
NGC 4531 | 12:34:15.8 | +13:04:31 | $3.1\times 2.0$ | $155^{\circ}$ | 24 | 0.0351 | 0.0014 | 0.0006 | 0.0009 | 0.0018
| | | | | 70 | 0.539 | 0.054 | 0.017 | 0.023 | 0.061
| | | | | 160 | 2.76f | 0.33 | 0.02 | 0.04 | 0.33
a These are galaxies that are not in the HRS but that appear in the 500
$\mu$m-selected sample published by Davies et al. (2012).
b Data are from NED.
c Data are from de Vaucouleurs et al. (1991) unless otherwise specified. If de
Vaucouleurs et al. (1991) specify both the minor/major axis ratio and the
position angle, then both axes and the position angle are listed. If de
Vaucouleurs et al. (1991) did not specify either of these data, then we
performed photometry on circular regions, and so only the major axis is
specified.
d The position angle is defined as degrees from north through east.
e Details on the sources of these uncertainties are given in Section 3.1.
f These 160 $\mu$m measurements are for galaxies that were covered in scan map
observations in which the final 160 $\mu$m images for these galaxies contain
NaN values within the optical disc as a consequence of incomplete coverage.
This typically occurs when scan maps are performed using the fast scan rate,
although NaN values within the optical discs of galaxies occasionally appear
in other data. The 160 $\mu$m measurements for these galaxies is based upon
interpolating over these pixels; see the text for details.
#### 3.1.1 Notes on photometry
Aside from typical issues described above with the data processing and
photometry, we encountered multiple problems that were unique to individual
targets. Notes on these issues (in the order in which the galaxies appear in
the table) are listed below.
Notes on the VNGS data
Arp 220 \- The centre of the galaxy, which is unresolved in the MIPS bands,
saturated the 24 $\mu$m detector, and so no 24 $\mu$m flux density is reported
for the source. The 160 $\mu$m error contains two anomalously high pixels
(pixels with error map values at least an order of magnitude higher than the
image map values) located off the peak of the emission. We ascertained that
the corresponding image map pixels did not look anomalous compatred to
adjacent pixels, so the unusually high values in the error map were probably
some type of artefact of the data reduction possibly related to a combination
of high surface brightness issues and coverage issues. We therefore excluded
these pixels when calculating the error map uncertainty.
NGC 891 \- This is an edge-on spiral galaxy in which the central plane is very
bright, and so features that look similar to Airy rings (except that they are
linear rather than ring-shaped) appear above and below the plane of the galaxy
in the 160 $\mu$m image. The measurement aperture we used for all three bands
has a major axis corresponding to 1.5 times the D25 isophote but a much
broader minor axis that encompasses the vertically-extended emission. Note
that this is the only edge-on galaxy where we have encountered this problem.
NGC 1068 \- This is another galaxy that is unresolved in the MIPS bands and
that saturatesd the 24 $\mu$m detector. It is not practical to perform 24
$\mu$m photometry measurements on this galaxy. The 160 $\mu$m error contains a
few anomalously high pixels (pixels with error map values at least an order of
magnitude higher than the image map values). This seemed similar to the
phenomenon described for the anomalous 160 $\mu$m error map pixels for Arp
220. We excluded these pixels when calculating the error map uncertainty.
NGC 3031 \- The 160 $\mu$m image includes residual cirrus emission between the
D25 isophote and the measurement aperture that was masked out when calculating
the 160 $\mu$m flux density. See Sollima et al. (2010) and Davies et al.
(2010b) for details on the features.
NGC 3034 \- The galaxy saturates the MIPS detectors in all three bands and
causes unusually severe artefacts to appear in the data, and so we report no
photometric measurements for this galaxy.
NGC 4038/4039 \- The 70 $\mu$m image is strongly affected by streaking from
latent image effects.
NGC 5128 \- The centre of the galaxy produced latent image effects that appear
as a broad streak in the final image. The artefact was masked out when
photometry was performed.
NGC 5236 \- The central 8 arcsec of the galaxy saturated the 24 $\mu$m and 160
$\mu$m data, but this region appears to contribute a relatively small fraction
of the total emission from NGC 5236. We think the 24 $\mu$m measuements should
still be reliable to within the calibration uncertainty of 4%. As for the 160
$\mu$m image, we interpolated across the single central saturated pixel to
estimate the flux density for the pixel; the correction is much smaller than
the calibration uncertainty.
Notes on the DGS data
HS 0052+2536 \- The 24 $\mu$m image shows an unresolved 24 $\mu$m source at
the central position of HS 0052+2536 and an unresolved 24 $\mu$m source with a
similar surface brightness at the central position of HS 0052+2537, which is
located $\sim 15$ arcsec to the north. We masked out HS 0052+2537 when
performing photometry.
IC 10 \- This galaxy was observed with MIPS only in the photometry map mode.
However, the photometry map mode is intended for objects smaller than 5
arcmin, while the optical disc of IC 10 and the infrared emission from it are
much more extended than this. While $~{}\hbox to0.0pt{$>$\hss}{\lower
4.30554pt\hbox{$\sim$}}90$ % of the optical disc was covered at 24 $\mu$m,
only part of the galaxy was observed at 70 and 160 $\mu$m, and a significant
fraction of the infrared emission may have fallen outside the observed
regions. Given this, we will not report 70 and 160 $\mu$m measurements for
this galaxy.
Mrk 153 \- In the 160 $\mu$m image, the galaxy becomes blended with another
galaxy to the east. We therefore do not report 160 $\mu$m flux densities for
this galaxy.
NGC 5253 \- This is another case where the galaxy is unresolved in the MIPS
bands and where the galaxy saturated the 24 $\mu$m detector, which is why we
report no 24 $\mu$m flux density for this galaxy.
NGC 6822 \- The galaxy has an extension to the south (Cannon et al., 2006)
that is not included within the optical disk given by de Vaucouleurs et al.
(1991), so for photometry, we used a 30 arcmin diameter circle centered on the
optical position of the galaxy given by NED. This galaxy also lies in a field
with cirrus structure on the same size as the galaxy. The version of the 70
and 160 $\mu$m data processing that we applied has removed the gradient in the
cirrus emission present in this part of the sky, which causes the final map to
appear significantly different from the SINGS version of the map for this
specific galaxy.
SBS 1249+493 \- The 24 $\mu$m image includes a bright central source and a
fainter source $\sim 12$ arcsec to the south. It is unclear as to whether this
source is associated with the galaxy; we masked it out before performing flux
density measurements.
Tol 0618-402 \- The brightest feature in the 160 $\mu$m photometry map image
is a streak-like feature running from northwest to southeast near the location
of the galaxy. It is unclear from this image alone if this is an artefact of
the data processing or a real large-scale feature, although based on what we
have seen in similar 160 $\mu$m photometry map data, the latter may be more
likely. No feature in the image appears to correspond to the source itself,
and so we reported the integrated 160 $\mu$m flux density within the 3 arcmin
diameter aperture on the source as the upper limit on the emission, using
regions flanking this region as the best background measurements available.
Tol 1214-277 \- We excldued a marginally-resolved source at approximately
right ascension 12:17:17.7 and declination -28:02:56 from the 24 and 70 $\mu$m
measurements, as this is likely to be a background galaxy. However, the source
became blended with Tol 1214-277 at 160 $\mu$m, so we do not report 160 $\mu$m
flux density measurements for Tol 1214-277.
II Zw 40 \- The 160 $\mu$m image contains only a few square arcmin of
background. The 160 $\mu$m background appears to contain a signficant surface
brightness gradient, which may be expected given that the galaxy lies at a
galactic latitude of $\sim-11$. Additionally, we had difficulty reproducing
the 160 $\mu$m flux density published by Engelbracht et al. (2008). Given
this, we did not feel confident reporting a 160 $\mu$m flux density for this
source.
Notes on the HRS data
NGC 4356 \- The galaxy falls near a 24 $\mu$m artefact we describe as also
affecting the NGC 4472 data (see below). However, the feature appears
relatively faint and broad in the viscinity of NGC 4356, and so we treat it as
part of the background.
NGC 4472 \- The 24 $\mu$m image in the scan map data from AORs 22484480,
22484736, 22484992, and 22455248 were affected by two streak-like regions that
run roughly perpendicular to the scan map direction. These features do not
appear in overlapping maps taken on other dates during the mission. We were
unable to identify the origin of this line. All we can say is that the
positions of these streaks vary with respect to the scan leg position and that
the width of the features is variable. One of these streak-like regions runs
across the optical disc of NGC 4472, and we masked it out before making 24
$\mu$m flux density measurements.
NGC 4486 \- The 160 $\mu$m data within the optical disc of NGC 4486 were
notably affected by residual striping in the images. Two strips approximately
3 arcmin in width to the north and south of the nucleus were affected and were
masked out when the 160 $\mu$m flux density was measured.
NGC 4526 \- Both the 70 and 160 $\mu$m images cover only the central 3 arcmin
of the galaxy, and the 160 $\mu$m image does not include a section on the
western side of the optical disc that is 2 arcmin in width. However, the
emission is relatively centralised, so these problems may not significantly
affect the photometry.
NGC 4552 \- In the 160 $\mu$m data, a cirrus feature oriented roughly east-
west can be seen crossing through the optical disc of this galaxy. We
otherwise detect no 160 $\mu$m emission; we found no 160 $\mu$m counterparts
to the 24 and 70 $\mu$m central source in this galaxy. Hence, we are reporting
the integrated flux density as an upper limit even though we get a $>5\sigma$
detection for the integrated flux densty within the optical disc and we detect
surface brightness features at $>5\sigma$ level.
NGC 4567/4568 \- The 70 $\mu$m data near this galaxy are heavily affected by
latent image effects.
NGC 4636 \- This is an elliptical galaxy with an optical disc with a size of
$6.0\times 4.7$ arcmin (de Vaucouleurs et al., 1991). At 160 $\mu$m, we detect
multiple off-center point sources within the optical disc of the galaxy that
are approximately half the brightness of the central source and that do not
appear to correspond to structure within the galaxy. We assume that the
central source is associated with the galaxy and the off-central sources are
background galaxies, but masking out the off-central sources was equivalent to
masking out the equivalent of most of the optical disc. We therefore perform a
160 $\mu$m measurement within a circle with a diameter of 80 arcsec and then
apply the multiplicative aperture correction of 1.745 given by Stansberry et
al. (2007) for a 30 K source (which, among the spectra used to calculate
aperture corrections, is the closest to the expected spectrum for this
object).
NGC 4647/4649 \- While the optical disc of these two galaxies overlap, NGC
4649 produces relatively compact 24 $\mu$m emission and no detectable 70 or
160 $\mu$m emission. We assume that the optical disc of NGC 4647 contains
negligible emission from NGC 4649. Hence, we are able to report separate flux
densities for each source at 24 $\mu$m, flux densities for NGC 4647 at 70 and
160 $\mu$m, and upper limits for the 70 and 160 $\mu$m flux densities for NGC
4649 using the part of NGC 4649 that does not include NGC 4647. Also, the 70
$\mu$m image is strongly affected by latent image artefacts.
NGC 4666 \- This galaxy was observed in photometry map mode. The galaxy is
observed in such a way that the latent image removal in the 24 $\mu$m data
processing leaves a couple of NaN values near the center of the galaxy. These
pixels correspond to locations between peaked emission, so it is clear that
the data are not related to saturation of the detectors. We interpolated over
these pixels before performing photometric measurements.
### 3.2 Comparisons of photometry to previously-published results
The MIPS calibration at this point is very well established, and comparisons
between MIPS and IRAS photometry have already been performed (Engelbracht et
al., 2007). Therefore, we believe that the most appropriate check of our
photometry would be to compare our measurements to other published MIPS
photometry measurements. As indicated above, MIPS photometric measurements
have previously been published for a significant fraction of the data that we
used. While it is impractical to cite every paper that has been published
based on the MIPS data for these galaxies, three papers have published MIPS
data for significant subsets of galaxies in the SAG2 and HeViCS samples. We
use these papers to check our data processing.
#### 3.2.1 Comparisons with SINGS data
SINGS was a survey with all of the Spitzer instruments that observed a cross-
section of a representative sample of galaxies within 30 Mpc. A total of 15
galaxies from the SAG2 surveys and in HeViCS were originally observed with
MIPS in SINGS. Preliminary photometry for the survey was published by Dale et
al. (2005), while the final photometry was published by Dale et al. (2007). We
compared our data to the data from Dale et al. (2007). However, we exclude NGC
5194/5195 because we are reporting one set of measurements for the system
while Dale et al. report separate flux densities for each galaxy.
The ratio of the Dale et al. (2007) 24 $\mu$m flux densities to ours is
$0.97\pm 0.08$, which is very good. The largest outlier is NGC 6822, where we
measure a $\sim 30$% higher flux density than Dale et al. However, as we
indicated above, this is a galaxy that is large in angular size and that has
infrared emission that extends outside its optical disk. Additionally, the
emission from foreground cirrus structure is relatively strong compared to the
diffuse emission from the galaxy itself. Ultimately, this may be a case where
measuring the diffuse emission from the target galaxy is simply frought with
uncertainty. Aside from this case, however, the comparison has produced very
pleasing results.
In comparing the Dale et al. (2007) 70 $\mu$m flux densities to our own, we
found one galaxy with a factor of $\sim 5$ difference in the flux densities.
This was NGC 4552, an elliptical galaxy with relatively weak emission from a
central source. Dale et al. reported a flux density of $0.52\pm 0.11$ Jy for
this galaxy, which is a factor of 5 higher than our measurement. The Dale et
al. number could be a factor of 10 too high because of a typographical error;
when we measured the flux density the SINGS Data Release 5 (DR5)
data111Available at
http://data.spitzer.caltech.edu/popular/sings/20070410_enhanced_v1/ . using
the same apertures that we used for our data, we obtained $0.04\pm 0.02$ Jy.
This measurement from the SINGS data is a factor of 2 lower than the
measurement from our mosaic. However, our image of this galaxy was made using
both SINGS data and additional 70 $\mu$m data that was taken after the SINGS
photometry was published, and so the measurement from our new mosaic may be
more reliable.
At 160 $\mu$m for NGC 4552, we reported an upper limit that is a factor of
$\sim 1.5$ lower than the Dale et al. (2007) measurement. Again, we think our
measurement could be more reliable because we combined SINGS data with other
scan map data not available to Dale et al., and so the signal-to-noise in our
data should be better.
Excluding NGC 4552, the ratio of the Dale et al. (2007) 70 $\mu$m flux
densities to ours is $1.11\pm 0.07$. At 160 $\mu$m, the ratio of the Dale et
al. flux densities to ours is $1.20\pm 0.07$. This shows that some systematic
effects cause the Dale et al. measurements to be slightly higher than ours,
although the agreement is close to the calibration uncertainty of the data,
and the scatter in the ratios is very small.
If Dale et al. used the data in DR5, then their 160 $\mu$m measurements would
have been based on data in which the flux calibration factor is 5% higher than
the one we used, which could explain part of the discrepancy at 160 $\mu$m.
However, this does not completely explain the discrepancy, and since the flux
calibration factor in the SINGS DR5 70 $\mu$m data is the same as ours,
differences in the factor cannot explain the discrepancies in that wave band.
Although we used data not available to Dale et al. to produce some of our
images, we still see the systematic effects in the cases where we used exactly
the same data as SINGS, so differences in the data used should not lead to
differences in the photometry.
One possible cause for the systematic offsets in the photometry could be the
differences in the way the short term drift was removed. The other possible
cause is differences in the way flux densities were measured and handled.
While we used relatively large apertures (1.5 times the D25 isophote) to
measure flux densities, Dale et al. used the D25 isophotes as apertures and
then applied aperture corrections. To check whether the data processing was
the primarily culprit for the discrepancy, we downloaded the SINGS DR5 data
and performed photometry on those data using the same software and apertures
that we had applied to our own (after correcting the 160 $\mu$m flux
calibration to match ours). The ratio of the measurements from the SINGS DR5
data to the measurements from our data is $0.95\pm 0.07$ at 70 $\mu$m and
$1.08\pm 0.04$ in the 160 $\mu$m data. This shows that the measurement
techniques are responsible for a significant part of the systematic offsets
between the Dale et al. measurements and ours, while the data processing
differences probably cause an additional offset in the 160 $\mu$m data.
Overall, we are satisfied with how our measurements compares to the data from
Dale et al. (2007). The scatter in the measurements is relatively small when
difficult cases are excluded. The remaining differences are at levels that are
comparable to the calibration uncertainties and that are in part related to
the measurement techniques, and these differences probably reflect limitations
in the photometric accuracy that can be achieved with MIPS data for nearby
galaxies in general.
#### 3.2.2 Comparisons with Engelbracht et al. (2008) data
Engelbracht et al. (2008) published data a survey of starburst galaxies that
spanned a broad range of metallicities. 22 of the 66 galaxies overlap with the
SAG2 sample: 21 of the galaxies are in the DGS, and NGC 5236 is in the VNGS.
Although Engelbracht et al. applied colour corrections while we have not, it
is still useful to compare the data.
The ratio of the Engelbracht et al. (2008) 24 $\mu$m measurements to our 24
$\mu$m measurements is $1.00\pm 0.13$, indicating that our measurements agree
with the Engelbracht et al. to within 13%. However, this includes some
infrared-faint galaxies where both Engelbracht et al. and we report $>10$%
uncertainties in the flux density measurements. If we use data where the 24
$\mu$m flux densities from both datasets are $>0.1$ Jy, the ratio becomes
$1.00\pm 0.05$. The remaining dispersion is equivalent to the uncertainty in
the flux calibration, which is very good.
Engelbracht et al report 24 $\mu$m flux densities for two objects for which we
do not report flux densities. For Tol 0618-402, we have reported an upper
limit of 0.0015 Jy, while Engelbracht et al. have reported a $\sim 4\sigma$
detection ($(4.4\pm 1.2)\times 10^{-4}$ Jy). We are reporting $<5\sigma$
detections as upper limits, so, given the signal-to-noise in the Engelbracht
et al. measurement, we would not report a flux density for this galaxy. None
the less, our upper limit for Tol 0618-402 is consistent with the Engelbracht
et al. flux density. The other object is NGC 5253, for which we reported no
flux density measurement because the 24 $\mu$m emission originates from an
unresolved source that saturates the 24 $\mu$m array. Engelbracht et al.
report a flux density for this galaxy but made no special notes about it.
Although the saturation may not be too difficult to deal with when measuring
the flux density, we prefer to be more conservative and report no flux density
for this object.
In comparing the Engelbracht et al. (2008) 70 $\mu$m data to ours, we found
one galaxy where the flux density measurements differ by a factor of 2. For
Tol 1214-277, our 70 $\mu$m flux density measurement is $0.073\pm 0.010$ Jy,
whereas Engelbracht et al. report $0.031\pm 0.003$ Jy. The signal from the
source is hardly $5\sigma$ above the noise in our image of this galaxy. We
also probably used a broader measurement aperture than Engelbracht et al.
Engelbracht et al. used apertures that were adjusted to radii that encompassed
all pixels with emission above a set signal-to-noise level, whereas we used a
3 arcmin diameter aperture, which was our standard aperture for point-like
sources. Our aperture may have included additional signal not included by
Engelbracht et al.
Excluding Tol 0618-402 (where we report an upper limit and Engelbracht et al.
(2008) report a $\sim 1.5$ detection) and Tol 1214-277 (discussed above), our
70 $\mu$m flux density measurements agree well with those from Engelbracht et
al. The ratio of the Engelbracht et al. (2008) 70 $\mu$m measurements to ours
is $1.04\pm 0.17$. For sources above 1 Jy, where the signal-to-noise is
primarily limited by the calibration uncertainty, the ratio is $1.02\pm 0.09$,
which is comparable to the calibration uncertainty of 10%.
A comparison of the Engelbracht et al. (2008) 160 $\mu$m measurements with
ours (for galaxies we detected above the $5\sigma$ level and where we did not
encounter problems with photometry) does not show agreement that is as good as
for the 24 and 70 $\mu$m data. Aside from non-detections, the ratio of the
Engelbracht 160 $\mu$m flux densities to ours is $0.88\pm 0.28$. Measurements
for UGC 4483 and UM 461 are particularly discrepant. We measure 160 $\mu$m
flux densities that are greater than a factor of 2 higher than the Engelbracht
et al. measurements. Howver, these are very faint galaxies; the flux densities
are $<0.2$ Jy. The Engelbracht et al. measurements are at the $<3\sigma$
level, and we used 160 $\mu$m data that would have been unavailable when the
Engelbracht et al. results were published, so the improved signal-to-noise in
our data could have allowed us to make more accurate measurements for these
faint galaxies. Excluding UGC 4483 and UM 461, the ratio of Engelbracht et al.
160 $\mu$m measurements to ours is $0.96\pm 0.19$. The scatter in the ratio is
still larger than the calibration uncertainty of 12%, but this may reflect
issues with simply measuring 160 $\mu$m flux densities in the MIPS data for
these dwarf galaxies, many of which are fainter than 1 Jy or in small fields.
Additionally, the colour correction applied by Engelbracht et al. could have
increased the dispersion in the ratios.
Overall, this comparison has shown excellent agreement between the 24 and 70
$\mu$m flux densities measured by us and by Engelbracht et al. (2008). In the
160 $\mu$m data, we found two discrepancies that cause some concern, but we
think these are unique cases. Our 160 $\mu$m flux densities for other DGS
sources were in general agreement with the Engelbracht et al. measurements,
thus demonstrating the reliability of our data reduction and photometry for
these data.
#### 3.2.3 Comparisons with Ashby et al. (2011) data
Ashby et al. (2011) published a multiwavelength survey of 369 nearby star-
forming galaxies that includes 24 $\mu$m data. 23 of the galaxies in the HRS
and 2 of the additional HeViCS galaxies overlap with the galaxies in the Ashby
et al. sample. Ashby et al. used SExtractor to measure flux densities and then
applied appropriate aperture corrections, which is notably different from the
aperture photometry that we applied.
We have one galaxy where our 24 $\mu$m measurements differ notably from Ashby
et al. (2011). For NGC 3430, we measured $0.4101\pm 0.0164$ Jy, but Ashby et
al. measured $0.17\pm 0.01$ Jy. We used the same data as Ashby et al. to
produce our image, so differences in the raw data cannot explain the
difference in flux densities. An examination of the image does not reveal any
indication of any problems with producing the image or making the photometric
measurement. The IRAS 25 $\mu$m flux density measurements of $0.27\pm 0.04$ Jy
given by the Faint Source Catalog Moshir et al. (1990) and $0.78\pm 0.05$ Jy
given by Surace et al. (2004) are also higher than the Ashby et al.
measurements numbers but still disagree with ours and with each other. We
ultimately suspect that the mismatching flux densities could be indicatinve of
a problem with the Ashby et al. measurement obtained using SExtractor for this
specific galaxy, as the Ashby et al. measurement is lower than all other
measurements at this wavelength. Unfortunately, we do not have access to the
final Ashby et al. images and cannot make any assessment of the differences
between their image and ours, which would help us to understand the problem
further.
Excluding NGC 3430, the ratio of the Ashby et al. (2011) measurements to ours
is $0.90\pm 0.09$. Ashby et al. assume that their uncertainties are 8%, so the
dispersion in the ratio of measurements is reasonably good. The systematic
offset may be a consequence of differences between the flux density
measurement methods. The second largest mismatch between our measurements and
the measurements from Ashby et al. is for NGC 4688, a late-type galaxy with
significant diffuse, low surface brightness 24 $\mu$m emission; Ashby et al.
measure a flux density $\sim 30$% lower than ours for this galaxy. Ashby et
al. also noted differences between the flux densities measured for NGC 4395 by
themselves and by Dale et al. (2009), which they thought could be the result
of incorrectly measuring diffuse emission in NGC 4395 using SExtractor. We
suspect that this could also be the reason for the mismatch between the flux
density measurements for NGC 4688 and may be the reason for the $\sim 10$%
offset in flux density measurements between the reported flux densities from
their catalog and ours.
## 4 Summary
We have gathered together raw MIPS 24, 70, and 160 $\mu$m MIPS data for
galaxies within the SAG2 and HeViCS surveys and reprocessed the data to
produce maps for the analysis of these galaxies. We have also performed
aperture photometry upon the galaxies in the surveys that can be used to study
the global spectral energy distributions of these sources. The flux density
measurements and the images will be distributed to the community through the
Herschel Database in Marseille181818Located at http://hedam.oamp.fr/ . so that
the broader astronomical community can benefit from these data.
As tests of our data processing and photometry, we have performed comparisons
between our photometric measurements and measurements published by Dale et al.
(2007), Engelbracht et al. (2008), and Ashby et al. (2011). Our measurements
generally agree well with the measurements from these other catalogs, and we
have documented and attempted to explain any major discrepancies or systematic
offsets between their measurements and ours. Given the good correspondence
between our measurements and the measurements from these other surveys, we are
confident about the reliability of our photometry measurements.
## Acknowledgments
We thank Laure Ciesla, Ali Dariush, Aurélie Remy, and Matthew W. L. Smith for
their assistance with either identifying data for galaxies within the Spitzer
archive, evaluating the final images and photometry, or proofreading the
manuscript. We also thank the anonymous reviewer for his/her comments. GJB is
funded by the STFC. This research has made use of the NASA/IPAC Extragalactic
Database (NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration.
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|
arxiv-papers
| 2012-02-21T13:12:15 |
2024-09-04T02:49:27.615673
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. J. Bendo, F. Galliano, S. C. Madden",
"submitter": "George J. Bendo",
"url": "https://arxiv.org/abs/1202.4629"
}
|
1202.4646
|
# Publication Trends in Astronomy: The Lone Author
Edwin A. Henneken
###### Abstract
In this short communication I highlight how the number of collaborators on
papers in the main astronomy journals has evolved over time. We see a trend of
moving away from single-author papers. This communication is based on data in
the holdings of the SAO/NASA Astrophysics Data System (ADS).
The ADS is funded by NASA Grant NNX09AB39G.
Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138
This communication illustrates the trend discussed by Mott Greene in the essay
“The demise of the lone author” (Greene (2007)). Trends are likely to be
different for different disciplines. As Mott observes: “In most fields outside
mathematics, fewer and fewer people know enough to work and write alone”. In
addition to this, in most disciplines large (and often multi-national)
collaborations have become more common and even unavoidable, because it is the
only way to get sufficient funding.
Figure 1 is an illustration of how the distribution of the number of authors
has changed over time in the main astronomy journals (The Astrophysical
Journal, The Astronomical Journal, Monthly Notices of the R.A.S. and Astronomy
& Astrophysics).
Figure 1.: The distribution of the relative frequency of the number of authors
per paper in the main astronomy journals for a number of years
Figure 2 highlights the “demise of the lone author” by showing the change in
the fraction of single author papers in the main astronomy journals. The
fraction in the main physics journals (Physical Review, Nuclear Physics,
Physics Letters) has been added for comparison.
Figure 2.: The fraction of papers by single authors in the main astronomy and
physics journals
The drop in the astronomy journals is more dramatic than for the physics
journals. A factor of about 10 versus a factor of about 3 or 4.
## References
* Greene (2007) Greene, Mott. 2007, Nature, 450, 1165 (doi:10.1038/4501165a)
|
arxiv-papers
| 2012-02-21T14:27:54 |
2024-09-04T02:49:27.645648
|
{
"license": "Public Domain",
"authors": "Edwin A. Henneken",
"submitter": "Edwin Henneken",
"url": "https://arxiv.org/abs/1202.4646"
}
|
1202.4711
|
# 1D Schrödinger operators with short range interactions: two-scale
regularization of distributional potentials
Yuriy Golovaty Department of Differential Equations, Ivan Franko National
University of Lviv
1 Universytetska str., 79000 Lviv, Ukraine
###### Abstract.
For real $L_{\infty}(\mathbb{R})$-functions $\Phi$ and $\Psi$ of compact
support, we prove the norm resolvent convergence, as $\varepsilon$ and $\nu$
tend to $0$, of a family $S_{\varepsilon\nu}$ of one-dimensional Schrödinger
operators on the line of the form
$S_{\varepsilon\nu}=-\frac{d^{2}}{dx^{2}}+\frac{\alpha}{\varepsilon^{2}}\Phi\left(\frac{x}{\varepsilon}\right)+\frac{\beta}{\nu}\Psi\left(\frac{x}{\nu}\right),$
provided the ratio $\nu/\varepsilon$ has a finite or infinite limit. The limit
operator $S_{0}$ depends on the shape of $\Phi$ and $\Psi$ as well as on the
limit of ratio $\nu/\varepsilon$. If the potential $\alpha\Phi$ possesses a
zero-energy resonance, then $S_{0}$ describes a non trivial point interaction
at the origin. Otherwise $S_{0}$ is the direct sum of the Dirichlet half-line
Schrödinger operators.
###### Key words and phrases:
1D Schrödinger operator, resonance, short range interaction, point
interaction, $\delta$-potential, $\delta^{\prime}$-potential, distributional
potential, solvable model, norm resolvent convergence
###### 2000 Mathematics Subject Classification:
Primary 34L40, 34B09; Secondary 81Q10
## 1\. Introduction
The present paper is concerned with convergence of the family of one-
dimensional Schrödinger operators of the form
$S_{\varepsilon\nu}=-\frac{d^{2}}{dx^{2}}+\frac{\alpha}{\varepsilon^{2}}\Phi\left(\frac{x}{\varepsilon}\right)+\frac{\beta}{\nu}\Psi\left(\frac{x}{\nu}\right),\quad\mathop{\rm
dom}S_{\varepsilon\nu}=W_{2}^{2}(\mathbb{R})$ (1.1)
as the positive parameters $\nu$ and $\varepsilon$ tend to zero
simultaneously. Here $\Phi$ and $\Psi$ are real potentials of compact
supports, and $\alpha$ and $\beta$ are real coupling constants.
Our motivation of the study on this convergence comes from an application to
the scattering of quantum particles by $\delta$\- and $\delta^{\prime}$-shaped
potentials, where $\delta$ is the Dirac delta-function. The potential in (1.1)
is a two-scale regularization of the distribution
$\alpha\delta^{\prime}(x)+\beta\delta(x)$ provided that the conditions
$\int_{\mathbb{R}}\Phi(t)\,dt=0,\qquad\int_{\mathbb{R}}t\Phi(t)\,dt=-1\quad\text{and}\quad\int_{\mathbb{R}}\Psi(t)\,dt=1$
(1.2)
hold. Our purpose is to construct the so-called solvable models describing
with admissible fidelity the real quantum interactions governed by the
Hamiltonian $S_{\varepsilon\nu}$. The quantum mechanical models that are based
on the concept of point interactions reveal an undoubted effectiveness
whenever solvability together with non triviality is required. It is an
extensive subject with a large literature (see e.g. [4, 7], and the references
given therein).
We emphasize that all results presented here concern arbitrary potentials
$\Phi$ and $\Psi$ of compact support, and the
$(\alpha\delta^{\prime}+\beta\delta)$-like potentials satisfying conditions
(1.2) are only a special case in our considerations, the title of paper
notwithstanding. It is interesting to observe that if the first condition in
(1.2) is not fulfilled, then these potentials do not converge even in the
distributional sense. However, surprisingly enough, the resolvents of
$S_{\varepsilon\nu}$ still converge in norm.
We say that the Schrödinger operator $-\frac{d^{2}}{dt^{2}}+\alpha\Phi$ in
$L_{2}(\mathbb{R})$ possesses a _half-bound state_ (or _zero-energy
resonance_) if there exists a non trivial solution $u_{\alpha}$ to the
equation $-u^{\prime\prime}+\alpha\Phi u=0$ that is bounded on the whole line.
The potential $\alpha\Phi$ is then called _resonant_. In this case, we also
say that $\alpha$ is a resonant coupling constant for the potential $\Phi$.
Such a solution $u_{\alpha}$ is unique up to a scalar factor and has nonzero
limits $u_{\alpha}(\pm\infty)=\lim_{x\to\pm\infty}u_{\alpha}(x)$ (see [9,
27]). Our main result reads as follows.
Let $\Phi$ and $\Psi$ be bounded real functions of compact support. Then the
operator family $S_{\varepsilon\nu}$ given by (1.1) converges as
$\nu,\varepsilon\to 0$ in the norm resolvent sense, i.e., the resolvents
$(S_{\varepsilon\nu}-z)^{-1}$ converge in the uniform operator topology,
provided the ratio $\nu/\varepsilon$ has a finite or infinite limit.
_Non-resonant case._ If the potential $\alpha\Phi$ does not possess a zero-
energy resonance, then the operators $S_{\varepsilon\nu}$ converge to the
direct sum $S_{-}\oplus S_{+}$ of the Dirichlet half-line Schrödinger
operators $S_{\pm}$.
_Resonant case._ If the potential $\alpha\Phi$ is resonant with the half-bound
state $u_{\alpha}$, then the limit operator $S$ is a perturbation of the free
Schrödinger operator defined by $S\phi=-\phi^{\prime\prime}$ on functions
$\phi$ in $W_{2}^{2}(\mathbb{R}\setminus\\{0\\})$, subject to the boundary
conditions at the origin
$\begin{pmatrix}\phi(+0)\\\
\phi^{\prime}(+0)\end{pmatrix}=\begin{pmatrix}\theta_{\alpha}(\Phi)&0\\\
\beta\,\omega_{\alpha}(\Phi,\Psi)&\theta_{\alpha}(\Phi)^{-1}\end{pmatrix}\begin{pmatrix}\phi(-0)\\\
\phi^{\prime}(-0)\end{pmatrix}.$ (1.3)
The diagonal matrix element $\theta_{\alpha}(\Phi)$ is specified by the half-
bound state of potential $\alpha\Phi$, and is defined by
$\theta_{\alpha}(\Phi)=\frac{u_{\alpha}^{+}}{u_{\alpha}^{-}},$ (1.4)
where $u_{\alpha}^{\pm}=u_{\alpha}(\pm\infty)$. The value
$\omega_{\alpha}(\Phi,\Psi)$ depends on both potentials $\Phi$ and $\Psi$ as
well as on the limit of ratio $\nu/\varepsilon$ as $\nu,\varepsilon\to 0$, and
describes different kinds of the resonance interaction between the potentials
$\Phi$ and $\Psi$. Three cases are to be distinguished:
* (i)
if $\nu/\varepsilon\to\infty$ as $\nu,\varepsilon\to 0$, then
$\omega_{\alpha}(\Phi,\Psi)=\frac{u_{\alpha}^{+}}{u_{\alpha}^{-}}\,\int_{\mathbb{R}_{+}}\kern-4.0pt\Psi(t)\,dt+\frac{u_{\alpha}^{-}}{u_{\alpha}^{+}}\,\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi(t)\,dt;$
(1.5)
* (ii)
if the ratio $\nu/\varepsilon$ converges to a finite positive number $\lambda$
as $\nu,\varepsilon\to 0$, then
$\omega_{\alpha}(\Phi,\Psi)=\frac{1}{u_{\alpha}^{-}\,u_{\alpha}^{+}}\,\int_{\mathbb{R}}\Psi(t)\,u^{2}_{\alpha}(\lambda
t)\,dt;$ (1.6)
* (iii)
if $\nu/\varepsilon\to 0$ as $\nu$ and $\varepsilon$ go to zero, then
$\omega_{\alpha}(\Phi,\Psi)=\frac{u^{2}_{\alpha}(0)}{u_{\alpha}^{-}\,u_{\alpha}^{+}}\,\int_{\mathbb{R}}\Psi(t)\,dt.$
(1.7)
The point interaction generated by conditions (1.3) may be regarded as the
first approximation to the real interaction governed by the Hamiltonian
$S_{\varepsilon\nu}$ with coupling constants $\alpha$ lying in vicinity of the
resonant values. The explicit relations between the matrix entries
$\theta_{\alpha}(\Phi)$, $\omega_{\alpha}(\Phi,\Psi)$ and the potentials
$\Phi$, $\Psi$ make it possible to carry out a quantitative analysis of this
quantum system, e.g. to compute approximate values of the scattering data. Of
course the same conclusion holds in the non-resonant case, but then the
quantum dynamics is asymptotically trivial.
It is natural to ask what happens if one of the coupling constants is zero,
and the family $S_{\varepsilon\nu}$ becomes one-parametric. For if $\beta=0$,
and so the $\delta$-like component of the short range potential is absent,
then the results are in agreement with the results obtained recently in [21,
22]: the operators
$S_{\varepsilon}=-\frac{d^{2}}{dx^{2}}+\frac{\alpha}{\varepsilon^{2}}\Phi\left(\frac{x}{\varepsilon}\right),\quad\mathop{\rm
dom}S_{\varepsilon}=W_{2}^{2}(\mathbb{R})$ (1.8)
converge as $\varepsilon\to 0$ in the norm resolvent sense to the operator $S$
defined by conditions (1.3) with $\beta=0$, if $\alpha\Phi$ possesses a zero-
energy resonance, and to the direct sum $S_{-}\oplus S_{+}$ otherwise. As for
the case $\alpha=0$, the limit Hamiltonian, as $\nu\to 0$, must be associated
with the $\beta\delta(x)$-interaction. However, we see at once that zero is a
resonant coupling constant for any potential $\Phi$, and the half-bound state
$u_{0}$ is a constant function. Therefore $\theta_{0}(\Phi)=1$, and
$\omega_{0}(\Phi,\Psi)=\int_{\mathbb{R}}\Psi\,dt$, no matter which a formula
of (1.5)–(1.7) we use. Hence, the operator $S$ is defined by the boundary
conditions
$\phi(+0)=\phi(-0),\qquad\phi^{\prime}(+0)=\phi^{\prime}(-0)+\beta\phi(0)\int_{\mathbb{R}}\Psi\,dt,$
as one should expect.
It has been believed for a long time [37] that the Hamiltonians
$S_{\varepsilon}$ given by (1.8) with $\alpha\neq 0$ converge as
$\varepsilon\to 0$ in the norm resolvent sense to the direct sum $S_{-}\oplus
S_{+}$ of the Dirichlet half-line Schrödinger operators for any potential
$\Phi$ having zero mean. If so, the $\delta^{\prime}$-shaped potential defined
through the regularization $\varepsilon^{-2}\Phi(\varepsilon^{-1}\,\cdot\,)$
must be opaque, i.e., acts as a perfect wall, in the limit $\varepsilon\to 0$.
However, the numerical analysis of exactly solvable models of
$S_{\varepsilon}$ with piece-wise constant $\Phi$ of compact support performed
recently by Zolotaryuk a.o. [16, 40, 41, 42] gives rise to doubts that the
limit $S_{-}\oplus S_{+}$ is correct. The authors demonstrated that for a
resonant $\Phi$, the limiting value of the transmission coefficient of
$S_{\varepsilon}$ is different from zero. The operators $S_{\varepsilon}$ also
arose in [2, 13, 14] in connection with the approximation of smooth planar
quantum waveguides by quantum graphs. Under the assumption that the mean value
of $\Phi$ is different from zero, the authors singled out the set of resonant
potentials $\Phi$ producing a “non-trivial” (i.e., different from $S_{-}\oplus
S_{+}$) limit of $S_{\varepsilon}$ in the norm resolvent sense (see also the
recent preprint [15]). A similar resonance phenomenon was also obtained in
[20], where the asymptotic behaviour of eigenvalues for the Schrödinger
operators perturbed by $\delta^{\prime}$-like short range potentials was
treated (see also [32]). The situation with these controversial results was
clarified in [21, 22]. Note that Šeba was the first [36] who discovered the
“resonant convergence” for a similar family of the Dirichlet Schrödinger
operators on the half-line.
There is a connection between the results presented here and the low energy
behaviour of Schrödinger operators, in particular the low-energy scattering
theory. Generally, the zero-energy resonances are the reason for different
“exceptional” cases of the asymptotic behaviour. Albeverio and Høegh-Krohn [6]
considered the family of Hamiltonians
$H_{\varepsilon}=-\Delta+\lambda(\varepsilon)\varepsilon^{-2}V(\varepsilon^{-1}x)$
in dimension three, where $\lambda(\varepsilon)$ was a smooth function with
$\lambda(0)=1$ and $\lambda^{\prime}(0)\neq 0$. It was shown that
$H_{\varepsilon}$ converge in the strong resolvent sense, as $\varepsilon\to
0$, to the operator that is either the free Hamiltonian $-\Delta$ or its
perturbation by a delta-function depending on whether or not there is a zero-
energy resonance for $-\Delta+V$. In [3], the low-energy scattering was
discussed; the authors used the results of [6] and the connection between the
low-energy behaviour of scattering matrix for the Hamiltonian $-\Delta+V$ in
$L_{2}(\mathbb{R}^{3})$ and for the corresponding scaled Hamiltonians
$-\Delta+\varepsilon^{-2}V(\varepsilon^{-1}x)$ as $\varepsilon\to 0$ to study
in detail possible resonant and non-resonant cases. Similar problem for
Hamiltonians including the Coulomb-type interaction was treated in [5]. The
low-energy scattering for the one-dimensional Schrödinger operator $S_{1}$ and
its connection to the behaviour of the corresponding scaled operators
$S_{\varepsilon}$ as $\varepsilon\to 0$ was thoroughly investigated by Bollé,
Gesztesy, Klaus, and Wilk [10, 9], taking into account the possibility of
zero-energy resonances; in dimension two, the low-energy asymptotics was
discussed in [8]. Continuity of the scattering matrix at zero energy for one-
dimensional Schrödinger operators in the resonant case was established by
Klaus in [28]. Relevant references in this context are also [1, 18]. Simon and
Klaus [29, 30, 27] observed the connection between the zero-energy resonances
and the coupling constant thresholds, i.e., the absorbtion of eigenvalues.
These results depend on properties of the corresponding Birman-Schwinger
kernel.
Singular point interactions for the Schrödinger operators in dimensions one
and higher have widely been discussed in both the physical and mathematical
literature; see [11, 19, 26, 35, 12, 31]. It is worth to note that the
considerable progress in theory of Schrödinger operators with distributional
potentials belonging to the Sobolev space $W_{2}^{-1}$ is due to Shkalikov,
Savchuk [38, 39], and Mikhailets, Goriunov, and Molyboga [33, 34, 25, 24].
## 2\. Preliminaries
There is no loss of generality in supposing that the supports of both $\Phi$
and $\Psi$ are contained in the interval $\mathcal{I}=[-1,1]$. Denote by
$\mathcal{P}$ the class of real-valued bounded functions of compact support
contained in $\mathcal{I}$.
###### Definition 2.1.
The resonant set $\Lambda_{\Phi}$ of a potential $\Phi\in\mathcal{P}$ is the
set of all real value $\alpha$ for which the operator
$-\frac{d^{2}}{dt^{2}}+\alpha\Phi$ in $L_{2}(\mathbb{R})$ possesses a half-
bound state, i.e., for which there exists a non trivial
$L_{\infty}(\mathbb{R})$-solution $u_{\alpha}$ to the equation
$-u^{\prime\prime}+\alpha\Phi u=0.$ (2.1)
The half-bound state $u_{\alpha}$ is then constant outside the support of
$\Phi$. Moreover, the restriction of $u_{\alpha}$ to $\mathcal{I}$ is a
nontrivial solution of the Neumann boundary value problem
$-u^{\prime\prime}+\alpha\Phi u=0,\quad t\in\mathcal{I},\qquad
u^{\prime}(-1)=0,\quad u^{\prime}(1)=0.$ (2.2)
Consequently, for any $\Phi\in\mathcal{P}$ the resonant set $\Lambda_{\Phi}$
is not empty and coincides with the set of all eigenvalues of the latter
problem with respect to the spectral parameter $\alpha$. In the case of a
nonnegative (resp. nonpositive) potential $\Phi$ the spectrum of (2.2) is
discrete and simple with one accumulation point at $-\infty$ (resp.
$+\infty$). Otherwise, (2.2) is a problem with indefinite weight function
$\Phi$, and has a discrete and simple spectrum with two accumulation points at
$\pm\infty$ [17].
We introduce some characteristics of the potentials $\Phi$ and $\Psi$. Let
$\theta$ be the map of $\Lambda_{\Phi}$ to $\mathbb{R}$ defined by
$\theta(\alpha)=\frac{u_{\alpha}^{+}}{u_{\alpha}^{-}}=\frac{u_{\alpha}(+1)}{u_{\alpha}(-1)}.$
Since the half-bound state is unique up to a scalar factor, this map is well
defined. Throughout the paper, we choose the half-bound state so that
$u_{\alpha}(x)=1$ for $x\leq-1$. Then $\theta(\alpha)=u_{\alpha}^{+}$, and
$u_{\alpha}(x)=\theta(\alpha)$ for $x\geq 1$. Here and subsequently,
$\theta_{\alpha}$ stands for the value $\theta(\alpha)$. For our purposes it
is convenient to introduce the maps:
$\displaystyle\zeta\colon\Lambda_{\Phi}\to\mathbb{R},$
$\displaystyle\zeta(\alpha)=\theta_{\alpha}\int_{\mathbb{R}_{+}}\kern-4.0pt\Psi\,dt+\theta_{\alpha}^{-1}\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,dt;$
(2.3)
$\displaystyle\varkappa\colon\Lambda_{\Phi}\times\mathbb{R}_{+}\to\mathbb{R},$
$\displaystyle\varkappa(\alpha,\lambda)=\theta_{\alpha}^{-1}\int_{\mathbb{R}}\Psi(t)\,u^{2}_{\alpha}(\lambda
t)\,dt;$ (2.4) $\displaystyle\mu\colon\Lambda_{\Phi}\to\mathbb{R},$
$\displaystyle\mu(\alpha)=\theta_{\alpha}^{-1}u^{2}_{\alpha}(0)\int_{\mathbb{R}}\Psi\,dt$
(2.5)
(compare with (1.5)–(1.7)).
Denote by $S(\gamma_{1},\gamma_{2})$ a perturbation of the free Schrödinger
operator acting via $S(\gamma_{1},\gamma_{2})\phi=-\phi^{\prime\prime}$ on
functions $\phi$ in $W_{2}^{2}(\mathbb{R}\setminus\\{0\\})$ obeying the
interface conditions $\phi(+0)=\gamma_{1}\phi(-0)$ and
$\phi^{\prime}(+0)=\gamma_{1}^{-1}\phi^{\prime}(-0)+\gamma_{2}\phi(-0)$ at the
origin. For every real $\gamma_{1}$ and $\gamma_{2}$, this operator is self-
adjoint provided $\gamma_{1}\neq 0$. Let $S_{\pm}$ denote the unperturbed
half-line Schrödinger operator $S_{\pm}=-d^{2}/dx^{2}$ on $\mathbb{R}_{\pm}$,
subject to the Dirichlet boundary condition at the origin, i.e.,
$\mathop{\rm dom}S_{\pm}=\\{\phi\in
W_{2}^{2}(\mathbb{R}_{\pm})\colon\phi(0)=0\\}.$
In the sequel, letters $C_{j}$ and $c_{j}$ denote various positive constants
independent of $\varepsilon$ and $\nu$, whose values might be different in
different proofs. Throughout the paper, $W_{2}^{l}(\Omega)$ stands for the
Sobolev space and $\|f\|$ stands for the $L_{2}(\mathbb{R})$-norm of a
function $f$.
We start with an easy auxiliary result, which will be often used below.
###### Proposition 2.2.
Assume $f\in L_{2}(\mathbb{R})$, $z\in\mathbb{C}\setminus\mathbb{R}$, and set
$y=(S(\gamma_{1},\gamma_{2})-z)^{-1}f$. Then the following holds for some
constants $C_{k}$ independent of $f$ and $t$:
$\displaystyle|y(\pm 0)|\leq C_{1}\|f\|,$ $\displaystyle|y^{\prime}(\pm
0)|\leq C_{2}\|f\|$ (2.6) $\displaystyle\bigr{|}y(\pm t)-y(\pm 0)\bigl{|}\leq
C_{3}t\|f\|,$ $\displaystyle\bigr{|}y^{\prime}(\pm t)-y^{\prime}(\pm
0)\bigl{|}\leq C_{4}t^{1/2}\|f\|$ (2.7)
for $t>0$. These inequalities hold also for $y=(S_{-}\oplus S_{+}-z)^{-1}f$.
###### Proof.
We first observe that $(S(\gamma_{1},\gamma_{2})-z)^{-1}$ is a bounded
operator from $L_{2}(\mathbb{R})$ to the domain of $S(\gamma_{1},\gamma_{2})$
equipped with the graph norm. The latter space is continuously embedded
subspace into $W_{2}^{2}(\mathbb{R}\setminus\\{0\\})$. Then
$\|y\|_{W_{2}^{2}(\mathbb{R}\setminus\\{0\\})}\leq c_{1}\|f\|$. Owing to the
Sobolev embedding theorem, we have
$\|y\|_{C^{1}(\mathbb{R}\setminus\\{0\\})}\leq c_{2}\|f\|$, which establishes
(2.6). Combining the previous estimates for $y$ with the inequalities
$\bigr{|}y^{(j)}(\pm t)-y^{(j)}(\pm 0)\bigl{|}\leq\left|\int_{0}^{\pm
t}|y^{(j+1)}(s)|\,ds\right|,\quad j=0,1,$
we obtain (2.7). For the case of $S_{-}\oplus S_{+}$, the proof is similar. ∎
Apparently, some versions of the next proposition are known, but we are at a
loss to give a precise reference.
###### Proposition 2.3.
Let $J$ be a finite interval in $\mathbb{R}$, and $t_{0}\in J$. Then the
solution to the Cauchy problem $v^{\prime\prime}+qv=f$ in $J$, $v(t_{0})=a$,
$v^{\prime}(t_{0})=b$ obeys the estimate
$\|v\|_{C^{1}(J)}\leq C(|a|+|b|+\|f\|_{L_{\infty}(J)})$
for some $C>0$ being independent of the initial data and right-hand side,
whenever $q,f\in L_{\infty}(J)$.
###### Proof.
Let $v_{1}$ and $v_{2}$ be the linear independent solutions to
$v^{\prime\prime}+qv=0$ such that $v_{1}(t_{0})=1$, $v^{\prime}_{1}(t_{0})=0$,
$v_{2}(t_{0})=0$ and $v^{\prime}_{2}(t_{0})=1$. Under the assumptions made on
$q$ and $f$, these solutions belong to $W_{2}^{2}(J)$; and consequently
$v_{j}\in C^{1}(J)$ by the Sobolev embedding theorem. Application of the
variation of parameters method yields
$v(t)=av_{1}(t)+bv_{2}(t)+\int_{t_{0}}^{t}k(t,s)f(s)\,ds,$ (2.8)
where $k(t,s)=v_{1}(s)v_{2}(t)-v_{1}(t)v_{2}(s)$. From this and the
representation of the first derivative
$v^{\prime}(t)=av^{\prime}_{1}(t)+bv^{\prime}_{2}(t)+\int_{t_{0}}^{t}\frac{\partial
k}{\partial t}(t,s)f(s)\,ds$
we have
$|v(t)|+|v^{\prime}(t)|\leq|a|\|v_{1}\|_{C^{1}(J)}+|b|\|v_{2}\|_{C^{1}(J)}+2|J|\,\|k\|_{C^{1}(J\times
J)}\|f\|_{L_{\infty}(J)}$
for $t\in J$, which completes the proof. ∎
We end this section with a proposition which will be useful in Sections 3 and
5. Denote by $[\,\cdot\,]_{b}$ the jump of a function at the point $x=b$.
###### Proposition 2.4.
Let $\mathbb{R}_{a}$ be the real line with two removed points $-a$ and $a$,
i.e., $\mathbb{R}_{a}=\mathbb{R}\setminus\\{-a,a\\}$. Assume $w\in
W_{2}^{2}(\mathbb{R}_{a})$. There exists a function $r\in
C^{\infty}(\mathbb{R}_{a})$ such that $w+r$ belongs to
$W_{2}^{2}(\mathbb{R})$, $r$ is zero in $(-a,a)$, and
$\max_{x\in\mathbb{R}_{a}}|r^{(k)}(x)|\leq
C\Bigl{(}\left|[w]_{-a}\right|+\left|[w]_{a}\right|+\left|[w^{\prime}]_{-a}\right|+\left|[w^{\prime}]_{a}\right|\Bigr{)}$
(2.9)
for $k=0,1,2$, where the constant $C$ does not depend on $w$ and $a$.
###### Proof.
Let us introduce functions $\varphi$ and $\psi$ that are smooth outside the
origin, have compact supports contained in $[0,\infty)$, and $\varphi(+0)=1$,
$\varphi^{\prime}(+0)=0$, $\psi(+0)=0$, $\psi^{\prime}(+0)=1$. Set
$r(x)=[w]_{-a}\,\varphi(-x-a)-[w^{\prime}]_{-a}\,\psi(-x-a)-[w]_{a}\,\varphi(x-a)-[w^{\prime}]_{a}\,\psi(x-a).$
(2.10)
All jumps are well defined, since $w\in C^{1}(\mathbb{R}_{a})$. Next, the
function $r$ is zero in $(-a,a)$ by construction. An easy computation shows
that $w+r$ is continuous on $\mathbb{R}$ along with its derivative and
consequently belongs to $W_{2}^{2}(\mathbb{R})$. Finally, (2.10) makes it
obvious that inequality (2.9) holds. ∎
## 3\. Convergence of the operators $S_{\varepsilon\nu}$. The case
$\nu\varepsilon^{-1}\to\infty$.
In this section, we analyze the case of a “$\delta$-like” sequence that is
slowly contracting relative to “$\delta^{\prime}$-like” one. The relations
between two parameters $\varepsilon$ and $\nu$ that lead to this case are,
roughly speaking, as follows: $\varepsilon\ll 1$, $\nu\ll 1$, but
$\nu/\varepsilon\gg 1$. It will be convenient to introduce the large parameter
$\eta=\nu/\varepsilon$. The first trivial observation is the following: if
$\nu\to 0$ and $\eta\to\infty$, then $\varepsilon\to 0$. The resonant and non-
resonant cases will be considered separately.
### 3.1. Resonant case
We start with the analysis of the more difficult resonant case. Suppose that
$\alpha\in\Lambda_{\Phi}$ and set $\zeta_{\alpha}=\zeta(\alpha)$, where
$\zeta$ is given by (2.3).
###### Theorem 3.1.
Assume $\Phi,\Psi\in\mathcal{P}$ and $\alpha$ belongs to the resonant set
$\Lambda_{\Phi}$. Then the operator family $S_{\varepsilon\nu}$ defined by
(1.1) converges to the operator $S(\theta_{\alpha},\beta\zeta_{\alpha})$ as
$\nu\to 0$ and $\eta\to\infty$ in the norm resolvent sense.
We have divided the proof into a sequence of lemmas.
Let us fix a function $f\in L_{2}(\mathbb{R})$ and a number $z\in\mathbb{C}$
with $\mathop{\rm Im}z\neq 0$. For abbreviation, in this section we let $S$
stand for $S(\theta_{\alpha},\beta\zeta_{\alpha})$. Our aim is to approximate
both vectors $(S_{\varepsilon\nu}-z)^{-1}f$ and $(S-z)^{-1}f$ in
$L_{2}(\mathbb{R})$ by the same element $y_{\varepsilon\nu}$ from the domain
of $S_{\varepsilon\nu}$. Of course, such an approximation must be uniform in
$f$ in bounded subsets of $L_{2}(\mathbb{R})$. We construct the vector
$y_{\varepsilon\nu}$ in the explicit form, which allows us to estimate
$L_{2}(\mathbb{R})$-norms of the differences
$(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}$ and
$(S-z)^{-1}f-y_{\varepsilon\nu}$. This is the aim of the next lemmas.
First we construct a candidate for the approximation as follows. Let us set
$y=(S-z)^{-1}f$. Write $w_{\varepsilon\nu}(x)=y(x)$ for $|x|>\nu$ and
$w_{\varepsilon\nu}(x)=y(-0)\bigl{(}u_{\alpha}(x/\varepsilon)+\beta\nu
h_{\varepsilon\nu}(x/\nu)\bigr{)}+\varepsilon
g_{\varepsilon\nu}(x/\varepsilon)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)\quad\text{for
}|x|\leq\nu.$
Here $h_{\varepsilon\nu}$, $g_{\varepsilon\nu}$, and $v_{\varepsilon\nu}$ are
solutions to the Cauchy problems
$\displaystyle\hskip 12.0pth^{\prime\prime}=\Psi(t)u_{\alpha}\left(\eta
t\right),\quad t\in\mathbb{R},\qquad h(0)=0,\quad h^{\prime}(0)=0;$ (3.1)
$\displaystyle\begin{cases}\displaystyle
g^{\prime\prime}-\alpha\Phi(t)g=\alpha\beta\eta
y(-0)\Phi(t)h_{\varepsilon\nu}\left(\eta t\right),\quad t\in\mathbb{R},\\\
\displaystyle g(-1)=0,\quad g^{\prime}(-1)=y^{\prime}(-0)+\beta
y(-0)\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds;\end{cases}$ (3.2)
$\displaystyle\hskip 8.0pt-v^{\prime\prime}+\alpha\Phi(t)v=f(\varepsilon
t)\chi_{\eta}(t),\quad t\in\mathbb{R},\quad v(0)=0,\;v^{\prime}(0)=0$ (3.3)
respectively, and $u_{\alpha}$ is the half-bound state corresponding to the
resonant coupling constant $\alpha$. Here and subsequently, $\chi_{a}$ is the
characteristic function of interval $(-a,a)$. Hence we can surely expect that
$y$ is a very satisfactory approximation to $(S_{\varepsilon\nu}-z)^{-1}f$ for
$|x|>\nu$, but the approximation on the support of $\Psi$ is more subtle.
###### Lemma 3.2.
The function $h_{\varepsilon\nu}$ possesses the following properties:
(i) there exist constants $C_{1}$ and $C_{2}$ such that
$\|h_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq
C_{1},\qquad|h_{\varepsilon\nu}(t)|\leq C_{2}\,t^{2}$ (3.4)
for $t\in\mathbb{R}$ and all $\varepsilon,\nu\in(0,1)$;
(ii) the asymptotic relations
$h_{\varepsilon\nu}^{\prime}(-1)=-\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds+O(\eta^{-1}),\qquad
h_{\varepsilon\nu}^{\prime}(1)=\theta_{\alpha}\int_{\mathbb{R}_{+}}\kern-4.0pt\Psi\,ds+O(\eta^{-1})$
(3.5)
hold as $\nu\to 0$ and $\eta\to\infty$.
###### Proof.
The solution $h_{\varepsilon\nu}$ and its derivative can be represented as
$h_{\varepsilon\nu}(t)=\int_{0}^{t}(t-s)\Psi(s)u_{\alpha}(\eta s)\,ds,\qquad
h^{\prime}_{\varepsilon\nu}(t)=\int_{0}^{t}\Psi(s)u_{\alpha}(\eta s)\,ds.$
(3.6)
The first estimate in (3.4) follows immediately from these relations, because
$\Psi$ and $u_{\alpha}$ belong to $L_{\infty}(\mathbb{R})$. By the same
reason,
$|h_{\varepsilon\nu}(t)|\leq c_{1}\left|\int_{0}^{t}|t-s|\,ds\right|\leq
C_{2}t^{2}.$
Now according to our choice of the half-bound state, we see that
$u_{\alpha}(\eta t)\to u_{\alpha}^{*}(t)=\begin{cases}1&\text{if }t<0,\\\
\theta_{\alpha}&\text{if }t>0\end{cases}$
in $L_{1,loc}(\mathbb{R})$, as $\eta\to\infty$. In addition, the difference
$u_{\alpha}(\eta t)-u_{\alpha}^{*}(t)$ is zero outside the interval
$[-\eta^{-1},\eta^{-1}]$ and bounded on this interval. In view of the second
relation in (3.6), this establishes the asymptotic formulas (3.5). ∎
###### Lemma 3.3.
There exist constants $C_{1}$ and $C_{2}$, independent of $f$, such that
$\displaystyle|g_{\varepsilon\nu}(t)|\leq C_{1}(1+|t|)\|f\|,$ $\displaystyle
t\in\mathbb{R},$ (3.7) $\displaystyle|g^{\prime}_{\varepsilon\nu}(t)|\leq
C_{2}\|f\|,$ $\displaystyle t\in\mathbb{R}$ (3.8)
for all $\varepsilon$ and $\nu$ whenever the ratio of $\varepsilon$ to $\nu$
remains bounded as $\varepsilon,\nu\to 0$. In addition, the value
$g^{\prime}_{\varepsilon\nu}(1)$ admits the asymptotics
$g^{\prime}_{\varepsilon\nu}(1)=\theta_{\alpha}^{-1}\left(y^{\prime}(-0)+\beta
y(-0)\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds\right)+O(\eta^{-1})\|f\|$ (3.9)
as $\nu\to 0$, $\eta\to\infty$.
###### Proof.
From Proposition 2.3 it follows that
$\|g_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq
c_{1}(|y(-0)|+|y^{\prime}(-0)|)+c_{2}\eta|y(-0)|\,\|h_{\varepsilon\nu}(\eta^{-1}\,\cdot\,)\|_{C(\mathcal{I})}.$
Next, in light of (3.4), we have
$\|h_{\varepsilon\nu}(\eta^{-1}\,\cdot\,)\|_{C(\mathcal{I})}=\max_{\phantom{1}|t|\leq\eta^{-1}}|h_{\varepsilon\nu}(t)|\leq
c_{3}\eta^{-2}.$ (3.10)
Combining this estimate with (2.6), we deduce
$\|g_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq
c_{4}(|y(-0)|+|y^{\prime}(-0)|)\leq c_{5}\|f\|.$ (3.11)
Since the support of $\Phi$ lies in $\mathcal{I}$, the function
$g_{\varepsilon\nu}$ is linear outside $\mathcal{I}$, namely
$g_{\varepsilon\nu}(t)=g_{\varepsilon\nu}^{\prime}(-1)(t+1)$ for $t\leq-1$ and
$g_{\varepsilon\nu}(t)=g_{\varepsilon\nu}(1)+g_{\varepsilon\nu}^{\prime}(1)(t-1)$
for $t\geq 1$. Therefore estimates (3.7), (3.8) follow easily from these
relations and (3.11).
Next, multiplying equation (3.2) by $u_{\alpha}$ and integrating on
$\mathcal{I}$ by parts yield
$\theta_{\alpha}g^{\prime}_{\varepsilon\nu}(1)-g^{\prime}_{\varepsilon\nu}(-1)=\alpha\beta\eta\,y(-0)\int_{-1}^{1}\Phi(s)\,h_{\varepsilon\nu}\left(\eta^{-1}s\right)u_{\alpha}(s)\,ds.$
The right-hand side can be estimated by $c_{6}\eta^{-1}\|f\|$ provided
$|\eta|\geq 1$, in view of (3.10) and Proposition 2.2. Recalling the initial
conditions (3.2), we obtain (3.9). ∎
###### Lemma 3.4.
There exist constants $C_{1}$ and $C_{2}$, independent of $f$, such that
$|v_{\varepsilon\nu}(t)|\leq
C_{1}\varepsilon^{-2}\nu^{3/2}\|f\|,\qquad|v^{\prime}_{\varepsilon\nu}(t)|\leq
C_{2}\varepsilon^{-1}\nu^{1/2}\|f\|$ (3.12)
for $t\in[-\eta,\eta]$, as $\nu\to 0$ and $\eta\to\infty$.
###### Proof.
The proof consists in the careful analysis of representation (2.8) for the
case of problem (3.3). In fact,
$v_{\varepsilon\nu}(t)=\int_{0}^{t}k(t,s)f(\varepsilon s)\chi_{\eta}(s)\,ds,$
where $k(t,s)=v_{1}(s)v_{2}(t)-v_{1}(t)v_{2}(s)$, and $v_{1}$, $v_{2}$ are
solutions of $-v^{\prime\prime}+\alpha\Phi v=0$ subject to the initial
conditions $v_{1}(0)=1$, $v^{\prime}_{1}(0)=0$ and $v_{2}(0)=0$,
$v^{\prime}_{2}(0)=1$ respectively.
The kernel $k$ admits the following estimates
$|k(t,s)|\leq c_{1}(|t|+|s|)+c_{2},\quad\left|\frac{\partial k}{\partial
t}(t,s)\right|\leq c_{3},\quad(t,s)\in\mathbb{R}^{2}$ (3.13)
with some positive constants $c_{j}$. Indeed, both solutions $v_{1}$ and
$v_{2}$ are linear functions outside the interval $\mathcal{I}$, since
$\mathop{\rm supp}\Phi\subset\mathcal{I}$. Set
$v_{j}(t)=a_{j}^{\pm}t+b_{j}^{\pm}$ for $\pm t>1$. Suppose that $t>1$ and
$s>1$; then
$k(t,s)=(b_{1}^{+}a_{2}^{+}-b_{2}^{+}a_{1}^{+})(t-s),\quad\frac{\partial
k}{\partial t}(t,s)=b_{1}^{+}a_{2}^{+}-b_{2}^{+}a_{1}^{+},$
which implies (3.13) for such $t$ and $s$. Next, if $t>1$ and $|s|<1$, then
$k(t,s)=v_{1}(s)(a_{2}^{+}t+b_{2}^{+})-v_{2}(s)(a_{1}^{+}t+b_{1}^{+}),\quad\frac{\partial
k}{\partial t}(t,s)=a_{2}^{+}v_{1}(s)-a_{1}^{+}v_{2}(s).$
That (3.13) for such $t$ and $s$ follows from the estimates
$\|v_{j}\|_{C(-1,1)}\leq c_{4}$, $j=1,2$. The other cases (such as $|t|<1$ and
$s>1$; $t<-1$ and $s<-1$, and so on) can be treated in a similar way.
Therefore, for $\eta$ large enough, we have
$\displaystyle\begin{aligned}
\max_{t\in[-\eta,\eta]}|v_{\varepsilon\nu}(t)|\leq\int_{-\eta}^{\eta}\max_{t\in[-\eta,\eta]}|k(t,s)||f(\varepsilon
s)|\,ds\leq\int_{-\eta}^{\eta}(c_{5}(\eta+|s|)+c_{6})|f(\varepsilon s)|\,ds\\\
\leq c_{7}\eta\int_{-\eta}^{\eta}|f(\varepsilon
s)|\,ds=c_{7}\eta\varepsilon^{-1}\int_{-\nu}^{\nu}|f(\tau)|\,d\tau\leq
c_{8}\eta\varepsilon^{-1}\nu^{1/2}\|f\|=c_{8}\eta\varepsilon^{-2}\nu^{3/2}\|f\|,\end{aligned}$
$\displaystyle\begin{aligned}
\max_{t\in[-\eta,\eta]}|v^{\prime}_{\varepsilon\nu}(t)|&\leq\int_{-\eta}^{\eta}\max_{t\in[-\eta,\eta]}\left|\frac{\partial
k}{\partial t}(t,s)\right||f(\varepsilon s)|\,ds\\\ &\leq
c_{9}\int_{-\eta}^{\eta}|f(\varepsilon s)|\,ds\leq
c_{10}\varepsilon^{-1}\int_{-\nu}^{\nu}|f(\tau)|\,d\tau\leq
c_{11}\varepsilon^{-1}\nu^{1/2}\|f\|,\end{aligned}$
which proves the lemma. ∎
###### Corollary 3.5.
The function $w_{\varepsilon\nu}$ is bounded in $[-\nu,\nu]$ uniformly in
$\varepsilon$ and $\nu$ provided the ratio $\varepsilon/\nu$ remains bounded
as $\varepsilon,\nu\to 0$, and there exists a constant $C$ such that
$\max_{|x|\leq\nu}|w_{\varepsilon\nu}(x)|\leq C\|f\|$.
###### Proof.
The corollary is a direct consequence of Lemmas 3.2–3.4. We only note that
$\max_{|x|\leq\nu}|\varepsilon
g_{\varepsilon\nu}(x/\varepsilon)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)|\leq\bigl{(}c_{1}\varepsilon(1+\nu/\varepsilon)+c_{2}\nu^{3/2}\bigr{)}\|f\|\\\
\leq c_{3}(\varepsilon+\nu)\|f\|\leq c_{4}\nu\|f\|,$ (3.14)
in view of (3.7), (3.12), and the assumption that $\varepsilon\leq c\nu$. ∎
By construction, $w_{\varepsilon\nu}$ belongs to
$W_{2}^{2}(\mathbb{R}\setminus\\{-\nu,\nu\\})$. In general, due to the
discontinuity at the points $x=\pm\nu$, $w_{\varepsilon\nu}$ is not an element
of $\mathop{\rm dom}S_{\varepsilon\nu}$. However, the jumps of
$w_{\varepsilon\nu}$ and the jumps of its first derivative at these points are
small enough, as shown below. By Proposition 2.4, there exists the corrector
function $r_{\varepsilon\nu}$ of the form (2.10) such that
$w_{\varepsilon\nu}+r_{\varepsilon\nu}$ belongs to
$W_{2}^{2}(\mathbb{R})=\mathop{\rm dom}S_{\varepsilon\nu}$. Set
$y_{\varepsilon\nu}=w_{\varepsilon\nu}+r_{\varepsilon\nu}$.
###### Lemma 3.6.
The corrector $r_{\varepsilon\nu}$ is small as $\nu\to 0$, $\eta\to\infty$,
and satisfies the inequality
$\max_{x\in\mathbb{R}\setminus\\{-\nu,\nu\\}}\bigl{|}r^{(k)}_{\varepsilon\nu}(x)\bigr{|}\leq
C\varrho(\nu,\eta)\|f\|$
for $k=0,1,2$, where $\varrho(\nu,\eta)=\nu^{1/2}+\eta^{-1}$.
###### Proof.
Assume $\varepsilon$ and $\nu$ are small enough, and $\eta\geq 1$. From our
choice of $u_{\alpha}$, we have that $u_{\alpha}(-\eta)=1$,
$u_{\alpha}(\eta)=\theta_{\alpha}$, and $u^{\prime}_{\alpha}(\pm\eta)=0$. Also
$g_{\varepsilon\nu}^{\prime}(\pm\eta)=g_{\varepsilon\nu}^{\prime}(\pm 1)$, and
the bounds
$\varepsilon|g_{\varepsilon\nu}(\pm\eta)|\leq c_{1}\nu\|f\|$ (3.15)
hold, owing to (3.14). These relations will be used repeatedly in the proof.
According to Proposition 2.4, it is sufficient to estimate the jumps of
$w_{\varepsilon\nu}$ and $w^{\prime}_{\varepsilon\nu}$. At the point $x=-\nu$
we have
$\displaystyle[w_{\varepsilon\nu}]_{-\nu}$ $\displaystyle=y(-0)+\beta\nu
y(-0)h_{\varepsilon\nu}(-1)+\varepsilon
g_{\varepsilon\nu}(-\eta)+\varepsilon^{2}v_{\varepsilon\nu}(-\eta)-y(-\nu),$
$\displaystyle[w^{\prime}_{\varepsilon\nu}]_{-\nu}$ $\displaystyle=\beta
y(-0)h^{\prime}_{\varepsilon\nu}(-1)+g^{\prime}_{\varepsilon\nu}(-1)+\varepsilon
v^{\prime}_{\varepsilon\nu}(-\eta)-y^{\prime}(-\nu).$
The first of these jumps can be bounded as follows:
$|[w_{\varepsilon\nu}]_{-\nu}|\leq|y(-0)-y(-\nu)|+\nu|\beta||y(-0)||h_{\varepsilon\nu}(-1)|\\\
+\varepsilon|g_{\varepsilon\nu}(-\eta)|+\varepsilon^{2}|v_{\varepsilon\nu}(-\eta)|\leq
c_{2}\nu\|f\|,$
by (3.4), (3.15), Proposition 2.2, and Lemma 3.4. Next, taking into account
(3.5) and the initial conditions for $g_{\varepsilon\nu}$, we see that
$\displaystyle[w^{\prime}_{\varepsilon\nu}]_{-\nu}$ $\displaystyle=\beta
y(-0)\Bigl{(}-\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds+O(\eta^{-1})\Bigr{)}+y^{\prime}(-0)+\beta
y(-0)\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds-y^{\prime}(-\nu)$
$\displaystyle+\varepsilon
v^{\prime}_{\varepsilon\nu}(-\eta)=y^{\prime}(-0)-y^{\prime}(-\nu)+O(\eta^{-1})y(-0)+O(\nu^{1/2})\|f\|,$
as $\eta\to\infty$ and $\nu\to 0$. We can now repeatedly apply Proposition 2.2
to deduce $\left|[w^{\prime}_{\varepsilon\nu}]_{-\nu}\right|\leq
c_{3}\varrho(\nu,\eta)\|f\|$.
Let us turn to the jumps at the point $x=\nu$. We get
$\displaystyle[w_{\varepsilon\nu}]_{\nu}$
$\displaystyle=y(\nu)-\theta_{\alpha}y(-0)-\beta\nu
y(-0)h_{\varepsilon\nu}(1)-\varepsilon
g_{\varepsilon\nu}(\eta)-\varepsilon^{2}v_{\varepsilon\nu}(\eta),$
$\displaystyle[w^{\prime}_{\varepsilon\nu}]_{\nu}$
$\displaystyle=y^{\prime}(\nu)-\beta
y(-0)h^{\prime}_{\varepsilon\nu}(1)-g^{\prime}_{\varepsilon\nu}(1)-\varepsilon
v^{\prime}_{\varepsilon\nu}(\eta).$
Recall that $y(+0)=\theta_{\alpha}y(-0)$, since $y\in\mathop{\rm dom}S$. This
gives
$\left|[w_{\varepsilon\nu}]_{\nu}\right|\leq|y(\nu)-y(+0)|+c_{4}\nu|y(-0)|+\varepsilon|g_{\varepsilon\nu}(\eta)|+\varepsilon^{2}|v_{\varepsilon\nu}(\eta)|\leq
c_{5}\nu\|f\|$
by (2.7), (3.12), and (3.15). Also, combining the relation
$y^{\prime}(+0)=\theta_{\alpha}^{-1}y^{\prime}(-0)+\beta\zeta_{\alpha}y(-0)$
and asymptotic formulas (3.5), (3.9), we deduce that
$\displaystyle[w^{\prime}_{\varepsilon\nu}]_{\nu}$
$\displaystyle=y^{\prime}(\nu)-\beta
y(-0)\Bigl{(}\theta_{\alpha}\int_{\mathbb{R}_{+}}\kern-4.0pt\Psi\,ds+O(\eta^{-1})\Bigr{)}$
$\displaystyle\phantom{=y^{\prime}(\nu)\,}-\Bigl{(}\theta_{\alpha}^{-1}y^{\prime}(-0)+\theta_{\alpha}^{-1}\beta
y(-0)\int_{\mathbb{R}_{-}}\kern-4.0pt\Psi\,ds+O(\eta^{-1})\|f\|\Bigr{)}-\varepsilon
v^{\prime}_{\varepsilon\nu}(\eta)$
$\displaystyle=y^{\prime}(\nu)-\theta_{\alpha}^{-1}y^{\prime}(-0)-\beta\zeta_{\alpha}y(-0)+O(\eta^{-1})\|f\|+O(\nu^{1/2})\|f\|$
$\displaystyle=y^{\prime}(\nu)-y^{\prime}(+0)+O(\eta^{-1}+\nu^{1/2})\|f\|,$
hence that $\left|[w^{\prime}_{\varepsilon\nu}]_{\nu}\right|\leq
c_{6}\varrho(\nu,\eta)\|f\|$. This inequality completes the proof. ∎
###### Proof of Theorem 3.1.
We first compute $(S_{\varepsilon\nu}-z)y_{\varepsilon\nu}$. For the
convenience of the reader we write
$y_{\varepsilon\nu}=w_{\varepsilon\nu}+r_{\varepsilon\nu}$ in the detailed
form
$y_{\varepsilon\nu}(x)=\begin{cases}y(x)+r_{\varepsilon\nu}(x)&\text{if
}|x|>\nu,\\\ y(-0)\bigl{(}u_{\alpha}(x/\varepsilon)+\nu\beta
h_{\varepsilon\nu}(x/\nu)\bigr{)}+\varepsilon
g_{\varepsilon\nu}(x/\varepsilon)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)&\text{if
}|x|\leq\nu.\end{cases}$ (3.16)
Recall that $r_{\varepsilon\nu}$ is zero in $(-\nu,\nu)$, by construction. Set
$f_{\varepsilon\nu}=(S_{\varepsilon\nu}-z)y_{\varepsilon\nu}$. If $|x|>\nu$,
then
$f_{\varepsilon\nu}(x)=\left(-\tfrac{d^{2}}{dx^{2}}-z\right)y_{\varepsilon\nu}(x)=f(x)-r^{\prime\prime}_{\varepsilon\nu}(x)-zr_{\varepsilon\nu}(x).$
Next, for $|x|<\nu$, we have
$\displaystyle\textstyle f_{\varepsilon\nu}(x)=$
$\displaystyle\left(-\tfrac{d^{2}}{dx^{2}}+\alpha\varepsilon^{-2}\Phi\left(\tfrac{x}{\varepsilon}\right)+\beta\nu^{-1}\Psi\left(\tfrac{x}{\nu}\right)-z\right)y_{\varepsilon\nu}(x)$
$\displaystyle=$
$\displaystyle\varepsilon^{-2}\,y(-0)\Bigl{\\{}-u^{\prime\prime}_{\alpha}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)u_{\alpha}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$ $\displaystyle\nu^{-1}\,\beta
y(-0)\Bigl{\\{}-h^{\prime\prime}_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)+\Psi\left(\tfrac{x}{\nu}\right)u_{\alpha}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\varepsilon^{-1}\,\Bigl{\\{}-g_{\varepsilon\nu}^{\prime\prime}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\eta\alpha\beta
y(-0)\Phi\left(\tfrac{x}{\varepsilon}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\Bigl{\\{}-v^{\prime\prime}_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\beta\Psi\left(\tfrac{x}{\nu}\right)\Bigl{\\{}\beta
y(-0)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)+\eta^{-1}g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\varepsilon\eta^{-1}v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}-zy_{\varepsilon\nu}(x)$
$\displaystyle=$ $\displaystyle
f(x)+\beta\Psi\left(\tfrac{x}{\nu}\right)\Bigl{\\{}\beta
y(-0)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)+\eta^{-1}g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\varepsilon\eta^{-1}v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}-zy_{\varepsilon\nu}(x),$
since $u_{\alpha}$, $h_{\varepsilon\nu}$, $g_{\varepsilon\nu}$, and
$v_{\varepsilon\nu}$ are solutions to equations (2.1), (3.1)–(3.3)
respectively.
Thus $(S_{\varepsilon\nu}-z)y_{\varepsilon\nu}=f-q_{\varepsilon\nu}$, and
consequently
$y_{\varepsilon\nu}=(S_{\varepsilon\nu}-z)^{-1}(f-q_{\varepsilon\nu})$, where
$q_{\varepsilon\nu}=r^{\prime\prime}_{\varepsilon\nu}+zr_{\varepsilon\nu}+zy_{\varepsilon\nu}\chi_{\nu}\\\
-\beta\Psi(\nu^{-1}\,\cdot\,)\bigl{(}\beta
y(-0)h_{\varepsilon\nu}(\nu^{-1}\,\cdot\,)+\eta^{-1}g_{\varepsilon\nu}(\varepsilon^{-1}\,\cdot\,)+\varepsilon\eta^{-1}v_{\varepsilon\nu}(\varepsilon^{-1}\,\cdot\,)\bigr{)}.$
(3.17)
Recall that $\chi_{\nu}$ is the characteristic function of $[-\nu,\nu]$. Owing
to Lemmas 3.2–3.4, we have
$\displaystyle|y(-0)|\,\left|\Psi\left(\tfrac{x}{\nu}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)\right|\leq
c_{1}\|h_{\varepsilon\nu}\|_{C(\mathcal{I})}\|f\|\,\chi_{\nu}(x)\leq
c_{2}\|f\|\,\chi_{\nu}(x),$ $\displaystyle\begin{aligned}
\eta^{-1}\left|\Psi\left(\tfrac{x}{\nu}\right)g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\right|\leq
c_{3}\eta^{-1}&\chi_{\nu}(x)\max_{x\in[-\nu,\nu]}|g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)|\\\
&\leq c_{4}\eta^{-1}(1+\eta)\|f\|\,\chi_{\nu}(x)\leq
c_{5}\|f\|\,\chi_{\nu}(x),\end{aligned}$ (3.18)
$\displaystyle\varepsilon\eta^{-1}|\Psi\left(\tfrac{x}{\nu}\right)v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)|\leq
c_{6}\varepsilon\eta^{-1}\chi_{\nu}(x)\max_{x\in[-\nu,\nu]}|v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)|\leq
c_{7}\nu^{1/2}\|f\|\,\chi_{\nu}(x),$ (3.19)
and hence $\|q_{\varepsilon\nu}\|\leq c\varrho(\nu,\eta)\|f\|$, in view of
Corollary 3.5 and Lemma 3.6. Note also that $\|\chi_{\nu}\|=(2\nu)^{1/2}$.
Therefore
$\|(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}\|=\|(S_{\varepsilon\nu}-z)^{-1}q_{\varepsilon\nu}\|\\\
\leq\|(S_{\varepsilon\nu}-z)^{-1}\|\,\|q_{\varepsilon\nu}\|\leq
C\varrho(\nu,\eta)\|f\|.$ (3.20)
Note that the resolvents $(S_{\varepsilon\nu}-z)^{-1}$ are uniformly bounded
with respect to $\varepsilon$ and $\nu$, because the operators
$S_{\varepsilon\nu}$ are self-adjoint.
We next observe that
$y_{\varepsilon\nu}-y=r_{\varepsilon\nu}+(w_{\varepsilon\nu}-y)\chi_{\nu}$.
Thus
$\|y_{\varepsilon\nu}-y\|\leq c\varrho(\nu,\eta)\|f\|,$ (3.21)
in view of Corollary 3.5 and Lemma 3.6. Form this we deduce for
$z\in\mathbb{C}\setminus\mathbb{R}$ that
$\displaystyle\|(S_{\varepsilon\nu}-z)^{-1}f-(S-z)^{-1}f\|$
$\displaystyle\leq\|(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}\|+\|y_{\varepsilon\nu}-(S-z)^{-1}f\|$
$\displaystyle\leq\|(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}\|+\|y_{\varepsilon\nu}-y\|\leq
C\varrho(\nu,\eta)\|f\|,$
for all $f\in L_{2}(\mathbb{R})$, by (3.20) and (3.21). The proof is completed
by noting that $\varrho(\nu,\eta)$ tends to zero as $\nu\to 0$ and
$\eta\to\infty$, that is to say, as $\nu\to 0$ and $\varepsilon\to 0$. ∎
### 3.2. Non-resonant case
Here we prove the following theorem:
###### Theorem 3.7.
Suppose the potential $\alpha\Phi$ is not resonant; then the operators
$S_{\varepsilon\nu}$ converge to the direct sum $S_{-}\oplus S_{+}$ of the
Dirichlet half-line Schrödinger operators as $\nu\to 0$ and $\eta\to\infty$ in
the norm resolvent sense.
As a matter of fact, this result is implicitly contained in the previous
proof. In the non-resonant case, equation (2.1) admits only one
$L_{\infty}(\mathbb{R})$-solution which is trivial. Additionally, for each
$f\in L_{2}(\mathbb{R})$, the function $y=(S_{-}\oplus S_{+}-z)^{-1}f$
satisfies the condition $y(0)=0$. Roughly speaking, the proof of Theorem 3.7
can be derived from the previous one with $u_{\alpha}$ and
$h_{\varepsilon\nu}$ replacing the zero functions and $y(\pm 0)$ replacing $0$
in the corresponding formulas.
###### Proof.
In this case the approximation $y_{\varepsilon\nu}$ is rather simpler than
(3.16). Whereas $y(0)=0$, we set
$y_{\varepsilon\nu}(x)=\begin{cases}y(x)+r_{\varepsilon\nu}(x)&\text{if
}|x|>\nu,\\\ \varepsilon
g(x/\varepsilon)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)&\text{if
}|x|\leq\nu.\end{cases}$
Here $y=(S_{-}\oplus S_{+}-z)^{-1}f$. As above, $r_{\varepsilon\nu}$ is a
$W_{2}^{2}$-corrector of the form (2.10) and $v_{\varepsilon\nu}$ is a
solutions of (3.3). The function $g$ is a solutions to the boundary value
problem
$g^{\prime\prime}-\alpha\Phi(t)g=0,\quad t\in\mathbb{R},\qquad
g^{\prime}(-1)=y^{\prime}(-0),\quad g^{\prime}(1)=y^{\prime}(+0).$
Such a solution exists, since $\alpha$ is not an eigenvalue of (2.2). In
addition, $g$ is linear outside $\mathcal{I}$, so it satisfies the
inequalities of the form (3.7), (3.8) and (3.18).
Reasoning as in the proof of Lemma 3.6 we deduce that
$\displaystyle|y(\pm\nu)-\varepsilon
g(\pm\eta)|\leq|y(\pm\nu)|+\varepsilon|g(\pm\eta)|+\varepsilon^{2}|v_{\varepsilon\nu}(\pm\eta)|\leq
c_{1}\nu\|f\|,$
$\displaystyle|y^{\prime}(\pm\nu)-g^{\prime}(\pm\eta)|\leq|y^{\prime}(\pm\nu)-y^{\prime}(\pm
0)|+\varepsilon|v^{\prime}_{\varepsilon\nu}(\pm\eta)|\leq
c_{2}\nu^{1/2}\|f\|,$
provided $\eta\gg 1$, and hence that
$\max_{x\in\mathbb{R}\setminus\\{-\nu,\nu\\}}\bigl{|}r^{(k)}_{\varepsilon\nu}(x)\bigr{|}\leq
C\nu^{1/2}\|f\|,\qquad k=0,1,2,$ (3.22)
by Proposition 2.4. Furthermore
$(S_{\varepsilon\nu}-z)y_{\varepsilon\nu}=f-q_{\varepsilon\nu}$ with
$q_{\varepsilon\nu}(x)=r^{\prime\prime}_{\varepsilon\nu}(x)+zr_{\varepsilon\nu}(x)+\varepsilon
z\chi_{\nu}(x)g(\tfrac{x}{\varepsilon})-\beta\Psi(\tfrac{x}{\nu})\left(\eta^{-1}g(\tfrac{x}{\varepsilon})+\eta^{-1}\varepsilon
v_{\varepsilon\nu}(\varepsilon^{-1}\,\cdot\,)\right),$
by calculations as in the proof of Theorem 3.1. Also
$\|q_{\varepsilon\nu}\|\leq c_{3}\nu^{1/2}\|f\|$, in view of (3.18), (3.19),
and (3.22). This implies
$\|(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}\|\leq c_{4}\nu^{1/2}\|f\|$.
The norm resolvent convergence of $S_{\varepsilon\nu}$ towards $S_{-}\oplus
S_{+}$ now follows precisely as in the proof of Theorem 3.1. ∎
## 4\. Convergence of the operators $S_{\varepsilon\nu}$. The case $\nu\sim
c\varepsilon$.
In this short section we apply the results of our recent work [23] to the case
$\nu\varepsilon^{-1}\to\lambda$ and $\lambda>0$. The parameters $\varepsilon$
and $\nu$ are in this case connected by the asymptotic relation
$\nu_{\varepsilon}=\lambda\varepsilon+o(\varepsilon)$ as $\varepsilon\to 0$.
Let us consider the operator family
$H_{\lambda}=\begin{cases}S(\theta_{\alpha},\beta\varkappa(\alpha,\lambda))&\text{if
}\alpha\in\Lambda_{\Phi},\\\ S_{-}\oplus S_{+}&\text{otherwise}\end{cases}$
(4.1)
for $\lambda>0$, where $\varkappa$ is given by (2.4). For convenience, we
shall write $S_{\varepsilon\nu}(\Phi,\Psi)$ for $S_{\varepsilon\nu}$, and
$\varkappa(\alpha,\lambda;\Phi,\Psi)$ for $\varkappa(\alpha,\lambda)$
indicating the dependence of $S_{\varepsilon\nu}$ and $\varkappa$ on
potentials $\Phi$ and $\Psi$.
For the case $\nu=\varepsilon$, it was proved in [23] that operators
$S_{\varepsilon\varepsilon}(\Phi,\Psi)$ converge to $H_{1}$ in the norm
resolvent sense, as $\varepsilon\to 0$. Moreover, this result is stable under
a small perturbation the potential $\Psi$. If a sequence of potentials
$\Psi_{\varepsilon}$ of compact support is uniformly bounded in
$L_{\infty}(\mathbb{R})$ and $\Psi_{\varepsilon}\to\Psi$ in
$L_{1}(\mathbb{R})$ as $\varepsilon\to 0$, then
$S_{\varepsilon\varepsilon}(\Phi,\Psi_{\varepsilon})\to H_{1}$ in the sense of
the norm resolvent convergence. Note that all estimates containing $\Psi$ in
the proofs of Theorems 4.1 and 5.1 in [23] remain true with $\Psi$ replaced by
$\Psi_{\varepsilon}$ due to the uniform boundedness of $\Psi_{\varepsilon}$ in
$L_{\infty}(\mathbb{R})$. Next, the $L_{1}$-convergence of
$\Psi_{\varepsilon}$ implies
$\varkappa(\alpha,1;\Phi,\Psi_{\varepsilon})\to\varkappa(\alpha,1;\Phi,\Psi)$,
as $\varepsilon\to 0$, for all $\alpha\in\Lambda_{\Phi}$. Observe also that
$S_{\varepsilon,\lambda\varepsilon}(\Phi,\Psi)=-\frac{d^{2}}{dx^{2}}+\frac{\alpha}{\varepsilon^{2}}\Phi\left(\frac{x}{\varepsilon}\right)+\frac{\beta}{\lambda\varepsilon}\Psi\left(\frac{x}{\lambda\varepsilon}\right)=S_{\varepsilon,\varepsilon}(\Phi,\Upsilon)$
with $\Upsilon=\frac{1}{\lambda}\Psi(\frac{1}{\lambda}\,\cdot\,)$. Next, we
see that
$\varkappa(\alpha,1;\Phi,\Upsilon)=\theta_{\alpha}^{-1}\int_{\mathbb{R}}\frac{1}{\lambda}\Psi\left(\frac{t}{\lambda}\right)u^{2}_{\alpha}(t)\,dt\\\
=\theta_{\alpha}^{-1}\int_{\mathbb{R}}\Psi\left(\tau\right)u^{2}_{\alpha}(\lambda\tau)\,d\tau=\varkappa(\alpha,\lambda;\Phi,\Psi).$
Therefore $S_{\varepsilon,\lambda\varepsilon}(\Phi,\Psi)\to H_{\lambda}$ as
$\varepsilon\to 0$ in the sense of uniform convergence of resolvents.
Repeating the previous scaling arguments leads to
$S_{\varepsilon\nu}(\Phi,\Psi)=S_{\varepsilon,\lambda\varepsilon}(\Phi,\Psi_{\varepsilon})$,
where
$\Psi_{\varepsilon}=\gamma_{\varepsilon}\Psi(\gamma_{\varepsilon}\,\cdot\,)$
and $\gamma_{\varepsilon}=\lambda\varepsilon/\nu_{\varepsilon}$. Since
$\gamma_{\varepsilon}\to 1$ as $\varepsilon$ goes to $0$,
$\Psi_{\varepsilon}\to\Psi$ in $L_{1}(\mathbb{R})$ as $\varepsilon\to 0$.
Hence both operators $S_{\varepsilon\nu}(\Phi,\Psi)$ and
$S_{\varepsilon,\lambda\varepsilon}(\Phi,\Psi)$ converge to the same limit
$H_{\lambda}$. We have proved:
###### Theorem 4.1.
If the ratio $\nu/\varepsilon$ tends to a finite positive number $\lambda$ as
$\nu,\varepsilon\to 0$, then $S_{\varepsilon\nu}$ converge to the operator
$H_{\lambda}$ defined by (4.1) in the norm resolvent sense.
## 5\. Convergence of the operators $S_{\varepsilon\nu}$. The case
$\nu\varepsilon^{-1}\to 0$.
We discuss in this section the case of the fast contracting $\Psi$-shaped
potential relative to the $\Phi$-shaped one. Therefore that
$\nu\varepsilon^{-1}\to 0$ as $\nu,\varepsilon\to 0$. First we note that if
$\varepsilon\to 0$ and $\eta\to 0$, then $\nu\to 0$. As in Section 3, the
resonant and non-resonant cases will be treated separately.
### 5.1. Resonant case
Let us consider the operator $S(\theta_{\alpha},\beta\mu_{\alpha})$, where
$\mu_{\alpha}=\mu(\alpha)$ and the mapping
$\mu\colon\Lambda_{\Phi}\to\mathbb{R}$ is given by (2.5).
###### Theorem 5.1.
Suppose $\Phi,\Psi\in\mathcal{P}$ and $\alpha\in\Lambda_{\Phi}$; then the
operator family $S_{\varepsilon\nu}$ converges to
$S(\theta_{\alpha},\beta\mu_{\alpha})$ in the norm resolvent sense, as
$\varepsilon,\eta\to 0$.
Given $f\in L_{2}(\mathbb{R})$ and $z\in\mathbb{C}\setminus\mathbb{R}$, we
write $y=(S-z)^{-1}f$, where $S=S(\theta_{\alpha},\beta\mu_{\alpha})$. Note
that $y$ satisfies the conditions
$y(+0)=\theta_{\alpha}y(-0),\quad
y^{\prime}(+0)=\theta_{\alpha}^{-1}y^{\prime}(-0)+\beta\mu_{\alpha}y(-0).$
(5.1)
Let us next guess $y_{\varepsilon\nu}$ has the form
$y_{\varepsilon\nu}(x)=\begin{cases}y(x)+r_{\varepsilon\nu}(x)&\text{for
}|x|>\varepsilon,\\\ y(-0)u_{\alpha}(x/\varepsilon)+\varepsilon
g_{\varepsilon\nu}(x/\varepsilon)+\beta\nu\varepsilon
h_{\varepsilon\nu}(x/\nu)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)&\text{for
}|x|\leq\varepsilon,\end{cases}$ (5.2)
where $g_{\varepsilon\nu}$, $h_{\varepsilon\nu}$, and $v_{\varepsilon\nu}$ are
solutions to the Cauchy problems
$\displaystyle\begin{cases}\displaystyle g^{\prime\prime}-\alpha\Phi(t)g=\beta
y(-0)\,\eta^{-1}\Psi(\eta^{-1}t)u_{\alpha}(t),\qquad t\in\mathbb{R},\\\
\displaystyle g(-1)=0,\quad g^{\prime}(-1)=y^{\prime}(-0);\end{cases}$ (5.3)
$\displaystyle\hskip 12.0pth^{\prime\prime}=\Psi(t)g_{\varepsilon\nu}(\eta
t),\quad t\in\mathbb{R},\qquad h(-1)=0,\quad h^{\prime}(-1)=0;$ (5.4)
$\displaystyle\begin{cases}-v^{\prime\prime}+\alpha\Phi(t)v+\beta\varepsilon\eta^{-1}\,\Psi(\eta^{-1}t)v=f(\varepsilon
t),\quad t\in\mathbb{R},\\\ \phantom{-}v(-1)=0,\quad
v^{\prime}(-1)=0\end{cases}$ (5.5)
respectively. As above, $u_{\alpha}$ is the half-bound state for the potential
$\alpha\Phi$, and $r_{\varepsilon\nu}$ adjusts this approximation so as to
obtain an element of $\mathop{\rm dom}S_{\varepsilon\nu}$. According to
Proposition 2.4, there exists a corrector function $r_{\varepsilon\nu}$ that
vanishes in $(-\varepsilon,\varepsilon)$.
###### Lemma 5.2.
If the ratio of $\nu$ to $\varepsilon$ remains bounded as $\nu,\varepsilon\to
0$, then there exists a constant $C$ such that for all $f\in
L_{2}(\mathbb{R})$
$\|g_{\varepsilon\nu}\|_{C(\mathcal{I})}\leq C\|f\|.$ (5.6)
In addition, $g^{\prime}_{\varepsilon\nu}(1)=y^{\prime}(+0)+O(\eta)\|f\|$ as
$\varepsilon,\eta\to 0$.
###### Proof.
Our proof starts with the observation that the right-hand side of equation
(5.3) contains a $\delta$-like sequence, namely
$\eta^{-1}\Psi(\eta^{-1}t)\to\left(\int_{\mathbb{R}}\Psi\,dt\right)\delta(x)\quad\text{in
}W_{2}^{-1}(\mathcal{I})$ (5.7)
as $\eta\to 0$. Let $g_{0}$ be the solution of (2.1) obeying the initial
conditions $g_{0}(-1)=0$ and $g_{0}^{\prime}(-1)=1$. Then $g_{\varepsilon\nu}$
can be represented as $g_{\varepsilon\nu}=y^{\prime}(-0)g_{0}+\beta
y(-0)\hat{g}_{\varepsilon\nu}$, where $\hat{g}_{\varepsilon\nu}$ solves the
equation $g^{\prime\prime}-\alpha\Phi
g=\eta^{-1}\Psi(\eta^{-1}\cdot\,)u_{\alpha}$ and satisfies zero initial
conditions at $t=-1$. Next, $\hat{g}_{\varepsilon\nu}$ converges in
$W_{2}^{1}(\mathcal{I})$ to the solution $\hat{g}$ of the problem
$g^{\prime\prime}-\alpha\Phi(t)g=u_{\alpha}(0)\,\left(\int_{\mathbb{R}}\Psi\,dt\right)\delta(x),\quad
t\in\mathcal{I},\qquad g(-1)=0,\quad g^{\prime}(-1)=0,$
which is clear from the explicit representation of $\hat{g}_{\varepsilon\nu}$
of the form (2.8). Thus the convergence in $W_{2}^{1}(\mathcal{I})$ implies
the uniform convergence of $\hat{g}_{\varepsilon\nu}$ to $\hat{g}$ in
$\mathcal{I}$, and consequently $\hat{g}_{\varepsilon\nu}$ is uniformly
bounded in $\varepsilon$ and $\nu$ provided $\eta<c$. From this we see that
$\|g_{\varepsilon\nu}\|_{C(\mathcal{I})}\leq|y^{\prime}(-0)|\,\|g_{0}\|_{C(\mathcal{I})}+|\beta|\,|y(-0)|\,\|\hat{g}_{\varepsilon\nu}\|_{C(\mathcal{I})}\leq
C\|f\|$, by (2.6).
Multiplying equation (5.3) by $u_{\alpha}$ and integrating on $\mathcal{I}$ by
parts yield
$\theta_{\alpha}g^{\prime}_{\varepsilon\nu}(1)-y^{\prime}(-0)=\beta
y(-0)\eta^{-1}\int_{-1}^{1}\Psi(\eta^{-1}s)u^{2}_{\alpha}(s)\,ds.$
Since $u_{\alpha}(t)=u_{\alpha}(0)+O(t)$ as $t\to 0$, we have
$g^{\prime}_{\varepsilon\nu}(1)=\theta_{\alpha}^{-1}\left(y^{\prime}(-0)+\beta
y(-0)u^{2}_{\alpha}(0)\int_{\mathbb{R}}\Psi\,ds\right)+O(\eta)\|f\|\\\
=\theta_{\alpha}^{-1}y^{\prime}(-0)+\beta\mu_{\alpha}y(-0)+O(\eta)\|f\|,\quad\eta\to
0,$
by (5.7) and (2.5). Therefore the asymptotic relation for
$g^{\prime}_{\varepsilon\nu}(1)$ follows from (5.1). ∎
###### Lemma 5.3.
There exist constants $C_{1}$ and $C_{2}$, independent of $f$, such that
$\displaystyle|h_{\varepsilon\nu}(t)|\leq C_{1}(1+|t|)\|f\|,$ $\displaystyle
t\in\mathbb{R},$ (5.8) $\displaystyle|h^{\prime}_{\varepsilon\nu}(t)|\leq
C_{2}\|f\|,$ $\displaystyle t\in\mathbb{R}$ (5.9)
for all $\varepsilon$ and $\nu$ whenever the ratio of $\nu$ to $\varepsilon$
is small enough.
###### Proof.
As in the proof of Lemma 3.3, equation (5.4) gives
$h_{\varepsilon\nu}(t)=t\int_{-1}^{1}\Psi(s)g_{\varepsilon\nu}(\eta
s)\,ds-\int_{-1}^{1}s\Psi(s)g_{\varepsilon\nu}(\eta s)\,ds\quad\text{for
}t\geq 1$
and $h_{\varepsilon\nu}(t)=0$ for $t\leq-1$. If $|\eta|\leq 1$, then (5.8),
(5.9) follow from (5.6). ∎
###### Lemma 5.4.
There exist constants $C$ independent of $f$ such that
$\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq C\varepsilon^{-1/2}\|f\|$
(5.10)
for all $\varepsilon$ and $\nu$ small enough.
###### Proof.
Let $v_{\varepsilon}$ be a solution of the auxiliary Cauchy problem
$-v^{\prime\prime}_{\varepsilon}+\alpha\Phi(t)v_{\varepsilon}=f(\varepsilon
t),\quad t\in\mathbb{R},\quad v_{\varepsilon}(-1)=0,\quad
v^{\prime}_{\varepsilon}(-1)=0.$
In view of Proposition 2.3 we have
$v_{\varepsilon}(t)=\int_{-1}^{t}k(t,s)f(\varepsilon s)\,ds,$
where $k=k(t,s)$ is a continuously differentiable function on
$\mathbb{R}^{2}$. Therefore
$\|v_{\varepsilon}\|_{C^{1}(\mathcal{I})}\leq
c_{1}\|k\|_{C^{1}(\mathcal{I}\times\mathcal{I})}\int_{-1}^{1}|f(\varepsilon
s)|\,ds\leq
c_{2}\varepsilon^{-1}\int_{-\varepsilon}^{\varepsilon}|f(\tau)|\,d\tau\leq
c_{3}\varepsilon^{-1/2}\|f\|.$ (5.11)
Next, the function
$\vartheta_{\varepsilon\nu}=v_{\varepsilon\nu}-v_{\varepsilon}$ solves the
problem
$-\vartheta^{\prime\prime}_{\varepsilon}+\alpha\Phi(t)\vartheta_{\varepsilon}=-\beta\varepsilon\eta^{-1}\,\Psi(\eta^{-1}t)v_{\varepsilon\nu},\quad
t\in\mathbb{R},\quad\vartheta_{\varepsilon}(-1)=0,\quad\vartheta^{\prime}_{\varepsilon}(-1)=0.$
We conclude from this that
$\|\vartheta_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq
c_{4}\varepsilon\eta^{-1}\|k\|_{C^{1}(\mathcal{I}\times\mathcal{I})}\int_{-1}^{1}|\Psi(\eta^{-1}s)||v_{\varepsilon\nu}(s)|\,ds\\\
\leq
c_{5}\varepsilon\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\,\eta^{-1}\int_{-1}^{1}|\Psi(\eta^{-1}s)|\,ds\\\
\leq
c_{5}\varepsilon\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\int_{\mathbb{R}}|\Psi(\tau)|\,d\tau\leq
c_{6}\varepsilon\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}.$
Hence, $\|v_{\varepsilon\nu}-v_{\varepsilon}\|_{C^{1}(\mathcal{I})}\leq
c_{6}\varepsilon\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}$, and consequently
$(1-c_{6}\varepsilon)\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq\|v_{\varepsilon}\|_{C^{1}(\mathcal{I})}.$
That $\|v_{\varepsilon\nu}\|_{C^{1}(\mathcal{I})}\leq
C\varepsilon^{-1/2}\|f\|$ follows from estimate (5.11) for $\varepsilon$ small
enough. ∎
Lemmas 5.2–5.4 have the following corollary.
###### Corollary 5.5.
The function $y_{\varepsilon\nu}$ is bounded in $[-\varepsilon,\varepsilon]$
uniformly in $\varepsilon$ and $\nu$ provided $\nu/\varepsilon\leq 1$, and
$\max_{|x|\leq\varepsilon}|y_{\varepsilon\nu}(x)|\leq C\|f\|$ with some
constant $C$ being independent of $f$.
The function $w_{\varepsilon\nu}=y_{\varepsilon\nu}-r_{\varepsilon\nu}$ and
its first derivative have the jumps at $x=\pm\varepsilon$:
$\displaystyle[w_{\varepsilon\nu}]_{-\varepsilon}=y(-0)-y(-\varepsilon),\qquad[w^{\prime}_{\varepsilon\nu}]_{-\varepsilon}=y^{\prime}(-0)-y^{\prime}(-\varepsilon),$
$\displaystyle[w_{\varepsilon\nu}]_{\varepsilon}=y(\varepsilon)-\theta_{\alpha}y(-0)-\varepsilon
g_{\varepsilon\nu}(1)-\beta\nu\varepsilon\,h_{\varepsilon\nu}(\eta^{-1})-\varepsilon^{2}v_{\varepsilon\nu}(1),$
$\displaystyle[w^{\prime}_{\varepsilon\nu}]_{\varepsilon}=y^{\prime}(\varepsilon)-g^{\prime}_{\varepsilon\nu}(1)-\varepsilon(\beta\,h^{\prime}_{\varepsilon\nu}(\eta^{-1})+v^{\prime}_{\varepsilon\nu}(1)).$
In view of (2.7), (5.6), (5.8), (5.10), and (5.1), we conclude that three of
the jumps can be bounded by $c_{1}\varepsilon^{1/2}\|f\|$. As for the last
one, we have
$\left|[w^{\prime}_{\varepsilon\nu}]_{\varepsilon}\right|\leq|y^{\prime}(\varepsilon)-y^{\prime}(+0)|+c_{2}\eta\|f\|+c_{3}\varepsilon(|h^{\prime}_{\varepsilon\nu}(\eta)|+|v^{\prime}_{\varepsilon\nu}(1)|)\leq
c_{2}(\varepsilon^{1/2}+\eta)\|f\|,$
by (5.9), (5.10), and Lemma 5.2. We can now repeatedly apply Proposition 2.4
to deduce
$\max_{x\in\mathbb{R}\setminus\\{-\varepsilon,\varepsilon\\}}\bigl{|}r^{(k)}_{\varepsilon\nu}(x)\bigr{|}\leq
C\sigma(\varepsilon,\eta)\|f\|$ (5.12)
for $k=0,1,2$, where $\sigma(\varepsilon,\eta)=\varepsilon^{1/2}+\eta$.
###### Proof of Theorem 5.1.
As in the proof of Theorem 3.1 we introduce the notation
$f_{\varepsilon\nu}=(S_{\varepsilon\nu}-z)y_{\varepsilon\nu}$. It is easy to
check that
$f_{\varepsilon\nu}(x)=f(x)-r^{\prime\prime}_{\varepsilon\nu}(x)-zr_{\varepsilon\nu}(x)$
for $|x|>\varepsilon$. Next, for $|x|<\varepsilon$, we have
$\displaystyle\textstyle f_{\varepsilon\nu}(x)=$
$\displaystyle\left(-\tfrac{d^{2}}{dx^{2}}+\alpha\varepsilon^{-2}\Phi\left(\tfrac{x}{\varepsilon}\right)+\beta\nu^{-1}\Psi\left(\tfrac{x}{\nu}\right)-z\right)y_{\varepsilon\nu}(x)$
$\displaystyle=$
$\displaystyle\varepsilon^{-2}y(-0)\Bigl{\\{}-u^{\prime\prime}_{\alpha}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)u_{\alpha}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\varepsilon^{-1}\Bigl{\\{}-g_{\varepsilon\nu}^{\prime\prime}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\beta\eta^{-1}y(-0)\Psi\left(\tfrac{x}{\nu}\right)u_{\alpha}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\beta\eta^{-1}\Bigl{\\{}-h^{\prime\prime}_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)+\Psi\left(\tfrac{x}{\nu}\right)g_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\Bigl{\\}}$
$\displaystyle+$
$\displaystyle\Bigl{\\{}-v^{\prime\prime}_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\alpha\Phi\left(\tfrac{x}{\varepsilon}\right)v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)+\beta\varepsilon^{2}\nu^{-1}\,\Psi\left(\tfrac{x}{\nu}\right)v_{\varepsilon\nu}\left(\tfrac{x}{\varepsilon}\right)\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\alpha\beta\eta\,\Phi\left(\tfrac{x}{\varepsilon}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)+\beta^{2}\varepsilon\,\Psi\left(\tfrac{x}{\nu}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)-zy_{\varepsilon\nu}(x)$
$\displaystyle=$ $\displaystyle
f(x)+\Bigl{\\{}\alpha\eta\,\Phi\left(\tfrac{x}{\varepsilon}\right)+\beta\varepsilon\,\Psi\left(\tfrac{x}{\nu}\right)\Bigr{\\}}\beta
h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)-zy_{\varepsilon\nu}(x),$
since $u_{\alpha}$, $g_{\varepsilon\nu}$, $h_{\varepsilon\nu}$, and
$v_{\varepsilon\nu}$ are solutions to equations (2.1) and (5.3)–(5.5)
respectively. Then $f_{\varepsilon\nu}=f-q_{\varepsilon\nu}$, where
$q_{\varepsilon\nu}=r^{\prime\prime}_{\varepsilon\nu}+zr_{\varepsilon\nu}+zy_{\varepsilon\nu}\chi_{\varepsilon}-\left(\alpha\eta\Phi(\varepsilon^{-1}\,\cdot\,)+\beta\varepsilon\Psi(\nu^{-1}\,\cdot\,)\right)\beta
h_{\varepsilon\nu}(\nu^{-1}\,\cdot\,).$
As above, $\chi_{\varepsilon}$ is the characteristic function of
$[-\varepsilon,\varepsilon]$. Consequently, we conclude from Lemma 5.3 that
$\displaystyle\begin{aligned}
\eta\left|\Phi\left(\tfrac{x}{\varepsilon}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)\right|\leq
c_{1}\eta\chi_{\varepsilon}(x)\max_{|x|\leq\varepsilon}&|h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)|\\\
&\leq c_{2}\eta(1+\eta^{-1})\|f\|\,\chi_{\varepsilon}(x)\leq
c_{3}\|f\|\,\chi_{\varepsilon}(x),\end{aligned}$
$\displaystyle\varepsilon\left|\Psi\left(\tfrac{x}{\nu}\right)h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)\right|\leq
c_{4}\varepsilon\chi_{\nu}(x)\max_{|x|\leq\nu}|h_{\varepsilon\nu}\left(\tfrac{x}{\nu}\right)|\leq
c_{5}\varepsilon\|f\|\,\chi_{\nu}(x),$
hence that $\|q_{\varepsilon\nu}\|\leq c\sigma(\varepsilon,\eta)\|f\|$, in
view of Corollary 5.5 and estimate (5.12). Thus
$y_{\varepsilon\nu}=(S_{\varepsilon\nu}-z)^{-1}f+(S_{\varepsilon\nu}-z)^{-1}q_{\varepsilon\nu}$,
and therefore
$\|(S_{\varepsilon\nu}-z)^{-1}f-y_{\varepsilon\nu}\|\leq\|(S_{\varepsilon\nu}-z)^{-1}\|\|q_{\varepsilon\nu}\|\leq
c_{6}\sigma(\varepsilon,\eta)\|f\|.$
By arguments that are completely analogous to those presented in the proof of
Theorem 3.1 we conclude that
$\|(S(\theta_{\alpha},\beta\mu_{\alpha})-z)^{-1}f-y_{\varepsilon\nu}\|\leq
C\sigma(\varepsilon,\eta)\|f\|$, and finally that operators
$S_{\varepsilon\nu}$ converge to $S(\theta_{\alpha},\beta\mu_{\alpha})$ in the
norm resolvent sense as $\varepsilon$ and $\eta$ tend to zero. ∎
### 5.2. Non-resonant case
Assume $\alpha$ does not belongs to the resonant set $\Lambda_{\Phi}$, and
write $y=(S_{-}\oplus S_{+}-z)^{-1}f$ for $f\in L_{2}(\mathbb{R})$.
###### Theorem 5.6.
If $\alpha\not\in\Lambda_{\Phi}$, then the operator family
$S_{\varepsilon\nu}$ defined by (1.1) converges to the direct sum $S_{-}\oplus
S_{+}$ in the norm resolvent sense as $\varepsilon,\eta\to 0$.
###### Proof.
In this case the approximation $y_{\varepsilon\nu}$ may be greatly simplified,
since $y(0)=0$. Looking at asymptotics (5.2), we set
$y_{\varepsilon\nu}(x)=\begin{cases}y(x)+r_{\varepsilon\nu}(x)\quad&\text{for
}|x|>\varepsilon,\\\ \varepsilon
g(x/\varepsilon)+\beta\nu\varepsilon\,h_{\varepsilon\nu}(x/\nu)+\varepsilon^{2}v_{\varepsilon\nu}(x/\varepsilon)\quad&\text{for
}|x|\leq\varepsilon,\end{cases}$
where $g$ and $h_{\varepsilon\nu}$ are solutions to the problems
$\displaystyle g^{\prime\prime}-\alpha\Phi(t)g=0,\quad t\in\mathbb{R},$
$\displaystyle g^{\prime}(-1)=y^{\prime}(-0),\quad
g^{\prime}(1)=y^{\prime}(0);$ $\displaystyle h^{\prime\prime}=\Psi(t)g(\eta
t),\quad t\in\mathbb{R},$ $\displaystyle h(-1)=0,\quad h^{\prime}(-1)=0$
respectively. As above, $v_{\varepsilon\nu}$ is a solution of (5.5), and the
corrector function $r_{\varepsilon\nu}$ is of the form (2.10) and provides the
inclusion $y_{\varepsilon\nu}\in W_{2}^{2}(\mathbb{R})$. The rest of the proof
is similar to the proof of Theorem 5.1. ∎
_Acknowledgements._ I would like to thank Rostyslav Hryniv and Alexander
Zolotaryuk for stimulating discussions.
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|
arxiv-papers
| 2012-02-21T17:38:04 |
2024-09-04T02:49:27.652331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuriy Golovaty",
"submitter": "Yuriy Golovaty",
"url": "https://arxiv.org/abs/1202.4711"
}
|
1202.4717
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-050
LHCb-PAPER-2011-028
Determination of the sign of the decay width difference in the $B^{0}_{s}$
system
The LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
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Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
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M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
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Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
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Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
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Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
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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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
The interference between the $K^{+}K^{-}$ S-wave and P-wave amplitudes in
$B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ decays with the $K^{+}K^{-}$ pairs in
the region around the $\phi(1020)$ resonance is used to determine the
variation of the difference of the strong phase between these amplitudes as a
function of $K^{+}K^{-}$ invariant mass. Combined with the results from our
$C\\!P$ asymmetry measurement in $B_{s}^{0}\rightarrow J/\psi\phi$ decays, we
conclude that the $B_{s}^{0}$ mass eigenstate that is almost $C\\!P=+1$ is
lighter and decays faster than the mass eigenstate that is almost $C\\!P=-1$.
This determines the sign of the decay width difference
$\Delta\Gamma_{s}\equiv\Gamma_{\rm L}-\Gamma_{\rm H}$ to be positive. Our
result also resolves the ambiguity in the past measurements of the $C\\!P$
violating phase $\phi_{s}$ to be close to zero rather than $\pi$. These
conclusions are in agreement with the Standard Model expectations.
Published on Physical Review Letters
The decay time distributions of $B^{0}_{s}$ mesons decaying into the
$J/\psi\phi$ final state have been used to measure the parameters $\phi_{s}$
and $\Delta\Gamma_{s}\equiv\Gamma_{\rm L}-\Gamma_{\rm H}$ of the $B^{0}_{s}$
system Abazov:2011ry ; CDF:2011af ; LHCb:2011aa . Here $\phi_{s}$ is the
$C\\!P$ violating phase equal to the phase difference between the amplitude
for the direct decay and the amplitude for the decay after oscillation.
$\Gamma_{\rm L}$ and $\Gamma_{\rm H}$ are the decay widths of the light and
heavy $B_{s}^{0}$ mass eigenstates, respectively. The most precise results,
presented recently by the LHCb experiment LHCb:2011aa ,
$\begin{array}[]{lcl}\phi_{s}&=&0.15\phantom{0}\pm 0.18\phantom{0}\,({\rm
stat})\pm 0.06\phantom{0}\,{\rm(syst)~{}rad},\\\ \Delta\Gamma_{s}&=&0.123\pm
0.029\,({\rm stat})\pm 0.011{\rm\,(syst)~{}ps}^{-1},\end{array}$ (1)
show no evidence of $C\\!P$ violation yet, indicating that $C\\!P$ violation
is rather small in the $B^{0}_{s}$ system. There is clear evidence for the
decay width difference $\Delta\Gamma_{s}$ being non-zero. It must be noted
that there exists another solution
$\begin{array}[]{lcl}\phi_{s}&=&\phantom{-}2.99\phantom{0}\pm
0.18\phantom{0}\,({\rm stat})\pm 0.06\phantom{0}\,{\rm(syst)~{}rad},\\\
\Delta\Gamma_{s}&=&-0.123\pm 0.029\,({\rm stat})\pm
0.011\,{\rm(syst)~{}ps}^{-1},\end{array}$ (2)
arising from the fact that the time dependent differential decay rates are
invariant under the transformation
$(\phi_{s},~{}\Delta\Gamma_{s})\leftrightarrow(\pi-\phi_{s},~{}-\Delta\Gamma_{s})$
together with an appropriate transformation for the strong phases. In the
absence of $C\\!P$ violation, $\sin\phi_{s}=0$, i.e. $\phi_{s}=0$ or
$\phi_{s}=\pi$, the two mass eigenstates also become $C\\!P$ eigenstates with
$C\\!P=+1$ and $C\\!P=-1$, according to the relationship between $B^{0}_{s}$
mass eigenstates and $C\\!P$ eigenstates given in Ref. Dunietz:2000cr . They
can be identified by the decays into final states which are $C\\!P$
eigenstates. In $B_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ decays, the final
state is a superposition of $C\\!P=+1$ and $C\\!P=-1$ for the $K^{+}K^{-}$
pair in the P-wave configuration and $C\\!P=-1$ for the $K^{+}K^{-}$ pair in
the S-wave configuration. Higher order partial waves are neglected. These
decays have different angular distributions of the final state particles and
are distinguishable.
Solution I is close to the case $\phi_{s}=0$ and leads to the light (heavy)
mass eigenstate being almost aligned with the $C\\!P=+1$ $(C\\!P=-1)$ state.
Similarly, solution II is close to the case $\phi_{s}=\pi$ and leads to the
heavy (light) mass eigenstate being almost aligned with the $C\\!P=+1$
$(C\\!P=-1)$ state. In Fig. 2 of Ref. LHCb:2011aa , a fit to the observed
decay time distribution shows that it can be well described by a superposition
of two exponential functions corresponding to $C\\!P=+1$ and $C\\!P=-1$,
compatible with no $C\\!P$ violation LHCb:2011aa . In this fit the lifetime of
the decay to the $C\\!P=+1$ final state is found to be smaller than that of
the decay to $C\\!P=-1$. Thus the mass eigenstate that is predominantly
$C\\!P$ even decays faster than the $C\\!P$ odd state. For solution I, we find
$\Delta\Gamma_{s}>0$, i.e. $\Gamma_{\rm L}>\Gamma_{\rm H}$, and for solution
II, $\Delta\Gamma_{s}<0$, i.e. $\Gamma_{\rm L}<\Gamma_{\rm H}$. In order to
determine if the decay width difference $\Delta\Gamma_{s}$ is positive or
negative, it is necessary to resolve the ambiguity between the two solutions.
Since each solution corresponds to a different set of strong phases, one may
attempt to resolve the ambiguity by using the strong phases either as
predicted by factorisation or as measured in $B^{0}\rightarrow J/\psi K^{*0}$
decays. Unfortunately these two possibilities lead to opposite answers
Nandi:2008rg . A direct experimental resolution of the ambiguity is therefore
desirable.
In this Letter, we resolve this ambiguity using the decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ with $J\\!/\\!\psi\rightarrow\mu^{+}\mu^{-}$. The total
decay amplitude is a coherent sum of S-wave and P-wave contributions. The
phase of the P-wave amplitude, which can be described by a spin-1 Breit-Wigner
function of the invariant mass of the $K^{+}\kern-1.60004ptK^{-}$ pair,
denoted by $m_{KK}$, rises rapidly through the $\rm\phi(1020)$ mass region. On
the other hand, the phase of the S-wave amplitude should vary relatively
slowly for either an $f_{0}(980)$ contribution or a nonresonant contribution.
As a result, the phase difference between the S-wave and P-wave amplitudes
falls rapidly with increasing $m_{KK}$. By measuring this phase difference as
a function of $m_{KK}$ and taking the solution with a decreasing trend around
the $\rm\phi(1020)$ mass as the physical solution, the sign of
$\Delta\Gamma_{s}$ is determined and the ambiguity in $\phi_{s}$ is resolved
Xie:2009fs . This is similar to the way the BaBar collaboration measured the
sign of $\cos 2\beta$ using the decay $B^{0}\rightarrow J/\psi
K^{0}_{\rm\scriptscriptstyle S}\pi^{0}$ Aubert:2004cp , where $2\beta$ is the
weak phase characterizing mixing-induced $C\\!P$ asymmetry in this decay.
The analysis is based on the same data sample as used in Ref. LHCb:2011aa ,
which corresponds to an integrated luminosity of 0.37 $\mbox{\,fb}^{-1}$ of
$pp$ collisions collected by the LHCb experiment at the Large Hadron Collider
at the centre of mass energy of $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The
LHCb detector is a forward spectrometer and is described in detail in Ref.
Alves:2008zz . The trigger, event selection criteria and analysis method are
very similar to those in Ref. LHCb:2011aa , and here we discuss only the
differences. The fraction of $K^{+}\kern-1.60004ptK^{-}$ S-wave contribution
measured within $\pm$12 MeV of the nominal $\rm\phi(1020)$ mass is $0.042\pm
0.015\pm 0.018$ LHCb:2011aa . (We adopt units such that $c=1$ and $\hbar=1$.)
The S-wave fraction depends on the mass range taken around the
$\rm\phi(1020)$. The result of Ref. LHCb:2011aa is consistent with the CDF
limit on the S-wave fraction of less than $6\%$ at $95\%$ CL (in the range
1009–1028 MeV) CDF:2011af , smaller than the DØ result of $(12\pm 3)\%$ (in
1010–1030 MeV) Abazov:2011hv and consistent with phenomenological
expectations Stone:2008ak . In order to apply the ambiguity resolution method
described above, the range of $m_{KK}$ is extended to 988–1050 MeV. Figure 1
shows the $\mu^{+}\mu^{-}K^{+}\kern-1.60004ptK^{-}$ mass distribution where
the mass of the $\mu^{+}\mu^{-}$ pair is constrained to the nominal
$J\\!/\\!\psi$ mass. We perform an unbinned maximum likelihood fit to the
invariant mass distribution of the selected $B^{0}_{s}$ candidates. The
probability density function (PDF) for the signal $B^{0}_{s}$ invariant mass
$m_{J/\psi KK}$ is modelled by two Gaussian functions with a common mean. The
fraction of the wide Gaussian and its width relative to that of the narrow
Gaussian are fixed to values obtained from simulated events. A linear function
describes the $m_{J/\psi KK}$ distribution of the background, which is
dominated by combinatorial background.
This analysis uses the sWeight technique Pivk:2004ty for background
subtraction. The signal weight, denoted by $W_{\rm s}(m_{J/\psi KK})$, is
obtained using $m_{J/\psi KK}$ as the discriminating variable. The
correlations between $m_{J/\psi KK}$ and other variables used in the analysis,
including $m_{KK}$, decay time $t$ and the angular variables $\Omega$ defined
in Ref. LHCb:2011aa , are found to be negligible for both the signal and
background components in the data. Figure 2 shows the $m_{KK}$ distribution
where the background is subtracted statistically using the sWeight technique.
The range of $m_{KK}$ is divided into four intervals: 988–1008, 1008–1020,
1020–1032 and 1032–1050 MeV. Table 1 gives the number of $B^{0}_{s}$ signal
and background candidates in each interval.
Figure 1: Invariant mass distribution for
$B^{0}_{s}\rightarrow\mu^{+}\mu^{-}K^{+}K^{-}$ candidates, with the mass of
the $\mu^{+}\mu^{-}$ pair constrained to the nominal $J/\psi$ mass. The result
of the fit is shown with signal (dashed curve) and combinatorial background
(dotted curve) components and their sum (solid curve).
Figure 2: Background subtracted $K^{+}\kern-1.60004ptK^{-}$ invariant mass
distribution for $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ candidates. The vertical dotted lines separate the four
intervals. Table 1: Numbers of signal and background events in the $m_{J/\psi
KK}$ range of 5200–5550 MeV and statistical power per signal event in four
intervals of $m_{KK}$.
$k$ | $m_{KK}$ interval (MeV) | $N_{{\rm sig};k}$ | $N_{{\rm bkg};k}$ | $W_{{\rm p};k}$
---|---|---|---|---
1 | 988–1008 | $\phantom{0}251\pm 21$ | $1675\pm 43$ | 0.700
2 | 1008–1020 | $4569\pm 70$ | $2002\pm 49$ | 0.952
3 | 1020–1032 | $3952\pm 66$ | $2244\pm 51$ | 0.938
4 | 1032–1050 | $\phantom{0}726\pm 34$ | $3442\pm 62$ | 0.764
Figure 3: Distribution of (a) $K^{+}\kern-1.60004ptK^{-}$ S-wave signal
events, and (b) $K^{+}\kern-1.60004ptK^{-}$ P-wave signal events, both in four
invariant mass intervals. In (b), the distribution of simulated
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
events in the four intervals assuming the same total number of P-wave events
is also shown (dashed lines). Note the interference between the
$K^{+}\kern-1.60004ptK^{-}$ S-wave and P-wave amplitudes integrated over the
angular variables has vanishing contribution in these distributions. Table 2:
Results from a simultaneous fit of the four intervals of $m_{KK}$, where the
uncertainties are statistical only. Only parameters which are needed for the
ambiguity resolution are shown.
Parameter | Solution I | Solution II
---|---|---
$\phi_{s}$ (rad) | 0.167 $\pm$ 0.175 | $-$2.975 $\pm$ 0.175
$\Delta\Gamma$ (${\rm\,ps}^{-1}$) | 0.120 $\pm$ 0.028 | $-0.120$ $\pm$ 0.028
${F_{\mathrm{S};1}}$ | 0.283 $\pm$ 0.113 | $-$0.283 $\pm$ 0.113
${F_{\mathrm{S};2}}$ | 0.061 $\pm$ 0.022 | $-$0.061 $\pm$ 0.022
${F_{\mathrm{S};3}}$ | 0.044 $\pm$ 0.022 | $-$0.044 $\pm$ 0.022
${F_{\mathrm{S};4}}$ | 0.269 $\pm$ 0.067 | $-$0.269 $\pm$ 0.067
$\delta_{\mathrm{S}\perp;1}$ (rad) | $-$$2.68\,\,_{-\,0.42}^{+\,0.35}$ | $0.46\,\,^{+\,0.42}_{-\,0.35}$
$\delta_{\mathrm{S}\perp;2}$ (rad) | $-$$0.22\,\,_{-\,0.13}^{+\,0.15}$ | $2.92\,\,^{+\,0.13}_{-\,0.15}$
$\delta_{\mathrm{S}\perp;3}$ (rad) | $-0.11\,\,_{-\,0.18}^{+\,0.16}$ | $3.25\,\,^{+\,0.18}_{-\,0.16}$
$\delta_{\mathrm{S}\perp;4}$ (rad) | $-0.97\,\,_{-\,0.43}^{+\,0.28}$ | $4.11\,\,^{+\,0.43}_{-\,0.28}$
Figure 4: Measured phase differences between S-wave and perpendicular P-wave
amplitudes in four intervals of $m_{KK}$ for solution I (full blue circles)
and solution II (full black squares). The asymmetric error bars correspond to
$\Delta\ln L=-0.5$ (solid lines) and $\Delta\ln L=-2$ (dash-dotted lines).
In this analysis we perform an unbinned maximum likelihood fit to the data
using the sFit method 2009arXiv0905.0724X , an extension of the sWeight
technique, that simplifies fitting in the presence of background. In this
method, it is only necessary to model the signal PDF, as background is
cancelled statistically using the signal weights.
The parameters of the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decay time distribution are estimated from a simultaneous
fit to the four intervals of $m_{KK}$ by maximizing the log-likelihood
function
$\displaystyle\ln L$
$\displaystyle({\bf\Theta_{P},\Theta_{\mathrm{S}}})=\sum_{k=1}^{4}W_{{\rm
p};k}\sum_{i=1}^{N_{k}}W_{\rm s}(m_{J/\psi KK;i})\times$ $\displaystyle\ln
P_{\rm
sig}(t_{i},\Omega_{i},q_{i},\omega_{i};{\bf\Theta_{P}},{\bf\Theta_{\mathrm{S}}}),\,$
where $N_{k}=N_{{\rm sig};k}+N_{{\rm bkg};k}$ is the number of candidates in
the $m_{J/\psi KK}$ range of 5200–5550 MeV for the $k$th interval.
$\bf\Theta_{P}$ represents the physics parameters independent of $m_{KK}$,
including $\phi_{s}$, $\Delta\Gamma_{s}$ and the magnitudes and phases of the
P-wave amplitudes. Note that the P-wave amplitudes for different polarizations
share the same dependence on $m_{KK}$. $\bf\Theta_{\mathrm{S}}$ denotes the
values of the $m_{KK}$-dependent parameters averaged over each interval,
namely the average fraction of S-wave contribution for the $k$th interval,
$F_{\mathrm{S};k}$, and the average phase difference between the S-wave
amplitude and the perpendicular P-wave amplitude for the $k$th interval,
$\delta_{\mathrm{S}\perp;k}$. $P_{\rm sig}$ is the signal PDF of the decay
time $t$, angular variables $\Omega$, initial flavour tag $q$ and the mistag
probability $\omega$. It is based on the theoretical differential decay rates
Xie:2009fs and includes experimental effects such as decay time resolution
and acceptance, angular acceptance and imperfect identification of the initial
flavour of the $B^{0}_{s}$ particle, as described in Ref. LHCb:2011aa . The
factors $W_{{\rm p};k}$ account for loss of statistical precision in parameter
estimation due to background dilution and are necessary to obtain the correct
error coverage. Their values are given in Table 1.
The fit results for $\phi_{s}$, $\Delta\Gamma_{s}$, $F_{\mathrm{S};k}$ and
$\delta_{\mathrm{S}\perp;k}$ are given in Table 2. Figure 3 shows the
estimated $K^{+}\kern-1.60004ptK^{-}$ S-wave and P-wave contributions in the
four $m_{KK}$ intervals. The shape of the measured P-wave $m_{KK}$
distribution is in good agreement with that of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
events simulated using a spin-1 relativistic Breit-Wigner function for the
$\rm\phi(1020)$ amplitude. In Fig. 4, the phase difference between the S-wave
and the perpendicular P-wave amplitude is plotted in four $m_{KK}$ intervals
for solution I and solution II.
Figure 4 shows a clear decreasing trend of the phase difference between the
S-wave and P-wave amplitudes in the $\rm\phi(1020)$ mass region for solution
I, as expected for the physical solution. To estimate the significance of the
result, we perform an unbinned maximum likelihood fit to the data by
parameterizing the phase difference $\delta_{{\mathrm{S}}\perp;k}$ as a linear
function of the average $m_{KK}$ value in the $k$th interval. This leads to a
slope of $-0.050_{-0.020}^{+0.013}$ rad/MeV for solution I and the opposite
sign for solution II, where the uncertainties are statistical only. The
difference of the $\ln L$ value between this fit and a fit in which the slope
is fixed to be zero is 11.0. Hence, the negative trend of solution I has a
significance of 4.7 standard deviations. Therefore, we conclude that solution
I, which has $\Delta\Gamma_{s}>0$, is the physical solution. The trend of
solution I is also qualitatively consistent with that of the phase difference
between the $K^{+}\kern-1.60004ptK^{-}$ S-wave and P-wave amplitudes versus
$m_{KK}$ measured in the decay $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ by the
BaBar collaboration delAmoSanchez:2010yp .
Several possible sources of systematic uncertainty on the phase variation
versus $m_{KK}$ have been considered. A possible background from decays with
similar final states such as
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ could
have a small effect. From simulation, the contamination to the signal from
such decays is estimated to be $1.1\%$ in the $m_{KK}$ range of 988–1050 MeV.
We add a $2.2\%$ contribution of simulated
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
to the data and repeat the analysis. The largest observed change is a shift of
$\delta_{\mathrm{S}\perp;4}$ by $0.06$ rad, which is only 20$\%$ of its
statistical uncertainty and has negligible effect on the slope of
$\delta_{\mathrm{S}\perp}$ versus $m_{KK}$. The effect of neglecting the
variation of the values of $F_{\mathrm{S}}$ and $\delta_{\mathrm{S}\perp}$ in
each $m_{KK}$ interval is determined to change the significance of the
negative trend of solution I by less than 0.1 standard deviations. We also
repeat the analysis for different $m_{KK}$ ranges, different ways of dividing
the $m_{KK}$ range, or different shapes of the signal and background
$m_{J/\psi KK}$ distributions. The significance of the negative trend of
solution I is not affected. To measure precisely the S-wave line shape and
determine its resonance structure, more data are needed. However, the results
presented here do not depend on such detailed knowledge.
In conclusion the analysis of the strong interaction phase shift resolves the
ambiguity between solution I and solution II. Values of $\phi_{s}$ close to
zero and positive $\Delta\Gamma_{s}$ are preferred. It follows that in the
$B^{0}_{s}$ system, the mass eigenstate that is almost $C\\!P$ even is lighter
and decays faster than the state that is almost $C\\!P$ odd. This is in
agreement with the Standard Model expectations (e.g., Lenz:2010gu ). It is
also interesting to note that this situation is similar to that in the neutral
kaon system.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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* (12) Y. Xie, sFit: a method for background subtraction in maximum likelihood fit, arXiv:0905.0724
* (13) BaBar collaboration, P. del Amo Sanchez et al., Dalitz plot analysis of $D^{+}_{s}\rightarrow K^{+}\kern-1.60004ptK^{-}\pi^{+}$, Phys. Rev. D83 (2011) 052001, arXiv:1011.4190
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arxiv-papers
| 2012-02-21T17:57:58 |
2024-09-04T02:49:27.664243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, K. de\n Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A.\n Comerma-Montells, F. Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, B.\n Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N. Y. David,\n I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De\n Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P.\n Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, 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, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, C.\n Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson,\n V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es,\n G. Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, 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, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, R. F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, D. Martinez\n Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B.\n Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J.\n Merkel, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n 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, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F.\n Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud,\n I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang,\n W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Yuehong Xie",
"url": "https://arxiv.org/abs/1202.4717"
}
|
1202.4812
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-047 LHCb-PAPER-2011-043
Measurement of the $\mathbfi{B^{\pm}}$ production cross-section in
$\mathbfi{pp}$ collisions at $\mathbfi{\sqrt{s}=7}$ TeV
The LHCb collaboration †††Authors are listed on the following pages.
The production of $B^{\pm}$ mesons in proton-proton collisions at
$\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ is studied using 35 pb-1 of data
collected by the LHCb detector. The $B^{\pm}$ mesons are reconstructed
exclusively in the $B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}$ mode, with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$. The differential production cross-section is
measured as a function of the $B^{\pm}$ transverse momentum in the fiducial
region $0<p_{\rm T}<40$ GeV/$c$ and with rapidity $2.0<y<4.5$. The total
cross-section, summing up $B^{+}$ and $B^{-}$, is measured to be
$\sigma(pp\rightarrow B^{\pm}X,\;\mbox{$0<p_{\rm T}<40$\;
GeV/$c$},\;2.0<y<4.5)=41.4\pm 1.5\,({\rm stat.})\pm 3.1\,({\rm
syst.})\,\rm\,\upmu b$.
Submitted to JHEP
The LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S.
Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, E. Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38,
S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30,
C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2,
M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44,
V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V.
Niess5, N. Nikitin29, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, K. Pal53,
J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M.
Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M.
Patel50, S.K. Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-
Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M.
Schiller39, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O.
Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A.
Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N.
Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P.
Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31, O. Shevchenko40, V.
Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49,
E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The study of the $b\overline{}b$ production cross-section is a powerful test
of perturbative quantum chromodynamics (pQCD) calculations. These are
available at next-to-leading order (NLO) [1] and with the fixed-order plus
next-to-leading logarithms (FONLL) [2, 3] approximations. In the NLO and FONLL
calculations, the theoretical predictions have large uncertainties arising
from the choice of the renormalisation and factorisation scales and the
b-quark mass [4]. Accurate measurements provide tests of the validity of the
different production models. Recently, the LHCb collaboration measured the
$b\bar{b}$ production cross-section in hadron collisions using
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $b$ decays [5] and
$b\rightarrow D\mu X$ decays [6]. The two most recent measurements of the
$B^{\pm}$ production cross-section in hadron collisions have been performed by
the CDF collaboration in the range $p_{\rm
T}>6\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $|y|<1$ [7], where $p_{\rm T}$
is the transverse momentum and $y$ is rapidity, and by the CMS collaboration
in the range $p_{\rm T}>~{}5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$|y|<2.4$ [8]. This paper presents a measurement of the $B^{\pm}$ production
cross-section in $pp$ collisions at a centre-of-mass energy of
$\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ using $34.6\pm
1.2\,\mbox{\,pb}^{-1}$ of data collected by the LHCb detector in 2010. The
$B^{\pm}$ mesons are reconstructed exclusively in the
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ mode,
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$.
Both the total production cross-section and the differential cross-section,
${\rm d}\sigma/{\rm d}p_{\rm T}$, as a function of the $B^{\pm}$ transverse
momentum for $0<p_{\rm T}<40$ GeV/$c$ and $2.0<y<4.5$, are measured.
The LHCb detector [9] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers.
The LHCb detector uses a two-level trigger system, the first level (L0) is
hardware based, and the second level is software based high level trigger
(HLT). Here only the triggers used in this analysis are described. At the L0
either a single muon candidate with $p_{\rm T}$ larger than
$1.4\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or a pair of muon candidates, one
with $p_{\rm T}$ larger than $0.56\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
the other with $p_{\rm T}$ larger than
$0.48\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, is required. Events passing
these requirements are read out and sent to an event filter farm for further
selection. In the first stage of the HLT, events satisfying one of the
following three selections are kept: the first one confirms the single-muon
candidates from L0 and applies a harder $p_{\rm T}$ selection at
$1.8\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; the second one confirms the
single-muon from L0 and looks for another muon in the event, and the third one
confirms the dimuon candidates from L0. Both the second and third selections
require the dimuon invariant mass to be greater than
$2.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The second stage of the HLT
selects events that pass any selections of previous stage and contain two muon
candidates with an invariant mass within 120
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. To reject high-
multiplicity events with a large number of $pp$ interactions, a set of global
event cuts (GEC) is applied on the hit multiplicities of sub-detectors.
## 2 Event selection
Candidates for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay are formed from pairs of particles with
opposite charge. Both particles are required to have a good track fit quality
($\chi^{2}/{\rm ndf}<4$, where ndf represents the number of degrees of freedom
in the fit), a transverse momentum $p_{\rm T}>0.7$ GeV/$c$ and to be
identified as a muon. In addition, the muon pair is required to originate from
a common vertex ($\chi^{2}/{\rm ndf}<9$). The mass of the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is required to be in the range
$3.04-3.14\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
The bachelor kaon candidates used to form
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$
candidates are required to have $p_{\rm T}$ larger than 0.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to have a good track fit quality
($\chi^{2}/{\rm ndf}<4$). No particle identification is used in the selection
of the kaon. A vertex fit is performed that constrains the three daughter
particles to originate from a common point and the mass of the muon pair to
match the nominal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. It is
required that $\chi^{2}/{\rm ndf}<9$ for this fit. To further reduce the
combinatorial background due to particles produced in the primary $pp$
interaction, only candidates with a decay time larger than 0.3 ${\rm ps}$ are
accepted. Finally, the fiducial requirement $0<p_{\rm T}<40$ GeV/$c$ and
$2.0<y<4.5$ is applied to the $B^{\pm}$ candidates.
## 3 Cross-section determination
The differential production cross-section is measured as
$\frac{{\rm d}\sigma}{{\rm d}p_{\rm T}}=\frac{N_{B^{\pm}}(p_{\rm T})}{{\cal
L}\;\epsilon_{\rm tot}(p_{\rm T})\;{\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm})\;{\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})\;\Delta p_{\rm T}},$ (1)
where $N_{B^{\pm}}(p_{\rm T})$ is the number of reconstructed
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ signal
events in a given $p_{\rm T}$ bin, ${\cal L}$ is the integrated luminosity,
$\epsilon_{\rm tot}(p_{\rm T})$ is the total efficiency, including geometrical
acceptance, reconstruction, selection and trigger effects, ${\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})$ and
${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})$ are the branching fractions of the
reconstructed decay chain [10], and $\Delta p_{\rm T}$ is the $p_{\rm T}$ bin
width.
Considering that the efficiencies depend on $p_{\rm T}$ and $y$, we calculate
the event yield in bins of these variables using an extended unbinned maximum
likelihood fit to the invariant mass distribution of the reconstructed
$B^{\pm}$ candidates in the interval
$5.15<M_{B^{\pm}}<5.55\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. We assume
that the signal and background shapes only depend on $p_{\rm T}$. Three
components are included in the fit procedure: a Crystal Ball function [11] to
model the signal, an exponential function to model the combinatorial
background and a double-Crystal Ball function 111A double-Crystal Ball
function has tails on both the low and high mass side of the peak with
separate parameters for the two. to model the Cabibbo suppressed decay
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$. The
shape of the latter component is found to fit well the distribution of
simulated events. The ratio of the number of
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$
candidates to that of the signal is fixed to ${\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm})/{\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})$
from Ref. [10]. The invariant mass distribution of the selected
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$
candidates and the fit result for one bin ($5.0<p_{\rm
T}<5.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) are shown in Fig. 1. The fit
returns a mass resolution of $9.14\pm 0.49$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and a mean of $5279.05\pm 0.56$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the uncertainties are
statistical only. Summing over all $p_{\rm T}$ bins, the total number of
signal events is about 9100.
Figure 1: Invariant mass distribution of the selected
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$
candidates for one bin ($5.0<p_{\rm
T}<5.5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$). The result of the fit to the
model described in the text is superimposed.
The geometrical acceptance and the reconstruction and selection efficiencies
are determined using simulated signal events. The simulation is based on
Pythia 6.4 generator [12] with parameters configured for LHCb [13]. The EvtGen
package [14] is used to describe the decays of the $B^{\pm}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$. QED radiative corrections are
modelled using Photos [15]. The Geant4 [16] simulation package is used to
trace the decay products through the detector. Since we select events passing
trigger selections that depend on ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ properties only, the trigger efficiency is obtained from a trigger-
unbiased data sample of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events
that would still be triggered if the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ candidate were removed. The efficiency of GEC is determined from data
to be $(92.6\pm 0.3)\%$, and assumed to be independent of the $B^{\pm}$
$p_{\rm T}$ and $y$. The total trigger efficiency is then the product of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ trigger efficiency and the GEC
efficiency. The luminosity is measured using Van der Meer scans and a beam-gas
imaging method [17]. The knowledge of the absolute luminosity scale is used to
calibrate the number of tracks in the vertex detector, which is found to be
stable throughout the data-taking period and can therefore be used to monitor
the instantaneous luminosity of the entire data sample. The integrated
luminosity of the data sample used in this analysis is determined to be $34.6$
pb-1.
The measurement is affected by the systematic uncertainty on the determination
of signal yields, efficiencies, branching fractions and luminosity.
The uncertainty on the determination of the signal yields mainly arises from
the description of final state radiation in the signal fit. The fitted signal
yield is corrected by 3.0%, which is estimated by comparing the fitted and
generated signal yields in the Monte Carlo simulation, and an uncertainty of
1.5% is assigned. The uncertainties from the effects of the Cabibbo-suppressed
background, multiple candidates and mass fit range are found to be negligible.
The uncertainties on the efficiencies arise from trigger ($0.5-6.0$% depending
on the bin), tracking ($3.9-4.4$% depending on the bin), muon identification
(2.5%) [5] and the vertex fit quality cut (1.0%). The trigger systematic
uncertainty has been evaluated by measuring the trigger efficiency in the
simulation using a trigger-unbiased data sample of simulated
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events. The tracking
uncertainty includes two components: the first one is the differences in track
reconstruction efficiency between data and simulation, estimated with a tag
and probe method [18] using ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ events; the second is due to the 2%
uncertainty on the hadronic interaction length of the detector used in the
simulation. The uncertainties from the effects of GEC,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass window cut and inter-bin
cross-feed are found to be negligible. The uncertainty due to the choice of
$p_{\rm T}$ binning is estimated to be smaller than 2.0%.
The product of ${\cal
B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})$ and
${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-})$ is calculated to be $(6.01\pm 0.20)\times
10^{-5}$, by taking their values from Ref. [10] with their correlations taken
into account.
The absolute luminosity scale is measured with a $3.5\%$ uncertainty [17],
dominated by the beam current uncertainty.
## 4 Results and conclusion
The measured $B^{\pm}$ differential production cross-section in bins of
$p_{\rm T}$ for $2.0<y<4.5$ is given in Table 1. This result is compared with
a FONLL prediction [2, 3] in Fig. 2. A hadronisation fraction
$f_{\bar{b}\rightarrow B^{+}}$ of $(40.1\pm 1.3)$% [10] is assumed to fix the
overall scale of FONLL. The uncertainty of the FONLL computation includes the
uncertainties on the $b$-quark mass, renormalisation and factorisation scales,
and CTEQ 6.6 [19] Parton Density Functions (PDF). Good agreement is observed
between data and the FONLL prediction. The integrated cross-section is
$\sigma(pp\rightarrow B^{\pm}X,\;\mbox{$0<p_{\rm T}<40$\;
GeV/$c$},\;2.0<y<4.5)=41.4\pm 1.5\,({\rm stat.})\pm 3.1\,({\rm
syst.})\,\rm\,\upmu b.$
This is the first measurement of $B^{\pm}$ production in the forward region at
$\sqrt{s}$ = 7 TeV.
Figure 2: Differential production cross-section as a function of the $B^{\pm}$ transverse momentum. The left plot shows the full $p_{\rm T}$ range, the right plot shows a zoom of the $p_{\rm T}$ range of $0-12$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The histogram (left) and the open circles with error bars (right) are the measurements. The red dashed lines in both plots are the upper and lower uncertainty limits of the FONLL computation. A hadronisation fraction $f_{\bar{b}\rightarrow B^{+}}$ of $(40.1\pm 1.3)$% [10] is assumed to fix the overall scale. The uncertainty of the FONLL computation includes the uncertainties of the $b$-quark mass, renormalisation and factorisation scales, and CTEQ 6.6 PDF. Table 1: Differential $B^{\pm}$ production cross-section in bins of $p_{\rm T}$ for $2.0<y<4.5$. The first and second quoted uncertainties are statistical and systematic, respectively. $p_{\rm T}$ $({\mathrm{\,Ge\kern-1.00006ptV\\!/}c})$ | ${\rm d\sigma}/{{\rm d}p_{\rm T}}$ $({\rm\,\upmu b}/({\mathrm{\,Ge\kern-1.00006ptV\\!/}c}))$ | $p_{\rm T}$ $({\mathrm{\,Ge\kern-1.00006ptV\\!/}c})$ | ${\rm d\sigma}/{{\rm d}p_{\rm T}}$ $({\rm\,\upmu b}/({\mathrm{\,Ge\kern-1.00006ptV\\!/}c}))$
---|---|---|---
$0.0-0.5$ | 1.37 $\pm$ 0.68 $\pm$ 0.13 | $7.0-7.5$ | 2.42 $\pm$ 0.20 $\pm$ 0.18
$0.5-1.0$ | 3.12 $\pm$ 0.82 $\pm$ 0.24 | $7.5-8.0$ | 2.09 $\pm$ 0.16 $\pm$ 0.15
$1.0-1.5$ | 3.90 $\pm$ 0.57 $\pm$ 0.29 | $8.0-8.5$ | 1.44 $\pm$ 0.11 $\pm$ 0.11
$1.5-2.0$ | 5.67 $\pm$ 1.05 $\pm$ 0.43 | $8.5-9.0$ | 1.33 $\pm$ 0.11 $\pm$ 0.10
$2.0-2.5$ | 8.44 $\pm$ 1.00 $\pm$ 0.64 | $9.0-9.5$ | 1.22 $\pm$ 0.10 $\pm$ 0.09
$2.5-3.0$ | 6.33 $\pm$ 0.66 $\pm$ 0.48 | $9.5-10.0$ | 0.83 $\pm$ 0.08 $\pm$ 0.06
$3.0-3.5$ | 5.04 $\pm$ 0.45 $\pm$ 0.38 | $10.0-10.5$ | 0.80 $\pm$ 0.08 $\pm$ 0.06
$3.5-4.0$ | 6.99 $\pm$ 0.68 $\pm$ 0.52 | $10.5-11.0$ | 0.65 $\pm$ 0.07 $\pm$ 0.05
$4.0-4.5$ | 5.48 $\pm$ 0.47 $\pm$ 0.41 | $11.0-12.0$ | 0.54 $\pm$ 0.04 $\pm$ 0.04
$4.5-5.0$ | 6.54 $\pm$ 0.79 $\pm$ 0.49 | $12.0-13.0$ | 0.41 $\pm$ 0.04 $\pm$ 0.03
$5.0-5.5$ | 4.42 $\pm$ 0.44 $\pm$ 0.33 | $13.0-14.5$ | 0.28 $\pm$ 0.02 $\pm$ 0.02
$5.5-6.0$ | 4.16 $\pm$ 0.37 $\pm$ 0.31 | $14.5-16.5$ | 0.17 $\pm$ 0.02 $\pm$ 0.01
$6.0-6.5$ | 3.40 $\pm$ 0.24 $\pm$ 0.25 | $16.5-21.5$ | 0.062 $\pm$ 0.005 $\pm$ 0.005
$6.5-7.0$ | 2.82 $\pm$ 0.22 $\pm$ 0.21 | $21.5-40.0$ | 0.011 $\pm$ 0.001 $\pm$ 0.001
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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|
arxiv-papers
| 2012-02-22T03:24:31 |
2024-09-04T02:49:27.675398
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y.\n Amhis, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L.\n Arrabito, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann,\n J.J. Back, D.S. Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H.\n Brown, K. de Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P.\n Collins, A. Comerma-Montells, F. Constantin, A. Contu, A. Cook, M. Coombes,\n G. Corti, B. Couturier, G.A. Cowan, R. Currie, C. D'Ambrosio, P. David,\n P.N.Y. David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J.M. De\n Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D.\n Esperante Pereira, A. Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C.\n Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S.\n Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L.\n Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon,\n Ph. 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, E. Greening, S. Gregson, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen,\n S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy,\n L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li\n Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac\n Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S.\n Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, S.K. Paterson, G.N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n 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, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S. Redford, M.M. Reid,\n A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N.A. Smith, E. Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F.\n Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D.R.\n Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang,\n W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Bo Liu",
"url": "https://arxiv.org/abs/1202.4812"
}
|
1202.4857
|
# On the Energy and Centrality Dependence of Higher Order Moments of Net-
Proton Distributions in Relativistic Heavy Ion Collisions
X. Wang and C. B. Yang Institute of Particle Physics, Central China Normal
University,Wuhan 430079, People’s Republic of China Key Laboratory of Quark
and Lepton Physics (CCNU), Ministry of Education, People’s Republic of China
###### Abstract
The higher order moments of the net-baryon distributions in relativistic heavy
ion collisions are useful probes for the QCD critical point and fluctuations.
Within a simple model we study the colliding energy and centrality dependence
of the net-proton distributions in the central rapidity region. The model is
based on considering the baryon stopping and pair production effects in the
processes. Based on some physical reasoning, the dependence is parameterized.
Predictions for the net-proton distributions for Au+Au and Pb+Pb collisions at
different centralities at $\sqrt{s_{NN}}$=39 and 2760 GeV, respectively, are
presented from the parameterizations for the model parameters. A possible test
of our model is proposed from investigating the net-proton distributions in
the non-central rapidity region for different colliding centralities and
energies.
###### pacs:
25.75.Gz, 21.65.Qr
## I Introduction
The investigation of QCD phase diagram is of crucial importance for our
understanding of the properties of matter with strong interactions. Lattice
QCD calculations have predicted, at vanishing baryon chemical potential, the
occurrence of a cross-over from hadronic phase to the deconfined quark-gluon
plasma phase above a critical temperature of about 170-190 MeV YA ; JB . A
distinct singular feature of the phase diagram is the QCD critical point MAS
which is located at the end of the transition boundary. A characteristic
feature of the QCD critical point for systems in the thermodynamical limit is
the divergence of the correlation length $\xi$ and extremely large critical
fluctuations. In ultra-relativistic heavy ion collisions, however, because of
finite size and rapid expansion of the produced system, those divergence may
be washed out. As estimated in MAS , the critical correlation length in heavy
ion collisions is not divergent but only about 2-3 fm. So the signals for the
critical point of the system produced in heavy ion collisions cannot be
observed as clearly as in the condensed matter physics. However, remnants of
those critical large fluctuations may become accessible in heavy ion
collisions through an event-by-event analysis of fluctuations in various
channels of conservative hadron quantum numbers, for example, baryon number,
electric charge, and strangeness FLUC . Particularly there would be a non-
monotonic behavior of non-Gaussian multiplicity fluctuations in an energy
scan, which would be a clear signature for the existence of a critical point.
In fact, at vanishing chemical potential it has been shown that moments of
conservative charge distributions are sensitive indicators for the occurrence
of a transition from hadronic to partonic matter SE .
Recently, great interest both experimentally STAR and theoretically THEO1 ;
THEO2 ; THEO3 has been aroused on the higher order moments of net-baryon
distributions in heavy ion collisions at the BNL Relativistic Heavy Ion
Collider (RHIC) energies. The theoretical interest on these higher order
moments comes from the discovery of the relation between the moments and the
thermal fluctuations near the critical points for the produced quark matter.
If some memory of the large correlation length in the quark matter sate
persists in the thermal medium in hadronization process, this must be
reflected in higher order moments of the distributions. Theoretical prediction
MAS2 showed that the third moment, called skewness, is proportional to
$\xi^{4.5}$ and that the fourth moment, or kurtosis, proportional to $\xi^{7}$
while the second moment proportional to $\xi^{2}$. More importantly, the
moments are closely related to the susceptibilities of the thermal medium.
Thus the higher order moments have stronger dependence on the correlation
length $\xi$ and are therefore more sensitive to the critical fluctuations.
Recently STAR Collaboration has published experimental data on the higher
order moments mom for different colliding systems at different colliding
energies for different colliding centralities. It has been argued that the
net-proton distribution can be a meaningful observable for the purpose of
detecting the critical fluctuations of net baryons in heavy ion collisions NP
. This statement makes experimental investigation of net-baryon fluctuations
much easier, because neutrons and strange baryons cannot been detected
easily/effectively in experiments. Based on theoretical and experimental
investigations, it has been argued that information of QCD phase diagram and
the critical point can be obtained from the energy dependence of those moments
THEO1 .
The moments of net-proton distributions have been studied recently by quite a
few groups with different event generators such as A Multi-Phase Transport
(AMPT) and Ultrarelativistic Quantum Molecular Dynamics (UrQMD) THEO3 , Heavy
Ion Jet INteraction Generator (HIJING) LUO , and hadron resonance gas model
HRG ; FK etc. Some other authors tried to search the statistical and
dynamical components in the net-proton distributions in chen , where the
statistical distributions for both proton and anti-proton are assumed
Poissonian and the departure from Poisson distributions is regarded as the
dynamical influence. One should pay attention that the use of independent
Poisson distributions for proton and anti-proton implies that protons and
anti-protons are produced completely uncorrelated. Therefore the baryon number
conservation may be violated in any event. On the multiplicity distribution of
hadrons, a canonical ensemble is employed in begun to derive the number
distribution for $\pi$ systems. This is reasonable because there are a lot of
$\pi$ particles in the final state of heavy ion collisions. But a simple
transportation of the method to the case for baryon production may be
problematic, because the relevant baryon particle number may be not large
enough for an equilibrium statistical description. In Ref. yw the net-proton
distributions in Au+Au collisions at $\sqrt{s_{NN}}=200{\rm GeV}$ are studied
from very simple but well established physics considerations: baryon stopping
and baryon pair production. For a given mean net-proton number from initial
nuclear stopping, the initial proton number is assumed to satisfy a Poisson
distribution. From the produced baryon pairs, the joint distribution for the
newly produced proton and anti-proton can be derived. Then one can obtain the
net-proton distribution in the final state of heavy ion collisions. Good
agreement with experimental data has been obtained for Au+Au collisions at
three colliding centralities at $\sqrt{s_{NN}}=200{\rm GeV}$.
This paper is an extension of the work in Ref. yw by studying the moments of
net-proton distributions in a given central rapidity window in Au+Au
collisions at lower RHIC energies at different centralities. This paper is
organized as follows. In next section, we will address our model and the
physics points for the centrality dependence of parameters. Using an
analytical expressions for the net-proton distribution derived in Ref. yw we
will show that the moments of the distribution up to fourth order can all be
well described by our model with suitably chosen parameters for four colliding
energies in Au+Au collisions at RHIC. In section III, our model results for
the moments are compared with the experimental data from STAR Collaboration.
The net-proton distributions for Au+Au collisions at $\sqrt{s_{\rm NN}}$=39
GeV are presented for three centralities. Then in section IV, we discuss the
energy dependence of the parameters and give parameterizations for such
dependence. Then we extrapolate the dependence to the CERN Large Hadron
Collider (LHC) energy and predict the net-proton distributions for Pb+Pb
collisions at different colliding centralities for $\sqrt{s_{\rm NN}}$ =2.76
TeV. The last section will be for a brief summary.
## II Model consideration for the centrality dependence
Nuclear stopping plays an important role in heavy ion collisions and the study
of such effect is a fundamental issue, since this effect is related to the
amount of energy and baryon number that get transferred from the beam nucleons
into the reaction zone. We denote $B$ the mean net-proton number in the final
state distribution. As can be seen from our model consideration, $B$ comes
only from the initially stopped protons. In nuclear-nuclear collisions the
mean net-baryon number $B$ in central rapidity region would be zero if there
were no nuclear stopping in the processes. Another physics point we consider
in Ref. yw and here is baryon pair production in the interactions. We denote
$\mu$ the mean number of produced baryon pairs within a given kinematic region
in the collisions at given colliding centrality. By assuming that baryon pairs
are produced independently, the pair number distribution is of Poissonian.
With isospin conservation, one can derive a simple analytical formula for the
net-proton number $\Delta p$ distribution $P(\Delta p)$ as a function of $B$
and $\mu$ as yw
$\displaystyle P(\Delta p)$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\frac{dx}{\pi}e^{-(2B+\mu)\sin^{2}\frac{x}{2}}\cos(x\Delta
p-B\sin x)\ .$ (1)
Now we study the centrality dependence of $B$ and $\mu$. Because the nuclear
stopping effect results from the interactions between a passing nucleon with
other nucleons on its way in a nuclear-nuclear collision, the baryon number
stopped in a given kinematic region is closely related to the number of
participant nucleons $N_{\rm p}$. In more central collisions the stopped net
baryon number will be larger. In nuclear-nuclear collisions, if every nucleon
from a nucleus suffers exactly the same interactions, the stopped proton
number would be $B\propto N_{\rm p}$. Of course, the real case is not so
simple. Because multiple scattering effect is more important for more central
collisions, a little larger $B/N_{\rm p}$ can be expected for more central
collisions. On the other hand, the stopped net-proton can be detected only in
the final state, or in other words, after evolving with the system for some
time. During the evolution of the system, the net-proton may diffuse into a
kinematic region out of our interested window. For central collisions, the
evolution time is longer and such diffusion effect is more obvious. This
effect would reduce $B/N_{\rm p}$ for central collisions. The real
proportional factor $B/N_{\rm p}$ is a result from the competition of the two
effects: initial multiple scattering and later baryon number diffusing.
Therefore, the factor $B/N_{\rm p}$ should have a weak centrality dependence.
Thus one may parameterize the centrality dependence of $B$ as
$B=a_{1}N_{p}(1-a_{2}N_{p})\ ,$ (2)
with $a_{1}$ and $a_{2}$ depending on the colliding energy of the system.
While $a_{1}$ is always positive, the magnitude of $a_{2}$ should be small and
$a_{2}$ can be negative or positive, depending on whether multiple scattering
is more important than baryon number diffusion or the opposite. For the baryon
pair production, similar physics considerations apply also. The probability
for a baryon pair production near a point in the system is determined by the
energy density at that point. If the energy density of the hot strong
interacting matter is uniform and the same for all colliding centralities, one
may expect $\mu$ proportional to the volume of the system, thus $\mu\propto
N_{\rm p}$. But the initial energy density is higher for more central
collisions due to stronger nuclear stopping effect, a larger $\mu/N_{\rm p}$
is expected for more central collisions, if the nuclear stopping is the only
physics in the process. On the other hand, as in the consideration for $B$,
the energy diffusion from central to non-central rapidity region will reduce
the value of $\mu/N_{\rm p}$. The competition of these two effect results in a
behavior of $\mu$ as a function of $N_{\rm p}$ similar to that of $B$.
Therefore we parameterize the colliding centrality dependence of $\mu$ as
$\mu=b_{1}N_{p}(1-b_{2}N_{p})\ ,$ (3)
with $b_{1}$ and $b_{2}$ also depending on the colliding energy. As for
$a_{2}$, the value of $b_{2}$ should be very small but can be positive or
negative due to the competition of initial energy stopping and diffusion in
the evolution of the system.
With the above four parameters, one can calculate the net-proton distribution
and all the associate moments for any colliding centrality for given center of
mass energy of the colliding system.
## III Comparison with the experimental Data
For Au+Au collisions at RHIC energies, we investigate the moments up to fourth
order for the distribution of net-proton in the central rapidity window
$|y|<0.5$ as functions of $N_{\rm p}$ by using our model described in the last
section. The fitted results from our model for the moments are shown in Figs.
1-4 for the mean, variance, skewness and kurtosis for four colliding energies
$\sqrt{s_{\rm NN}}$=19.6, 39, 62.4 and 200 GeV. The fitted parameters are
tabulated in TABLE I. It can be seen that our simple model can describe quite
well the centrality dependence of moments for the four energies with the
parameters chosen. From the excellent agreement with the experimental data,
one can conclude that our model contains the necessary physics for the net-
proton distributions.
Figure 1: Mean value of the net-proton distributions as a function of $N_{\rm p}$ for Au+Au collisions at four $\sqrt{s_{\rm NN}}$, (a) 200; (b) 62.4; (c) 39 and (d) 19.6 GeV. The points are from RHIC/STAR data mom . Figure 2: variance of the net-proton distributions as a function of $N_{\rm p}$ for Au+Au collisions at four $\sqrt{s_{\rm NN}}$, (a) 200; (b) 62.4; (c) 39 and (d) 19.6 GeV. The points are from RHIC/STAR data mom . Figure 3: Skewness of the net-proton distributions as a function of $N_{\rm p}$ for Au+Au collisions at four $\sqrt{s_{\rm NN}}$, (a) 200; (b) 62.4; (c) 39 and (d) 19.6 GeV. The points are from RHIC/STAR data mom . Figure 4: Kurtosis of the net-proton distributions as a function of $N_{\rm p}$ for Au+Au collisions at four $\sqrt{s_{\rm NN}}$, (a) 200; (b) 62.4; (c) 39 and (d) 19.6 GeV. The points are from RHIC/STAR data mom . $\sqrt{s_{\rm NN}}$ (GeV) | $a_{1}$ | $10^{4}a_{2}$ | $b_{1}$ | $10^{4}b_{2}$
---|---|---|---|---
19.6 | 0.022 | $-0.126$ | 0.0132 | $0.121$
39 | 0.0012 | $-1.97$ | 0.0300 | $4.90$
62.4 | 0.0096 | $-0.159$ | 0.0383 | $5.10$
200 | 0.0052 | $1.88$ | 0.0585 | $5.00$
Table 1: Fitted parameters for four colliding energies.
With those parameters in TABLE 1, one can calculate the net-proton
distributions quite easily for different centralities for the four energies as
in TABLE 1. Our new parametrization for $B$ and $\mu$ can give distributions
for Au+Au collisions at $\sqrt{s_{\rm NN}}$=200 GeV. The newly obtained
distributions have no visible difference from those in Ref. yw , in good
agreement with the STAR data, so will not be presented here. As an example to
show the distributions, we present here only the distributions at
$\sqrt{s_{\rm NN}}=39$ GeV for Au+Au collisions at different colliding
centralities, for later comparison with experimental results.
Figure 5: Net-proton distributions for Au+Au collisions at $\sqrt{s_{\rm
NN}}=39$ GeV at different colliding centralities.
## IV Colliding energy dependence of net-proton distribution
After discussing the net-proton distributions for Au+Au collisions at four
different colliding energies, one can discuss the colliding energy dependence
of parameters in our model. As we discussed in Sec. II, the values and their
dependence on the colliding energy can tell us some physics in the colliding
process, especially the competition of effects from the initial multiple
scattering and later baryon number transportation. With the increase of
colliding energy, the stopped baryon number in the central rapidity region
will decrease. Thus the value of $a_{1}$ will decrease with $\sqrt{s_{\rm
NN}}$. Though the energy fraction stopped in the central rapidity region
becomes smaller at higher colliding energy, the energy density in that region
still increases with $\sqrt{s_{\rm NN}}$. Then $b_{1}$ will increase with
$\sqrt{s_{\rm NN}}$. For $\sqrt{s_{\rm NN}}$ high enough, one may expect
$a_{1}$ and $b_{1}$ saturates at some limiting values. The behaviors of
$a_{2}$ and $b_{2}$ can be quite different from those of $a_{1}$ and $b_{1}$.
Since $a_{2}$ and $b_{2}$ depend on the competition of effects from initial
multiple nucleon-nucleon scattering and later baryon/energy diffusion, the
behaviors of $a_{2}$ and $b_{2}$ with the increase of $\sqrt{s_{\rm NN}}$ can
be complicated. Both the effects from the initial multiple nucleon-nucleon
scattering and later baryon/energy diffusion become stronger with the increase
of $\sqrt{s_{\rm NN}}$, but their rates of increase may be different. For a
given increase of $\sqrt{s_{\rm NN}}$, if the initial multiple scattering
becomes more important, $|a_{2}|$ ($|b_{2}|$) will increase with $\sqrt{s_{\rm
NN}}$ in the region with negative $a_{2}$ ($b_{2}$). Otherwise, $|a_{2}|$
($|b_{2}|$) will decrease in the same region. For $\sqrt{s_{\rm
NN}}\to\infty$, the interaction duration in the produced matter will very long
and the diffusion effect will be much stronger than that from initial multiple
scattering. Then one can expect $a_{2}$ and $b_{2}$ approaching some positive
saturating values, implying that $B/N_{\rm p}$ and $\mu/N_{\rm p}$ is smaller
for central collisions when $\sqrt{s_{\rm NN}}$ is high enough.
To see the colliding energy dependence of the four parameters, we plot the
fitted parameters shown in TABLE I as functions of $\sqrt{s_{\rm NN}}$ in GeV.
The plots are shown in Figs. 6 and 7. The shown dependence of the parameters
on the colliding energy can be described by the following expressions
$\begin{array}[]{ccl}a_{1}&=&14.55(1+9.23\times 10^{-3}\sqrt{s_{\rm
NN}})/(1+40.2\sqrt{s_{\rm NN}})\ ,\\\ 10^{4}a_{2}&=&1.87-1.21\times
10^{-4}(\sqrt{s_{\rm NN}})^{4}\exp(-0.107\sqrt{s_{\rm NN}})\ ,\\\
b_{1}&=&-0.02(1-0.105\sqrt{s_{\rm NN}})/(1+0.0295\sqrt{s_{\rm NN}})\ ,\\\
10^{4}b_{2}&=&5.0-0.626(\sqrt{s_{\rm NN}})^{7.14}\exp(-0.979\sqrt{s_{\rm
NN}})\ .\\\ \end{array}$ (4)
The functional form for $a_{i}$ and $b_{i}$ ($i=1,2$) are chosen to satisfy
the demands from the physics considerations in the last paragraph. With the
above expressions for the energy dependence of the parameters, it is
straightforward to calculate the values of those parameter at the LHC energy
$\sqrt{s_{\rm NN}}$=2760 GeV. By assuming that our model can be applied to
that energy, one can calculate the net-proton distributions at that energy for
different colliding centralities. The obtained distributions are shown in Fig.
8. The values of $N_{\rm p}$ used in the calculation are from LHC .
Figure 6: Colliding energy dependence of parameters $a_{1}$ and $a_{2}$ for
the initially stopped proton number in the given central rapidity window.
Points marked by star are from our model fitting, and the points marked by
triangle are calculated from Eq. (4) at $\sqrt{s_{\rm NN}}$=2760 GeV. Figure
7: Colliding energy dependence of parameters $b_{1}$ and $b_{2}$ for the mean
number of produced baryon pairs in the given central rapidity window. Points
marked by star are from our model fitting, and the points marked by triangle
are calculated from Eq. (4) at $\sqrt{s_{\rm NN}}$=2760 GeV. Figure 8: Net-
proton distributions for Pb+Pb collisions at $\sqrt{s_{\rm NN}}=2760$ GeV at
different colliding centralities.
In all considerations up to now, we only discussed the net-proton
distributions in the central rapidity region $|y|<0.5$. Because of the
diffusion of baryon number and energy from central to non-central rapidity
region, the parameters $a_{2}$ and $b_{2}$ show complicated behaviors as
functions of colliding energy. If we consider the net-proton distributions in
the non-central region, $|y|>0.5$ for say, the same physics arguments apply
and one can expect that the distributions can also be described by Eq. (1)
with centrality dependence of parameters $B$ and $\mu$ being given by Eqs. (2)
and (3). One can also expect that the colliding energy dependence of $a_{1}$
and $b_{1}$ is similar to that for the case in central rapidity region. For
$a_{2}$ and $b_{2}$, the baryon number and energy diffusion from central to
non-central region will not compete to but cooperate with the multiple
scattering effect. Then $a_{2}$ and $b_{2}$ will be negative for all colliding
energies and all A+A collisions. This statement can be tested experimentally.
## V Conclusion
We studied the net-proton distributions for Au+Au collisions at four colliding
energies for $\sqrt{s_{\rm NN}}$ from 19.6 to 200 GeV at different
centralities. Based on some physical arguments, the parameters in our model
are parameterized as functions of centrality and energy. The higher order
moments for the distributions are in good agreement with the experimental
data. Prediction for the net-proton distributions at LHC energy $\sqrt{s_{\rm
NN}}$=2.76 TeV is presented for different centralities. The net-proton
distribution in a non-central rapidity region and its dependence on centrality
are discussed.
It should be mentioned that nothing else is assumed in this model except an
initial stopped net-proton and a finite probability for producing baryon pairs
from the produced matter. Therefore, our model has nothing to do with thermal
equilibrium and/or critical fluctuations. Because our model consideration is
based on normal physics effects, our results can be used as a baseline for
detecting novel physics in the processes.
###### Acknowledgements.
This work was supported in part by the National Natural Science Foundation of
China under Grant No. 11075061 and by the Programme of Introducing Talents of
Discipline to Universities under No. B08033. The authors thank Dr. X.F. Luo
for sending us the experimental data. We are grateful to N. Xu and X.F. Luo
for valuable discussions.
## References
* (1) Y. Aoki et al., Nature 443, 675 (2006); M. Cheng et al., Phys. Rev. D 74, 054507 (2006).
* (2) J. Berges, K. Rajagopal, Nucl. Phys. B 538, 215 (1999).
* (3) M. Stephanov, K. Rajagopal and E. Shuryak, Phys. Rev. D 60, 114028 (1999).
* (4) M.A. Stephanov,K. Rajagopal and E.V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998); S. Jeon, V. Koch, Phys. Rev. Lett. 85, 2076 (2000); M. Asakawa, U.W. Heinz and B. Müller, Nucl. Phys. A 698, 519 (2002); V. Koch, J. Phys. G 35, 104030 (2008).
* (5) S. Ejiri, F. Karsch and K. Redlich, Phys. Lett. B 633, 275 (2006).
* (6) M.M. Aggarwal et al., (STAR Collaboration), Phys. Rev. Lett. 105, 022302 (2010).
* (7) S. Gupta et al., Science 332, 1525 (2011); X.F. Luo, B. Mohanty, H.G. Ritter and N. Xu, arXiv:1105.5049.
* (8) F. Karsh and K. Redlich, Phys. Lett. B 695, 136 (2011); M. A. Stephanov, Phys. Rev. Lett. 107, 052301 (2011); B. Friman et al., Eur. Phys. J. C 71, 1694 (2011).
* (9) Y. Zhou et al., Phys. Rev. C 82, 014905 (2010); K. Xiao et al., Chin. Phys. C 35, 467 (2011).
* (10) Y. Hatta and M.A. Stephanov, Phys. Rev. Lett. 91, 102003 (2003).
* (11) M.A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009).
* (12) X.F. Luo (for the STAR Collaboration), J. Phys: Conf. Ser. 316, 012003 (2011).
* (13) X.F. Luo et al., J. Phys.G 37, 094061 (2010).
* (14) P. Braun-Munzinger, K. Redlich and J. Stachel, in Quark Gluon Plasma 3, edited by R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2004); A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167 (2006).
* (15) F. Karch and K. Redlich, Phys. Lett. B 695, 136 (2011).
* (16) L.Z. Chen et al., J. Phys. G 38, 115005 (2011).
* (17) V.V. Begun et al., Phys. Rev. C 70, 034901 (2004).
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* (19) K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 106, 032301 (2011).
|
arxiv-papers
| 2012-02-22T08:51:33 |
2024-09-04T02:49:27.684645
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "X. Wang and C. B. Yang",
"submitter": "Chunbin Yang",
"url": "https://arxiv.org/abs/1202.4857"
}
|
1202.4876
|
# Observability inequalities and measurable sets
J. Apraiz Universidad del País Vasco/Euskal Herriko Unibertsitatea
Departamento de Matemática Aplicada
Escuela Universitaria Politécnica de Donostia-San Sebastián
Plaza de Europa 1
20018 Donostia-San Sebastián, Spain. jone.apraiz@ehu.es , L. Escauriaza
Universidad del País Vasco/Euskal Herriko Unibertsitatea
Dpto. de Matemáticas
Apto. 644, 48080 Bilbao, Spain. luis.escauriaza@ehu.es , G. Wang Department
of Mathematics and Statistics, Wuhan University, Wuhan, China
wanggs62@yeah.net and C. Zhang Department of Mathematics and Statistics,
Wuhan University, Wuhan, China zhangcansx@163.com
###### Abstract.
This paper presents two observability inequalities for the heat equation over
$\Omega\times(0,T)$. In the first one, the observation is from a subset of
positive measure in $\Omega\times(0,T)$, while in the second, the observation
is from a subset of positive surface measure on $\partial\Omega\times(0,T)$.
It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a
bounded Lipschitz and locally star-shaped domain. Some applications for the
above-mentioned observability inequalities are provided.
###### Key words and phrases:
observability inequality, heat equation, measurable set, spectral inequality
###### 1991 Mathematics Subject Classification:
Primary: 35B37
The first two authors are supported by Ministerio de Ciencia e Innovación
grants, MTM2004-03029 and MTM2011-2405.
The last two authors are supported by the National Natural Science Foundation
of China under grants 11161130003 and 11171264 and partially by National Basis
Research Program of China (973 Program) under grant 2011CB808002.
## 1\. Introduction
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$ and $T$ be a
fixed positive time. Consider the heat equation:
(1.1) $\begin{cases}\partial_{t}u-\Delta u=0,\ &\text{in}\
\Omega\times(0,T),\\\ u=0,\ &\text{on}\ \partial\Omega\times(0,T),\\\
u(0)=u_{0},\ &\text{in}\ \Omega,\end{cases}$
with $u_{0}$ in $L^{2}(\Omega)$. The solution of (1.1) will be treated as
either a function from $[0,T]$ to $L^{2}(\Omega)$ or a function of two
variables $x$ and $t$. Two important apriori estimates for the above equation
are as follows:
(1.2) $\|u(T)\|_{L^{2}(\Omega)}\leq
N(\Omega,T,\mathcal{D})\int_{\mathcal{D}}|u(x,t)|\,dxdt,\;\;\mbox{for
all}\;\;u_{0}\in L^{2}(\Omega),$
where $\mathcal{D}$ is a subset of $\Omega\times(0,T)$, and
(1.3) $\|u(T)\|_{L^{2}(\Omega)}\leq
N(\Omega,T,\mathcal{J})\int_{\mathcal{J}}|\tfrac{\partial}{\partial\nu}u(x,t)|\,d\sigma
dt,\;\;\mbox{for all}\;\;u_{0}\in L^{2}(\Omega),$
where $\mathcal{J}$ is a subset of $\partial\Omega\times(0,T)$. Such apriori
estimates are called observability inequalities.
In the case that $\mathcal{D}=\omega\times(0,T)$ and
$\mathcal{J}=\Gamma\times(0,T)$ with $\omega$ and $\Gamma$ accordingly open
and nonempty subsets of $\Omega$ and $\partial\Omega$, both inequalities (1.2)
and (1.3) (where $\partial\Omega$ is smooth) were essentially first
established, via the Lebeau-Robbiano spectral inequalities in
[G.LebeauL.Robbiano] (See also [G.LebeauE.Zuazua, Miller2, Fernandez-
CaraZuazua1]). These two estimates were set up to the linear parabolic
equations (where $\partial\Omega$ is of class $C^{2}$), based on the Carleman
inequality provided in [FursikovOImanuvilov]. In the case when
$\mathcal{D}=\omega\times(0,T)$ and $\mathcal{J}=\Gamma\times(0,T)$ with
$\omega$ and $\Gamma$ accordingly subsets of positive measure and positive
surface measure in $\Omega$ and $\partial\Omega$, both inequalities (1.2) and
(1.3) were built up in [ApraizEscauriaza1] with the help of a propagation of
smallness estimate from measurable sets for real-analytic functions first
established in [Vessella] (See also Theorem 4). For $\mathcal{D}=\omega\times
E$, with $\omega$ and $E$ accordingly an open subset of $\Omega$ and a subset
of positive measure in $(0,T)$, the inequality (1.2) (with $\partial\Omega$ is
smooth) was proved in [gengshengwang1] with the aid of the Lebeau-Robbiano
spectral inequality, and it was then verified for heat equations (where
$\Omega$ is convex) with lower terms depending on the time variable, through a
frequency function method in [PhungWang1]. When $\mathcal{D}=\omega\times E$,
with $\omega$ and $E$ accordingly subsets of positive measure in $\Omega$ and
$(0,T)$, the estimate (1.2) (with $\partial\Omega$ is real-analytic) was
obtained in [canzhang].
The purpose of this study is to establish inequalities (1.2) and (1.3), when
$\mathcal{D}$ and $\mathcal{J}$ are arbitrary subsets of positive measure and
of positive surface measure in $\Omega\times(0,T)$ and
$\partial\Omega\times(0,T)$ respectively. Such inequalities not only are
mathematically interesting but also have important applications in the control
theory of the heat equation, such as the bang-bang control, the time optimal
control, the null controllability over a measurable set and so on (See Section
LABEL:S:3 for the applications).
The starting point we choose here to prove the above-mentioned two
inequalities is to assume that the Lebeau-Robbiano spectral inequality stands
on $\Omega$. To introduce it, we write
$0<\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{j}\leq\cdots$
for the eigenvalues of $-\Delta$ with the zero Dirichlet boundary condition
over $\partial\Omega$, and $\\{e_{j}:j\geq 1\\}$ for the set of
$L^{2}(\Omega)$-normalized eigenfunctions, i.e.,
$\begin{cases}\Delta e_{j}+\lambda_{j}e_{j}=0,\ &\text{in}\ \Omega,\\\
e_{j}=0,\ &\text{on}\ \partial\Omega.\end{cases}$
For $\lambda>0$ we define
$\mathcal{E}_{\lambda}f=\sum_{\lambda_{j}\leq\lambda}(f,e_{j})\,e_{j}\quad\text{and}\quad\mathcal{E}_{\lambda}^{\perp}f=\sum_{\lambda_{j}>\lambda}(f,e_{j})\,e_{j},$
where
$(f,e_{j})=\int_{\Omega}f\,e_{j}\,dx,\ \text{when}\ f\in L^{2}(\Omega),\ j\geq
1.$
Throughout this paper the following notations are effective:
$(f,g)=\int_{\Omega}fg\,dx\ \text{and}\
\|f\|_{L^{2}(\Omega)}=\left(f,f\right)^{\frac{1}{2}};$
$\nu$ is the unit exterior normal vector to $\partial\Omega$; $d\sigma$ is
surface measure on $\partial\Omega$; $B_{R}(x_{0})$ stands for the ball
centered at $x_{0}$ in $\mathbb{R}^{n}$ of radius $R$; $\triangle_{R}(x_{0})$
denotes $B_{R}(x_{0})\cap\partial\Omega$; $B_{R}=B_{R}(0)$,
$\triangle_{R}=\triangle_{R}(0)$; for measurable sets
$\omega\subset\mathbb{R}^{n}$ and
$\mathcal{D}\subset\mathbb{R}^{n}\times(0,T)$, $|\omega|$ and $|\mathcal{D}|$
stand for the Lebesgue measures of the sets; for each measurable set
$\mathcal{J}$ in $\partial\Omega\times(0,T)$, $|\mathcal{J}|$ denotes its
surface measure on the lateral boundary of $\Omega\times\mathbb{R}$;
$\\{e^{t\Delta}:t\geq 0\\}$ is the semigroup generated by $\Delta$ with zero
Dirichlet boundary condition over $\partial\Omega$. Consequently,
$e^{t\Delta}f$ is the solution of Equation (1.1) with the initial state $f$ in
$L^{2}(\Omega)$. The Lebeau-Robbiano spectral inequality is as follows:
_For each $0<R\leq 1$, there is $N=N(\Omega,R)$, such that the inequality_
(1.4) $\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}\leq
Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{2}(B_{R}(x_{0}))}$
_holds, when $B_{4R}(x_{0})\subset\Omega$, $f\in L^{2}(\Omega)$ and
$\lambda>0$._
To our best knowledge, the inequality (1.4) has been proved under condition
that $\partial\Omega$ is at least $C^{2}$ [G.LebeauL.Robbiano,
G.LebeauE.Zuazua, RousseauRobbiano2, luqi]. In the current work, we obtain
this inequality when $\Omega$ is a bounded Lipschitz and locally star-shaped
domain in $\mathbb{R}^{n}$ (See Definitions 1 and LABEL:D:_contractable in
Section 3). It can be observed from Section 3 that bounded $C^{1}$ domains,
polygons in the plane, Lipschitz polyhedrons in $\mathbb{R}^{n}$, with $n\geq
3$, and bounded convex domains in $\mathbb{R}^{n}$ are always bounded
Lipschitz and locally star-shaped (See Remarks
LABEL:R:_algobastanteimportantene and LABEL:R:_algoasombroso in Section 3).
Our main results related to the observability inequalities are stated as
follows:
###### Theorem 1.
Suppose that a bounded domain $\Omega$ verifies the condition (1.4) and $T>0$.
Let $x_{0}\in\Omega$ and $R\in(0,1]$ be such that
$B_{4R}(x_{0})\subset\Omega$. Then, for each measurable set
$\mathcal{D}\subset B_{R}(x_{0})\times(0,T)$ with $|\mathcal{D}|>0$, there is
a positive constant $B=B(\Omega,T,R,\mathcal{D})$, such that
(1.5) $\|e^{T\Delta}f\|_{L^{2}(\Omega)}\leq
e^{B}\int_{\mathcal{D}}|e^{t\Delta}f(x)|\,dxdt,\;\;\mbox{when}\;\;f\in
L^{2}(\Omega).$
###### Theorem 2.
Suppose that a bounded Lipschitz domain $\Omega$ verifies the condition (1.4)
and $T>0$. Let $q_{0}\in\partial\Omega$ and $R\in(0,1]$ be such that
$\triangle_{4R}(q_{0})$ is real-analytic. Then, for each measurable set
$\mathcal{J}\subset\triangle_{R}(q_{0})\times(0,T)$ with $|\mathcal{J}|>0$,
there is a positive constant $B=B(\Omega,T,R,\mathcal{J})$, such that
(1.6) $\|e^{T\Delta}f\|_{L^{2}(\Omega)}\leq
e^{B}\int_{\mathcal{J}}|\tfrac{\partial}{\partial\nu}\,e^{t\Delta}f(x)|\,d\sigma
dt,\;\;\mbox{when}\;\;f\in L^{2}(\Omega).$
The definition of the real analyticity for $\triangle_{4R}(q_{0})$ is given in
Section 4 (See Definition LABEL:D:_fromteralocalrealanalitica).
###### Theorem 3.
Let $\Omega$ be a bounded Lispchitz and locally star-shaped domain in
$\mathbb{R}^{n}$. Then, $\Omega$ verifies the condition (1.4).
It deserves mentioning that Theorem 2 also holds when $\Omega$ is a Lipschitz
polyhedron in $\mathbb{R}^{n}$ and $\mathcal{J}$ is a measurable subset with
positive surface measure of $\partial\Omega\times(0,T)$ (See the part $(ii)$
in Remark LABEL:section4remark11).
In this work we use the new strategy developed in [PhungWang1] to prove
parabolic observability inequalities: a mixing of ideas from [Miller2], the
global interpolation inequalitiy in Theorems 6 and LABEL:interpolation and the
telescoping series method. This new strategy can also be extended to more
general parabolic evolutions with variable time-dependent second order
coefficients and with unbounded lower order time-dependent coefficients. To do
it one must prove the global interpolation inequalities in Theorems 6 and
LABEL:interpolation for the corresponding parabolic evolutions. These can be
derived in the more general setting from the Carleman inequalities in
[Escauriaza1, EscauriazaFernandez1, EscauriazaVega, Fernandez1, KochTataru] or
from local versions of frequency function arguments
[EscauriazaFernandezVessella, PhungWang1]. Here we choose to derive the
interpolation inequalities only for the heat equation and from the condition
(1.4) because it is technically less involved and helps to make the
presentation of the basic ideas more clear.
The rest of the paper is organized as follows: Section 2 proves Theorem 1;
Section 3 shows Theorem 3; Section LABEL:S:5 verifies Theorem 2; Section
LABEL:S:3 presents some applications of Theorem 1 and Theorem 2 in the control
theory of the heat equation and Section LABEL:S:8 is an Appendix completeting
some of the technical details in the work.
## 2\. Interior observability
Throughout this section $\Omega$ denotes a bounded domain and $T$ is a
positive time. First of all, we recall the following observability estimate or
propagation of smallness inequality from measurable sets:
###### Theorem 4.
Assume that $f:B_{2R}\subset\mathbb{R}^{n}\longrightarrow\mathbb{R}$ is real-
analytic in $B_{2R}$ verifying
$|\partial^{\alpha}f(x)|\leq\frac{M|\alpha|!}{(\rho R)^{|\alpha|}}\ ,\
\text{when}\ \ x\in B_{2R},\ \alpha\in\mathbb{N}^{n},$
for some $M>0$ and $0<\rho\leq 1$. Let $E\subset B_{R}$ be a measurable set
with positive measure. Then, there are positive constants
$N=N(\rho,|E|/|B_{R}|)$ and $\theta=\theta(\rho,|E|/|B_{R}|)$ such that
(2.1) $\|f\|_{L^{\infty}(B_{R})}\leq N\left(\text{\hbox
to0.0pt{|\hss}{$\int_{E}$}}\,|f|\,dx\right)^{\theta}M^{1-\theta}.$
The estimate (2.1) is first established in [Vessella] (See also
[Nadirashvili2] and [Nadirashvili] for other close results). The reader may
find a simpler proof of Theorem 4 in [ApraizEscauriaza1, §3], the proof there
was built with ideas taken from [Malinnikova], [Nadirashvili2] and [Vessella].
Theorem 4 and the condition (1.4) imply the following:
###### Theorem 5.
Assume that $\Omega$ verifies (1.4), $\omega$ is a subset of positive measure
such that $\omega\subset B_{R}(x_{0})$, with $B_{4R}(x_{0})\subset\Omega$, for
some $R\in(0,1]$. Then, there is a positive constant
$N=N(\Omega,R,|\omega|/|B_{R}|)$ such that
(2.2) $\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}\leq
Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{1}(\omega)},\;\;\mbox{when}\;\;f\in
L^{2}(\Omega)\;\;\mbox{and}\;\;\lambda>0.$
###### Proof.
Without loss of generality we may assume $x_{0}=0$. Because
$B_{4R}\subset\Omega$ and (1.4) stands, there is $N=N(\Omega,R)$ such that
(2.3) $\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}\leq
Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{2}(B_{R})},\;\;\mbox{when}\;\;f\in
L^{2}(\Omega)\;\;\mbox{and}\;\;\lambda>0.$
For $f\in L^{2}(\Omega)$ arbitrarily given, define
$u(x,y)=\sum_{\lambda_{j}\leq\lambda}(f,e_{j})e^{\sqrt{\lambda_{j}}y}e_{j}.$
One can verify that $\Delta u+\partial^{2}_{y}u=0$ in
$B_{4R}(0,0)\subset\Omega\times\mathbb{R}$. Hence, there are $N=N(n)$ and
$\rho=\rho(n)$ such that
$\|\partial^{\alpha}_{x}\partial_{y}^{\beta}u\|_{L^{\infty}(B_{2R}(0,0))}\leq\frac{N(|\alpha|+\beta)!}{(R\rho)^{|\alpha|+\beta}}\left(\text{\hbox
to0.0pt{|\hss}{$\int_{B_{4R}(0,0)}$}}|u|^{2}\,dxdy\right)^{\frac{1}{2}},\
\text{when}\ \alpha\in\mathbb{N}^{n},\beta\geq 1.$
For the later see [Morrey, Chapter 5], [FJohn2, Chapter 3]. Thus,
$\mathcal{E}_{\lambda}f$ is a real-analytic function in $B_{2R}$, with the
estimates:
$\|\partial^{\alpha}_{x}\mathcal{E}_{\lambda}f\|_{L^{\infty}(B_{2R})}\leq
N|\alpha|!(R\rho)^{-|\alpha|}\|u\|_{L^{\infty}(\Omega\times(-4,4))},\
\text{for}\ \alpha\in\mathbb{N}^{n}.$
By either extending $|u|$ as zero outside of $\Omega\times\mathbb{R}$, which
turns $|u|$ into a subharmonic function in $\mathbb{R}^{n+1}$ or the local
properties of solutions to elliptic equations [GilbargTrudinger, Theorems
8.17, 8.25] and the orthonormality of $\\{e_{j}:j\geq 1\\}$ in $\Omega$, there
is $N=N(\Omega)$ such that
$\|u\|_{L^{\infty}(\Omega\times(-4,4))}\leq
N\|u\|_{L^{2}(\Omega\times(-5,5))}\leq
Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}.$
The last two inequalities show that
$\|\partial^{\alpha}_{x}\mathcal{E}_{\lambda}f\|_{L^{\infty}(B_{2R})}\leq
Ne^{N\sqrt{\lambda}}|\alpha|!\left(R\rho\right)^{-|\alpha|}\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)},\
\text{for}\ \alpha\in\mathbb{N}^{n},$
with $N$ and $\rho$ as above. In particular, $\mathcal{E}_{\lambda}f$ verifies
the hypothesis in Theorem 4 with
$M=Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)},$
and there are $N=N(\Omega,R,|\omega|/|B_{R}|)$ and
$\theta=\theta(\Omega,R,|\omega|/|B_{R}|)$ with
(2.4) $\|\mathcal{E}_{\lambda}f\|_{L^{\infty}(B_{R})}\leq
Ne^{N\sqrt{\lambda}}\|\mathcal{E}_{\lambda}f\|_{L^{1}(\omega)}^{\theta}\|\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}^{1-\theta}.$
Now, the estimate (2.2) follows from (2.3) and (2.4). ∎
###### Theorem 6.
Let $\Omega$, $x_{0}$, $R$ and $\omega$ be as in Theorem 5. Then, there are
$N=N(\Omega,R,|\omega|/|B_{R}|)$ and
$\theta=\theta(\Omega,R,|\omega|/|B_{R}|)\in(0,1)$, such that
(2.5)
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq\left(Ne^{\frac{N}{t-s}}\|e^{t\Delta}f\|_{L^{1}(\omega)}\right)^{\theta}\|e^{s\Delta}f\|_{L^{2}(\Omega)}^{1-\theta},$
when $0\leq s<t$ and $f\in L^{2}(\Omega)$.
###### Proof.
Let $0\leq s<t$ and $f\in L^{2}(\Omega)$. Since
$\|e^{t\Delta}\mathcal{E}_{\lambda}^{\perp}f\|_{L^{2}(\Omega)}\leq
e^{-\lambda(t-s)}\|e^{s\Delta}f\|_{L^{2}(\Omega)},\ \text{when}\ f\in
L^{2}(\Omega),$
it follows from Theorem 5 that
$\begin{split}\|e^{t\Delta}f\|_{L^{2}(\Omega)}&\leq\|e^{t\Delta}\mathcal{E}_{\lambda}f\|_{L^{2}(\Omega)}+\|e^{t\Delta}{\mathcal{E}}_{\lambda}^{\perp}f\|_{L^{2}(\Omega)}\\\
&\leq
Ne^{N\sqrt{\lambda}}\|e^{t\Delta}\mathcal{E}_{\lambda}f\|_{L^{1}(\omega)}+e^{-\lambda(t-s)}\|e^{s\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
Ne^{N\sqrt{\lambda}}\left[\|e^{t\Delta}f\|_{L^{1}(\omega)}+\|e^{t\Delta}\mathcal{E}_{\lambda}^{\perp}f\|_{L^{2}(\omega)}\right]+e^{-\lambda(t-s)}\|e^{s\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
Ne^{N\sqrt{\lambda}}\left[\|e^{t\Delta}f\|_{L^{1}(\omega)}+e^{-\lambda(t-s)}\|e^{s\Delta}f\|_{L^{2}(\Omega)}\right].\end{split}$
Consequently, it holds that
(2.6) $\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq
Ne^{N\sqrt{\lambda}}\left[\|e^{t\Delta}f\|_{L^{1}(\omega)}+e^{-\lambda(t-s)}\|e^{s\Delta}f\|_{L^{2}(\Omega)}\right].$
Because
$\max_{\lambda\geq 0}e^{A\sqrt{\lambda}-\lambda(t-s)}\leq
e^{\frac{N(A)}{t-s}},\ \text{for all}\ A>0,$
it follows from (2.6) that for each $\lambda>0$,
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq\\\
Ne^{\frac{N}{t-s}}\left[e^{N\lambda(t-s)}\|e^{t\Delta}f\|_{L^{1}(\omega)}+e^{-\lambda(t-s)/N}\|e^{s\Delta}f\|_{L^{2}(\Omega)}\right].$
Setting $\epsilon=e^{-\lambda(t-s)}$ in the above estimate shows that the
inequality
(2.7) $\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq
Ne^{\frac{N}{t-s}}\left[\epsilon^{-N}\|e^{t\Delta}f\|_{L^{1}(\omega)}\,+\epsilon\|e^{s\Delta}f\|_{L^{2}(\Omega)}\right],$
holds, for all $0<\epsilon\leq 1$. The minimization of the right hand in (2.7)
for $\epsilon$ in $(0,1)$, as well as the fact that
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq\|e^{s\Delta}f\|_{L^{2}(\Omega)},\
\text{when}\ t>s,$
implies Theorem 6. ∎
###### Remark 1.
Theorem 6 shows that the observability or spectral elliptic inequality (2.2)
implies the inequality (2.5). In particular, the elliptic spectral inequality
(1.4) implies the inequality:
(2.8)
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq\left(Ne^{\frac{N}{t-s}}\|e^{t\Delta}f\|_{L^{2}(B_{R}(x_{0}))}\right)^{\theta}\|e^{s\Delta}f\|_{L^{2}(\Omega)}^{1-\theta},$
when $0\leq s<t$, $B_{4R}(x_{0})\subset\Omega$ and $f\in L^{2}(\Omega)$. In
fact, both (2.2) and (2.5) or (1.4) and (2.8) are equivalent, for if (2.5)
holds, take
$f=\sum_{\lambda_{j}\leq\lambda}e^{\lambda_{j}/\sqrt{\lambda}}a_{j}e_{j}$,
$s=0$ and $t=1/\sqrt{\lambda}$ in (2.5) to derive that
$\Big{(}\sum_{\lambda_{j}\leq\lambda}a_{j}^{2}\Big{)}^{\frac{1}{2}}\leq
Ne^{N\sqrt{\lambda}}\Big{\|}\sum_{\lambda_{j}\leq\lambda}a_{j}e_{j}\Big{\|}_{L^{1}(\omega)},\
\text{when}\ a_{j}\in\mathbb{R},\ j\geq 1,\lambda>0.$
The interested reader may want here to compare the previous claims, Theorem 3
and [PhungWang1, Proposition 2.2].
###### Lemma 1.
Let $B_{R}(x_{0})\subset\Omega$ and $\mathcal{D}\subset
B_{R}(x_{0})\times(0,T)$ be a subset of positive measure. Set
$\mathcal{D}_{t}=\\{x\in\Omega:(x,t)\in\mathcal{D}\\},\
E=\\{t\in(0,T):|\mathcal{D}_{t}|\geq|\mathcal{D}|/(2T)\\},\ t\in(0,T).$
Then, $\mathcal{D}_{t}\subset\Omega$ is measurable for a.e. $t\in(0,T)$, $E$
is measurable in $(0,T)$, $|E|\geq|\mathcal{D}|/2|B_{R}|$ and
(2.9) $\chi_{E}(t)\chi_{\mathcal{D}_{t}}(x)\leq\chi_{\mathcal{D}}(x,t),\
\text{in}\ \Omega\times(0,T).$
###### Proof.
From Fubini’s theorem,
$|\mathcal{D}|=\int_{0}^{T}|\mathcal{D}_{t}|\,dt=\int_{E}|\mathcal{D}_{t}|\,dt+\int_{[0,T]\setminus
E}|\mathcal{D}_{t}|\,dt\leq|B_{R}||E|+|\mathcal{D}|/2.$
∎
###### Theorem 7.
Let $x_{0}\in\Omega$ and $R\in(0,1]$ be such that
$B_{4R}(x_{0})\subset\Omega$. Let $\mathcal{D}\subset B_{R}(x_{0})\times(0,T)$
be a measurable set with $|\mathcal{D}|>0$. Write $E$ and $\mathcal{D}_{t}$
for the sets associated to $\mathcal{D}$ in Lemma 1. Then, for each
$\eta\in(0,1)$, there are
$N=N(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right),\eta)$ and
$\theta=\theta(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right),\eta)$ with
$\theta\in(0,1)$, such that
(2.10)
$\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}\leq\left(Ne^{N/(t_{2}-t_{1})}\int_{t_{1}}^{t_{2}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds\right)^{\theta}\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}^{1-\theta},$
when $0\leq t_{1}<t_{2}\leq T$, $|E\cap(t_{1},t_{2})|\geq\eta(t_{2}-t_{1})$
and $f\in L^{2}(\Omega)$. Moreover,
(2.11)
$\begin{split}&e^{-\frac{N+1-\theta}{t_{2}-t_{1}}}\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}-e^{-\frac{N+1-\theta}{q\left(t_{2}-t_{1}\right)}}\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
N\int_{t_{1}}^{t_{2}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;q\geq(N+1-\theta)/(N+1).\end{split}$
###### Proof.
After removing from $E$ a set with zero Lebesgue measure, we may assume that
$\mathcal{D}_{t}$ is measurable for all $t$ in $E$. From Lemma 1,
$\mathcal{D}_{t}\subset B_{R}(x_{0})$, $B_{4R}(x_{0})\subset\Omega$ and
$|\mathcal{D}_{t}|/|B_{R}|\geq|\mathcal{D}|/(2T|B_{R}|)$, when $t$ is in $E$.
From Theorem 6, there are $N=N(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right))$
and $\theta=\theta(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right))$ such that
(2.12)
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq\left(Ne^{\frac{N}{t-s}}\|e^{t\Delta}f\|_{L^{1}(\mathcal{D}_{t})}\right)^{\theta}\|e^{s\Delta}f\|_{L^{2}(\Omega)}^{1-\theta},$
when $0\leq s<t$, $t\in E$ and $f\in L^{2}(\Omega)$. Let $\eta\in(0,1)$ and
$0\leq t_{1}<t_{2}\leq T$ satisfy $|E\cap(t_{1},t_{2})|\geq\eta(t_{2}-t_{1})$.
Set $\tau=t_{1}+\frac{\eta}{2}\,\left(t_{2}-t_{1}\right)$. Then
(2.13)
$|E\cap(\tau,t_{2})|=|E\cap(t_{1},t_{2})|-|E\cap(t_{1},\tau)|\geq\frac{\eta}{2}(t_{2}-t_{1}).$
From (2.12) with $s=t_{1}$ and the decay property of
$\|e^{t\Delta}f\|_{L^{2}(\Omega)}$, we get
(2.14)
$\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}\leq\left(Ne^{\frac{N}{t_{2}-t_{1}}}\|e^{t\Delta}f\|_{L^{1}(\mathcal{D}_{t})}\right)^{\theta}\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}^{1-\theta},\
t\in E\cap(\tau,t_{2}).$
The inequality (2.10) follows from the integration with respect to $t$ of
(2.14) over $E\cap(\tau,t_{2})$, Hölder’s inequality with $p=1/\theta$ and
(2.13).
The inequality (2.10) and Young’s inequality imply that
(2.15) $\begin{split}&\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}\leq\\\
&\epsilon\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}+\epsilon^{-\frac{1-\theta}{\theta}}Ne^{\frac{N}{t_{2}-t_{1}}}\int_{t_{1}}^{t_{2}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;\epsilon>0.\end{split}$
Multiplying first (2.15) by
$\epsilon^{\frac{1-\theta}{\theta}}e^{-\frac{N}{t_{2}-t_{1}}}$ and then
replacing $\epsilon$ by $\epsilon^{\theta}$, we get that
$\begin{split}&\epsilon^{1-\theta}e^{-\frac{N}{\left(t_{2}-t_{1}\right)}}\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}-\epsilon\,e^{-\frac{N}{\left(t_{2}-t_{1}\right)}}\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
N\int_{t_{1}}^{t_{2}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;\epsilon>0.\end{split}$
Choosing $\epsilon=e^{-\frac{1}{t_{2}-t_{1}}}$ in the above inequality leads
to
$\begin{split}&e^{-\frac{N+1-\theta}{\left(t_{2}-t_{1}\right)}}\|e^{t_{2}\Delta}f\|_{L^{2}(\Omega)}-e^{-\frac{N+1}{\left(t_{2}-t_{1}\right)}}\|e^{t_{1}\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
N\int_{t_{1}}^{t_{2}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds.\end{split}$
This implies (2.11), for $q\geq\frac{N+1-\theta}{N+1}$. ∎
The reader can find the proof of the following Lemma 2 in either [JLLions, pp.
256-257] or [PhungWang1, Proposition 2.1].
###### Lemma 2.
Let $E$ be a subset of positive measure in $(0,T)$. Let $l$ be a density point
of E. Then, for each $z>1$, there is $l_{1}=l_{1}(z,E)$ in $(l,T)$ such that,
the sequence $\\{l_{m}\\}$ defined as
$l_{m+1}=l+z^{-m}\left(l_{1}-l\right),\ m=1,2,\cdots,$
verifies
(2.16) $|E\cap(l_{m+1},l_{m})|\geq\frac{1}{3}\left(l_{m}-l_{m+1}\right),\
\text{when}\ m\geq 1.$
###### Proof of Theorem 1.
Let $E$ and $\mathcal{D}_{t}$ be the sets associated to $\mathcal{D}$ in Lemma
1 and $l$ be a density point in $E$. For $z>1$ to be fixed later,
$\\{l_{m}\\}$ denotes the sequence associated to $l$ and $z$ in Lemma 2.
Because (2.16) holds, we may apply Theorem 7, with $\eta=1/3$, $t_{1}=l_{m+1}$
and $t_{2}=l_{m}$, for each $m\geq 1$, to get that there are
$N=N(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right))>0$ and
$\theta=\theta(\Omega,R,|\mathcal{D}|/\left(T|B_{R}|\right))$, with
$\theta\in(0,1)$, such that
(2.17)
$\begin{split}&e^{-\frac{N+1-\theta}{l_{m}-l_{m+1}}}\|e^{l_{m}\Delta}f\|_{L^{2}(\Omega)}-e^{-\frac{N+1-\theta}{q\left(l_{m}-l_{m+1}\right)}}\|e^{l_{m+1}\Delta}f\|_{L^{2}(\Omega)}\\\
&\leq
N\int_{l_{m+1}}^{l_{m}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;q\geq\frac{N+1-\theta}{N+1}\;\;\mbox{and}\;\;m\geq
1.\end{split}$
Setting $z=1/q$ in (2.17) (which leads to $1<z\leq\frac{N+1}{N+1-\theta}$) and
$\gamma_{z}(t)=e^{-\frac{N+1-\theta}{\left(z-1\right)\left(l_{1}-l\right)t}},\
t>0,$
recalling that
$l_{m}-l_{m+1}=z^{-m}\left(z-1\right)\left(l_{1}-l\right),\ \text{for}\ m\geq
1,$
we have
(2.18)
$\begin{split}\gamma_{z}(z^{-m})\|e^{l_{m}\Delta}f\|_{L^{2}(\Omega)}-\gamma_{z}(z^{-m-1})\|e^{l_{m+1}\Delta}f\|_{L^{2}(\Omega)}\\\
\leq
N\int_{l_{m+1}}^{l_{m}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;m\geq
1.\end{split}$
Choose now
$z=\frac{1}{2}\left(1+\frac{N+1}{N+1-\theta}\right).$
The choice of $z$ and Lemma 2 determines $l_{1}$ in $(l,T)$ and from (2.18),
(2.19)
$\begin{split}\gamma(z^{-m})\|e^{l_{m}\Delta}f\|_{L^{2}(\Omega)}-\gamma(z^{-m-1})\|e^{l_{m+1}\Delta}f\|_{L^{2}(\Omega)}\\\
\leq
N\int_{l_{m+1}}^{l_{m}}\chi_{E}(s)\|e^{s\Delta}f\|_{L^{1}(\mathcal{D}_{s})}\,ds,\;\;\mbox{when}\;\;m\geq
1.\end{split}$
with
$\gamma(t)=e^{-A/t}\ \text{and}\
A=A(\Omega,R,E,|\mathcal{D}|/\left(T|B_{R}|\right))=\frac{2\left(N+1-\theta\right)^{2}}{\theta\left(l_{1}-l\right)}\,.$
Finally, because of
$\|e^{T\Delta}f\|_{L^{2}(\Omega)}\leq\|e^{l_{1}\Delta}f\|_{L^{2}(\Omega)},\
\sup_{t\geq 0}\|e^{t\Delta}f\|_{L^{2}(\Omega)}<+\infty,\ \lim_{t\to
0+}\gamma(t)=0,$
and (2.9), the addition of the telescoping series in (2.19) gives
$\|e^{T\Delta}f\|_{L^{2}(\Omega)}\leq
Ne^{zA}\int_{\mathcal{D}\cap(\Omega\times[l,l_{1}])}|e^{t\Delta}f(x)|\,dxdt,\;\;\mbox{for}\;\;f\in
L^{2}(\Omega),$
which proves (1.5) with $B=zA+\log N$.
∎
###### Remark 2.
The constant $B$ in Theorem 1 depends on $E$ because the choice of
$l_{1}=l_{1}(z,E)$ in Lemma 2 depends on the possible complex structure of the
measurable set $E$ (See the proof of Lemma 2 in [PhungWang1, Proposition
2.1]). When $\mathcal{D}=\omega\times(0,T)$, one may take $l=T/2$, $l_{1}=T$,
$z=2$ and then,
$B=A(\Omega,R,|\omega|/|B_{R}|)/T.$
###### Remark 3.
The proof of Theorem 1 also implies the following observability estimate:
$\sup_{m\geq 0}\sup_{l_{m+1}\leq t\leq
l_{m}}e^{-z^{m+1}A}\|e^{t\Delta}f\|_{L^{2}(\Omega)}\leq
N\int_{\mathcal{D}\cap(\Omega\times[l,l_{1}])}|e^{t\Delta}f(x)|\,dxdt,$
for $f$ in $L^{2}(\Omega)$, and with $z$, $N$ and $A$ as defined along the
proof of Theorem 1. Here, $l_{0}=T$.
## 3\. Spectral inequalities
Throughout this section, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$.
###### Definition 1.
$\Omega$ is a Lipschitz domain (sometimes called strongly Lipschitz or
Lipschitz graph domains) with constants $m$ and $\varrho$ when for each point
$p$ on the boundary of $\Omega$ there is a rectangular coordinate system
$x=(x^{\prime},x_{n})$ and a Lipschitz function
$\phi:\mathbb{R}^{n-1}\longrightarrow\mathbb{R}$ verifying
(3.1) $\phi(0^{\prime})=0,\quad|\phi(x_{1}^{\prime})-\phi(x_{2}^{\prime})|\leq
m|x_{1}^{\prime}-x_{2}^{\prime}|,\ \text{for all}\
x_{1}^{\prime},x_{2}^{\prime}\in\mathbb{R}^{n-1},$
$p=(0^{\prime},0)$ on this coordinate system and
(3.2)
$\begin{split}&Z_{m,\varrho}\cap\Omega=\\{(x^{\prime},x_{n}):|x^{\prime}|<\varrho,\
\phi(x^{\prime})<x_{n}<2m\varrho\\},\\\
&Z_{m,\varrho}\cap\partial\Omega=\\{(x^{\prime},\phi(x^{\prime})):|x^{\prime}|<\varrho\\},\end{split}$
where $Z_{m,\varrho}=B^{\prime}_{\varrho}\times(-2m\varrho,2m\varrho)$.
|
arxiv-papers
| 2012-02-22T10:49:34 |
2024-09-04T02:49:27.692366
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. Apraiz, L. Escauriaza, G. Wang, and C. Zhang",
"submitter": "Can Zhang",
"url": "https://arxiv.org/abs/1202.4876"
}
|
1202.4961
|
# Strongly universal string hashing is fast
Owen Kaser o.kaser@computer.org Daniel Lemire lemire@gmail.com Dept. of
CSAS, University of New Brunswick, 100 Tucker Park Road, Saint John, NB,
Canada LICEF, Université du Québec à Montréal (UQAM), 100 Sherbrooke West,
Montreal, QC, H2X 3P2 Canada
###### Abstract
We present fast strongly universal string hashing families: they can process
data at a rate of 0.2 CPU cycle per byte. Maybe surprisingly, we find that
these families—though they requires a large buffer of random numbers—are often
faster than popular hash functions with weaker theoretical guarantees.
Moreover, conventional wisdom is that hash functions with fewer
multiplications are faster. Yet we find that they may fail to be faster due to
operation pipelining. We present experimental results on several processors
including low-powered processors. Our tests include hash functions designed
for processors with the Carry-Less Multiplication (CLMUL) instruction set. We
also prove, using accessible proofs, the strong universality of our families.
###### keywords:
String Hashing , Superscalar Computing , Barrett Reduction , Carry-less
multiplications , Binary Finite Fields
## 1 Introduction
For 32-bit numbers, random hashing with good theoretical guarantees can be
just as fast as popular alternatives [1]. In turn, these guarantees ensure the
reliability of various algorithms and data structures: frequent-item mining
[2], count estimation [3, 4], and hash tables [5, 6]. We want to show that we
can also get good theoretical guarantees over larger objects (such as strings)
without sacrificing speed. For example, we consider variable-length strings
made of 32-bit characters: all data structures can be represented as such
strings, up to some padding.
We restrict our attention to hash functions mapping strings to $L$-bit
integers, that is, integer in $[0,2^{L})$ for some positive integer $L$. In
random hashing, we select a hash function at random from a family [7, 8]. The
hash function can be chosen whenever the software is initialized. While random
hashing is not yet commonplace, it can have significant security benefits [9]
in a hash table: without randomness, an attacker can more easily exploit the
fact that adding $n$ keys hashing to the same value typically takes quadratic
time ($\Theta(n^{2})$). For this reason, random hashing was adopted in the
Ruby language as of version 1.9 [10] and in the Perl language as of version
5.8.1.
A family of hash functions is $k$-wise independent if the hash values of any
$k$ distinct elements are independent. For example, a family is pairwise
independent—or strongly universal—if given any two distinct elements $s$ and
$s^{\prime}$, their hash values $h(s)$ and $h(s^{\prime})$ are independent:
$\displaystyle P(h(s)=y|h(s^{\prime})=y^{\prime})=P(h(s)=y)$
for any two hash values $y,y^{\prime}$. When a hashing family is not strongly
universal, it can still be universal if the probability of a collision is no
larger than if it were strongly universal: $P(h(s)=h(s^{\prime}))\leq 1/2^{L}$
when $2^{L}$ is the number of hash values. If the collision probability is
merely bounded by some $\epsilon$ larger than $1/2^{L}$ but smaller than $1$
($P(h(s)=h(s^{\prime}))\leq\epsilon<1$), we have an almost universal family.
However, strong universality might be more desirable than universality or
almost universality:
* 1.
We say that a family is uniform if all hash values are equiprobable
($P(h(s)=y)=1/2^{L}$ for all $y$ and $s$): strongly universal families are
uniform, but universal or almost universal families may fail to be uniform. To
see that universality fails to imply uniformity, consider the family made of
the two functions over 1-bit integers (0,1): the identity and a function
mapping all values to zero. The probability of a collision between two
distinct values is exactly $1/2$ which ensures universality even though we do
not have uniformity since $P(h(0)=0)=1$.
* 2.
Moreover, if we have strong universality over $L$ bits, then we also have it
over any subset of bits. The corresponding result may fail for universal and
almost universal families: we might have universality over $L$ bits, but fail
to have almost universality over some subset of bits. Consider the non-uniform
but universal family $\\{h(x)=x\\}$ over $L$-bit integers: if we keep only the
least significant $L^{\prime}$ bits ($0<L^{\prime}<L$), universality is lost
since $h(0)\bmod{2^{L^{\prime}}}=h(2^{L^{\prime}})\bmod{2^{L^{\prime}}}$.
There is no need to use slow operations such as modulo operations, divisions
or operations in finite fields to have strong universality. In fact, for short
strings having few distinct characters, Zobrist hashing requires nothing more
than table look-ups and bitwise exclusive-or operations, and it is more than
strongly universal (3-wise independent) [11, 12]. Unfortunately, it becomes
prohibitive for long strings as it requires the storage of $nc$ random numbers
where $n$ is the maximal length of a string and $c$ is the number of distinct
characters.
A more practical approach to strong universality is Multilinear hashing (§ 2).
Unfortunately, it normally requires that the computations be executed in a
finite field. Some processors have instructions for finite fields (§ 4) or
they can be emulated with a software library (§ 5.3). However, if we are
willing to double the number of random bits, we can implement it using regular
integer arithmetic. Indeed, using an idea from Dietzfelbinger [13], we
implement it using only one multiplication and one addition per character (§
3). We further attempt to speed it up by reducing the number of
multiplications by half. We believe that these families are the fastest
strongly universal hashing families on current computers. We evaluate these
hash families experimentally (§ 5):
* 1.
Using fewer multiplications has often improved performance, especially on low-
power processors [14]. Yet trading away the number of multiplications fails to
improve (and may even degrade) performance on several processors according to
our experiments—which include low-power processors. However, reducing the
number of multiplications is beneficial on other processors (e.g., AMD), even
server-class processors.
* 2.
We also find that strongly universal hashing may be computationally
inexpensive compared to common hashing functions, as long as we ignore the
overhead of generating long strings of random numbers. In effect—if memory is
abundant compared to the length of the strings—the strongly universal
Multilinear family is faster than many of the commonly used alternatives.
* 3.
We consider hash functions designed for hardware supported carry-less
multiplications (§ 4). This support should drastically improve the speed of
some operations over binary finite fields ($GF(2^{L})$). Unfortunately, we
find that the carry-less hash functions fail to be competitive (§ 5.4).
## 2 The Multilinear family
The Multilinear hash family is one of the simplest strongly universal family
[7]. It takes the form of a scalar product between random values (sometimes
called keys) and string components, where operations are over a finite field:
$\displaystyle h(s)=m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}.$
The hash function $h$ is defined by the randomly generated values
$m_{1},m_{2},\dots$ It is strongly universal over fixed-length strings. We can
also apply it to variable-length strings as long as we forbid strings ending
with zero. To ensure that strings never end with zero, we can append a
character value of one to all variable-length strings.
An apparent limitation of this approach is that strings cannot exceed the
number of random values. In effect, to hash 32-bit strings of length $n$, we
need to generate and store $32(n+1)$ random bits using a finite field of
cardinality $2^{32}$. However, Stinson [15] showed that strong universality
requires at least $1+a(b-1)$ hash functions where $a$ is the number of strings
and $b$ is the number of hash values. Thus, if we have 32-bit strings mapped
to 32-bit hash values, we need at least $\approx 2^{32(n+1)}$ hash functions:
Multilinear is almost optimal.
Hence, the requirement to store many random numbers cannot be waived without
sacrificing strong universality. Note that Stinson’s bound is not affected by
manipulations such as treating a length $n$ string of $W$-bit words as a
length $n/2$ string of $2W$-bit words.
If multiplications are expensive and we have long strings, we can attempt to
improve speed by reducing the number of multiplications by half [16, 17]:
$\displaystyle h(s)=m_{1}+\sum_{i=1}^{n/2}(m_{2i}+s_{2i-1})(m_{2i+1}+s_{2i}).$
(1)
While this new form assumes that the number of characters in the string is
even, we can simply pad the odd-length strings with an extra character with
value zero. With variable-length strings, the padding to even length must
follow the addition of a character value of one.
Could we reduce the number of multiplications further? Not in general: the
computation of a scalar product between two vectors of length $n$ requires at
least $\lceil n/2\rceil$ multiplications [18, Corollary 4]. However, we could
try to avoid generic multiplications altogether and replace them by squares
[14]:
$\displaystyle h(s)=m_{1}+\sum_{i=1}^{n}(m_{i+1}+s_{i})^{2}.$
Indeed, squares can be sometimes be computed faster. Unfortunately, this
approach fails in binary finite fields ($GF(2^{L})$) because
$\displaystyle(m_{i+1}+s_{i})^{2}$ $\displaystyle=$ $\displaystyle
m_{i+1}^{2}+m_{i+1}s_{i}+m_{i+1}s_{i}+s_{i}^{2}$ $\displaystyle=$
$\displaystyle m_{i+1}^{2}+s_{i}^{2}$
since every element is its own additive inverse. Thus, we get
$\displaystyle h(s)=m_{1}+\sum_{i=1}^{n}m_{i+1}^{2}+\sum_{i=1}^{n}s_{i}^{2}$
which is a poor hash function (e.g., $h(\texttt{ab})=h(\texttt{ba})$).
There are fast algorithms to compute multiplications [19, 20, 21] in binary
finite fields. Yet these operations remain much slower than a native operation
(e.g., a regular 32-bit integer multiplication). However, some recent
processors have support for finite fields. In such cases, the penalty could be
small for using finite fields, as opposed to regular integer arithmetic (see §
4 and § 5.4). (Though they are outside our scope, there are also fast
techniques for computing hash functions over a finite field having prime
cardinality [22].)
## 3 Making Multilinear strongly universal in the ring
$\mathbb{Z}/2^{K}\mathbb{Z}$
On processors without support for binary finite fields, we can trade memory
for speed to essentially get the same properties as finite fields on _some_ of
the bits using fast integer arithmetic. For example, Dietzfelbinger [13]
showed that the family of hash functions of the form
$\displaystyle h_{A,B}(x)=\left(Ax+B\mod{2^{K}}\right) \div 2^{L-1}$
where the integers $A,B\in[0,2^{K})$ and $x\in[0,2^{L})$ is strongly universal
for $K>L-1$. (For fewer parentheses, we adopt the convention that
$Ax+B\mod{2^{K}}\equiv(Ax+B)\bmod{2^{K}}$. The symbol $\div$ denotes integer
division: $x\div y=\lfloor x/y\rfloor$ for positive integers.) We generalize
Dietzfelbinger hashing from the linear to the multilinear case.
The main difference between a finite field and common integer arithmetic (in
the integer ring $\mathbb{Z}/2^{K}\mathbb{Z}$) is that elements of fields have
inverses: given the equation $ax=b$, there is a unique solution $x=a^{-1}b$
when $a\neq 0$. However, the same is “almost” true in integer rings used for
computer arithmetic as long as the variable $a$ is small. For example, when
$a=1$, we can solve for $ax=b$ exactly ($x=b$). When $a=2$, then there are at
most two solutions to the equation $ax=b$. We build on these observations to
derive a stronger result.
We let $\tau=\textrm{trailing}(a)$ be the number of trailing zeros of the
integer $a$ in binary notation. For example, we have that
$\textrm{trailing}(2^{j})=j$.
###### Proposition 1
Given integers $K,L$ satisfying $K\geq L-1\geq 0$, consider the equation
$\displaystyle\left(ax+c\mod{2^{K}}\right)\div 2^{L-1}=b$
where $a$ is an integer in $[1,2^{L})$, $b$ is an integer in $[0,2^{K-L+1})$
and $c$ an integer in $[0,2^{K})$. Given $a$, $b$ and $c$, there are exactly
$2^{L-1}$ integers $x$ in $[0,2^{K})$ satisfying the equation.
Proof. Let $\tau=\textrm{trailing}(a)$. We have $\tau\leq L-1$ since
$a\in[1,2^{L})$. Because $a$ is non-zero, we have that $a^{\prime}=a\div
2^{\tau}$ is odd and $a=2^{\tau}a^{\prime}$.
We have
$\left((ax+c)\bmod{2^{K}}\right)\div
2^{L-1}=\left((2^{\tau}a^{\prime}x+c)\bmod{2^{K}}\right)\div 2^{L-1}$
$\displaystyle=$ $\displaystyle\left(2^{\tau}[a^{\prime}x+(c\div
2^{\tau})]+(c\bmod{2^{\tau}})\mod{2^{K}}\right)\div 2^{L-1}.$
We show that the term $(c\bmod{2^{\tau}})$ can be removed. Indeed, the $\tau$
least significant bits of $2^{\tau}[a^{\prime}x+(c\div 2^{\tau})]+(c\bmod
2^{\tau})$ are those of $c\mod{2^{\tau}}$ whereas the more significant bits
are those of $2^{\tau}[a^{\prime}x+(c\div 2^{\tau})]$. The final division by
${2^{L-1}}$ will dismiss the $L-1$ least significant bits, and $\tau\leq L-1$,
so that the term $(c\bmod{2^{\tau}})$ can be ignored.
Hence, we have
$\left(ax+c\mod{2^{K}}\right)\div 2^{L-1}=(2^{\tau}[a^{\prime}x+(c\div
2^{\tau})]\mod{2^{K}})\div 2^{L-1}$
$\displaystyle=$ $\displaystyle\left(2^{\tau}\left[a^{\prime}x+(c\div
2^{\tau})\mod{2^{K-\tau}}\right]\right)\div 2^{L-1}$ $\displaystyle=$
$\displaystyle\left(a^{\prime}x+(c\div 2^{\tau})\mod{2^{K-\tau}}\right)\div
2^{L-1-\tau}$ $\displaystyle=$
$\displaystyle\left(a^{\prime}(x\bmod{2^{K-\tau}})+(c\div
2^{\tau})\mod{2^{K-\tau}}\right)\div 2^{L-1-\tau}.$
Setting $x^{\prime}=x\bmod{2^{K-\tau}}$ and $c^{\prime}=c\div 2^{\tau}$, we
finally have
$\displaystyle\left(ax+c\mod{2^{K}}\right)\div
2^{L-1}=\left(a^{\prime}x^{\prime}+c^{\prime}\mod{2^{K-\tau}}\right)\div
2^{L-1-\tau}.$
Let $z$ be an integer such that $z\div 2^{L-1-\tau}=b$. Consider
$a^{\prime}x^{\prime}+c^{\prime}\mod{2^{K-\tau}}=z$. We can rewrite it as
$a^{\prime}x^{\prime}\bmod{2^{K-\tau}}=z-c^{\prime}\bmod{2^{K-\tau}}$. Because
$a^{\prime}$ is odd, $a^{\prime}$ and $2^{K-\tau}$ are coprime (their greatest
common divisor is 1). Hence, there is a unique integer
$x^{\prime}\in[0,2^{K-\tau})$ such that
$a^{\prime}x^{\prime}\bmod{2^{K-\tau}}=z-c^{\prime}\bmod{2^{K-\tau}}$ [23,
Cor. 31.25].
Given $b$, there are $2^{L-1-\tau}$ integers $z$ such that $z\div
2^{L-1-\tau}=b$. Given $x^{\prime}$, there are $2^{\tau}$ integers $x$ in
$[0,2^{K})$ such that $x^{\prime}=x\bmod{2^{K-\tau}}$. It follows that there
are $2^{L-1-\tau}\times 2^{\tau}=2^{L-1}$ integers $x$ in $[0,2^{K})$ such
that $\left(a^{\prime}x^{\prime}+c^{\prime}\mod{2^{K-\tau}}\right)\div
2^{L-1-\tau}=b$ holds. $\square$
###### Example 1
Consider the equation $(6x+10\bmod 64)\div 4=5$. By Proposition 1, there must
be exactly 4 solutions to this equation (setting $K=6,L=3$). We can find them
using the proof of the lemma. The integer $6$ has 1 trailing zero in binary
notation ($110$) so that $\tau=1$. We can write $6=2\times 3$ so that
$a^{\prime}=3$. Similarly, $c^{\prime}=10\div 2=5$. Hence we must consider the
equation $3x^{\prime}+5\bmod{2^{5}}=z$ for values of $z$ such that $z\div
2=5$. There are two such values: $z=10$ and $z=11$. We have that
$\displaystyle 3x^{\prime}+5\bmod{32}=10\Rightarrow
3x^{\prime}\bmod{32}=5\Rightarrow x^{\prime}=23\text{~{}and}$ $\displaystyle
3x^{\prime}+5\bmod{32}=11\Rightarrow 3x^{\prime}\bmod{32}=6\Rightarrow
x^{\prime}=2.$
It remains to solve for $x$ in $x^{\prime}=x\bmod 32$ with the constraint that
$x$ is an integer in $[0,64)$. When $x^{\prime}=2$, we have that
$x\in\\{2,34\\}$. When $x^{\prime}=23$, we have that $x\in\\{23,55\\}$. Hence,
the solutions are 2, 23, 34 and 55.
Using Proposition 1, we can show that fast variations of Multilinear are
strongly universal even though we use regular integer arithmetic, not finite
fields.
###### Theorem 1
Given integers $K,L$ satisfying $K\geq L-1\geq 0$, consider the families of
$(K-L+1)$-bit hash functions
* 1.
Multilinear:
$\displaystyle
h(s)=\left(\left(m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}\right)\mod{2^{K}}\right)\div
2^{L-1}$
* 2.
Multilinear-HM:
$\displaystyle
h(s)=\left(\left(m_{1}+\sum_{i=1}^{n/2}(m_{2i}+s_{2i-1})(m_{2i+1}+s_{2i})\right)\mod{2^{K}}\right)\div
2^{L-1}$
which assumes that $n$ is even.
Here the $m_{i}$’s are random integers in $[0,2^{K})$ and the string
characters $s_{i}$ are integers in $[0,2^{L})$. These two families are
strongly universal over fixed-length strings, or over variable-length strings
that do not end with the zero character. We can apply the second family to
strings of odd length by appending an extra zero element so that all strings
have an even length.
Proof. We begin with the first family (Multilinear). Given any two distinct
strings $s$ and $s^{\prime}$, consider the equations $h(s)=y$ and
$h(s^{\prime})=y^{\prime}$ for any two hash values $y$ and $y^{\prime}$.
Without loss of generality, we can assume that the strings have the same
length. If not, we can pad the shortest string with zeros without changing its
hash value. We need to show that $P(h(s)=y\land
h(s^{\prime})=y^{\prime})=2^{2(L-K-1)}$. Because the two strings are distinct,
we can find $j$ such that $s_{j}\neq s^{\prime}_{j}$. Without loss of
generality, assume that $s^{\prime}_{j}-s_{j}\in[0,2^{L})$.
We want to solve the equations
$\displaystyle\left(\left(m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}\right)\mod{2^{K}}\right)\div
2^{L-1} $ $\displaystyle=$ $\displaystyle y,$ (2)
$\displaystyle\left(\left(m_{1}+\sum_{i=1}^{n}m_{i+1}s^{\prime}_{i}\right)\mod{2^{K}}\right)\div
2^{L-1} $ $\displaystyle=$ $\displaystyle y^{\prime}$ (3)
for integers $m_{1},m_{2},\ldots$ in $[0,2^{K})$.
Consider the following equation
$\displaystyle\left(m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}\right)\mod{2^{K}} $
$\displaystyle=$ $\displaystyle z.$
There is a bijection between $m_{1}$ and $z\in[0,2^{K})$. That is, for every
value of $m_{1}$, there is a unique $z$, and vice versa. Specifically, we have
$\displaystyle m_{1} $ $\displaystyle=$ $\displaystyle
z-\sum_{i=1}^{n}m_{i+1}s_{i}\mod{2^{K}}.$
If we choose $z$ such that $z\div 2^{L-1}=y$, we effectively solve Equation 2.
By substitution in Equation 3, we have
$\displaystyle\left(m_{j+1} (s^{\prime}_{j}-s_{j})+z+\sum_{i\neq
j,i=1}^{n}m_{i+1}(s^{\prime}_{i}-s_{i})\mod{2^{K}}\right)\div
2^{L-1}=y^{\prime}.$
This equation is independent of $m_{1}$. By Proposition 1, there are exactly
$2^{L-1}$ solutions $m_{j+1}$ to this last equation. (Indeed, in the statement
of Proposition 1, substitute $m_{j+1}$ for $x$, $s^{\prime}_{j}-s_{j}$ for
$a$, $z+\sum_{i\neq j,i=1}^{n}m_{i+1}(s^{\prime}_{i}-s_{i})\mod{2^{K}}$ for
$b$ and $y^{\prime}$ for $b$.)
Meanwhile, there are $2^{L-1}$ possible values $z$ such that $z\div
2^{L-1}=y$. Because there is a bijection between $m_{1}$ and $z$, there are
also $2^{L-1}$ possible values for $m_{1}$.
So, focusing only on $m_{1}$ and $m_{j+1}$, there are $2^{L-1}\times 2^{L-1}$
values satisfying $h(s)=y$ and $h(s^{\prime})=y^{\prime}$. Yet there are
$2^{K}\times 2^{K}$ possible pairs $m_{1},m_{j+1}$. Thus the probability that
$h(s)=y$ and $h(s^{\prime})=y^{\prime}$ is $\frac{2^{L-1}\times
2^{L-1}}{2^{K}\times 2^{K}}=2^{2(L-K-1)}$ which completes the proof for the
first family.
The proof that the second family (Multilinear-HM) is strongly universal is
similar. As before, set $z$ in $[0,2^{K})$ such that $z\div 2^{L-1}=y$. Solve
for $m_{1}$ from the first equation:
$\displaystyle
m_{1}=\left(z-\sum_{i=1}^{n/2}(m_{2i}+s_{2i-1})(m_{2i+1}+s_{2i})\right)\mod{2^{K}}.$
Then by substitution, we get
$\displaystyle\Bigg{(}\Bigg{(}\sum_{i=1}^{n/2}(m_{2i}+s^{\prime}_{2i-1})(m_{2i+1}+s^{\prime}_{2i})$
$\displaystyle-$ $\displaystyle(m_{2i}+s_{2i-1})(m_{2i+1}+s_{2i})$
$\displaystyle+$ $\displaystyle z\Bigg{)}\mod 2^{K}\Bigg{)}\div
2^{L-1}=y^{\prime}.$
We can rewrite this last equation, as either
$((m_{j}(s^{\prime}_{j}-s_{j})+\rho+z\mod 2^{K})\div 2^{L-1}=y^{\prime}$ if
$j$ is even or as $((m_{j+1}(s^{\prime}_{j}-s_{j})+\rho+z\mod 2^{K})\div
2^{L-1}=y^{\prime}$ if $j$ is odd, where $\rho$ is independent of either
$m_{j}$ ($j$ even) or $m_{j+2}$ ($j$ odd). As before, by Proposition 1, there
are exactly $2^{L-1}$ solutions for $m_{j}$ ($j$ even) or $m_{j+2}$ ($j$ odd)
if $z$ is fixed. As before, there are $2^{L-1}$ distinct possible values for
$z$, and $2^{L-1}$ distinct corresponding values for $m_{1}$. Hence, the pair
$m_{1},m_{j}$ can take $2^{L-1}\times 2^{L-1}$ distinct values out of
$2^{K}\times 2^{K}$ values, which completes the proof. $\square$
To apply Theorem 1 to variable-length strings, we can append the character
value one to all strings so that they never end with the character value zero,
as in § 2. If we use Multilinear-HM, we should add the character value one
before padding odd-length strings to an even length.
Theorem 1 is both more general (because it includes strings) and more specific
(because the cardinality of the set of hash values is a power of two) than a
similar result by Dietzfelbinger [13, Theorem 4]. However, we believe our
proof is more straightforward: we mostly use elementary mathematics.
While Dietzfelbinger did not consider the multilinear case, others proposed
variations suited to string hashing. Pǎtraşcu and Thorup [24] state without
proof that Multilinear-HM over strings of length two is strongly universal for
$K=64,L=32$. They extend this approach to strings, taking characters two by
two:
$\displaystyle
h(s)=\left(\left(\bigoplus_{i=1}^{n/2}(m_{3i-2}+s_{2i-1})(m_{3i-1}+s_{2i})+m_{3i}\right)\mod{2^{K}}\right)\div
2^{L}$
where $\bigoplus$ is the bitwise exclusive-or operation and $n$ is even.
Unfortunately, their approach uses more operations and requires 50% more
random numbers than Multilinear-HM. They also refer to an earlier reference
[25] where a similar scheme was erroneously described as universal, and
presented as folklore:
$\displaystyle
h(s)=\left(\left(\bigoplus_{i=1}^{n/2}(m_{2i+1}+s_{2i+1})(m_{2i+2}+s_{2i+2})\right)\mod{2^{K}}\right)\div
2^{L}$
where $n$ is even. To falsify the universality of this last family, we can
verify numerically that for $K=6,L=3$, the strings $0,0$ and $2,6$ collide
with probability $\frac{576}{4096}>\frac{1}{2^{3}}$. In any case, we see no
benefit to this last approach for long strings because Multilinear-HM is
likely just as fast, and it is strongly universal.
### 3.1 Implementing Multilinear
If 32-bit values are required, we can generate a large buffer of 64-bit
unsigned random integers $m_{i}$. The computation of either
$\displaystyle
h(s)=\left(m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}\mod{2^{64}}\right)\div 2^{32}$
or
$\displaystyle
h(s)=\left(m_{1}+\sum_{i=1}^{n/2}(m_{2i}+s_{2i-1})(m_{2i+1}+s_{2i})\mod{2^{64}}\right)\div
2^{32}$
is then a simple matter using unsigned integer arithmetic common to most
modern processors. The division by $2^{32}$ can be implemented efficiently by
a right shift (`>>`32).
One might object that according to Theorem 1, 63-bit random numbers are
sufficient if we wish to hash 32-bit characters to a 32-bit hash value. The
division by $2^{32}$ should then be replaced by a division by $2^{31}$.
However, we feel that such an optimization is unlikely to either save memory
or improve speed.
Multilinear is essentially an inner product and thus can benefit from
multiply-accumulate CPU instructions: by processing the multiplication and the
subsequent addition as one machine operation, the processor may be able to do
the computation faster than if the computations were done separately. Several
processors have such integer multiply-accumulate instructions (ARM, MIPS,
Nvidia and PowerPC). Comparatively, we do not know of any multiply-xor-
accumulate instruction in popular processors.
Unfortunately, some languages—such as Java—fail to support unsigned integers.
With a two’s complement representation, the de facto standard in modern
processors, additions and multiplications give identical results, up to
overflow flags, as long as no promotion is involved: e.g., multiplying 32-bit
integers using 32-bit arithmetic, or 64-bit integers using 64-bit arithmetic.
However, we must still be careful: promotions and divisions differ when we use
signed integers:
* 1.
If we store string characters using 32-bit integers (int) and random values as
64-bit integers (long), Java will sign-extend the 32-bit integer to a 64-bit
integer when computing $\texttt{m}_{i+1}*\texttt{s}_{i}$, giving an unintended
result for negative string characters. Use
$\texttt{m}_{i+1}*(\texttt{s}_{i}\&\texttt{0xFFFFFFFFl})$ instead.
* 2.
Unsigned and signed divisions differ. Correspondingly, for the division by
$2^{32}$—to retrieve the 32 most significant bits—the unsigned right-shift
operator (`>>>`) must be used in Java, and not the regular right shift (`>>`).
Because we assume that the number of bits is a constant, the computational
complexity of Multilinear is linear ($O(n)$). Multilinear uses $n$
multiplications, $n$ additions, and one shift, whereas Multilinear-HM uses
$n/2$ multiplications, $3n/2$ additions, and one shift. In both cases we use
$2n+1$ operations, although there may be benefits to having fewer
multiplications. (Admittedly, Single Instruction, Multiple Data (SIMD)
processors can do several instructions at once.)
Consider that we need at least $\approx 32(n+1)$ random bits for strongly
universal 32-bit hashing of $32n$ bits [15]. That is, we must aggregate
$\approx 64n+32$ bits into a 32-bit hash value. Assume that we only allow
unary and binary operations. A 32-bit binary operation maps 64 bits to 32
bits, a reduction of 32 bits. Hence, we require at least $2n$ 32-bit
operations for strongly universal hashing. Alternatively, we require at least
$n$ 64-bit operations. Hence, for $n$ large, Multilinear and Multilinear-HM
use at most twice the minimal number of operations.
### 3.2 Word size optimization
The number of required bits is application dependent: for a hash table, one
may be able to bound the maximum table size. In several languages such as
Java, 32-bit hash values are expected. Meanwhile the key parameters of our
hash functions Multilinear and Multilinear-HM are $L$ (the size of characters)
and $K$ (the size of the operations), and these two hash functions deliver
$K-L+1$ usable bits.
However, both $K$ and $L$ can be adjusted given a desired number of usable
random bits. Indeed, a length $n$ string of $L$-bit characters can be
reinterpreted as a length $n\lceil L/L^{\prime}\rceil$ string of
$L^{\prime}$-bit characters, for any non-zero $L^{\prime}$. Thus, we can
either grow $L$ and $K$ or reduce $L$ and $K$, for the same number of usable
bits.
To reduce the need for random bits, we should use large values of $K$.
Consider a long input string that we can represent as a string of 32-bit or
96-bit characters. Assume we want 32-bit hash values. Assume also that our
random data only comes in strings of 64-bit or 128-bit characters. If we
process the string as a 32-bit string, we require 64 random bits per
character. The ratio of random strings to hashed strings is two. If we process
the string as a 96-bit string, we require 128 random bits per character and
the ratio of random strings to hashed strings is $128/96=4/3\approx 1.33$.
What if we could represent the string using 224-bit characters and have random
bits packaged into characters of 256 bits? We would then have a ratio of
$8/7\approx 1.14$.
We can formalize this result. Suppose we require $z$ pairwise independent bits
and that we have $M$ input bits. Stinson [15] showed that this requires at
least $1+2^{M}(2^{z}-1)$ hash functions or, equivalently,
$\log(1+2^{M}(2^{z}-1))$ random bits. Thus, given any hashing family, the
ratio of its required number of random bits to the Stinson limit (henceforth
Stinson ratio) must be greater or equal to one. The $M$ input bits can be
represented as an $L$-bit $n$-character string for $M=nL$. Under Multilinear
(and Multilinear-HM), we must have $z=K-L+1$. Thus we use
$K(n+1)=(z+L-1)(\lceil M/L\rceil+1)$ random bits. We have that $(z+L-1)(\lceil
M/L\rceil+1)\leq(z+L-1)(M/L+2)$ which is minimized when
$L=\sqrt{(z-1)\frac{M}{2}}.$ (4)
Rounding $L=\sqrt{(z-1)M/2}$ up and substituting it back into $(z+L-1)(\lceil
M/L\rceil+1)$, we get an upper bound on the number of random bits required by
Multilinear. This bound is nearly optimal when $\lceil M/L\rceil\approx M/L$,
that is, when $M$ is large. Unfortunately, this estimate fails to consider
that word sizes are usually prescribed. For example, we could be required to
choose $K\in\\{8,16,32,64\\}$. That is, we have to choose
$L\in\\{9-z,17-z,33-z,65-z\\}$. Fig. 1 shows the corresponding Stinson ratios.
When there are many input bits ($M\gg 1$), the ratio of Multilinear converges
to one. That is, as long as we can decompose input data into strings of large
characters (having $\approx\sqrt{(z-1)M/2}$ bits), Multilinear requires almost
a minimal number of bits. This may translate into an optimal memory usage.
(The result also holds for Multilinear-HM except that it is slightly less
efficient for strings having an odd number of characters.) If we restrict the
word sizes to common machine word sizes ($K\in\\{8,16,32,64\\}$), the ratio is
$\approx 2$ for large input strings. We also consider the case where we could
use 128-bit words ($K\in\\{8,16,32,64,128\\}$). It improves the ratio
noticeably ($\approx 1.33$), as expected.
Figure 1: For large inputs, Multilinear requires an almost optimal number of
random bits when arbitrary word sizes ($K$) are allowed. It has lower
efficiency when the word size is constrained. The plot was generated for
32-bit hash values ($z=32$).
We can also choose the word size ($K$) to optimize speed. On a 64-bit
processor, setting $K=64$ would make sense. We can compare this default with
two alternatives:
1. 1.
We can try to support much larger words using fast multiplication algorithms
such as Karatsuba’s. We could merely try to minimize the number of random
bits. However, this ignores the growing computation cost of multiplications
over many bits, e.g., Karatsuba’s algorithm is in $\Omega(n^{1.58})$. For
simplicity, suppose that the cost of $K$-bit multiplication costs $K^{a}$ time
for $a>1$. Roughly speaking, to hash $M$ bits, we require $\lceil M/L\rceil$
multiplications. When we have long strings ($M\gg L$), we can simplify $\lceil
M/L\rceil\approx M/L$. If we desire $z$-bit hash values, then we need to use
multiplication on $K=z+L-1$ bits. Thus, the processing cost can be (roughly)
approximated as $\frac{M(z+L-1)^{a}}{L}$. Starting from $L=1$, this function
initially decreases to a minimum at
$L=\frac{z-1}{a-1}$ (5)
before increasing again as $L^{a-1}$. (When $a=1.5$ and $z=32$, we have
$\frac{z-1}{a-1}=62$.) See Fig. 2. Hence, while we can minimize the total
number of random bits by using many bits per character ($L$ large), we may
want to keep $L$ relatively small to take into account the superlinear cost of
multiplications.
2. 2.
We can support 128-bit words on a 64-bit processor, with some overhead.
(Recent GNU GCC compilers have the __uint128 type, as a C-language extension.)
A single 128-bit multiplication may require up to three 64-bit
multiplications. However, it processes more data: with $z=32$ hashed bits,
each 128-bit multiplication hashes 97 input bits. Comparatively, setting
$K=64$, we require a single 64-bit multiplication, but we process only 32 bits
of data. (Formally, we could process 33 bits of data, but for convenient
implementation, we process data in powers of two.) Hence, it is unclear which
approach is faster: three 64-bit multiplications and 128 bits of random data
to process 97 input bits, or a single 64-bit multiplication and 64 bits of
random data, to process 33 input bits. However, the 128-bit approach will use
33% fewer random bits. Going to 256-bit word sizes would only reduce the
number of random bits by 14%: using larger and larger words leads to
diminishing returns.
We assess these two alternatives experimentally in § 5.5.
Figure 2: Modeled computational cost per bit as a function of the number of
bits per character ($\frac{(z+L-1)^{a}}{L}$) for 32-bit hashing values
($z=32$) and $a=1.5$.
## 4 Fast Multilinear with carry-less multiplications
To help support fast operations over binary finite fields ($GF(2^{L})$), AMD
and Intel introduced the Carry-less Multiplication (CLMUL) instruction set
[26]. Given the binary representations of two numbers,
$a=\sum_{i=1}^{L}a_{i}2^{i-1}$ and $b=\sum_{i=1}^{L}b_{i}2^{i-1}$, the carry-
less multiplication is given by $c=\sum_{i=1}^{2L-1}c_{i}2^{i-1}$ where
$c_{i}=\bigoplus_{j=1+i}^{2L-1}a_{j}b_{j-i}$. (We write $a\star b=c$.) If we
represent the two $L$-bit integers $a$ and $b$ as polynomials in GF(2)[$x$],
then the carry-less multiplication is equivalent to the usual polynomial
multiplication:
$\displaystyle\left(\sum_{i=1}^{L}a_{i}x^{i-1}\right)\left(\sum_{i=1}^{L}b_{i}x^{i-1}\right)=\sum_{i=1}^{2L-1}c_{i}2^{i-1}.$
With a fast carry-less computation, we can compute Multilinear efficiently.
Given any irreducible polynomial $p(x)$ of degree $L$, the field
GF(2)[$x$]/$p(x)$ is isomorphic to GF$(2^{L})$. Hence, we want to compute
$h(s)=m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}$ over GF(2)[$x$]/$p(x)$. (Similarly, we
can use Equation 1 to reduce the number of multiplications by half.) Computing
all multiplications over GF(2)[$x$]/$p(x)$ would still be expensive given fast
carry-less multiplication. Instead, we first compute
$m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}$ over GF(2)[$x$] and then return the
remainder of the division of the final result by $p(x)$. Indeed, think of the
values $m_{1},m_{2},\ldots$ and $s_{1},s_{2},\ldots$ as polynomials of degree
at most $L$ in GF(2)[$x$]. Each of the $n$ multiplications in GF(2)[$x$] is
equivalent to a carry-less multiplication over $L$-bit integers which results
in a $2L-1$-bit value. Similarly, each of the $n$ additions in GF(2)[$x$] is
an exclusive-or operation. That is, we want to compute the $2L-1$-bit integer
$\displaystyle\bar{h}(s)=m_{1}\oplus\left(\bigoplus_{i=1}^{n}m_{i+1}\star
s_{i}\right).$ (6)
Finally, considering $\bar{h}(s)$ as an element of GF(2)[$x$], noted $q(x)$,
we must compute $q(x)/p(x)$. The remainder (as an $L$-bit integer) is the
final hash value $h(s)$.
If done naively, computing the remainder of the division by an irreducible
polynomial may remain relatively expensive, especially for short strings since
they require few multiplications. A common technique to quickly compute the
remainder is the Barrett reduction algorithm [27]. The adaptation of this
reduction to GF(2)[$x$] is especially convenient [28] when we choose the
irreducible polynomial $p(x)$ such that $\textrm{degree}(p(x)-x^{L})\leq L/2$,
that is, when we can write it as $p(x)=\sum_{i=0}^{\lfloor
L/2\rfloor}a_{i}x_{i}+x^{L}$. (There are such irreducible polynomials for
$L\in\\{1,2,\ldots,400\\}$ [29] and we conjecture that such a polynomial can
be found for any $L$ [30].) In this case, the remainder of $q(x)/p(x)$ is
given by
$\displaystyle((((q\div 2^{L})\star p)\div 2^{L})\star p)\oplus q)\mod 2^{L}$
where $q$ and $p$ are the $2L-1$-bit and $L+1$-bit integers representing
$q(x)$ and $p(x)$. (See B for implementation details.) We expect the two
carry-less multiplications to account for most of the running time of the
reduction.
Unfortunately, in its current Intel implementation, carry-less multiplications
have significantly reduced throughput compared to regular integer
multiplications. Indeed, with pipelining, it is possible to complete one
regular multiplication per cycle, but only one carry-less multiplication every
8 cycles [31]. However, using a result from § 2, we can reduce the number of
multiplications by half if we compute
$\displaystyle\bar{h}(s)=m_{1}\oplus\left(\bigoplus_{i=1}^{n/2}(m_{2i}+s_{2i-1})\star(m_{2i+1}+s_{2i})\right)$
instead. (Henceforth, we refer to last variation as GF Multilinear-HM whereas
we refer to version based on Equation 6 as GF Multilinear.) Yet even a fast
implementation of Barrett reduction will still be much slower than merely
selecting the left-most $L$ bits as in Multilinear.
However, the carry-less approach might still be preferable to the schemes
described in § 3 (e.g., Multilinear) because fewer random bits are required.
Indeed, to generate $L$-bit hash values from $n$-character strings, the carry-
less scheme used $(n+1)L$ random bits, whereas Multilinear requires
$2L+n(2L-1)$ random bits.
## 5 Experiments
Our experiments show the following results:
* 1.
It is best to implement Multilinear with loop unrolling. With this
optimization, Multilinear is just as fast (on Intel processors) as
Multilinear-HM, even though it has twice the number of multiplications. In
general, processor microarchitectural differences are important in determining
which method is fastest. (§ 5.2)
* 2.
In the absence of processor support for carry-less multiplication (see § 4),
hashing using Multilinear over binary finite fields is an order of magnitude
slower than Multilinear even when using a highly optimized library. (§ 5.3)
* 3.
Even with hardware support for carry-less multiplication, hashing using
Multilinear over binary finite fields remains nearly an order of magnitude
slower than Multilinear. (§ 5.4)
* 4.
Given a 64-bit processor, it is noticeably faster to use a word size of 64
bits even though a larger word size (128 bits) uses fewer random bits (33%
less). Use of multiprecision arithmetic libraries can further reduce the
overhead from accessing random bits, but they are also not competitive with
respect to speed, though they can halve the number of required random bits. (§
5.5)
* 5.
Multilinear is generally faster than popular string-hashing algorithms. (§
5.6)
### 5.1 Experimental setup
We evaluated the hashing functions on the platforms shown in Table 1. Our
software is freely available online [32]. For Intel and AMD processors, we
used the processor’s time stamp counter (rdtsc instruction [33]) to estimate
the number of cycles required to hash each byte. Unfortunately, the ARM
instruction set does not provide access to such a counter. Hence, for ARM
processors (Apple A4 and Nvidia Tegra), we estimated the number of cycles
required by dividing the wall-clock time by the documented processor clock
rate (1 GHz).
Table 1: Platforms used. Processor | Bits | GCC version | Flags, besides -O3 -funroll-loops
---|---|---|---
64-bit processors
Intel Core 2 Duo | 64 | GNU GCC 4.6.2 | -march=core2 -mno-sse2
Intel Xeon X5260 | 64 | GNU GCC 4.1.2 | -march=nocona
Intel Core i7-860 | 64 | GNU GCC 4.6.2 | -march=corei7 -mno-sse2
Intel Core i7-2600 | 64 | GNU GCC 4.6.2 | -march=corei7 -mno-sse2
Intel Core i7-2677M | 64 | GNU GCC 4.6.2 | -march=corei7 -mno-sse2
AMD Sempron 3500+ | 64 | GNU GCC 4.4.3 | -march=k8 -mno-sse2
AMD V120 | 64 | GNU GCC 4.4.3 | –march=amdfam10 -mno-sse2
32-bit processors
Intel Atom N270 | 32 | GNU GCC 4.5.2 | -march=atom
Apple A4 | 32 | GNU GCC 4.2.1 | -march=armv7
Nvidia Tegra 2 | 32 | GNU GCC 4.4.3111From the Android NDK, configured for the android-9 platform, and used on a Motorola XOOM. |
VIA Nehemiah | 32 | GNU GCC 3.3.4 | -march=i686
For the 64-bit machines, 64-bit executables were produced and all operations
were executed using 64-bit arithmetic except where noted. All timings were
repeated three times. For the 32-bit processors, 32-bit operations were used
to process 16-bit strings. Therefore, results between 32- and 64-bit
processors are not directly comparable. Good optimization flags were found by
a trial-and-error process. We note that using profile-guided optimizations did
not improve this code any more than simply enabling loop unrolling (-funroll-
loops). With (only) versions 4.4 and higher of GCC, it was important for speed
to forbid use of SSE2 instructions when compiling Multilinear and Multilinear-
HM.
We found that the speed is insensitive to the content of the string: in our
tests we hashed randomly generated strings. We reuse the same string for all
tests. Unless otherwise specified, we hash randomly generated 32-bit strings
of 1024 characters.
In addition to Multilinear and Multilinear-HM we also implemented Multilinear
(2-by-2) which is merely Multilinear with 2-by-2 loop unrolling (see A for
representative C implementations).
Our timings should represent the best possible performance: the performance of
a function may degrade [21] when it is included in an application because of
bandwidth and caching.
### 5.2 Reducing the multiplications or unrolling may fail to improve the
speed
We ran our experiments over both the 32-bit and 64-bit processors. For the
32-bit processors, we generated both 16-bit and 32-bit hash values. Our
experimental results are given in Table 2.
Table 2: Estimated CPU cycles per byte for fast Multilinear hashing | Multilinear | 2-by-2 | Multilinear-HM
---|---|---|---
64-bit processors and 32-bit hash values and characters
Intel Core 2 Duo | 0.54 | 0.52 | 0.52
Intel Xeon X5260 | 0.50 | 0.50 | 0.50
Intel Core i7-860 | 0.42 | 0.42 | 0.42
Intel Core i7-2600 | 0.34 | 0.27 | 0.28
Intel Core i7-2677M | 0.25 | 0.20 | 0.20
AMD Sempron 3500+ | 0.63 | 0.60 | 0.40
AMD V120 | 0.63 | 0.63 | 0.40
64-bit arithmetic and 32-bit hash values and characters on 32-bit processors
Intel Atom N270 | 4.2 | 4.2 | 3.6
Apple A4 | 3.0 | 2.7 | 3.3
Nvidia Tegra 2 | 3.3 | 3.0 | 4.9
VIA Nehemiah | 12 | 12 | 8.2
32-bit processors and 16-bit hash values and characters
Intel Atom N270 | 2.1 | 3.5 | 2.6
Apple A4 | 1.9 | 2.6 | 1.7
Nvidia Tegra 2 | 1.8 | 2.2 | 1.9
VIA Nehemiah | 5.2 | 5.2 | 3.6
We see that over 64-bit Intel processors ( except for the i7-2600 ), there is
little difference between Multilinear, Multilinear (2-by-2) and Multilinear-
HM, even though Multilinear and Multilinear (2-by-2) have twice the number of
multiplications. We believe that Intel processors use aggressive pipelining
techniques well suited to these computations. On AMD processors, Multilinear-
HM is the clear winner, being at least 33% faster.
For the 32-bit processors, we get different vastly different results depending
on whether we generate 16-bit or 32-bit hash values.
* 1.
As expected, it is roughly twice as expensive to generate 32-bit hash values
than to generate 16-bit values.
* 2.
For the VIA processor, Multilinear-HM is 45% faster than Multilinear and
Multilinear (2-by-2). We suspect that the computational cost is tightly tied
to the number of multiplications.
* 3.
When the 32-bit ARM-based processors generate 32-bit hash values, Multilinear
(2-by-2) is preferable. We are surprised that Multilinear-HM is the worse
choice. We believe that this is related to the presence of a multiply-
accumulate instruction in ARM processors. When generating 16-bit hash values,
Multilinear (2-by-2) becomes the worse choice. There is no significant benefit
to using Multilinear-HM as opposed to Multilinear.
* 4.
The Intel Atom processor benefits from Multilinear-HM when generating 32-bit
hash value, but Multilinear is preferable to generate 16-bit hash values. As
with the ARM-based processors, Multilinear (2-by-2) is a poor choice for
generating 16-bit hash values.
### 5.3 Binary-finite-field libraries are not competitive
We obtained the $\texttt{mp}\mathbb{F}_{b}$ library from INRIA. This code is
reported [34] to be generally faster than popular alternatives such as NTL and
Zen, and our own tests found it to be more than twice as fast as Plank’s
library [35].
We computed Multilinear in $GF(2^{32})$, using the version with half the
number of multiplications (see Equation 1) because the library does much more
work in multiplication than addition. Even so, on our Core 2 Duo, hashing
32-bit strings of 1024 characters was an order of magnitude slower than
Multilinear: averaged over a million attempts, the code using
$\texttt{mp}\mathbb{F}_{b}$ required an average of 7.69 $\mu s$ per string,
compared with 0.78 $\mu s$ for Multilinear. While our implementation of
Multilinear uses twice as many random bits as Multilinear in $GF(2^{32})$,
this gain is offset by the memory usage of the finite-field library.
### 5.4 Hardware-supported carry-less multiplications are not fast enough
Intel reports a throughput of one carry-less product every 8 cycles [31] on a
processor such as the Intel Core i7-2600. Consider GF Multilinear-MH: it uses
one carry-less multiplication for every two 32-bit characters. Hence, it
requires at least 4 cycles to process each character. Hence, in the best
scenario possible, GF Multilinear-MH will be four times slower than
Multilinear-MH which requires only 1.1 cycles per 32-bit character ( 0.28
cycle per byte).
To assess the actual performance, we implemented both GF Multilinear and GF
Multilinear-MH in C (§ B). Of the processors we tested, only the i7-2600 has
support for the CLMUL instruction set. If we use the flags -O3 -funroll-loops
-Wall -maes -msse4 -mpclmul, we get 12.5 CPU cycles per 32-byte character with
GF Multilinear and only 7.2 CPU cycles with GF Multilinear-MH. We might be
able to improve our implementation: e.g., we expect that much time is spent
loading data into XMM registers. However, the throughput of the carry-less
multiplication limits the character throughput of GF Multilinear and GF
Multilinear-MH to 8 and 4 cycles. On the bright side, GF Multilinear and GF
Multilinear-MH require half the number of random bits.
### 5.5 The sweet-spot for multiprecision arithmetic is not sweet enough
To implement the techniques of § 3.2, we used the GMP library [36] version
5.0.2 to implement Multilinear (2-by-2). As usual, we are hashing 4 kB of
data, though data to be hashed are read in large chunks (up to 2048 bits). The
hash output is always 32 bits ($z=32$). Results show a benefit as the chunk
size $L$ goes from 32 to 512 bits, but thereafter the situation degrades. See
Fig. 3. In the best case, using 512-bit arithmetic, we require almost 13
$\mu$s per string on the Core 2 Duo platform. For comparison, we find that the
fewest random bits would be needed when $L=1024$ (§ 3.2). As expected, the
running time is minimized for a lower value of $L$ to account for the
superlinear running cost of multiplications.
Unfortunately, we can do 12 times better without the GMP library (0.78 $\mu s$
for 64-bit Multilinear) so it is not practical to use 512-bit arithmetic, even
though it uses fewer random bits (nearly half as many).
Figure 3: Microseconds to hash 4 kB using various word sizes and GMP.
As a lightweight alternative to a multiprecision library, we experimented with
the __uint128 type provided as a GCC extension for 64-bit machines. We used
128-bit random numbers and processed three 32-bit words with each 128-bit
operation. Since __uint128 multiplications are more expensive than __uint128
additions, we tested the Multilinear-HM scheme. On our Core 2 Duo machine, the
result was 38% slower than Multilinear (2-by-2) using 64-bit operations. This
poor results is mitigated by the fact that we use 128 random bits per 96 input
bits, versus 64 random bits per 32 input bits (a saving of nearly 33% for long
strings). Investigation using hardware performance counters showed many
“unaligned loads” from retrieving 128-bit quantities when we step through
memory with 96-bit steps. To reduce this, we tried processing only two 32-bit
words with each 128-bit operation, since we retrieved aligned 64-bit
quantities. However, the result was 61% slower than Multilinear (2-by-2) using
64-bit operations.
### 5.6 Strongly universal hashing is inexpensive?
In a survey, Thorup [1] concluded that strongly universal hash families are
just as efficient, or even more efficient, than popular hash functions with
weaker theoretical guarantees. However, he only considered 32-bit integer
inputs. We consider strings.
In Table 2, we compare the fastest Multilinear (Multilinear-HM) with two non-
universal fast 32-bit string hash functions, Rabin-Karp [37] and SAX [38].
(They are similar to hash functions found in programming languages such as
Java or Perl.) Even though these functions were designed for speed and lack
strong theoretical guarantees, they are far slower than Multilinear on desktop
processors (AMD and Intel). Only for ARM processors (Apple A4 and Nvidia Tegra
2) with 32-bit hash values are they much faster. We suspect that this good
result on ARM processors is due to the multiply-accumulate instruction.
Clearly, such a multiply-accumulate operation greatly benefits simple hashing
functions such as Rabin-Karp and SAX.
Table 3: A comparison of estimated CPU cycles per byte between fast Multilinear hashing and common hash functions | Rabin-Karp | SAX | best Multilinear
---|---|---|---
32-bit hash values and characters on 64-bit processors
Intel Core 2 Duo | 1.3 | 1.3 | 0.52
Intel Xeon X5260 | 1.4 | 1.6 | 0.50
Intel Core i7-860 | 1.4 | 1.6 | 0.42
Intel Core i7-2600 | 0.89 | 1.1 | 0.27
Intel Core i7-2677M | 0.64 | 0.82 | 0.20
AMD Sempron 3500+ | 1.0 | 1.5 | 0.40
AMD V120 | 1.0 | 1.5 | 0.40
32-bit hash values and characters on 32-bit processors
Intel Atom N270 | 1.1 | 2.0 | 4.2
Apple A4 | 0.88 | 1.2 | 2.7
Nvidia Tegra 2 | 0.85 | 1.2 | 3.0
VIA Nehemiah | 2.0 | 3.0 | 8.2
16-bit hash values and characters on 32-bit processors
Intel Atom N270 | 2.1 | 4.1 | 2.2
Apple A4 | 1.8 | 2.1 | 1.8
Nvidia Tegra 2 | 1.6 | 2.4 | 1.7
VIA Nehemiah | 5.0 | 6.6 | 3.6
Crosby and Wallach [9] showed that almost universal hashing could be as fast
as common deterministic hash functions. One of their most competitive almost
universal schemes is due to Black et al. [17]. Their fast family of hash
functions is called NH:
$\displaystyle h(s)=\sum_{i=1}^{n/2}(m_{2i-1}+s_{2i-1}\mod
{2^{L/2}})(m_{2i}+s_{2i}\mod {2^{L/2}})\mod{2^{L}}.$
NH is almost universal over fixed-length strings, or over variable-length
strings that do not end with the zero character; we can apply it to strings
having odd length by appending a character with value zero. It fails to be
uniform: the value
$\displaystyle(m_{1}+s_{1}\mod {2^{L/2}})(m_{2}+s_{2}\mod {2^{L/2}})$
is zero whenever either $m_{1}+s_{1}\mod {2^{L/2}}$ or $m_{2}+s_{2}\mod
{2^{L/2}}$ is zero, which occurs with probability
$\frac{2^{L/2+1}-1}{2^{L}}>\frac{1}{2^{L}}$ over all possible values of
$m_{1},m_{2}$. Moreover, the least significant bits may fail to be almost
universal: e.g., for $L=6$, there are 96 pairs of distinct strings colliding
with probability 1 over the least two significant bits. When processing 32-bit
characters, it generates 64-bit hash values with collision probability of
$1/2^{32}$. Hence, in our tests over 32-bit characters, NH generates 64-bit
hash values whereas the Multilinear families generate 32-bit hash values, but
both have a collision probability bounded by $1/2^{32}$. Thus, while NH saves
memory because it uses nearly half the number of random bits compared to our
fast Multilinear families, Multilinear families may save memory in a system
that stores hash values because their hash values have half the number of
bits. Table 4 shows that the 64-bit NH on 64-bit processors runs at about the
same speed as the best Multilinear on most processors. Only on some Intel Core
i7 processors (2600 and 2677M), NH’s running time is 60% of Multilinear when
we enable SSE support. In other words, sacrificing theoretical guarantees does
not always translate into better speed.
Table 4: A comparison of estimated CPU cycles per byte between fast Multilinear hashing and the almost universal hash function NH from Black et al. [17] for 32-bit hash values using 64-bit arithmetic. When running NH tests, we remove the -mno-sse2 flag where it is present for better results. | NH [17] | best Multilinear
---|---|---
Intel Core 2 Duo | 0.53 | 0.52
Intel Xeon X5260 | 0.50 | 0.50
Intel Core i7-860 | 0.42 | 0.42
Intel Core i7-2600 | 0.16222We use the -march=corei7-avx flag for best results. | 0.27
Intel Core i7-2677M | 0.12 | 0.20
AMD Sempron 3500+ | 0.38 | 0.40
AMD V120 | 0.38 | 0.40
Overall, these numbers indicate that strongly universal string hashing is
computationally inexpensive on most Intel and AMD processors. To gain good
results with the various 64-bit processors, we recommend Multilinear-HM.
Unfortunately—over long strings—strongly universal hashing requires many
random numbers. Generating and storing these random numbers is the main
difficulty. Whether this is a problem depends on the memory available, the CPU
cache, the application workload and the length of the strings. (Intel
researchers reported the generation of true random numbers in hardware at high
speed (4 Gbps) [39].) In practice, unexpectedly long strings may require the
generation of new random numbers while hashing a given string [9]. This
overhead should be relatively inexpensive if we know the length of each string
before we process it.
## 6 Conclusion
Over moderately long 32-bit strings ($\approx$1024 characters), current
desktop processors can achieve strongly universal hashing with no more than
0.5 CPU cycle per byte, and sometimes as little as 0.2 CPU cycle per byte.
Meanwhile, at least twice as many cycles are required for Rabin-Karp hashing
even though it is not even universal.
While it uses half the number of multiplications, we have found that
Multilinear-HM is often no faster than Multilinear on Intel processors.
Clearly, Intel’s pipelining architecture has some benefits.
For AMD processors, Multilinear-HM was faster ($\approx$ 33%), as expected
because it uses fewer multiplications. Yet another alternative, Multilinear
(2-by-2), was slightly faster ($\approx$ 15%) for 32-bit hashing on the mobile
ARM-based processors even though it requires twice as many multiplication as
Multilinear-HM. These mobile ARM-based processors also computed 32-bit Rabin-
Karp hashing with fewer cycles per byte than many desktop processors. We
believe that this is related to the presence of a multiply-accumulate in the
ARM instruction set.
## Acknowledgments
The authors are grateful for related online discussions with P. L. Bannister,
V. Khare, V. Venugopal, J. S. Culpepper. This work is supported by NSERC grant
261437.
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## Appendix A Implementations in C
We implemented the following hash functions:
* 1.
Multilinear: $h(s)=m_{1}+\sum_{i=1}^{n}m_{i+1}s_{i}$
* 2.
Multilinear (2-by-2):
$h(s)=m_{1}+\sum_{i=1}^{n/2}m_{2i}s_{2i-1}+s_{2i}m_{2i+1}$
* 3.
Multilinear-HM:
$h(s)=m_{1}+\sum_{i=1}^{n/2}(m_{2i}+s_{2i-1})(s_{2i}+m_{2i+1})$
For simplicity we assume that the number of characters ($n$) is even.
Following a common convention, we write the unsigned 32-bit and 64-bit integer
data types as uint32 and uint64. The variable p is a pointer to the initial
value of the string whereas endp is a pointer to the location right after the
last 32-bit character of the _string_. The variable m is a pointer to the
64-bit random numbers. (When using 63-bit random numbers as allowed by Theorem
1, the right shifts should be by 31 instead. In practice, we use 64-bit
numbers.) On some compilers and processors, it was useful to disable SSE2:
under GNU GCC we can achieve this result with function attributes (e.g. by
preceding the function declaration by `__attribute__ ((__target__ ("no-
sse2")))`).
#### Multilinear
⬇
uint32 hash(uint64 * m, uint32 * p, uint32 * endp) {
uint64 sum = *(m++);
for(;p!=endp;++m,++p) {
sum+= *m * *p;
}
return sum>>32;
}
#### Multilinear (2-by-2)
⬇
uint32 hash(uint64 * m, uint32 * p, uint32 * endp) {
uint64 sum = *(m++);
for(; p!= endp; m+=2,p+=2 ) {
sum+= (*m * *p) + (*(m + 1) * *(p+1));
}
return sum>>32;
}
#### Multilinear-HM
⬇
uint32 hash(uint64 * m, uint32 * p, uint32 * endp) {
uint64 sum = *(m++);
for(;p!=endp;m+=2,p+=2 ) {
sum += (*m + *p) * (*(m+1) + *(p+1));
}
return sum>>32;
}
## Appendix B Implementations with carry-less multiplications
We implemented Multilinear in $GF(2^{32})$ in C using the Carry-less
Multiplication (CLMUL) instruction set [26] supported by recent Intel and AMD
processors. We also implemented the counterpart to Multilinear-HM which
executes half the number of multiplications.
We use the same conventions as in A regarding the variables p and m except
that the later is a pointer to 32-bit random numbers. We wrote our C programs
using SSE intrinsics: they are functions supported by several major compilers
(including GNU GCC, Intel and Microsoft) that generate SIMD instructions.
The Barrett reduction algorithm is adapted from Knežević et al. [28]. The
variable C contains the chosen irreducible polynomial. We initialize it as
C $\displaystyle=$ _mm_set_epi64x(0,1UL\+ (1UL<<2)\+ (1UL<<6)
$\displaystyle\texttt{+~{}(1UL<<7)}\texttt{+~{}(1UL<<32));}.$
#### Barrett reduction
⬇
uint32 barrett( __m128i A) {
__m128i Q1 = _mm_srli_epi64 (A, n);
__m128i Q2 = _mm_clmulepi64_si128( Q1, C, 0x00);
__m128i Q3 = _mm_srli_epi64 (Q2, n);
__m128i f = _mm_xor_si128 (A, _mm_clmulepi64_si128( Q3, C, 0x00));
return _mm_cvtsi128_si64(f) ;
}
#### GF Multilinear
⬇
uint32 hash(uint32 * m, uint32 * p, uint32 * endp) {
__m128i sum = _mm_set_epi64x(0,*(m++));
for(;p!=endp;++m,++p ) {
__m128i t = _mm_set_epi64x(*m,*p);
__m128i c = _mm_clmulepi64_si128( t, t, 0x10);
sum = _mm_xor_si128 (c,sum);
}
return barret(sum);
}
#### GF Multilinear-MH
⬇
uint32 hash(uint32 * m, uint32 * p, uint32 * endp) {
__m128i sum = _mm_set_epi64x(0,*(m++));
for(;p!=endp;m+=2,p+=2 ) {
__m128i t1 = _mm_set_epi64x(*m,*(m+1));
__m128i t2 = _mm_set_epi64x(*p,*(p+1));
__m128i t = _mm_xor_si128(t1,t2);
__m128i c = _mm_clmulepi64_si128( t, t, 0x10);
sum = _mm_xor_si128 (c,sum);
}
return barret(sum);
}
|
arxiv-papers
| 2012-02-22T16:34:24 |
2024-09-04T02:49:27.701942
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Owen Kaser and Daniel Lemire",
"submitter": "Daniel Lemire",
"url": "https://arxiv.org/abs/1202.4961"
}
|
1202.4979
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-027 CERN-PH-EP-2012-039
Opposite-side flavour tagging of $B$ mesons at the LHCb experiment
The LHCb collaboration †††Authors are listed on the following pages.
The calibration and performance of the opposite-side flavour tagging
algorithms used for the measurements of time-dependent asymmetries at the LHCb
experiment are described. The algorithms have been developed using simulated
events and optimized and calibrated with
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ decay modes with 0.37
$\mbox{\,fb}^{-1}$ of data collected in $pp$ collisions at $\sqrt{s}=$
7$\mathrm{\,Te\kern-1.00006ptV}$ during the 2011 physics run. The opposite-
side tagging power is determined in the $B^{+}\rightarrow$ $J/\psi K^{+}$
channel to be (2.10$\pm$0.08$\pm$0.24)%, where the first uncertainty is
statistical and the second is systematic.
Submitted to Eur. Phys. J. C
The LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
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J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
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P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
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R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
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Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
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Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
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Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
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Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
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O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The identification of the flavour of reconstructed $B^{0}$ and $B^{0}_{s}$
mesons at production is necessary for the measurements of oscillations and
time-dependent $C\\!P$ asymmetries. This procedure is known as flavour tagging
and is performed at LHCb by means of several algorithms.
Opposite-side (OS) tagging algorithms rely on the pair production of $b$ and
$\bar{b}$ quarks and infer the flavour of a given $B$ meson (signal $B$) from
the identification of the flavour of the other $b$ hadron111Unless explicitly
stated, charge conjugate modes are always included throughout this paper.
(tagging $B$). The algorithms use the charge of the lepton ($\mu$, $e$) from
semileptonic $b$ decays, the charge of the kaon from the $b\rightarrow
c\rightarrow s$ decay chain or the charge of the inclusive secondary vertex
reconstructed from $b$-hadron decay products. All these methods have an
intrinsic dilution on the tagging decision, for example due to the possibility
of flavour oscillations of the tagging $B$. This paper describes the
optimization and calibration of the OS tagging algorithms which are performed
with the data used for the first measurements performed by LHCb on $B^{0}_{s}$
mixing and time-dependent $C\\!P$ violation [1, 2, 3].
Additional tagging power can be derived from same-side tagging algorithms
which determine the flavour of the signal $B$ by exploiting its correlation
with particles produced in the hadronization process. The use of these
algorithms at LHCb will be described in a forthcoming publication. The use of
flavour tagging in previous experiments at hadron colliders is described in
Refs. [4, 5].
The sensitivity of a measured $C\\!P$ asymmetry is directly related to the
effective tagging efficiency $\varepsilon_{\rm eff}$, or tagging power. The
tagging power represents the effective statistical reduction of the sample
size, and is defined as
$\varepsilon_{\rm eff}={{\varepsilon_{\rm tag}}{\cal
D}^{2}}={{\varepsilon_{\rm tag}}(1-2\omega)^{2}},$ (1)
where $\varepsilon_{\rm tag}$ is the tagging efficiency, $\omega$ is the
mistag fraction and ${\cal{D}}$ is the dilution. The tagging efficiency and
the mistag fraction are defined as
${\varepsilon_{\rm tag}}=\frac{R+W}{R+W+U}\qquad{\rm
and}~{}~{}~{}~{}~{}\omega=\frac{W}{R+W},$ (2)
where $R$, $W$, $U$ are the number of correctly tagged, incorrectly tagged and
untagged events, respectively.
The mistag fraction can be measured in data using flavour-specific decay
channels, i.e. those decays where the final state particles uniquely define
the quark/antiquark content of the signal $B$. In this paper, the decay
channels $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$, $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ are used. For
charged mesons, the mistag fraction is obtained by directly comparing the
tagging decision with the flavour of the signal $B$, while for neutral mesons
it is obtained by fitting the $B^{0}$ flavour oscillation as a function of the
decay time.
The probability of a given tag decision to be correct is estimated from the
kinematic properties of the tagging particle and the event itself by means of
a neural network trained on Monte Carlo (MC) simulated events to identify the
correct flavour of the signal $B$. When more than one tagging algorithm gives
a response for an event, the probabilities provided by each algorithm are
combined into a single probability and the decisions are combined into a
single decision. The combined probability can be exploited on an event-by-
event basis to assign larger weights to events with low mistag probability and
thus to increase the overall significance of an asymmetry measurement. In
order to get the best combination and a reliable estimate of the event weight,
the calculated probabilities are calibrated on data. The default calibration
parameters are extracted from the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel. The other two flavour-specific channels are used to perform
independent checks of the calibration procedure.
## 2 The LHCb detector and the data sample
The LHCb detector [6] is a single-arm forward spectrometer which measures
$C\\!P$ violation and rare decays of hadrons containing $b$ and $c$ quarks. A
vertex detector (VELO) determines with high precision the positions of the
primary and secondary vertices as well as the impact parameter (${\rm IP}$) of
the reconstructed tracks with respect to the primary vertex. The tracking
system also includes a silicon strip detector located in front of a dipole
magnet with integrated field about 4 Tm, and a combination of silicon strip
detectors and straw drift chambers placed behind the magnet. Charged hadron
identification is achieved through two ring-imaging Cherenkov (RICH)
detectors. The calorimeter system consists of a preshower detector, a
scintillator pad detector, an electromagnetic calorimeter and a hadronic
calorimeter. It identifies high transverse energy hadron, electron and photon
candidates and provides information for the trigger. Five muon stations
composed of multi-wire proportional chambers and triple-GEMs (gas electron
multipliers) provide fast information for the trigger and muon identification
capability.
The LHCb trigger consists of two levels. The first, hardware-based, level
selects leptons and hadrons with high transverse momentum, using the
calorimeters and the muon detectors. The hardware trigger is followed by a
software High Level Trigger (HLT), subdivided into two stages that use the
information from all parts of the detector. The first stage performs a partial
reconstruction of the event, reducing the rate further and allowing the next
stage to fully reconstruct and to select the events for storage up to a rate
of 3 kHz [7].
The majority of the events considered in this paper were triggered by a single
hadron or muon track with large momentum, transverse momentum and $\rm IP$. In
the HLT, the channels with a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
meson in the final state were selected by a dedicated di-muon decision that
does not apply any requirement on the $\rm IP$ of the muons.
The data used in this paper were taken between March and June 2011 and
correspond to an integrated luminosity of 0.37 $\mbox{\,fb}^{-1}$. The
polarity of the LHCb magnet was reversed several times during the data taking
period in order to minimize systematic biases due to possible detector
asymmetries.
## 3 Flavour tagging algorithms
Opposite-side tagging uses the identification of electrons, muons or kaons
that are attributed to the other $b$ hadron in the event. It also uses the
charge of tracks consistent with coming from a secondary vertex not associated
with either the primary or the signal $B$ vertex. These taggers are called
electron, muon, kaon and vertex charge taggers, respectively. The tagging
algorithms were developed and studied using simulated events [8].
Subsequently, the criteria to select the tagging particles and to reconstruct
the vertex charge are re-tuned, using the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and the
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ control channels. An iterative
procedure is used to find the selection criteria which maximize the tagging
power $\varepsilon_{\rm eff}$.
Only charged particles reconstructed with a good quality of the track fit are
used. In order to reject poorly reconstructed tracks, the track is required to
have a polar angle with respect to the beamline larger than 12 $\rm\,mrad$ and
a momentum larger than 2 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Moreover, in
order to avoid possible duplications of the signal tracks, the selected
particles are required to be outside a cone of 5 $\rm\,mrad$ formed around any
daughter of the signal $B$. To reject tracks coming from other primary
interactions in the same bunch crossing, the impact parameter significance
with respect to these pile-up ($\rm PU$) vertices, $\rm{IP_{PU}}/\sigma_{\rm
IP_{PU}}>3$, is required.
### 3.1 Single-particle taggers
The tagging particles are selected exploiting the properties of the $b$-hadron
decay. A large impact parameter significance with respect to the primary
vertex ($\rm{IP}/\sigma_{\rm IP}$) and a large transverse momentum $p_{\rm T}$
are required. Furthermore, particle identification cuts are used to define
each tagger based on the information from the RICH, calorimeter and muon
systems. For this purpose, the differences between the logarithm of the
likelihood for the muon, electron, kaon or proton and the pion hypotheses
(referred as ${\rm DLL}_{\mu-\pi}$, ${\rm DLL}_{e-\pi}$, ${\rm DLL}_{K-\pi}$
and ${\rm DLL}_{p-\pi}$) are used. The detailed list of selection criteria is
reported in Table 1. Additional criteria are used to identify the leptons.
Muons are required not to share hits in the muon chambers with other tracks,
in order to avoid mis-identification of tracks which are close to the real
muon. Electrons are required to be below a certain threshold in the ionization
charge deposited in the silicon layers of the VELO, in order to reduce the
number of candidates coming from photon conversions close to the interaction
point. An additional cut on the ratio of the particle energy $E$ as measured
in the electromagnetic calorimeter and the momentum $p$ of the candidate
electron measured with the tracking system, $E/p>0.6$, is applied.
In the case of multiple candidates from the same tagging algorithm, the
single-particle tagger with the highest $p_{\rm T}$ is chosen and its charge
is used to define the flavour of the signal $B$.
Table 1: Selection criteria for the OS muon, electron and kaon taggers. Tagger | min $p_{\rm T}$ | min $p$ | min (${\rm IP}/\sigma_{\rm IP}$) | Particle identification | min (${\rm IP_{PU}}/\sigma_{\rm IP_{PU}}$)
---|---|---|---|---|---
| [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | [${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ ] | | cuts |
$\mu$ | 1.2 | 2.0 | - | ${\rm DLL}_{\mu-\pi}>2.5$ | 3.0
$e$ | 1.0 | 2.0 | 2.0 | ${\rm DLL}_{e-\pi}>4.0$ | 3.0
$K$ | 0.8 | 5.9 | 4.0 | ${\rm DLL}_{K-\pi}>6.5$ | 4.7
| | | | ${\rm DLL}_{K-p}>-3.5$ |
### 3.2 Vertex charge tagger
The vertex charge tagger is based on the inclusive reconstruction of a
secondary vertex corresponding to the decay of the tagging $B$. The vertex
reconstruction consists of building a composite candidate from two tracks with
a transverse momentum $\mbox{$p_{\rm
T}$}>0.15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\rm{IP}/\sigma_{\rm
IP}>2.5$. The pion mass is attributed to the tracks. Moreover, good quality of
the vertex reconstruction is required and track pairs with an invariant mass
compatible with a $K^{0}_{\rm\scriptscriptstyle S}$ meson are excluded. For
each reconstructed candidate the probability that it originates from a
$b$-hadron decay is estimated from the quality of the vertex fit as well as
from the geometric and kinematic properties. Among the possible candidates the
one with the highest probability is used. Tracks that are compatible with
coming from the two track vertex but do not originate from the primary vertex
are added to form the final candidate. Additional requirements are applied to
the tracks asspociated to the reconstructed secondary vertex: total momentum
$>10$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, total $p_{\rm T}$ $>1.5$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, total invariant mass $>0.5$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and the sum of $\rm{IP}/\sigma_{\rm
IP}$ of all tracks $>10$.
Finally, the charge of the tagging $B$ is calculated as the sum of the charges
$Q_{i}$ of all the tracks associated to the vertex, weighted with their
transverse momentum to the power $\kappa$
$Q_{\rm vtx}=\frac{\Sigma_{i}Q_{i}p^{\kappa}_{\rm
Ti}}{\Sigma_{i}p^{\kappa}_{\rm Ti}},$ (3)
where the value $\kappa=0.4$ optimizes the tagging power. Events with $|Q_{\rm
vtx}|<0.275$ are rejected as untagged.
### 3.3 Mistag probabilities and combination of taggers
For each tagger $i$, the probability $\eta_{i}$ of the tag decision to be
wrong is estimated by using properties of the tagger and of the event itself.
This mistag probability is evaluated by means of a neural network trained on
simulated $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ events to identify the correct flavour of the signal $B$ and
subsequently calibrated on data as explained in Sect. 5.
The inputs to each of the neural networks are the signal $B$ transverse
momentum, the number of pile-up vertices, the number of tracks preselected as
tagging candidates and various geometrical and kinematic properties of the
tagging particle ($p$, $p_{\rm T}$ and ${\rm IP}/\sigma_{\rm IP}$ of the
particle), or of the tracks associated to the secondary vertex (the average
values of $p_{\rm T}$, of $\rm IP$, the reconstructed invariant mass and the
absolute value of the vertex charge).
If there is more than one tagger available per event, the decisions provided
by all available taggers are combined into a final decision on the initial
flavour of the signal $B$. The combined probability $P(b)$ that the meson
contains a $b$-quark is calculated as
$P(b)=\frac{p(b)}{p(b)+p(\bar{b})},\qquad\quad P(\bar{b})=1-P(b),$ (4)
where
$p(b)=\prod_{i}\left(\frac{1+d_{i}}{2}-d_{i}(1-\eta_{i})\right),\qquad\quad
p(\bar{b})=\prod_{i}\left(\frac{1-d_{i}}{2}+d_{i}(1-\eta_{i})\right).$ (5)
Here, $d_{i}$ is the decision taken by the $i$-th tagger based on the charge
of the particle with the convention $d_{i}=1(-1)$ for the signal $B$
containing a $\bar{b}$($b$) quark and $\eta_{i}$ the corresponding predicted
mistag probability. The combined tagging decision and the corresponding mistag
probability are $d=-1$ and $\eta=1-P(b)$ if $P(b)>P(\bar{b})$, otherwise
$d=+1$ and $\eta=1-P(\bar{b})$.
The contribution of taggers with a poor tagging power is limited by requiring
the mistag probabilities of the kaon and the vertex charge to be less than
0.46.
Due to the correlation among taggers, which is neglected in Eq. 5, the
combined probability is slightly overestimated. The largest correlation occurs
between the vertex charge tagger and the other OS taggers, since the secondary
vertex may include one of these particles. To correct for this overestimation,
the combined OS probability is calibrated on data, as described in Sect. 5.
## 4 Control channels
The flavour-specific $B$ decay modes
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ are used for the tagging
analysis. All three channels are useful to optimize the performance of the OS
tagging algorithm and to calibrate the mistag probability. The first two
channels are chosen as representative control channels for the decays
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}$,
which are used for the measurement of the $B^{0}_{s}$ mixing phase $\phi_{s}$
[2, 3], and the last channel allows detailed studies given the high event
yield of the semileptonic decay mode. All $B$ decay modes with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson in the final state share
the same trigger selection and common offline selection criteria, which
ensures a similar performance of the tagging algorithms. Two trigger
selections are considered, with or without requirements on the $\rm IP$ of the
tracks. They are labelled “lifetime biased” and “lifetime unbiased”
respectively.
### 4.1 Analysis of the $\boldsymbol{B^{+}\rightarrow J/\psi K^{+}}$ channel
The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
candidates are selected by combining $J/\psi\rightarrow\mu^{+}\mu^{-}$ and
$K^{+}$ candidates. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons
are selected by combining two muons with transverse momenta $\mbox{$p_{\rm
T}$}>$ 0.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ that form a common vertex of
good quality and have an invariant mass in the range $3030-3150$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $K^{+}$ candidates are
required to have transverse momenta $\mbox{$p_{\rm T}$}>1$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and momenta $\mbox{$p$}>10$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to form a common vertex of good
quality with the $J/\psi$ candidate with a resulting invariant mass in a
window $\pm 90$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the $B^{+}$
mass. Additional requirements on the particle identification of muons and
kaons are applied to suppress the background contamination. To enhance the
sample of signal events and reduce the dominant background contamination from
prompt $J/\psi$ mesons combined with random kaons, only the events with a
reconstructed decay time of the $B^{+}$ candidate $t>0.3$${\rm\,ps}$ are
selected. The decay time $t$ and the invariant mass $m$ of the $B^{+}$ meson
are extracted from a vertex fit that includes a constraint on the associated
primary vertex, and a constraint on the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mass for the evaluation of the $J/\psi K$ invariant mass. In case of
multiple $B$ candidates per event, only the one with the smallest vertex fit
$\chi^{2}$ is considered.
The signal events are statistically disentangled from the background, which is
dominated by partially reconstructed $b$-hadron decays to $J/\psi K^{+}X$
(where $X$ represents any other particle in the decay), by means of an
unbinned maximum likelihood fit to the reconstructed $B^{+}$ mass and decay
time. In total $\sim 85\,000$ signal events are selected with a background to
signal ratio $B/S\sim 0.035$, calculated in a window of $\pm 40$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ centred around the $B^{+}$ mass.
The mass fit model is based on a double Gaussian distribution peaking at the
$B^{+}$ mass for the signal and an exponential distribution for the
background. The time distributions of both the signal and the background are
assumed to be exponential, with separate decay constants. The fraction of
right, wrong or untagged events in the sample is determined according to a
probability density function (PDF), ${\cal P}(r)$, that depends on the tagging
response $r$, defined by
${\cal P}(r)=\left\\{\begin{array}[]{ll}{\varepsilon_{\rm
tag}}(1-\omega)&\mbox{$r$=``right tag decision''}\\\ {\varepsilon_{\rm
tag}}~{}\omega&\mbox{$r$=``wrong tag decision''}\\\ 1-{\varepsilon_{\rm
tag}}&\mbox{$r$=``no tag decision''.}\end{array}\right.$ (6)
The parameters $\omega$ and $\varepsilon_{\rm tag}$ (defined in Eq. 2) are
different for signal and background. Fig. 1 shows the mass distribution of the
selected and tagged events, together with the superimposed fit.
Figure 1: Mass distribution of OS tagged
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events.
Black points are data, the solid blue line, red dotted line and green area are
the overall fit, the signal and the background components, respectively.
### 4.2 Analysis of the $\boldsymbol{B^{0}\\!\rightarrow
D^{*-}\mu^{+}\nu_{\mu}}$ channel
The $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ channel is selected by
requiring that a muon and the decay $D^{*-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\rightarrow K^{+}\pi^{-})\pi^{-}$
originate from a common vertex, displaced with respect to the $pp$ interaction
point. The muon and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
transverse momenta are required to be larger than 0.8
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and 1.8
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ respectively. The selection criteria
exploit the long $B^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ lifetimes by applying cuts on the
impact parameters of the daughter tracks, on the pointing of the reconstructed
$B^{0}$ momentum to the primary vertex, on the difference between the $z$
coordinate of the $B^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ vertices, and on the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ flight distance. Additional cuts
are applied on the muon and kaon particle identification and on the quality of
the fits of all tracks and vertices. In case of multiple $B$ candidates per
event the one with the smallest impact parameter significance with respect to
the primary vertex is considered. Only events triggered in the HLT by a single
particle with large momentum, large transverse momentum and large $\rm IP$ are
used. In total, the sample consists of $\sim$482 000 signal events.
Even though the final state is only partially reconstructed due to the missing
neutrino, the contamination of background is small and the background to
signal ratio $B/S$ is measured to be $\sim 0.14$ in the signal mass region.
The main sources of background are events containing a $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ originating from a $b$-hadron
decay (referred to as $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$-from-$B$), events with a $D^{*-}$
not from a $b$-hadron decay, decays of $B^{+}$ mesons to the same particles as
the signal together with an additional pion (referred to as $B^{+}$) and
combinatorial background. The different background sources can be disentangled
from the signal by exploiting the different distributions of the observables
$m$$=$$m_{K\pi}$, $\Delta m$$=$$m_{K\pi\pi}$$-$$m_{K\pi}$, the reconstructed
$B^{0}$ decay time $t$ and the mixing state $q$. The mixing state is
determined by comparing the flavour of the reconstructed signal $B^{0}$ at
decay time with the flavour indicated by the tagging decision (flavour at
production time). For unmixed (mixed) events $q$$=$$+$$1$($-$$1$) while for
untagged events $q$$=$$0$. The decay time is calculated using the measured
$B^{0}$ decay length, the reconstructed $B^{0}$ momentum and a correction for
the missing neutrino determined from simulation. It is parametrized as a
function of the reconstructed $B^{0}$ invariant mass.
An extended unbinned maximum likelihood fit is performed by defining a PDF for
the observables ($m,\Delta m,t,q$) as a product of one PDF for the masses and
one for the $t$ and $q$ observables. For the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $D^{*-}$ mass peaks two double
Gaussian distributions with common mean are used, while a parametric function
motivated by available phase space is used to describe the $\Delta m$
distributions of the $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$-from-$B$, and combinatorial
background components. The decay time distribution of the signal consists of
mixed, unmixed and untagged events, and is given by
${\cal P}^{\rm s}(t,q)\propto\left\\{\begin{array}[]{ll}{\varepsilon_{\rm
tag}}~{}a(t)\left\\{e^{-t/\tau_{B^{0}}}\left[1+q(1-2\omega)\cos(\Delta
m_{d}t)\right]\otimes R(t-t^{\prime})\right\\}&\mbox{ if $q=\pm 1$}\\\
(1-{\varepsilon_{\rm tag}})a(t)\left\\{e^{-t/\tau_{B^{0}}}\otimes
R(t-t^{\prime})\right\\}&\mbox{ if $q=0$},\\\ \end{array}\right.$ (7)
where $\Delta m_{d}$ and $\tau_{B^{0}}$ are the $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing frequency and $B^{0}$
lifetime. The decay time acceptance function is denoted by $a(t)$ and
$R(t-t^{\prime})$ is the resolution model, both extracted from simulation. A
double Gaussian distribution with common mean is used for the decay time
resolution model. In Eq. 7 the tagging parameters are assumed to be the same
for $B$ and $\bar{B}$-mesons.
The decay time distributions for the $B^{+}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$-from-$B$ background components are
taken as exponentials convolved by the resolution model and multiplied by the
same acceptance function as used for the signal. For the prompt $D^{*}$ and
combinatorial background, Landau distributions with independent parameters are
used. The dependence on the mixing observable $q$ is the same as for the
signal. The tagging parameters $\varepsilon_{\rm tag}$ and $\omega$ of the
signal and of each background component are varied independently in the fit,
except for the $B^{+}$ background where they are assumed to be equal to the
parameters in the signal decay. Figure 2 shows the distributions of the mass
and decay time observables used in the maximum likelihood fit. The raw
asymmetry is defined as
${\cal A^{\rm raw}}(t)=\frac{N^{\rm unmix}(t)-N^{\rm mix}(t)}{N^{\rm
unmix}(t)+N^{\rm mix}(t)}$ (8)
where $N^{\rm mix}$ ($N^{\rm unmix}$) is the number of tagged events which
have (not) oscillated at decay time $t$. From Eq. 7 it follows that the
asymmetry for signal is given by
${\cal A}(t)=(1-2\omega)\cos(\Delta m_{d}\,t).$ (9)
Figure 3 shows the raw asymmetry for the subset of events in the signal mass
region that are tagged with the OS tagger combination. At small decay times
the asymmetry decreases due to the contribution of background events, ${\cal
A}\simeq 0$. The value of $\Delta m_{d}$ was fixed to $\Delta m_{d}$ $=0.507$
${\rm\,ps^{-1}}$ [9]. Letting the $\Delta m_{d}$ parameter vary in the fit
gives consistent results.
\begin{overpic}[width=432.48048pt]{Fig2a.pdf} \put(80.0,68.0){\small{(a)}}
\end{overpic}
\begin{overpic}[width=432.48048pt]{Fig2b.pdf} \put(80.0,68.0){\small{(b)}}
\end{overpic}
\begin{overpic}[width=432.48048pt]{Fig2c.pdf} \put(80.0,68.0){\small{(c)}}
\end{overpic}
Figure 2: Distributions of (a) $K^{+}\pi^{-}$ invariant mass, (b) mass
difference $m(K\pi\pi)$$-$$m(K\pi)$ and (c) decay time of the
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ events. Black points with errors
are data, the blue curve is the fit result. The other lines represent signal
(red dot-dashed), $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$-from-$B$
decay background (gray dashed), $B^{+}$ background (green short dashed),
$D^{*}$ prompt background (magenta solid). The combinatorial background is the
magenta filled area. Figure 3: Raw mixing asymmetry of $B^{0}\\!\rightarrow
D^{*-}\mu^{+}\nu_{\mu}$ events in the signal mass region when using the
combination of all OS taggers. Black points are data and the red solid line is
the result of the fit. The lower plot shows the pulls of the residuals with
respect to the fit.
### 4.3 Analysis of the $\boldsymbol{B^{0}\rightarrow J/\psi K^{*0}}$ channel
The $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
channel is used to extract the mistag rate through a fit of the flavour
oscillation of the $B^{0}$ mesons as a function of the decay time. The flavour
of the $B^{0}$ meson at production time is determined from the tagging
algorithms, while the flavour at the decay time is determined from the
$K^{*0}$ flavour, which is in turn defined by the kaon charge.
The $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
candidates are selected from ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
$\rightarrow$ $\mu^{+}\mu^{-}$ and $K^{*0}$ $\rightarrow$ $K^{+}\pi^{-}$
decays. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are selected
by the same selection as used for the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel, described in Sect. 4.1. The $K^{*0}$ candidates are reconstructed
from two good quality charged tracks identified as $K^{+}$ and $\pi^{-}$. The
reconstructed $K^{*0}$ meson is required to have a transverse momentum higher
than 1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a good quality vertex and an
invariant mass within $\pm$ 70${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of
the nominal $K^{*0}$ mass. Combinations of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $K^{*0}$ candidates are
accepted as $B^{0}$ candidates if they form a common vertex with good quality
and an invariant mass in the range
$5100-5450$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The $B^{0}$ transverse
momentum is required to be higher than 2
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The decay time and the invariant mass
of the $B^{0}$ are extracted from a vertex fit with an identical procedure as
for the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel, by applying a constraint to the associated primary vertex, and a
constraint to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. In case
of multiple $B$ candidates per event, only the candidate with the smallest
$\chi^{2}$ of the vertex is kept.
Only events that were triggered by the “lifetime unbiased” selection are kept.
The $B^{0}$ candidates are required to have a decay time higher than 0.3 ps to
remove the large combinatorial background due to prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production. The sample contains
$\sim 33\,000$ signal events.
The decay time distribution of signal events is parametrized as in Eq. 7,
without the acceptance correction. The background contribution, with a
background to signal ratio $B/S\sim 0.29$, is due to misreconstructed
$b$-hadron decays, where a dependence on the decay time is expected (labelled
“long-lived” background). We distinguish two long-lived components. The first
corresponds to events where one or more of the four tracks originate from a
long-lived particle decay, but where the flavour of the reconstructed $K^{*0}$
is not correlated with a true $b$-hadron. Its decay time distribution is
therefore modelled by a decreasing exponential. In the second long-lived
background component, one of the tracks used to build the $K^{*0}$ originated
from the primary vertex, hence the correlation between the $K^{*0}$ and the
$B$ flavour is partially lost. Its decay time distribution is more “signal-
like”, i.e. it is a decreasing exponential with an oscillation term, but with
different mistag fraction and lifetime, left as free parameters in the fit.
The signal and background decay time distributions are convolved with the same
resolution function, extracted from data. The mass distributions, shown in
Fig. 4, are described by a double Gaussian distribution peaking at the $B^{0}$
mass for the signal component, and by an exponential with the same exponent
for both long-lived backgrounds.
The OS mistag fraction is extracted from a fit to all tagged data, with the
values for the $B^{0}$ lifetime and $\Delta m_{d}$ fixed to the world average
[9]. Figure 5 shows the time-dependent mixing asymmetry in the signal mass
region, obtained using the information of the OS tag decision. Letting the
$\Delta m_{d}$ parameter vary in the fit gives consistent results.
Figure 4: Mass distribution of OS tagged
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
events. Black points are data, the solid blue line, red dotted line and green
area are the overall fit, the signal and the background components,
respectively. Figure 5: Raw mixing asymmetry of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
in the signal mass region, for all OS tagged events. Black points are data and
the red solid line is the result of the fit. The lower plot shows the pulls of
the residuals with respect to the fit.
## 5 Calibration of the mistag probability on data
For each individual tagger and for the combination of taggers, the calculated
mistag probability ($\eta$) is obtained on an event-by-event basis from the
neural network output. The values are calibrated in a fit using the measured
mistag fraction ($\omega$) from the self-tagged control channel
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$. A
linear dependence between the measured and the calculated mistag probability
for signal events is used, as suggested by the data distribution,
$\omega(\eta)=p_{0}+p_{1}(\eta-\langle\eta\rangle)\;,$ (10)
where $p_{0}$ and $p_{1}$ are parameters of the fit and $\langle\eta\rangle$
is the mean calculated mistag probability. This parametrization is chosen to
minimize the correlation between the two parameters. Deviations from
$p_{0}=\langle\eta\rangle$ and $p_{1}=1$ would indicate that the calculated
mistag probability should be corrected.
In order to extract the $p_{0}$ and $p_{1}$ calibration parameters, an
unbinned maximum likelihood fit to the mass, tagging decision and mistag
probability $\eta$ observable is performed. The fit parametrization takes into
account the probability density function of $\eta$, $\cal P(\eta)$, that is
extracted from data for signal and background separately, using events in
different mass regions. For example, the PDF for signal events from Eq. 6 then
becomes
${\cal P^{\rm s}}(r,\eta)=\left\\{\begin{array}[]{ll}{\varepsilon_{\rm
tag}}\left(1-\omega(\eta)\right)\cal P^{\rm s}(\eta)&\mbox{$r$=``right tag
decision''}\\\ {\varepsilon_{\rm tag}}~{}\omega(\eta)\cal P^{\rm
s}(\eta)&\mbox{$r$=``wrong tag decision''}\\\ 1-{\varepsilon_{\rm
tag}}&\mbox{$r$=``no tag decision''.}\end{array}\right.$ (11)
The measured mistag fraction of the background is assumed to be independent
from the calculated mistag probability, as confirmed by the distribution of
background events.
The calibration is performed on part of the data sample in a two-step
procedure. Each tagger is first calibrated individually. The results show
that, for each single tagger, only a minor adjustment of $p_{0}$ with respect
to the starting calibration of the neural network, performed on simulated
events, is required. In particular, the largest correction is $p_{0}-$
$\langle\eta\rangle=$ 0.033$\pm$0.005 in the case of the vertex charge tagger,
while the deviations from unity of the $p_{1}$ parameter are about 10%,
similar to the size of the corresponding statistical errors. In a second step
the calibrated mistag probabilities are combined and finally the combined
mistag probability is calibrated. This last step is necessary to correct for
the small underestimation ($p_{0}-\langle\eta\rangle=$ 0.022$\pm$0.003) of the
combined mistag probability due to the correlation among taggers neglected in
the combination procedure. The calibrated mistag is referred to as $\eta_{c}$
in the following.
Figure 6 shows the distribution of the mistag probability for each tagger and
for their combination, as obtained for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
selected in a $\pm 24$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window
around the $B^{+}$ mass.
Figure 6: Distribution of the calibrated mistag probability for the single OS
taggers and their combination for
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
selected in a $\pm 24$ ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ mass window
around the $B^{+}$ mass.
## 6 Tagging performance
The tagging performances of the single taggers and of the OS combination
measured after the calibration of the mistag probability are shown in Tables
2, 3 and 4 for the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$, $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ channels,
respectively.
The performance of the OS combination is evaluated in different ways. First
the average performance of the OS combination is calculated, giving the same
weight to each event. In this case, the best tagging power is obtained by
rejecting the events with a poor predicted mistag probability $\eta_{c}$
(larger than $0.42$), despite a lower $\varepsilon_{\rm tag}$. Additionally,
to better exploit the tagging information, the tagging performance is
determined on independent samples obtained by binning the data in bins of
$\eta_{c}$. The fits described in the previous sections are repeated for each
sub-sample, after which the tagging performances are determined. As the
samples are independent, the tagging efficiencies and the tagging powers are
summed and subsequently the effective mistag is extracted. The total tagging
power increases by about 30% with respect to the average value, as shown in
the last line of Tables 2-4.
The measured tagging performance is similar among the three channels. The
differences between the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
results are large in absolute values, but still compatible given the large
statistical uncertainties of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
results. Differences between the tagging efficiency in the
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ and the $B\rightarrow J/\psi X$
channels were shown in previous MC studies to be related to the different $B$
momentum spectra and to different contributions to the trigger decision [8].
Table 2: Tagging performance in the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ channel. Uncertainties are statistical only. Taggers | $\varepsilon_{\rm tag}$[%] | $\omega$ [%] | ${\varepsilon_{\rm tag}}(1-2\omega)^{2}$ [%]
---|---|---|---
$\mu$ | 4.8$\pm$0.1 | 29.9$\pm$0.7 | 0.77$\pm$0.07
$e$ | 2.2$\pm$0.1 | 33.2$\pm$1.1 | 0.25$\pm$0.04
$K$ | 11.6$\pm$0.1 | 38.3$\pm$0.5 | 0.63$\pm$0.06
$Q_{\mathrm{vtx}}$ | 15.1$\pm$0.1 | 40.0$\pm$0.4 | 0.60$\pm$0.06
OS average ($\eta_{c}<$0.42) | 17.8$\pm$0.1 | 34.6$\pm$0.4 | 1.69$\pm$0.10
OS sum of $\eta_{c}$ bins | 27.3$\pm$0.2 | 36.2$\pm$0.5 | 2.07$\pm$0.11
Table 3: Tagging performance in the $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ channel. Uncertainties are statistical only. Taggers | $\varepsilon_{\rm tag}$[%] | $\omega$ [%] | ${\varepsilon_{\rm tag}}(1-2\omega)^{2}$ [%]
---|---|---|---
$\mu$ | 4.8$\pm$0.1 | 34.3$\pm$1.9 | 0.48$\pm$0.12
$e$ | 2.2$\pm$0.1 | 32.4$\pm$2.8 | 0.27$\pm$0.10
$K$ | 11.4$\pm$0.2 | 39.6$\pm$1.2 | 0.49$\pm$0.13
$Q_{\mathrm{vtx}}$ | 14.9$\pm$0.2 | 41.7$\pm$1.1 | 0.41$\pm$0.11
OS average ($\eta_{c}<$0.42) | 17.9$\pm$0.2 | 36.8$\pm$1.0 | 1.24$\pm$0.20
OS sum of $\eta_{c}$ bins | 27.1$\pm$0.3 | 38.0$\pm$0.9 | 1.57$\pm$0.22
Table 4: Tagging performance in the $B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ channel. Uncertainties are statistical only. Taggers | $\varepsilon_{\rm tag}$[%] | $\omega$ [%] | ${\varepsilon_{\rm tag}}(1-2\omega)^{2}$ [%]
---|---|---|---
$\mu$ | 6.08$\pm$0.04 | 33.3$\pm$0.4 | 0.68$\pm$0.04
e | 2.49$\pm$0.02 | 34.3$\pm$0.7 | 0.25$\pm$0.02
K | 13.36$\pm$0.05 | 38.3$\pm$0.3 | 0.74$\pm$0.04
$Q_{\mathrm{vtx}}$ | 16.53$\pm$0.06 | 41.5$\pm$0.3 | 0.48$\pm$0.03
OS average ($\eta_{c}<$0.42) | 20.56$\pm$0.06 | 36.1$\pm$0.3 | 1.58$\pm$0.06
OS sum of $\eta_{c}$ bins | 30.48$\pm$0.08 | 37.0$\pm$0.3 | 2.06$\pm$0.06
## 7 Systematic uncertainties
The systematic uncertainties on the calibration parameters $p_{0}$ and $p_{1}$
are studied by repeating the calibration procedure on
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
for different conditions. The difference is evaluated between the value of the
fitted parameter and the reference value, and is reported in the first row of
Table 5. Several checks are performed of which the most relevant are reported
in Table 6 and are described below:
Table 5: Fit values and correlations of the OS combined mistag calibration parameters measured in the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ channels. The uncertainties are statistical only. Channel | $p_{0}$ | $p_{1}$ | $\langle\eta_{c}\rangle$ | $p_{0}-p_{1}\langle\eta_{c}\rangle$ | $\rho(p_{0},p_{1})$
---|---|---|---|---|---
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $0.384\pm 0.003$ | $1.037\pm 0.038$ | $0.379$ | $-0.009\pm 0.014$ | $0.14$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $0.399\pm 0.008$ | $1.016\pm 0.102$ | $0.378$ | $\;\;\;0.015\pm 0.039$ | $0.05$
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ | $0.395\pm 0.002$ | $1.022\pm 0.026$ | $0.375$ | $\;\;\;0.008\pm 0.010$ | $0.14$
Table 6: Systematic uncertainties on the calibration parameters $p_{0}$ and $p_{1}$ obtained with $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events. Systematic effect | $\delta p_{0}$ | $\delta p_{1}$ | $\delta(p_{0}-p_{1}\langle\eta_{c}\rangle)$
---|---|---|---
Run period | $\pm 0.005$ | $\pm 0.003$ | $\pm 0.004$
$B$-flavour | $\pm 0.008$ | $\pm 0.067$ | $\pm 0.020$
Fit model assumptions ${\cal P}(\eta)$ | $<\pm 0.001$ | $\pm 0.005$ | $\pm 0.002$
Total | $\pm 0.009$ | $\pm 0.07$ | $\pm 0.02$
* •
The data sample is split according to the run periods and to the magnet
polarity, in order to check whether possible asymmetries of the detector
efficiency, or of the alignment accuracy, or variations in the data-taking
conditions introduce a difference in the tagging calibration.
* •
The data sample is split according to the signal flavour, as determined by the
reconstructed final state. In fact, the calibration of the mistag probability
for different $B$ flavours might be different due to the different
particle/antiparticle interaction with matter or possible detector
asymmetries. In this case a systematic uncertainty has to be considered,
unless the difference is explicitly taken into account when fitting for
$C\\!P$ asymmetries.
* •
The distribution of the mistag probability in the fit model, ${\cal P}(\eta)$,
is varied either by assuming the signal and background distributions to be
equal or by swapping them. In this way possible uncertainties related to the
fit model are considered.
In addition, the stability of the calibration parameters is verified for
different bins of transverse momentum of the signal $B$.
The largest systematic uncertainty in Table 6 originates from the dependence
on the signal flavour. As a cross check this dependence is also measured with
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ events, repeating the calibration
after splitting the sample according to the signal decay flavour. The
differences in this case are $\delta p_{0}=\pm 0.009$ and $\delta p_{1}=\pm
0.009$, where the latter is smaller than in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel. Both for the run period dependence and for the signal flavour the
variations of $\delta p_{0}$ and $\delta p_{1}$ are not statistically
significant. However, as a conservative estimate of the total systematic
uncertainty on the calibration parameters, all the contributions in Table 6
are summed in quadrature. The tagging efficiencies do not depend on the
initial flavour of the signal $B$. In the case of the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ channel
the values are $(27.4\pm 0.2)$% for the $B^{+}$ and $(27.1\pm 0.2)$% for the
$B^{-}$.
## 8 Comparison of decay channels
The dependence of the calibration of the OS mistag probability on the decay
channel is studied. The values of $p_{0}$, $p_{1}$ and
$\langle\eta_{c}\rangle$ measured on the whole data sample for all the three
channels separately, are shown in Table 5. The parameters $p_{1}$ are
compatible with 1, within the statistical uncertainty. The differences
$p_{0}-p_{1}\langle\eta_{c}\rangle$, shown in the fifth column, are compatible
with zero, as expected. In the last column the correlation coefficients are
shown.
To extract the calibration parameters in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
channel an unbinned maximum likelihood fit to mass, time and $\eta_{c}$ is
performed. In analogy to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel, the fit uses the probability density functions of $\eta_{c}$,
extracted from data for signal and background separately by using the sPlot
[10] technique. The results confirm the calibration performed in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel, albeit with large uncertainties. The results for the
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ channel are obtained from a fit
to independent samples corresponding to different ranges of the calculated
mistag probability as shown in Fig. 7. The trigger and offline selections, as
well as signal spectra, differ for this decay channel with respect to the
channels containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson.
Therefore the agreement in the resulting parameters is a validation of the
calibration and its applicability to $B$ decays with different topologies. In
Fig. 8 the dependency of the measured OS mistag fraction as a function of the
mistag probability is shown for the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ signal events. The superimposed
linear fit corresponds to the parametrization of Eq. 10 and the parameters of
Table 5.
\begin{overpic}[width=341.43306pt]{Fig7a}
\put(35.0,68.0){\small{$0.43\leq\eta_{c}<0.50$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7b}
\put(35.0,68.0){\small{$0.38\leq\eta_{c}<0.43$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7c}
\put(35.0,68.0){\small{$0.35\leq\eta_{c}<0.38$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7d}
\put(35.0,68.0){\small{$0.31\leq\eta_{c}<0.35$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7e}
\put(35.0,68.0){\small{$0.24\leq\eta_{c}<0.31$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7f}
\put(35.0,68.0){\small{$0.17\leq\eta_{c}<0.24$}} \end{overpic}
\begin{overpic}[width=341.43306pt]{Fig7g}
\put(35.0,68.0){\small{$\eta_{c}<0.17$}} \end{overpic}
Figure 7: Raw mixing asymmetry as a function of $B$ decay time in
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ events, in the signal mass
region, using the OS tagger. Events are split into seven samples of decreasing
mistag probability $\eta_{c}$.
Figure 8: Measured mistag fraction ($\omega$) versus calculated mistag
probability ($\eta_{c}$) calibrated on
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ signal
events for the OS tagger, in background subtracted events. Left and right
plots correspond to $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ signal events.
Points with errors are data, the red lines represent the result of the mistag
calibration, corresponding to the parameters of Table 5.
The output of the calibrated flavour tagging algorithms will be used in a
large variety of time-dependent asymmetry measurements, involving different
$B$ decay channels. Figure 9 shows the calculated mistag distributions in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
channels. These events are tagged, triggered by the “lifetime unbiased” lines
and have an imposed cut of $t>0.3{\rm\,ps}$. The event selection for the decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
described elsewhere [3]. The distributions of the calculated OS mistag
fractions are similar among the channels and the average does not depend on
the $p_{\rm T}$ of the $B$. It has been also checked that the mistag
probability does not depend on the signal $B$ pseudorapidity.
\begin{overpic}[angle={0},width=450.69414pt]{Fig9a.pdf}
\put(85.0,68.0){\small{(a)}} \end{overpic}
\begin{overpic}[angle={0},width=450.69414pt]{Fig9b.pdf}
\put(85.0,68.0){\small{(b)}} \end{overpic}
\begin{overpic}[angle={0},width=450.69414pt]{Fig9c.pdf}
\put(85.0,68.0){\small{(c)}} \end{overpic}
\begin{overpic}[angle={0},width=450.69414pt]{Fig9d.pdf}
\put(85.0,68.0){\small{(d)}} \end{overpic}
\begin{overpic}[angle={0},width=450.69414pt]{Fig9e.pdf}
\put(85.0,68.0){\small{(e)}} \end{overpic}
\begin{overpic}[angle={0},width=450.69414pt]{Fig9f.pdf}
\put(85.0,68.0){\small{(f)}} \end{overpic}
Figure 9: Top: calibrated mistag probability distribution for (a)
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, (b)
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
(c) $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
events. Bottom: distributions of the mean calibrated OS mistag probability as
a function of signal $p_{\rm T}$ for the (d) $B^{+}$, (e) $B^{0}$ and (f)
$B^{0}_{s}$ channels. The plots show signal events extracted with the sPlot
technique and with the requirement $t>0.3$${\rm\,ps}$. The three $p_{\rm T}$
distributions are fitted with straight lines and the slopes are compatible
with zero.
## 9 Event-by-event results
In order to fully exploit the tagging information in the $C\\!P$ asymmetry
measurements, the event-by-event mistag probability is used to weight the
events accordingly. The effective efficiency is calculated by summing the
mistag probabilities on all signal events
$\sum_{i}{(1-2\omega(\eta^{i}_{c})^{2})}/N$. We underline that the use of the
per-event mistag probability allows the effective efficiency to be calculated
on any set of selected events, also for non flavour-specific channels. Table 7
reports the event-by-event tagging power obtained using the calibration
parameters determined with the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
as reported in Table 5. The uncertainties are obtained by propagating the
statistical and systematic uncertainties of the calibration parameters. In
addition to the values for the three control channels the result obtained for
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
events is shown. For all channels the signal is extracted using the sPlot
technique. The results for the tagging power are compatible among the channels
containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. The higher
value for $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ is related to the
higher tagging efficiency.
Table 7: Tagging efficiency, mistag probability and tagging power calculated from event-by-event probabilities for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, $B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ and $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ signal events. The quoted uncertainties are obtained propagating the statistical (first) and systematic (second) uncertainties on the calibration parameters determined from the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events. Channel | ${\varepsilon_{\rm tag}}$ [%] | $\omega\,$ [%] | ${\varepsilon_{\rm tag}}{\cal D}^{2}$ [%]
---|---|---|---
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | $27.3\pm 0.1$ | $36.1\pm 0.3\pm 0.8$ | $2.10\pm 0.08\pm 0.24$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | $27.3\pm 0.3$ | $36.2\pm 0.3\pm 0.8$ | $2.09\pm 0.09\pm 0.24$
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$ | $30.1\pm 0.1$ | $35.5\pm 0.3\pm 0.8$ | $2.53\pm 0.10\pm 0.27$
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ | $24.9\pm 0.5$ | $36.1\pm 0.3\pm 0.8$ | $1.91\pm 0.08\pm 0.22$
## 10 Summary
Flavour tagging algorithms were developed for the measurement of time-
dependent asymmetries at the LHCb experiment. The opposite-side algorithms
rely on the pair production of $b$ and $\bar{b}$ quarks and infer the flavour
of the signal $B$ meson from the identification of the flavour of the other
$b$ hadron. They use the charge of the lepton ($\mu$, $e$) from semileptonic
$B$ decays, the charge of the kaon from the $b\rightarrow c\rightarrow s$
decay chain or the charge of the inclusive secondary vertex reconstructed from
$b$-hadron decay products. The decision of each tagger and the probability of
the decision to be incorrect are combined into a single opposite side decision
and mistag probability. The use of the event-by-event mistag probability fully
exploits the tagging information and estimates the tagging power also in non
flavour-specific decay channels.
The performance of the flavour tagging algorithms were measured on data using
three flavour-specific decay modes
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow D^{*-}\mu^{+}\nu_{\mu}$. The
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ channel
was used to optimize the tagging power and to calibrate the mistag
probability. The calibration parameters measured in the three channels are
compatible within two standard deviations.
By using the calibration parameters determined from
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ events
the OS tagging power was determined to be ${\varepsilon_{\rm
tag}}(1-2\omega)^{2}$ = (2.10$\pm$0.08$\pm$0.24)% in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel, (2.09$\pm$0.09$\pm$0.24)% in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
channel and (2.53$\pm$0.10$\pm$0.27)% in the $B^{0}\\!\rightarrow
D^{*-}\mu^{+}\nu_{\mu}$ channel, where the first uncertainty is statistical
and the second is systematic. The evaluation of the systematic uncertainty is
currently limited by the size of the available data sample.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
## References
* [1] LHCb collaboration, R. Aaij et al., Measurement of the $B^{0}_{s}-\bar{B}^{0}_{s}$ oscillation frequency $\Delta m_{s}$ in $B^{0}_{s}\rightarrow D^{-}_{s}(3)\pi$ decays, arXiv:1112.4311. Submitted to Phys. Lett. B
* [2] LHCb collaboration, R. Aaij et al., Measurement of $\phi_{s}$ in $B^{0}_{s}\rightarrow J/\psi f_{0}(980)$, Phys. Lett. B707 (2012) 497, arXiv:1112.3056
* [3] LHCb collaboration, R. Aaij et al., Measurement of the CP violating phase $\phi_{s}$ in the decay $B^{0}_{s}\rightarrow J/\psi\phi$, arXiv:1112.3183. Submitted to Phys. Rev. Lett.
* [4] DØ collaboration, V. M. Abazov et al., Measurement of $B_{d}$ mixing using opposite-side flavor tagging, Phys.Rev. D74 (2006) 112002, arXiv:0609034v1
* [5] CDF collaboration, T. Aaltonen et al., Measurement of $B^{0}$ oscillations and calibration of flavor tagging in semileptonic decays, http://www-cdf.fnal.gov/physics/new/bottom/060406.blessed-semi_B0mix/. CDF Note 8235
* [6] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [7] V. Gligorov, C. Thomas, and M. Williams, The HLT inclusive B triggers, LHCb-2011-016
* [8] M. Calvi, O. Leroy, and M. Musy, Flavour tagging algorithms and performances in LHCb, LHCb-2007-058
* [9] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
* [10] M. Pivk and F. R. Le Diberder, sPlot: a statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
|
arxiv-papers
| 2012-02-22T17:26:40 |
2024-09-04T02:49:27.713978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw,\n S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi,\n A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, K. de\n Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A.\n Comerma-Montells, F. Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, B.\n Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N. Y. David,\n I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De\n Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P.\n Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, 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, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, C.\n Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson,\n V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es,\n G. Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, 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, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, R. F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, D. Martinez\n Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B.\n Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J.\n Merkel, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n 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, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F.\n Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud,\n I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang,\n W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Marta Calvi",
"url": "https://arxiv.org/abs/1202.4979"
}
|
1202.5087
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-004 LHCb-PAPER-2011-033 February 21, 2012
Search for the $X(4140)$ state in $B^{+}\rightarrow J/\psi\phi K^{+}$ decays
The LHCb collaboration †††Authors are listed on the following pages.
A search for the $X(4140)$ state in $B^{+}\rightarrow J/\psi\phi K^{+}$ decays
is performed with 0.37 fb-1 of $pp$ collisions at $\sqrt{s}=7$ TeV collected
by the LHCb experiment. No evidence for this state is found, in $2.4\,\sigma$
disagreement with a measurement by CDF. An upper limit on its production rate
is set, ${{\cal B}(B^{+}\rightarrow X(4140)K^{+})\times{\cal
B}(X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}/{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})}<0.07$ at 90% confidence level.
Submitted to Physical Review D Rapid Communications
The LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47,35, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, F. Constantin26, A. Contu52, A. Cook43, M. Coombes43, G.
Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De Cian37, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, L. Estève44, A. Falabella16,e,14, E. Fanchini20,j, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10, F.
Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, N. Gauvin36, M. Gersabeck35, T.
Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B. Gui53, E. Gushchin30,
Yu. Guz32, T. Gys35, G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S.
Hansmann-Menzemer11, R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45,
T. Hartmann56, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11,
M. Jahjah Hussein5, E. Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F.
Jing3, M. John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S.
Kandybei40, M. Karacson35, T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U.
Kerzel35, T. Ketel39, A. Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35,
G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N.
Nikitin29, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,35, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M. Plo
Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E. Rodrigues48,51,
F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S. Roiser35, V.
Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33, G.
Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B. Saitta15,d,
C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios34, R.
Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M. Schiller39, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A.
Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M.
Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6,
P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27,
T. Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, A.C. Smith35, N.A. Smith49, E. Smith52,46,
K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2,
B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T.
Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E.
Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-Joergensen52,
N. Torr52, E. Tournefier4,50, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38, M.
Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, H. Voss10, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42,
A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
In this article, results are presented from the search for the narrow
$X(4140)$ resonance decaying to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ using $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K^{+}$ events111Charge-conjugate states are implied in this paper.
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$\phi\rightarrow K^{+}K^{-}$), in a data sample corresponding to an integrated
luminosity of $0.37$ fb-1 collected in $pp$ collisions at the LHC at
$\sqrt{s}=7$ TeV using the LHCb detector. The CDF collaboration reported a
3.8$\,\sigma$ evidence for the $X(4140)$ state (also referred to as $Y(4140)$
in the literature) in these decays using $p\bar{p}$ data collected at the
Tevatron ($\sqrt{s}=1.96$ TeV) [1]. A preliminary update of the CDF analysis
with 6.0 fb-1 reported $115\pm 12$
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$
events and $19\pm 6$ $X(4140)$ candidates leading to a statistical
significance of more than 5$\,\sigma$ [2]. The mass and width were determined
to be $4143.4^{+2.9}_{-3.0}\pm 0.6$ MeV and $15.3^{+10.4}_{-6.1}\pm 2.5$ MeV,
respectively222Units in which $c=1$ are used.. The relative branching ratio
was measured to be ${\cal B}(B^{+}\rightarrow X(4140)K^{+})\times{\cal
B}(X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})=0.149\pm 0.039\pm 0.024$.
Charmonium states at this mass are expected to have much larger widths because
of open flavour decay channels [3]. Thus, their decay rate into the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ mode, which is near the
kinematic threshold, should be small and unobservable. Therefore, the
observation by CDF has triggered wide interest among model builders of exotic
hadronic states. It has been suggested that the $X(4140)$ resonance could be a
molecular state [4, 5, 6, 7, 8, 9, 10], a tetraquark state [11, 12], a hybrid
state [13, 14] or a rescattering effect [15, 16]. The Belle experiment found
no evidence for the $X(4140)$ state in the
$\gamma\gamma\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
process, which disfavoured the molecular interpretation [17]. The CDF data
also suggested that there could be a second state at a mass of
$4274.4^{+8.4}_{-6.4}\pm 1.9$ MeV with a width of $32.3^{+21.9}_{-15.3}\pm
7.6$ MeV [2]. In this case, the event yield was $22\pm 8$ with $3.1\,\sigma$
significance. This observation has also received attention in the literature
[18, 19].
The LHCb detector [20] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter (ECAL) and a hadronic calorimeter (HCAL). Muons are identified by
a muon system (MUON) composed of alternating layers of iron and multiwire
proportional chambers. The MUON, ECAL and HCAL provide the capability of
first-level hardware triggering. The single and dimuon hardware triggers
provide good efficiency for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
events. Events passing the hardware trigger are read out and sent to an event
filter farm for further processing. Here, a software based two-stage trigger
reduces the rate from 1 MHz to about 3 kHz. The most efficient software
triggers [21] for this analysis require a charged track with transverse
momentum ($p_{\rm T}$) of more than $1.7$ GeV ($p_{\rm T}>1.0$ GeV if
identified as muon) and with an IP to any primary $pp$-interaction vertex (PV)
larger than $100$ $\mu$m. A dimuon trigger requiring $p_{\rm T}(\mu)>0.5$ GeV,
large dimuon mass, $M(\mu^{+}\mu^{-})>2.7$ GeV, and with no IP requirement
complements the single track triggers. At final stage, we either require a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
candidate with $p_{\rm T}>1.5$ GeV or a muon-track pair with significant IP.
In the subsequent offline analysis, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ candidates are selected with the following
criteria: $p_{\rm T}(\mu)>0.9$ GeV, $\chi^{2}$ per degree of freedom of the
two muons forming a common vertex, $\chi^{2}_{\rm
vtx}(\mu^{+}\mu^{-})/\hbox{\rm ndf}<9$, and a mass window
$3.04<M(\mu^{+}\mu^{-})<3.14$ GeV. We then find $K^{+}K^{-}K^{+}$ combinations
consistent with originating from a common vertex with $\chi^{2}_{\rm
vtx}(K^{+}K^{-}K^{+})/\hbox{\rm ndf}<9$. Every charged track with $p_{\rm
T}>0.25$ GeV, missing all PVs by at least 3 standard deviations
($\chi^{2}_{\rm IP}(K)>9$) and classified more likely to be a kaon than a pion
according to the particle identification system, is considered a kaon
candidate. A five-track ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}K^{+}$ vertex is formed ($\chi^{2}_{\rm
vtx}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}K^{+})/\hbox{\rm
ndf}<9$). This $B^{+}$ candidate is required to have $p_{\rm T}>4.0$ GeV and a
decay time as measured with respect to the PV of at least $0.25$ ps. When more
than one PV is reconstructed, the one that gives the smallest IP significance
for the $B^{+}$ candidate is chosen. The invariant mass of a
$\mu^{+}\mu^{-}K^{+}K^{-}K^{+}$ combination is evaluated after the muon pair
is constrained to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass, and
all final state particles are constrained to a common vertex.
Further background suppression is provided by a likelihood ratio. In the case
of uncorrelated input variables this provides the most efficient
discrimination between signal and background. The overall likelihood is a
product of probability density functions, ${\cal P}(x_{i})$ (PDFs), for the
four sensitive variables ($x_{i}$): smallest $\chi^{2}_{\rm IP}(K)$ among the
kaon candidates, $\chi^{2}_{\rm vtx}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}K^{+})/\hbox{\rm ndf}$, the pointing of the $B^{+}$ candidate
to the closest primary vertex, $\chi^{2}_{\rm IP}(B)$, and the cosine of the
largest opening angle between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and kaon candidates in the plane transverse to the beam. The latter
peaks towards $+1$ for the signal as the $B^{+}$ meson has a high transverse
momentum. Backgrounds combining particles from two different $B$ mesons peak
at $-1$. Backgrounds including other random combinations are uniformly
distributed. The signal PDFs, ${\cal P}_{\rm sig}(x_{i})$, are obtained from
the Monte Carlo simulation (MC) of
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}K^{+}$
decays. The background PDFs, ${\cal P}_{\rm bkg}(x_{i})$, are obtained from
the data candidates with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}K^{+}$ invariant mass between 5.6 and 6.4 GeV (far-sideband).
A logarithm of the ratio of the signal and background PDFs is formed: ${\rm
DLL}_{\rm sig/bkg}=-2\sum_{i}^{4}\ln({\cal P}_{\rm sig}(x_{i})/{\cal P}_{\rm
bkg}(x_{i}))$. A requirement on the log-likelihood ratio, ${\rm DLL}_{\rm
sig/bkg}<-1$, has been chosen by maximizing $N_{\rm sig}/\sqrt{N_{\rm
sig}+N_{\rm bkg}}$, where $N_{\rm sig}$ is the expected
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}K^{+}$
signal yield and the $N_{\rm bkg}$ is the background yield in the $B^{+}$ peak
region ($\pm 2.5\,\sigma$). The absolute normalization of $N_{\rm sig}$ and
$N_{\rm bkg}$ comes from a fit to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K$ invariant mass distribution with ${\rm DLL}_{\rm sig/bkg}<0$,
while their dependence on the ${\rm DLL}_{\rm sig/bkg}$ requirement comes from
the signal simulation and the far-sideband, respectively.
Figure 1: Mass distribution for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$
candidates in the data after the $\pm 15$ MeV $\phi$ mass requirement. The fit
of a Gaussian signal with a quadratic background (dashed line) is superimposed
(solid red line).
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K$ invariant mass
distribution, with a requirement that at least one $K^{+}K^{-}$ combination
has an invariant mass within $\pm 15$ MeV of the $\phi$ mass, is shown in Fig.
1. A fit to a Gaussian and a quadratic function in the range $5.1-5.5$ GeV
results in $346\pm 20$ $B^{+}$ events with a mass resolution of $5.2\pm 0.3$
MeV. Alternatively requiring the invariant mass
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}K^{+})$ to be within
$\pm 2.5$ standard deviations of the observed $B^{+}$ peak position, we fit
the $M(K^{+}K^{-})$ mass distribution (two combinations per event) using a
binned maximum likelihood fit with a P-wave relativistic Breit-Wigner
representing the $\phi(1020)$ and a two-body phase-space distribution to
represent combinatorial background, both convolved with a Gaussian mass
resolution. The $\phi$ resonance width is fixed to the PDG value ($4.26$ MeV)
[22]. The $M(K^{+}K^{-})$ mass distribution is displayed in Fig. 2 with the
fit results overlaid. The fitted parameters are the $\phi$ yield, the $\phi$
mass ($1019.3\pm 0.2$ MeV), the background yield and the mass resolution
($1.4\pm 0.3$ MeV). Replacing the two-body phase-space function by a third-
order polynomial does not change the results. In order to subtract a non-$B$
contribution, we fit the $M(K^{+}K^{-})$ distribution from the $B$ mass near-
sidebands (from $4$ to $14$ standard deviations on either side) leaving only
the $\phi$ yield and the two-body phase-space background yield as free
parameters. After scaling to the signal region, this leads to $14\pm 3$
background events. The background subtracted
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$ yield
($N_{B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}}$)
is $382\pm 22$ events.
Figure 2: Invariant $M(K^{+}K^{-})$ mass distribution selecting
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}K^{+}$
events in the $\pm 2.5\,\sigma$ region around the $B^{+}$ mass peak. The
dashed line shows the two-body phase-space contribution. The small blue dotted
$\phi$ peak on top of it illustrates the amount of the background $\phi$
mesons estimated from the fit to the $B^{+}$ mass near-sidebands.
To search for the $X(4140)$ state, we select events within $\pm 15$ MeV of the
$\phi$ mass. According to the fit to the $M(K^{+}K^{-})$ distribution this
requirement is 85% efficient. Figure 3 shows the mass difference
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)-M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ distribution (no
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\phi$ mass constraints have
been used). No narrow structure is observed near the threshold. We employ the
fit model used by CDF [2] to quantify the compatibility of the two
measurements. The data are fitted with a spin-zero relativistic Breit-Wigner
shape together with a three-body phase-space function (${\cal F}^{\rm
bkg}_{1}$), both convolved with the detector resolution. The efficiency
dependence is extracted from simulation (Fig. 4) and applied as a shape
correction to the three-body phase-space and the Breit-Wigner function. The
mass and width of the $X(4140)$ peak are fixed to the central values obtained
by the CDF collaboration. The mass-difference resolution was determined from
the $B^{+}\rightarrow X(4140)K^{+}$ simulation to be $1.5\pm 0.1$ MeV. A
binned maximum likelihood fit of the signal and background yields is shown in
Fig. 3(a). The region above 1400 MeV is excluded since it is more likely to
contain non $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+}$ backgrounds. By excluding also the region below 1030 MeV, where the
three-body phase-space and signal yields are very small ($0.5\%$ and $3.5\%$
of the yields included in the fit, respectively), we make our results less
vulnerable to possible small contributions from the other sources. The fit
shown in Fig. 3(a) gives a $X(4140)$ yield of $6.9\pm 4.9$ events. Fitting the
second state at a mass of $4274.44$ MeV and with a width of $32.3$ MeV [2]
does not affect the $X(4140)$ yield. Reflections of $K\phi$ resonances [23,
24] and possible broad $J/\psi\phi$ resonances can also contribute near and
under the narrow $X(4140)$ resonance. To explore the sensitivity of our
results to the assumed background shape, we also fit the data in the
$1020-1400$ MeV range with a quadratic function multiplied by the efficiency-
corrected three-body phase-space factor (${\cal F}^{\rm bkg}_{2}$) to impose
the kinematic threshold. The preferred value of the $X(4140)$ yield is $0.6$
events with a positive error of $7.1$ events. This fit is shown in Fig. 3(b).
Figure 3: Distribution of the mass difference
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)-M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ for the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$ in
the $B^{+}$ ($\pm 2.5\,\sigma$) and $\phi$ ($\pm 15$ MeV) mass windows. Fit of
$X(4140)$ signal on top of a smooth background is superimposed (solid red
line). The dashed blue (dotted blue) line on top illustrates the expected
$X(4140)$ ($X(4274)$) signal yield from the CDF measurement [2]. The top and
bottom plots differ by the background function (dashed black line) used in the
fit: (a) an efficiency-corrected three-body phase-space (${\cal F}^{\rm
bkg}_{1}$); (b) a quadratic function multiplied by the efficiency-corrected
three-body phase-space factor (${\cal F}^{\rm bkg}_{2}$). The fit ranges are
1030–1400 and 1020–1400 MeV, respectively. Figure 4: Efficiency dependence on
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)-M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ as determined
from the simulation (points with error bars). The efficiency is normalized
with respect to the efficiency of the $\phi$ signal fit to the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$
events distributed according to the phase-space model. A cubic polynomial was
fitted to the simulated data (superimposed).
A similar fit was performed to simulated $B^{+}\rightarrow X(4140)K^{+}$ data
to estimate the efficiency for this channel. The efficiency ratio between this
fit and the $\phi$ signal fit to the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$
events distributed according to the phase-space model,
$\epsilon(B^{+}\rightarrow
X(4140)K^{+},X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)/\epsilon(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K^{+})$, was determined to be $0.62\pm 0.04$ and includes the
efficiency of the $\phi$ mass window requirement. Using our
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$ yield
multiplied by this efficiency ratio and by the CDF value for ${\cal
B}(B^{+}\rightarrow X(4140)K^{+})/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+})$
[2], leads to a prediction that we should have observed $35\pm 9\pm 6$ events,
where the first uncertainty is statistical from the CDF data and the second
includes both the CDF and LHCb systematic uncertainties. Given the $B^{+}$
yield and relative efficiency, our sensitivity to the $X(4140)$ signal is a
factor of two better than that of the CDF. The central value of this estimate
is shown as a dashed line in Fig. 3. Taking the statistical and systematic
errors from both experiments into account, our results disagree with the CDF
observation by 2.4$\,\sigma$ (2.7$\,\sigma$) when using ${\cal F}^{\rm
bkg}_{1}$ (${\cal F}^{\rm bkg}_{2}$) background shapes.
Since no evidence for the $X(4140)$ state is found, we set an upper limit on
its production. Using a Bayesian approach, we integrate the fit likelihood
determined as a function of the $X(4140)$ yield and find an upper limit on the
number of signal events of $16$ ($13$) at 90% confidence level (CL) for the
two background shapes. Dividing the least stringent limit on the signal yield
by the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+}$ yield and $\epsilon(B^{+}\rightarrow
X(4140)K^{+})/\epsilon(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K^{+})$ gives a limit on ${\cal B}(B^{+}\rightarrow
X(4140)K^{+})\times{\cal
B}(X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+})$.
The systematic uncertainty on $\epsilon(B^{+}\rightarrow
X(4140)K^{+})/\epsilon(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K^{+})$ is 6%. This uncertainty includes the statistical error from
the simulation as well as the observed differences in track reconstruction
efficiency between the simulation and data measured with the inclusive
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
signal. Fit systematics related to the detector resolution and the uncertainty
in the shape of the efficiency dependence on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ mass were also studied and
found to be small. We multiply our limit by 1.06 and obtain at 90% CL
$\frac{{\cal B}(B^{+}\rightarrow X(4140)K^{+})\times{\cal
B}(X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})}<0.07.$
We also set an upper limit on the $X(4274)$ state suggested by the CDF
collaboration [2]. The fit with ${\cal F}^{\rm bkg}_{1}$ background shape
gives $3.4^{+6.5}_{-3.4}$ events at this mass. The fit with the ${\cal F}^{\rm
bkg}_{2}$ background shape gives zero signal events with a positive error of
$10$. Integration of the fit likelihoods gives $<24$ and $<20$ events at 90%
CL, respectively. The relative efficiency at this mass is
$\epsilon(B^{+}\rightarrow
X(4274)K^{+},X(4274)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)/\epsilon(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi K^{+})=0.86\pm 0.10$. The least stringent limit on the signal
events yields an upper limit of
$\frac{{\cal B}(B^{+}\rightarrow X(4274)K^{+})\times{\cal
B}(X(4274)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)}{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})}<0.08$
at 90% CL, which includes the systematic uncertainty. CDF did not provide a
measurement of this ratio of branching fractions. Assuming the efficiency is
similar for the $X(4274)$ and $X(4140)$ resonances, their $X(4274)$ event
yield corresponds to ${\cal B}(B^{+}\rightarrow X(4274)K^{+})\times{\cal
B}(X(4274)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})=0.17\pm 0.06$ (statistical uncertainty only). Scaling to our data, we
should have observed $53\pm 19$ $X(4274)$ events, which is illustrated in Fig.
3.
In summary, the most sensitive search for the narrow
$X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ state
just above the kinematic threshold in
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi K^{+}$
decays has been performed using 0.37 fb-1 of data collected with the LHCb
detector. We do not confirm the existence of such a state. Our results
disagree at the $2.4\,\sigma$ level with the CDF measurement. An upper limit
on ${{\cal B}(B^{+}\rightarrow X(4140)K^{+})\times{\cal
B}(X(4140)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi))/}$
${{\cal B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi
K^{+})}$ of $<0.07$ at 90% CL is set.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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|
arxiv-papers
| 2012-02-23T04:56:25 |
2024-09-04T02:49:27.728275
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S.\n Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw,\n S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi,\n A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier,\n C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F. Constantin,\n A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, S. De Capua, M. De Cian,\n F. De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, 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, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y.M. Kim, M. Knecht, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, 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, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde,\n R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, 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, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, K.\n Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A.C. Smith, N.A.\n Smith, E. Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De\n Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp,\n S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah,\n S. Swientek, M. Szczekowski, P. Szczypka, T. 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"submitter": "Tomasz Skwarnicki",
"url": "https://arxiv.org/abs/1202.5087"
}
|
1202.5138
|
# Group properties and invariant solutions of a sixth-order thin film equation
in viscous fluid
Ding-jiang Huang1,2,3 djhuang@fudan.edu.cn Qin-min Yang1 Shuigeng Zhou2,3
1Department of Mathematics, East China University of Science and Technology,
Shanghai 200237, China 2School of Computer Science, Fudan University,
Shanghai 200433, China 3Shanghai Key Lab of Intelligent Information
Processing, Fudan University, Shanghai 200433, China
###### Abstract
Using group theoretical methods, we analyze the generalization of a one-
dimensional sixth-order thin film equation which arises in considering the
motion of a thin film of viscous fluid driven by an overlying elastic plate.
The most general Lie group classification of point symmetries, its Lie
algebra, and the equivalence group are obtained. Similar reductions are
performed and invariant solutions are constructed. It is found that some
similarity solutions are of great physical interest such as sink and source
solutions, travelling-wave solutions, waiting-time solutions, and blow-up
solutions.
PACS numbers
47.15.G-,02.20.Sv,02.30.Jr
###### pacs:
Valid PACS appear here
††preprint: APS/123-QED
## I Introduction
In the past several decades there is an increasing interest in physics and
mathematics literatures in higher-order nonlinear diffusion equations because
they are models of various interesting phenomenon in fluid physics and have
surprising mathematic structure and properties. Probably, one of the most
famous example is the fourth-order thin film equation in the form
$u_{t}=(u^{\alpha}u_{xxx})_{x},\quad\quad\alpha>0$ (1)
which was first introduced by Greenspan in 1978 Greenspan (1978). This
equation describes the surface-tension-dominated motion of thin viscous films
for the film height $u(t,x)$ and spreading droplets in the lubrication
approximation Greenspan (1978). In particular, for $\alpha=3$ it describes a
classical thin film of Newtonian fluid, as reviewed in Oron _et al._ (1997),
$\alpha=1$ occurs in the dynamics of a Hele-Shaw cell Constantin _et al._
(1993) and $\alpha=2$ arises in a study of wetting films with a free contact
line between film and substrate Bertozzi (1998). There also exist many
interesting generalizations of the famous equation (1) (see King (2001); Yarin
_et al._ (1993) and reference therein).
Apart from the fourth-order equations, another interesting higher-order
diffusion model is the sixth-order nonlinear thin film equation in the form
$u_{t}=(u^{m}u_{xxxxx})_{x},$ (2)
which appear in flow modeling. The case $m=3$, for instance, was first
introduced by King in King (1986) as a model of the oxidation of silicon in
semiconductor devices King (1989) or for a moving boundary given by a beam of
negligible mass on a surface of a thin film Smith _et al._ (1996a). Here
$u\geq 0$ will be treated as the thickness of a fluid film beneath an elastic
plate and $p=u_{xxxx}$ as the pressure within the film King (1986). The other
derivatives of $u$ can in the usual way be assigned different physical
meaning, for instance $\Gamma=-u_{xx}$ is the bending moment on the overlying
plate and $\Sigma=u_{xxx}$ is the shearing force Landau _et al._ (1986), here
all such expressions are dimensionless. An equation of this type can be used
to model the motion of a thin film of viscous fluid overlain by an elastic
plate King (1989); see also Hobart et al. Hobart _et al._ (2000) and Huang et
al. Huang and Suo (2002) for possible applications of such modelling
approaches to the wrinkling upon annealing of SiGe films bonded to Si
substrates. Other plausible applications of Eq. (2), and suitable
generalizations thereof, include a simple model for the influence of a crust
on a solidifying melt or for a microfluidic pump (see Koch et al. Koch _et
al._ (1997), for instance).
Eqs. (1) and (2) are also the second and the third member of a hierarchy
arising from the generalized Reynolds equation
$u_{t}=(u^{m}p_{x})_{x},$ (3)
under different driving forces respectively. For gravity driven flows, we have
$p=u$, giving the very widely studied porous-medium equation (see, for
example, Aronson Aronson (1986)). For surface-tension driven flows we have
$p=-u_{xx}$, leading to the fourth-order thin film equation (1). For elastic
plate driven flows, we have $p=u_{xxxx}$, which give the sixth-order thin film
equation (2) King (1989).
Up to now, the mathematic structure and properties of the fourth-order thin
film equation (1) have been widely investigated, including (non-)uniqueness,
wetting behaviour and contact line motion, in particular optimal propagation
rates, waiting time or dead core phenomena and self-similar solutions(see
Hulshof Hulshof (2001), for instance). Recent years, there are also many
researches devoted to symmetry group structure and exact solutions of the
fourth-order thin film equations (1) and their generalizationsSmyth and Hill
(1988); Bernis and McLeod (1991); Choudhury (1995); Bernis _et al._ (2000);
Gandarias and Bruzon (2000); Bruzon _et al._ (2003); Qu (2006); Gandarias and
Ibragimov (2008); Gandarias and Medina (2001); Cherniha _et al._ (2010), or
searching for special invariant finite vector spaces of solutions Galaktionov
and Svirshchevskii (2007).
However, the sixth-order thin film equation (2) has been much less extensively
investigated. It was only a few researches that were devoted to qualitative
mathematic properties such as the existence of weak solutions, initial
boundary value problems (see Bernis Friedman Bernis and Friedman (1990), King
et al Flitton and King (2004), Smith et al. Smith _et al._ (1996b), Evans et
al. Evans _et al._ (2000), Barrett et al. Barrett _et al._ (2004) for
existing studies, the first two being analytical and the others primarily
numerical), while the symmetry group properties and corresponding algebraic
structure as well as explicit exact solutions of Eq. (2) still remain open.
Therefore, the aim of the present work is to find such group properties,
algebraic structure and exact solutions. To do this, we investigate
alternatively a more general sixth-order nonlinear diffusion equation in the
form
$u_{t}=(f(u)u_{xxxxx})_{x},$ (4)
than the original equation (2), where $f(u)$ is an arbitrary smooth function
depending on the geometry of the problem and $f_{u}\neq 0$( i.e., (4) is a
nonlinear equation).
We use the method of Lie groups, one of the powerful tools available to solve
nonlinear PDEs, and which was discovered and applied firstly by S. Lie in the
nineteenth century, but only in the last decades has it become a common tool
for both mathematicians and physicists (see for examples Bluman and Kumei
(1989); Ovsiannikov (1982); Olver (1986); Bluman _et al._ (2010); Alfinito
_et al._ (1995); Carbonaro (1997); Pulov _et al._ (1998); Senthilvelan _et
al._ (2002); Struckmeier and Riedel (2002); Stewart and Momoniat (2004)). The
method consists of looking for the infinitesimal generators of a group of
point transformations which leave the equation under study invariant. An
important point of the Lie theory is that the conditions for an equation to
admit a group of transformations are represented by a set of linear equations,
the so-called “determining equations”, which are usually completely solvable.
Having once found the groups of transformations, one can obtain a number of
interesting results, which include the possibility to reduce a partial
differential equation with two independent variables to an ordinary
differential equation with one independent variable, etc.. Solving these
reduced equations, one can obtain some particular solutions for the original
equations. These particular solutions are usually called “similarity
solutions” or “invariant solutions” Bluman and Kumei (1989); Ovsiannikov
(1982); Olver (1986); Bluman _et al._ (2010). When the equation contains
“arbitrary elements”(a variable coefficient deriving from the particular
equation of state chosen to characterize the physical mechanism. “Arbitrary
elements” are functions or variable parameters whose form is not strictly
fixed and can be assigned freely on the grounds of physical hypotheses about
the nature of the medium under consideration.), the theory gives rise to the
problem of group classification of differential equations which is the core
stone of modern group analysis Ovsiannikov (1982); Bluman _et al._ (2010). In
particular, in the past several years, a numbers of novel techniques, such as
algebraic methods based on subgroup analysis of the equivalence group Basarab-
Horwath _et al._ (2001); Zhdanov and Lahno (1999); Gazeau and Winternitza
(1992); Popovych _et al._ (2010), compatibility and direct integration
Ovsiannikov (1982); Ibragimov (1994) (also referred as the Lie-Ovsiannikov
method) as well as their generalizations (eg. method of furcate split Nikitin
and Popovych (2001), additional and conditional equivalence transformations
Popovych and Ivanova (2004); Huang and Ivanova (2007), extended and
generalized equivalence transformation group, gauging of arbitrary elements by
equivalence transformations Ivanova _et al._ (2010); Huang and Zhou (2011))
have been proposed to solve group classification problem for numerous
nonlinear partial differential equations. Although a great deal of
classification was solved by these methods, almost all of them are limited to
the equations whose order are lower than four (see Huang and Zhou (2011) for
details).
In this paper we extend these new techniques, specific compatibility and
direct integration as well as equivalence transformation techniques, to sixth-
order nonlinear diffusion equations. We first carry out group classification
of Eq. (4) under the usual equivalence group. The Lie group of point
symmetries of Eq. (2), as a special case of Eq. (4), and its Lie algebra are
also obtained. Then similar reductions of the classification models are
performed and invariant solutions are also constructed. It is found that some
similarity solutions are solutions with physical interest: sink and source
solutions, travelling-wave solutions, waiting-time solutions and blow-up
solutions.
The rest of this paper is organized as follows: In Sec. II we derive the
equivalence group and perform the group classification related to Eq. (4). In
Sec. III, similar reductions of classification models are carried out. Sec. IV
contains examples of some specific exact solutions, including sink and source
solutions, travelling-wave solutions, waiting-time solutions and blow-up
solutions, while in Sec. V some concluding remarks are reported.
## II SYMMETRY CLASSIFICATION
Background and procedures of the modern Lie group theory are well described in
literature Bluman and Kumei (1989); Ovsiannikov (1982); Olver (1986); Bluman
_et al._ (2010); Popovych and Ivanova (2004); Huang and Ivanova (2007).
Without going into the details of the theory, we present only the results
below.
Let
${\bf
Q}=\tau(t,x,u)\partial_{t}+\xi(t,x,u)\partial_{x}+\phi(t,x,u)\partial_{u}$
be a vector field or infinitesimal operator on the space of independent and
dependent variables $t,~{}x,~{}u$. A local group of transformations $G$ is a
symmetry group of Eq. (4) if and only if
$\rm{pr}^{(6)}{\bf Q}|(\Delta)=0,$ (5)
whenever $\Delta=u_{t}-\big{[}f(u)u_{xxxxx}\big{]}_{x}=0$ for every generator
of $G$, where $\rm{pr}^{(6)}{\bf Q}$ is the sixth-order prolongation of ${\bf
Q}$.
Expanding Eq. (5) we get
$\displaystyle\phi^{t}=$ $\displaystyle\phi
f^{\prime\prime}(u)u_{x}u_{xxxxx}+f^{\prime}(u)u_{xxxxx}\phi^{x}+f^{\prime}(u)u_{x}\phi^{xxxxx}$
(6) $\displaystyle+\phi f^{\prime}(u)u_{xxxxxx}+f(u)\phi^{xxxxxx}$
which must be satisfied whenever Eq. (4) is satisfied. Substituting the
formulae of $\phi^{t}$, $\phi^{x}$, $\phi^{xxxxx}$ and $\phi^{xxxxxx}$ into
Eq. (6) we get an equation of $t,x,u$ and the derivatives of
$\tau,\xi,\phi,u$. Replacing $u_{t}$ by the right hand side of Eq. (4)
whenever it occurs, and equating the coefficients of the various independent
monomials to zero, we obtain the determining equations
$\begin{cases}\tau_{x}=\tau_{u}=\xi_{u}=\phi_{uu}=0\\\
3(2\phi_{xu}-5\xi_{xx})f(u)+\phi_{x}f^{\prime}(u)=0\\\
(\phi_{xu}-2\xi_{xx})f^{\prime}(u)=0\\\ 3\phi_{xxu}-4\xi_{xxx}=0\\\
(\phi_{xxu}-\xi_{xxx})f^{\prime}(u)=0\\\ (\tau_{t}-6\xi_{x})f(u)+\phi
f^{\prime}(u)=0\\\ 4\phi_{xxxu}-3\xi_{xxxx}=0\\\
\phi_{xxxxxx}f(u)-\phi_{t}=0\\\
\phi_{xxxxx}f^{\prime}(u)+\xi_{t}+6\phi_{xxxxxu}f(u)-\xi_{xxxxxx}f(u)=0\\\
5\phi_{xxxxu}-2\xi_{xxxxx}=0\\\ (5\phi_{xxxxu}-\xi_{xxxxx})f^{\prime}(u)=0\\\
(2\phi_{xxxu}-\xi_{xxxx})f^{\prime}(u)=0\end{cases}$
The first three equations imply that $\phi_{xu}=\xi_{xx}=0$, which together
with the sixth, the eighth and the ninth equations imply that
$\phi_{t}=\phi_{x}=\xi_{t}=0$, so the determining equations reduce to
$\begin{cases}\tau_{x}=\tau_{u}=0\\\ \xi_{t}=\xi_{u}=\xi_{xx}=0\\\
\phi_{t}=\phi_{x}=\phi_{uu}=0\\\ (\tau_{t}-6\xi_{x})f(u)+\phi
f^{\prime}(u)=0,\end{cases}$ (7)
which is equivalent to
$\begin{cases}\xi=ax+b\\\ \tau=ct+d\\\ \phi=pu+q\\\
(c-6a)f(u)+(pu+q)f^{\prime}(u)=0\end{cases}$ (8)
where $a$, $b$, $c$, $d$, $p$, and $q$ are arbitrary constants.
In order to make the classification as simple as possible, we next look for
equivalence transformations of class (4), and then solve system (8) under
these transformations. An equivalence transformation is a nondegenerate change
of the variables $t$, $x$ and $u$ taking any equation of the form (4) into an
equation of the same form, generally speaking, with different $f(u)$. The set
of all equivalence transformations forms the equivalence group $G^{\sim}$. To
find the connected component of the unity of $G^{\sim}$, we have to
investigate Lie symmetries of the system that consists of Eq. (4) and some
additional conditions, i.e.
$\begin{cases}u_{t}=f_{u}u_{x}u_{xxxxx}+fu_{xxxxxx},\\\ f_{t}=0,\\\
f_{x}=0.\end{cases}$ (9)
That is to say we must seek for an operator of the Lie algebra $A^{\sim}$ of
$G^{\sim}$ in the form
$\displaystyle{\bf X}=$
$\displaystyle\tau(t,x,u)\partial_{t}+\xi(t,x,u)\partial_{x}$ (10)
$\displaystyle+\phi(t,x,u)\partial_{u}+\psi(t,x,u,f)\partial_{f}.$
Here $u$ and $f$ are considered as different variables: $u$ is on the space
$(t,x)$ and $f$ is on the extended space $(t,x,u)$. The coordinates $\tau$,
$\xi$, $\phi$ of the operator (10) are sought as functions of $t$, $x$, $u$
while the coordinates $\psi$ are sought as functions of $t$, $x$, $u$ and $f$.
Applying $\rm{pr}^{(6)}{\bf X}$ to Eq. (9) we get the infinitesimal criterion
$\begin{cases}\phi^{t}=u_{x}u_{xxxxx}\psi^{u}+f_{u}u_{xxxxx}\phi^{x}\\\
\qquad+f_{u}u_{x}\phi^{xxxxx}+u_{xxxxxx}\psi+f\phi^{xxxxxx},\\\
\psi^{t}=0,\quad\psi^{x}=0\end{cases}$ (11)
which must be satisfied whenever Eq. (9) is satisfied. Substituting the
formulae of $\phi^{t}$, $\phi^{x}$, $\phi^{xxxxx}$, $\phi^{xxxxxx}$,
$\psi^{t}$, $\psi^{x}$, and $\psi^{u}$ into Eq. (11) we get equations of $t$,
$x$, $u$, $f$, and the partial derivatives of $\tau$, $\xi$, $\phi$, $u$, $f$,
and $\psi$. Replacing $u_{t}$, $f_{t}$ and $f_{x}$ by the right hand side of
Eq. (9) whenever they occur, and equating the coefficients of various
independent monomials to zero, we obtain
$\begin{cases}\xi_{xx}=0,\quad\xi_{t}=0,\quad\xi_{u}=0\\\
\tau_{x}=0,\quad\tau_{u}=0\\\ \phi_{x}=0,\quad\phi_{t}=0,\quad\phi_{uu}=0\\\
\psi_{x}=0,\quad\psi_{t}=0,\quad\psi_{u}=0\\\
\psi=(6\xi_{x}-\tau_{t})f\end{cases}$
which can be reduced to
$\begin{cases}\tau=c_{4}t+c_{1}\\\ \xi=c_{5}x+c_{2}\\\ \phi=c_{6}u+c_{3}\\\
\psi=(6c_{5}-c_{4})f\end{cases}$ (12)
where $c_{1},c_{2},\ldots,c_{6}$ are arbitrary constants.
Thus the Lie algebra of $G^{\sim}$ for class (4) is
$A^{\sim}=\langle\partial_{t},\partial_{x},\partial_{u},t\partial_{t}-f\partial_{f},x\partial_{x}+6f\partial_{f},u\partial_{u}\rangle.$
Continuous equivalence transformations of class (4) are generated by the
operators from $A^{\sim}$. In fact, $G^{\sim}$ contains the following
continuous transformations:
$\displaystyle\tilde{t}=t{\varepsilon_{4}}+\varepsilon_{1},\qquad\tilde{x}=x{\varepsilon_{5}}+\varepsilon_{2},$
$\displaystyle\tilde{u}=u{\varepsilon_{6}}+\varepsilon_{3},\qquad\tilde{f}=f\varepsilon_{4}^{-1}\varepsilon_{5}^{6},$
where $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{6}$ are arbitrary
constants.
Solve the last equation of system (8) under the above equivalence group
$G^{\sim}$, we can obtain three inequivalent equations of class (4) with
respect to the transformations from $G^{\sim}$:
Case 1: $f(u)$ is an arbitrary nonconstant function, the symmetry algebra of
class (4) is a three-dimensional Lie algebra which is generated by the
operators
$Q_{1}=\partial_{t},\quad Q_{2}=\partial_{x},\quad
Q_{3}=t\partial_{t}+\frac{1}{6}x\partial_{x};$ (13)
Case 2: $f=e^{\lambda u}\mod G^{\sim}(\lambda\neq 0)$, the symmetry algebra of
class (4) is a four-dimensional Lie algebra which is generated by the
operators
$\displaystyle Q_{1}=\frac{1}{\lambda}\partial_{u}-t\partial_{t},$
$\displaystyle Q_{2}=\partial_{t},$ (14) $\displaystyle
Q_{3}=-x\partial_{x}-\frac{6}{\lambda}\partial_{u},$ $\displaystyle
Q_{4}=\partial_{x};$
Case 3: $f=u^{m}\mod G^{\sim}(m\neq 0)$, the symmetry algebra of class (4) is
a four-dimensional Lie algebra which is generated by the operators
$\displaystyle Q_{1}=\frac{1}{m}u\partial_{u}-t\partial_{t},\quad$
$\displaystyle Q_{2}=\partial_{t},$ (15) $\displaystyle
Q_{3}=-x\partial_{x}-\frac{6}{m}u\partial_{u},\quad$ $\displaystyle
Q_{4}=\partial_{x}.$
From the above results, it is easy to see that equation (2) is exactly
corresponding to case 3, thus possess a four-dimensional symmetry algebra.
## III SIMILARITY REDUCTION
In order to obtain all the inequivalent reductions, we look for the one-
dimensional optimal systems (see Ovsiannikov (1982)). These systems,
similarity variables and reduced equations are listed below. In the following
tables III.1, III.2, III.3, each row shows the infinitesimal generators
$Q_{i}$ of each optimal system, as well as its similarity variable, similarity
solution and reduced equation. $\alpha$ is an arbitrary constant, while
$\lambda$ is a non-vanishing arbitrary constant. Note that in the case
$f(u)=u^{m}$ which corresponds to Eq. (2), we only consider $m\neq 0$,
otherwise the equation is linear.
### III.1 $f(u)$ is an arbitrary nonconstant function
In this case, the symmetry operators are Eq. (13). These operators satisfy the
commutation relations
$[Q_{1},\ Q_{3}]=Q_{1},\ \ [Q_{2},\ Q_{3}]=\frac{1}{6}Q_{2}$
and thus the corresponding symmetry algebra is a realization of the algebra
$A_{3,5}^{a}(0<|a|<1)$ Patera and Winternitz (1977). According to the results
of Patera and Winternitz Patera and Winternitz (1977), an optimal system of
one-dimensional subalgebras is those spanned by
$Q_{1},\quad Q_{2},\quad Q_{3},\quad Q_{1}+\alpha Q_{2}.$
Therefore, the corresponding similarity variables and reduced ODEs can be
easily calculated. Such results are listed in Table III.1.
Table 1. Reduced ODEs for arbitrary nonconstant $f(u)$
(let $E=\big{[}f(v)v_{yyyyy}\big{]}_{y}$). $i$ Subalgebra Ansatz $u=$ $y$
Reduced ODEi 1 $\langle Q_{1}\rangle$ $v(y)$ $x$ $E=0$ 2 $\langle
Q_{2}\rangle$ $v(y)$ $t$ $v_{y}=0$ 3 $\langle Q_{3}\rangle$ $v(y)$
$xt^{{-}1/6}$ $E=-yv_{y}/6$ 4 $\langle Q_{1}+\alpha Q_{2}\rangle$ $v(y)$
$x-\alpha t$ $E=-\alpha v_{y}$
### III.2 $f(u)=e^{\lambda u}$ ($\lambda\neq 0$)
In this case, the symmetry operators are given by Eq. (14), which satisfy the
commutation relations
$[Q_{1},\ Q_{2}]=Q_{2},\ \ [Q_{3},\ Q_{4}]=Q_{4}$ (16)
and thus the corresponding symmetry algebra is a realization of the algebra
$2A_{2}$. According to the results of Patera and Winternitz Patera and
Winternitz (1977) again, an optimal system of one-dimensional subalgebras is
those generated by
$Q_{2},Q_{3},Q_{4},Q_{1}+\alpha Q_{3},Q_{1}+\alpha Q_{4},Q_{2}+\alpha
Q_{4},Q_{2}+\alpha Q_{3}.$
The corresponding similarity variables and reduced ODEs are listed in Table
III.2.
Table 2. Reduced ODEs for $f(u)=e^{\lambda u}$
(let $\lambda\neq 0$, $E=\big{(}e^{\lambda v}v_{yyyyy}\big{)}_{y}$). $i$
Subalgebra Ansatz $u=$ $y$ Reduced ODEi 5 $\langle Q_{2}\rangle$ $v(y)$ $x$
$E=0$ 6 $\langle Q_{3}\rangle$ $v(y)+\frac{6}{\lambda}\ln x$ $t$
$144e^{\lambda v}-\lambda v_{y}=0$ 7 $\langle Q_{4}\rangle$ $v(y)$ $t$
$v_{y}=0$ 8 $\langle Q_{1}+\alpha Q_{3}\rangle$
$v(y)+\frac{6\alpha-1}{\lambda}\ln t$ $xt^{-\alpha}$
$E=\frac{6\alpha-1}{\lambda}-\alpha yv_{y}$ 9 $\langle Q_{1}+\alpha
Q_{4}\rangle$ $v(y)-\frac{1}{\lambda}\ln t$ $x+\alpha\ln t$ $E=\alpha
v_{y}-\frac{1}{\lambda}$ 10 $\langle Q_{2}+\alpha Q_{4}\rangle$ $v(y)$
$x-\alpha t$ $E=-\alpha v_{y}$ 11 $\langle Q_{2}+\alpha Q_{3}\rangle$
$v(y)-\frac{6\alpha t}{\lambda}$ $xe^{\alpha t}$ $E=\alpha
yv_{y}-\frac{6\alpha}{\lambda}$
### III.3 $f(u)=u^{m}$ ($m\neq 0$)
In this case, the symmetry operators are Eq. (15). These operators share the
same commutation relations Eq. (16). Hence an optimal system of one-
dimensional subalgebras is the same as the case $f(u)=e^{\lambda u}$. The
corresponding similarity variables and reduced ODEs are listed in Table III.3.
Table 3. Reduced ODEs for $f(u)=u^{m}$
(let $m\neq 0$, $E=\big{(}v^{m}v_{yyyyy}\big{)}_{y}$). $i$ Subalgebra Ansatz
$u=$ $y$ Reduced ODEi 12 $\langle Q_{2}\rangle$ $v(y)$ $x$ $E=0$ 13 $\langle
Q_{3}\rangle$ $v(y)x^{\frac{6}{m}}$ $t$
$v^{m+1}\prod\limits_{k=-4}^{1}(\frac{6}{m}{+}k)=v_{y}$ 14 $\langle
Q_{4}\rangle$ $v(y)$ $t$ $v_{y}=0$ 15 $\langle Q_{1}+\alpha Q_{3}\rangle$
$v(y)t^{\frac{6\alpha-1}{m}}$ $xt^{-\alpha}$ $E=\frac{6\alpha-1}{m}v-\alpha
yv_{y}$ 16 $\langle Q_{1}+\alpha Q_{4}\rangle$ $v(y)t^{{-}\frac{1}{m}}$
$x+\alpha\ln t$ $E=\alpha v_{y}-\frac{1}{m}v$ 17 $\langle Q_{2}+\alpha
Q_{4}\rangle$ $v(y)$ $x-\alpha t$ $E=-\alpha v_{y}$ 18 $\langle Q_{2}+\alpha
Q_{3}\rangle$ $v(y)e^{-\frac{6\alpha t}{m}}$ $xe^{\alpha t}$ $E=\alpha
yv_{y}-\frac{6\alpha}{m}v$
## IV INVARIANT SOLUTIONS
Using the above reduced ODEs, we can construct some invariant solutions for
the original equations (4). It is easy to see that some of the similarity
variables in the tables III.1, III.2 and III.3 have a clear physical
interpretation. Besides, for some higher order reduced ODEs, they can be
further reduced by using new symmetries. Below, we discuss some facts related
with some types of similarity solutions with physical interest and obtain some
particular solutions. Different types of solutions are separately analyzed.
### IV.1 Source and Sink Solutions
There are two ODEs, i.e., ODE${}_{\ref{PowerFunction4}}$ and
ODE${}_{\ref{PowerFunction7}}$, among our reduced equations are related to
this type of solutions. In fact, if we choose $\alpha=\frac{1}{m+6}$ in
ODE${}_{\ref{PowerFunction4}}$, then the similarity solution has the form
$u(t,x)=\frac{1}{t^{\frac{1}{m+6}}}v(\frac{x}{t^{\frac{1}{m+6}}}).$
Thus, if $m>-6$ it is clear that $u(t,x)\to\delta(x)$ as $t\to 0$ and the
similarity solution is a source solution; if $m<-6$ it is clear that
$u(t,x)\to\delta(x)$ as $t\to+\infty$ and the similarity solution is a sink
solution. Furthermore, we can also observe that, for the above choice of
$\alpha=\frac{1}{m+6}$, ODE${}_{\ref{PowerFunction4}}$ can be integrated once
to obtain
$v^{m}v_{yyyyy}+\frac{1}{m+6}yv=k,$
where $k$ is an arbitrary constant. Thus, we can obtain a class of source
solutions and sink solutions for the general thin-film equations (4) with
$f(u)=u^{m}$ (i.e., equation (2)) by solving the above fifth-order ODE. If we
further choose $k$ as zero, we have
$v_{yyyyy}=-\frac{1}{m+6}yv^{1-m}.$
This equation admits the symmetry group corresponding to the infinitesimal
generator ${\bf v}=y\partial_{y}+\frac{6}{m}v\partial_{v}$. Taking into
account that the invariants of its first prolongation and setting
$x_{1}=vy^{-\frac{6}{m}},\quad
u_{1}=y^{\frac{6}{m}}(yv^{\prime}-\frac{6}{m}v)^{-1},$
this equation becomes a fourth-order ODE:
$\displaystyle-m^{5}u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}}+5m^{4}(3mu_{1x_{1}}+2mu_{1}^{2}$
$\displaystyle-6u_{1}^{2})u_{1}^{2}u_{1x_{1}x_{1}x_{1}}+10m^{5}u_{1}^{2}u_{1x_{1}x_{1}}^{2}$
$\displaystyle-5m^{3}[21m^{2}u_{1x_{1}}^{2}+20m(m-3)u_{1}^{2}u_{1x_{1}}$
$\displaystyle+(7m^{2}-48m+72)u_{1}^{4}]u_{1}u_{1x_{1}x_{1}}+105m^{5}u_{1x_{1}}^{4}$
$\displaystyle+150m^{4}(m-3)u_{1}^{2}u_{1x_{1}}^{3}+15m^{3}(7m^{2}-48m$
$\displaystyle+72)u_{1}^{4}u_{1x_{1}}^{2}+10m^{2}(m-3)(5m^{2}-48m$
$\displaystyle+72)u_{1}^{6}u_{1x_{1}}+[m^{5}x_{1}^{-m+1}/(m+6)$
$\displaystyle+72(m-2)(m-3)(m-6)(2m-3)x_{1}]u_{1}^{9}$
$\displaystyle+12m(2m^{4}-50m^{3}+315m^{2}$
$\displaystyle-720m+540)u_{1}^{8}=0.$
Thus, source and sink solutions can be also obtained by solving the above
fourth-order ODE.
If we choose $m=-6$ in ODE${}_{\ref{PowerFunction7}}$, then the similarity
solution has the form
$u(t,x)=\frac{1}{e^{-\alpha t}}v(\frac{x}{e^{-\alpha t}}),$
thus, if $\alpha>0$ it is clear that $u(t,x)\to\delta(x)$ as $t\to+\infty$ and
the similarity solution is a sink solution; if $\alpha<0$ it is clear that
$u(t,x)\to\delta(x)$ as $t\to-\infty$ and the similarity solution is a source
solution. As in the previous case, for the choice of $m=-6$,
ODE${}_{\ref{PowerFunction7}}$ can be integrated once to obtain a fifth-order
ODE
$v^{-6}v_{yyyyy}-\alpha yv=k,$
where $k$ is an arbitrary constant. Consequently, source and sink solutions
can be computed by solving a fifth-order ODE.
### IV.2 Travelling-wave Solutions
This types of solution corresponds to the reductions III.1, III.2 and III.3.
In fact, in these three reductions the similarity variables are given by
$y=x-\alpha t$, $u=v$, so that $u(t,x)=v(x-\alpha t)$, thus the corresponding
solutions are travlling-wave solutions. Due to the physical interest of this
type of solutions, in what follows we study further symmetries of the
associated ODEs and then construct some such kinds of solutions. First of all,
we integrate these three equations once trivially and obtain
$\displaystyle\mbox{ODE}^{\prime}_{\ref{ArbitraryFunction4}}:f(v)v_{yyyyy}+\alpha
v=k,$ $\displaystyle\mbox{ODE}^{\prime}_{\ref{ExponentFunction6}}:e^{\lambda
v}v_{yyyyy}+\alpha v=k_{1},$
$\displaystyle\mbox{ODE}^{\prime}_{\ref{PowerFunction6}}:v^{m}v_{yyyyy}+\alpha
v=k_{2},$
where $k,k_{1},k_{2}$ are arbitrary constants. Because
ODE${}_{\ref{ExponentFunction6}}$ and ODE${}_{\ref{PowerFunction6}}$ are given
by ODE${}_{\ref{ArbitraryFunction4}}$ for $f(u)=e^{\lambda u}$ and
$f(u)=u^{m}$ respectively, so we will focus on the
ODE${}^{\prime}_{\ref{ArbitraryFunction4}}$ below. This equation is invariant
under the group of translations in the $y$-direction, with infinitesimal
generator $\frac{\partial}{\partial y}$. Set
$x_{1}=v,\quad u_{1}=v^{-1}_{y},$
then the equation becomes a fourth-order ODE:
$\displaystyle
f(x_{1})(105u^{4}_{1x_{1}}-105u_{1}u^{2}_{1x_{1}}u_{1x_{1}x_{1}}+10u_{1}^{2}u^{2}_{1x_{1}x_{1}}+$
(17) $\displaystyle
15u_{1}^{2}u_{1x_{1}}u_{1x_{1}x_{1}x_{1}}-u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}})+\alpha
x_{1}u_{1}^{9}=ku_{1}^{9}.$
We further suppose that
${\bf v}=\xi(x_{1},u_{1})\partial_{x_{1}}+\phi(x_{1},u_{1})\partial_{u_{1}}$
is an infinitesimal generator of the last equation, then the coefficients
$\xi(x_{1},u_{1})$ and $\phi(x_{1},u_{1})$ are satisfied with
$\begin{cases}\xi=ax_{1}+b,\\\ \phi=cu_{1},\\\ [5(a+c)\alpha
x_{1}+b\alpha-(4a+5c)k]f(x_{1})\\\ +[-a\alpha
x_{1}^{2}+(ak-b\alpha)x_{1}+kb]f^{\prime}(x_{1})=0.\end{cases}$ (18)
If $f(u)=e^{\lambda u}$, from the above system we can obtain $\xi=\phi=0$,
which means that ODE${}^{\prime}_{\ref{ExponentFunction6}}$ has no nontrivial
symmetry. Thus, it can not be reduced again. Consequently, the travelling wave
solutions for Eq. (4) with $f(u)=e^{\lambda u}$ can be computed by solving a
fourth-order ODE:
$\displaystyle e^{\lambda
x_{1}}(105u^{4}_{1x_{1}}-105u_{1}u^{2}_{1x_{1}}u_{1x_{1}x_{1}}+10u_{1}^{2}u^{2}_{1x_{1}x_{1}}+$
$\displaystyle
15u_{1}^{2}u_{1x_{1}}u_{1x_{1}x_{1}x_{1}}-u_{1}^{3}u_{1x_{1}x_{1}x_{1}x_{1}})+\alpha
x_{1}u_{1}^{9}=ku_{1}^{9}.$
If $f(u)=u^{m}$, then we have three cases from system (18)
$\begin{array}[]{ll}(i)\quad\ k\neq 0,\quad\xi=0,\quad\phi=0;\\\ (ii)\quad
k=0,\quad m=1,\quad\xi=ax_{1}+b,\quad\phi=-\frac{4}{5}au_{1};\\\ (iii)\quad
k=0,\quad m\neq 1,\quad\xi=ax_{1},\quad\phi=\frac{m-5}{5}au_{1}.\end{array}$
Due to the triviality, the first case is excluded from the consideration. From
the second case, we can get a rational travelling wave solution for Eq. (4)
with $f(u)=u$ in the form
$u(t,x)=-\frac{1}{120}\alpha(x-\alpha t)^{5}+\sum_{i=0}^{4}c_{i}(x-\alpha
t)^{i}.$
For the third case, we can set
$x_{2}=u_{1}x_{1}^{\frac{5-m}{5}},\quad
u_{2}=x_{1}^{\frac{m-5}{5}}(x_{1}u_{1}^{\prime}+\frac{5-m}{5}u_{1})^{-1},$
then Eq. (17) can be reduced to:
$\displaystyle
625x_{2}^{3}u_{2}^{2}u_{2x_{2}x_{2}x_{2}}-125[50x_{2}u_{2x_{2}}+(11m$
$\displaystyle-25)x_{2}u_{2}^{2}+75u_{2}]x_{2}^{2}u_{2}u_{2x_{2}x_{2}}+9375x_{2}^{3}u_{2x_{2}}^{3}$
$\displaystyle+125[3x_{2}u_{2}(11m-25)+275]x_{2}^{2}u_{2}u_{2x_{2}}^{2}$
$\displaystyle+25[125(5m-12)x_{2}u_{2}+(46m^{2}-225m$
$\displaystyle+250)x_{2}^{2}u_{2}^{2}+2625]x_{2}u_{2}^{2}u_{2x_{2}}+(24m^{4}+875m^{2}$
$\displaystyle-250m^{3}-1250m+625\alpha x_{2}^{5}+625)x_{2}^{4}u_{2}^{7}$
$\displaystyle+10(48m^{3}-375m^{2}+875m-625)x_{2}^{3}u_{2}^{6}$
$\displaystyle+125(38m^{2}-195m+225)x_{2}^{2}u_{2}^{5}+13125(2m$
$\displaystyle-5)x_{2}u_{2}^{4}+65625u_{2}^{3}=0.$
Consequently, the travelling wave solutions for Eq. (4) with $f(u)=u^{m}(m\neq
1)$ can be computed by solving a third-order ODE.
Finally, we consider a special situation when $f(u)=ue^{-u}$ and $k=0$, in
which system (18) infers that ${\bf
v}=-5\partial_{x_{1}}+u_{1}\partial_{u_{1}}$. Let
$x_{2}=u_{1}e^{\frac{x_{1}}{5}},\quad
u_{2}=-(5u_{1x_{1}}+u)^{-1}e^{-\frac{x_{1}}{5}},$
then Eq. (17) is reduced to:
$\displaystyle
x_{2}^{3}u_{2}^{2}u_{2x_{2}x_{2}x_{2}}-x_{2}^{2}u_{2}(10x_{2}u_{2x_{2}}+11u_{2}^{2}x_{2}$
$\displaystyle+15u_{2})u_{2x_{2}x_{2}}+15x_{2}^{3}u_{2x_{2}}^{3}+11x_{2}^{2}u_{2}(5$
$\displaystyle+3x_{2}u_{2})u_{2x_{2}}^{2}+x_{2}u_{2}^{2}(46x_{2}^{2}u_{2}^{2}+125x_{2}u_{2}$
$\displaystyle+105)u_{2x_{2}}+x_{2}^{4}(625\alpha
x_{2}^{5}+24)u_{2}^{7}+96x_{2}^{3}u_{2}^{6}$
$\displaystyle+190x_{2}^{2}u_{2}^{5}+210x_{2}u_{2}^{4}+105u_{2}^{3}=0$
Therefore, the travelling wave solutions for equation (4) with $f(u)=ue^{-u}$
can be computed by solving a third-order ODE too.
### IV.3 Waiting-time Solutions
ODE${}_{\ref{PowerFunction2}}$ is a first-order equation that can be easily
solved, in this way we obtain a family of waiting-time solutions for the
sixth-order thin film equation (4) corresponding to $f(u)=u^{m}$ (if $m\neq
3/2,2,3$ or $6$). These solutions are given by
$u(t,x)=\left\\{\begin{aligned}
x^{\frac{6}{m}}\big{[}m(t_{0}-t)\prod\limits_{k=-4}^{1}(\frac{6}{m}+k)\big{]}^{-\frac{1}{m}},\quad
x\geq 0,\\\ 0,\quad\quad\quad\quad\quad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ x<0.\end{aligned}\right.$
where $t_{0}$ being an arbitrary constant.
### IV.4 Blow-up Solutions
ODE${}_{\ref{ExponentFunction2}}$ is also a first-order equation. Solving it
we get for the sixth-order thin film equation (4) with $f(u)=e^{\lambda u}$
the corresponding similarity solution
$u(t,x)=\frac{1}{\lambda}\ln\frac{(x-x_{0})^{6}}{144(t_{0}-t)}$
where $t_{0}$ is an arbitrary constant. This solution describes a localized
blow-up at $x=x_{0}$. Note that the solution is only valid if
$\frac{(x-x_{0})^{6}}{144(t_{0}-t)}\leq 1$, then, it ceases before $t=t_{0}$.
## V CONCLUDING REMARKS
We have carried out a detailed group-theoretical analysis for the generalized
one-dimensional sixth-order thin film equation (4) which arises in considering
the motion of a thin film of viscous fluid driven by an overlying elastic
plate. A complete Lie point symmetry group classification for the class (4)
have been performed under the continuous equivalence transformation group.
Based on these, a complete list of symmetry reductions of the classification
cases have been derived by making use of the optimal system of one-dimensional
subalgebras of the corresponding Lie symmetry algebras. Furthermore, invariant
solutions of the Eq. (4) with different functional form of $f$ have been
constructed by solving the reduced ODEs. In particular, by focusing our
attention in those aspects with physical interest, we have found:
1. 1.
The thin film equation (4), for the case $f(u)=u^{m}$ (which corresponds to
equation (2)), $m>-6$ admits source solutions and $m<-6$ admits sink
solutions. These solutions are related to the solutions of a fourth-order ODE.
If $m=-6$, Eq. (4) admits source and sink solutions. In this case these
families of solutions are related to a fifth-order ODE.
2. 2.
The thin film equation (4) has travelling-wave solutions. In the case
$f(u)=u^{m}$, for $m=1$ the equation admits a rational travelling-wave
solutions, for $m\neq 1$ the problem of finding these solutions can be
transformed into the problem of solving third-order ODEs. In the case
$f(u)=e^{\lambda u}$, the travelling wave solutions can be computed by solving
a fourth-order ODE. While for the case $f(u)=ue^{-u}$, the travelling wave
solutions of equation (4) can be computed by solving a third-order ODE.
3. 3.
Waiting-time solutions in the case $f(u)=u^{m}$, and blow-up solutions in the
case $f(u)=e^{\lambda u}$ are obtained in the context of symmetry reductions.
However, it should be noted that these two types of solutions can also be
obtained by means of variable separation. In the first case one takes
$u(t,x)=T(t)X(x)$ and in the second case $u(t,x)=T(t)+X(x)$.
These results may lead to further applications in physics and engineering such
as tests in numerical solutions of Eq.(4) and as trial functions for
application of variational approach in the analysis of different perturbed
versions of Eq.(4). Other topics including nonclassical symmetry, non-Lie
exact solutions and physical applications of class (4) will be studied in
subsequent publication.
###### Acknowledgements.
This work was partially supported by the National Key Basic Research Project
of China under Grant No. 2010CB126600, the National Natural Science Foundation
of China under Grant No. 60873070, Shanghai Leading Academic Discipline
Project No. B114, the Postdoctoral Science Foundation of China under Grant
Nos. 20090450067, 201104247,, Shanghai Postdoctoral Science Foundation under
Grant No. 09R21410600 and the Fundamental Research Funds for the Central
Universities under Grant No. WM0911004.
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|
arxiv-papers
| 2012-02-23T10:04:28 |
2024-09-04T02:49:27.738003
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ding-jiang Huang, Qin-min Yang and Shuigeng Zhou",
"submitter": "Ding-jiang Huang",
"url": "https://arxiv.org/abs/1202.5138"
}
|
1202.5146
|
hep-th/yymmnnn
The holographic superconductors
in higher-dimensional AdS soliton
Chong Oh Lee
Department of Physics, Kunsan National University,
Kunsan 573-701, Korea
cohlee@kunsan.ac.kr
###### Abstract
We explore the behavior of the holographic superconductors at zero temperature
for a charged scalar field coupled to a Maxwell field in higher-dimensional
AdS soliton spacetime via analytical way. In the probe limit, we obtain the
critical chemical potentials increase linearly as a total dimension $d$ grows
up. We find that the critical exponent for condensation operator is obtained
as 1/2 independently of $d$, and the charge density is linearly related to the
chemical potential near the critical point. Furthermore, we consider a
slightly generalized setup the Einstein-Power-Maxwell field theory, and find
that the critical exponent for condensation operator is given as $1/(4-2n)$ in
terms of a power parameter $n$ of the Power-Maxwell field, and the charge
density is proportional to the chemical potential to the power of $1/(2-n)$.
## 1 Introduction
Nonlinear theory of electrodynamics has been suggested in Ref. [1] in search
for an improvement over Maxwell theory with a infinite electrostatic self-
energy of a point, and its extended form has been obtained in Ref. [2]. It has
been found in Ref. [3] through investigation of transition to state of virtual
charged particle in quantum electrodynamics. It has been also studied in
gravity theory. For example, black hole solutions are obtained from nonlinear
electrodynamics minimally coupled to gravity for a static and spherical
symmetric spacetime [4], and by nonlinear electrodynamics with power-law
function [5].
On the other hand, for asymptotically AdS spacetime, it is of interest to
attempt to study the phase transition in the model for holographic
superconductors [6, 7] since it allows new predictions through exploring the
proposed AdS/CFT correspondence [8, 9, 10], which relates a gravitational
theory on asymptotically in the bulk to a conformal field theory in the
boundary. Their behavior has been explored by a gravitational theory of a
charged scalar field coupled to a Maxwell field [11, 12, 13]. The gravity
model of the holographic superconductor has revived many investigations for
their potential applications along these directions [14]-[29]. A few phase
transition studies in a Stueckelberg form have been carried out [30]-[38].
Furthermore a superconducting phase dual to the AdS soliton configuration is
interesting case [33]-[37] since the AdS black hole in the
Poincar$\rm{\acute{e}}$ coordinate can exhibit a phase transition to the AdS
soliton even if the AdS black hole and the AdS soliton have the same boundary
topology in asymptotically AdS spacetimes [39].
Even if the model for holographic superconductors is well established in four-
and five-dimensional spacetime it is less explored in higher-dimensional
spacetime. Thus, one intriguing question is that of the higher-dimensional
behavior for holographic superconductors. Another is how they are affected
from the Power-Maxwell field since they are governed by the gravity theory
with electric field coupled to the charged scalar field.
In this paper we consider the Einstein-Maxwell field theory in higher-
dimensional AdS soliton and find the critical exponent for condensation
operator is 1/2 independently of $d$ in the limit of probe at zero
temperature, and the charge density is directly proportional to the chemical
potential.
The paper is organized as follows: In the next section we investigate the
model for holographic superconductors. We obtain the critical chemical
potentials for various dimensions of operators in $d$-dimensional spacetime,
and the relations between the charge density and the chemical potential near
the critical point. In the last section we give our conclusion.
## 2 Holographic Duality in the AdS soliton background
In this section, we will construct the phase transition model for the
Einstein-Power-Maxwell field theory in the AdS soliton background.
Considering a superconductor dual to a AdS soliton configuration in the probe
limit, the line element of $d$-dimensional AdS soliton is given by [33, 40,
41]
$\displaystyle
ds^{2}=\frac{dr^{2}}{f(r)}+\frac{r^{2}}{L^{2}}(-dt^{2}+h_{ij}dx^{i}dx^{j})+f(r)d\eta^{2},$
(2.1)
with
$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}\left(1-\frac{L^{d-1}r_{0}^{d-1}}{r^{d-1}}\right),$
(2.2)
where $L$ is AdS radius and $r_{0}$ is the tip of soliton. One must impose the
periodicity $\eta\sim\eta+\frac{\pi}{r_{0}}$ to avoid a conical singularity
[42]. The $d$-dimensional Power-Maxwell-scalar action with negative
cosmological constant is
$\displaystyle S=\int d^{d}x\sqrt{-g}$
$\displaystyle\bigg{\\{}R-2\Lambda-\alpha(F_{\mu\nu}F^{\mu\nu})^{n}-\partial_{\mu}\Psi\partial^{\mu}\Psi-m^{2}\Psi^{2}$
$\displaystyle-\Psi^{2}(\partial_{\mu}\Phi-qA_{\mu})(\partial^{\mu}\Phi-
qA^{\mu})\bigg{\\}},$
where $g$ denotes the determinant of the metric, $R$ the Ricci scalar, and
$\Lambda=(d-1)(d-2)/L^{2}$ the cosmological constant. $F^{\mu\nu}$ is the
strength of the Power-Maxwell (PM) field $F=dA$, the complex scalar field
$\Psi$, the coupling constant $\alpha$, and the power of PM field $n$. We may
take the solutions of $r$ only,
$\displaystyle A=\phi(r)dt,~{}~{}~{}~{}~{}~{}\Psi=|\Psi|=\psi(r),$ (2.4)
and impose the gauge choice $\Phi=0$, and set $L=1$ and $q=1$ through
appropriately scaling symmetries in as [22]. Then the equations of motion are
given by
$\displaystyle\ddot{\psi}+\left(\frac{\dot{f}}{f}+\frac{d-2}{r}\right)\dot{\psi}+\left(\frac{r^{2}\phi^{2}}{f}-\frac{m^{2}}{f}\right)\psi=0,$
(2.5)
$\displaystyle\ddot{\phi}+\left\\{\frac{\dot{f}}{f}+\left(\frac{d-4}{2n-1}\right)\frac{1}{r}\right\\}\dot{\phi}+\frac{1}{\alpha
n(2n-1)(-2)^{n}}\frac{\psi^{2}\phi}{\dot{\phi}^{2(n-1)}f}=0,$ (2.6)
which leads to
$\displaystyle\psi^{{}^{\prime\prime}}+\left(\frac{f^{{}^{\prime}}}{f}-\frac{d-4}{z}\right)\psi^{{}^{\prime}}+\frac{r_{0}^{2}}{z^{4}}\left(\frac{z^{2}\phi^{2}}{f}-\frac{m^{2}}{f}\right)\psi=0,$
(2.7)
$\displaystyle\phi^{{}^{\prime\prime}}+\left\\{\frac{f^{{}^{\prime}}}{f}-\left(\frac{d-4}{2n-1}-2\right)\frac{1}{z}\right\\}\phi^{{}^{\prime}}-\frac{r_{0}^{2n}}{\alpha
n(2n-1)(-1)^{3n+1}2^{n}z^{4n}}\frac{\psi^{2}\phi}{(\phi^{{}^{\prime}})^{2(n-1)}f}=0,$
(2.8)
by introducing a new coordinate $z=r_{0}/r$. Here a dot denotes the derivative
with respect to $r$ and a prime is the derivative with respect to $z$.
In order to solve the above equations, one needs to impose boundary condition
at the tip $z=1$ ($r=r_{0}$) and one at the origin $z=0$ ($r=\infty$). Thus,
at the tip one can do the expansion
$\displaystyle\psi(z)$ $\displaystyle=$ $\displaystyle
a_{1}+a_{2}(z-1)+a_{3}(z-1)^{2}+\cdots,$ (2.9) $\displaystyle\phi(z)$
$\displaystyle=$ $\displaystyle b_{1}+b_{2}(z-1)+b_{3}(z-1)^{2}+\cdots,$
(2.10) $\displaystyle f(z)$ $\displaystyle=$ $\displaystyle
c_{2}(z-1)+\cdots,$ (2.11)
whose solutions behave as
$\displaystyle\psi(z=1)$ $\displaystyle=$ $\displaystyle a_{1},$ (2.12)
$\displaystyle\phi(z=1)$ $\displaystyle=$ $\displaystyle b_{1},$ (2.13)
where $a_{1}$ and $b_{1}$ are constants. Since one can set $r_{0}=1$ through
appropriately scaling symmetries in as [22], at the origin, the solutions
behave as
$\displaystyle\psi$ $\displaystyle=$ $\displaystyle
z^{\lambda_{-}}\,\psi_{-}+z^{\lambda_{+}}\,\psi_{+},$ (2.14)
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\mu-\rho
z^{(d-2)/(2n-1)-1},$ (2.15)
with
$\displaystyle\lambda_{\pm}=\frac{1}{2}\left\\{(d-1)\pm\sqrt{(d-1)^{2}+4m^{2}}\right\\},$
(2.16)
and hereafter $r_{0}=1$. In light of AdS/CFT correspondence, $\psi_{\pm}$ can
be interpreted as the expectation value of the operator $\cal{O}_{\pm}$ dual
to the charged scalar field $\psi$
$\displaystyle\psi$ $\displaystyle=$ $\displaystyle z^{\lambda_{-}}\,<{\cal
O}_{-}>+z^{\lambda_{+}}\,<{\cal O}_{+}>,$ (2.17)
and the constants $\mu$ and $\rho$ are able to be considered as the chemical
potential and charge density in the dual field theory. Since the condensation
goes to zero ($\psi\rightarrow 0$) near the critical temperature, the Eq.
(2.8) reduces to
$\displaystyle\phi^{{}^{\prime\prime}}+\left\\{\frac{f^{{}^{\prime}}}{f}-\left(\frac{d-4}{2n-1}-2\right)\frac{1}{z}\right\\}\phi^{{}^{\prime}}=0,$
(2.18)
which yields the general solution
$\displaystyle\phi=\beta+\gamma g(z),$ (2.19)
whose integration constants $\beta$ and $\gamma$ are determined by the
boundary conditions (2.12), (2.13), (2.14), and (2.15)
$\displaystyle\phi=\mu,$ (2.20)
i.e. in order to render the gauge field finite near the tip, the Neumann
boundary condition near $z=1$ imposes $\gamma=0$ so that $\beta$ is obtained
as $\mu$. This means $\phi$ has only constant solution independent of the
power of the Power-Maxwell field $n$ for any dimension $d$ in as the Einstein-
Maxwell-scalar theory [34]. Near the origin $z=0$, one can introduce a trial
function $F(z)$ for $\psi(z)$ as in [28]
$\displaystyle\psi(z)|_{z\rightarrow 0}\sim<{\cal
O}_{\pm}>\,z^{\lambda_{\pm}}\,F(z),$ (2.21)
which satisfies $F(0)=1$ and $F^{{}^{\prime}}(0)=0$. Substituting Eqs. (2.20)
and (2.21) into Eq. (2.7) we get
$\displaystyle F^{{}^{\prime\prime}}(z)$ $\displaystyle+$
$\displaystyle\left\\{-\frac{(d-2)z^{d-1}+2}{z(1-z^{d-1})}+\frac{2\lambda_{\pm}}{z}-\frac{d-4}{z}\right\\}F^{{}^{\prime}}(z)$
$\displaystyle+$
$\displaystyle\left[\frac{\lambda_{\pm}(\lambda_{\pm}-1)}{z^{2}}-\frac{\lambda_{\pm}}{z}\left\\{\frac{(d-3)z^{d-1}+2}{z(1-z^{d-1})}+\frac{d-4}{z}\right\\}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\mu^{2}}{1-z^{d-1}}-\frac{m^{2}}{z^{2}(1-z^{d-1})}\right]F(z)=0,$
which leads to
$\displaystyle\left\\{T(z)F^{{}^{\prime}}(z)\right\\}^{{}^{\prime}}-P(z)F(z)+\mu^{2}Q(z)F(z)=0,$
(2.23)
via the following functions:
$\displaystyle T(z)$ $\displaystyle=$ $\displaystyle
z^{2\lambda_{\pm}-3}(z^{d-1}-1),$ (2.24) $\displaystyle P(z)$ $\displaystyle=$
$\displaystyle-T(z)\left[\frac{\lambda_{\pm}(\lambda_{\pm}-1)}{z^{2}}-\frac{\lambda_{\pm}}{z}\left\\{\frac{(d-3)z^{d-1}+2}{z(1-z^{d-1})}+\frac{d-4}{z}\right\\}-\frac{m^{2}}{z^{2}(1-z^{d-1})}\right],$
$\displaystyle Q(z)$ $\displaystyle=$ $\displaystyle\frac{T(z)}{1-z^{d-1}}$
After setting the trial function $F(z)=1-az^{2}$, the minimum eigenvalues of
$\mu^{2}$ is calculated from the variation of the following functional [28]
$\displaystyle\mu^{2}=\frac{\int_{0}^{1}dz\bigg{\\{}T(z)F^{{}^{\prime}2}(z)+P(z)F^{2}(z)\bigg{\\}}}{\int_{0}^{1}dzQ(z)F^{2}(z)}.$
(2.25)
After taking $m^{2}=d(d-2)/4$, from Eq.(2.16) we get the operator ${\cal
O}_{-}$ of conformal dimension
$\displaystyle\lambda_{-}=\frac{d-2}{2},$ (2.26)
Then, $\mu_{-}^{2}$ is explicitly given by
$\displaystyle\mu_{-}^{2}=\frac{s_{\mu_{-}}(a,d)}{t_{\mu_{-}}(a,d)},$ (2.27)
where
$\displaystyle s_{\mu_{-}}(a,d)$ $\displaystyle=$ $\displaystyle
d(d-4)\bigg{\\{}(2d-5)(2d-7)(d^{3}-6d^{2}+28d-24)a^{2}$
$\displaystyle-2(d-2)^{3}(2d-7)(2d-3)a+(d-2)^{3}(2d-5)(2d-3)\bigg{\\}},$
$\displaystyle t_{\mu_{-}}(a,d)$ $\displaystyle=$ $\displaystyle
4(2d-3)(2d-5)(2d-7)\bigg{\\{}(d-2)(d-4)a^{2}-2d(d-4)a+d(d-2)\bigg{\\}}.$
When the constant $a_{-}$ is
$\displaystyle a_{-}=\frac{s_{a_{-}}(d)}{t_{a_{-}}(d)},$ (2.29) $\displaystyle
s_{a_{-}}(d)$ $\displaystyle=$ $\displaystyle
2d^{6}-11d^{5}+7d^{4}+12d^{3}+132d^{2}-376d+240-2\bigg{(}53d^{10}-882d^{9}$
$\displaystyle+6094d^{8}-22310d^{7}+44985d^{6}-43972d^{5}+5624d^{4}+16608d^{3}+12448d^{2}$
$\displaystyle-33024d+14400\bigg{)}^{1/2},$ $\displaystyle t_{a_{-}}(d)$
$\displaystyle=$ $\displaystyle
2d^{6}-d^{5}-129d^{4}+578d^{3}-620d^{2}-472d+672,$
the minimum eigenvalue $\mu_{\rm min(-)}$ yields
$\displaystyle\mu_{\rm min(-)}=\frac{s_{\mu_{\rm min(-)}}(d)}{t_{\mu_{\rm
min(-)}}(d)},$ (2.31)
with
$\displaystyle s_{\mu_{\rm min(-)}}(d)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}11d^{5}-105d^{4}+371d^{3}-600d^{2}+440d-120-(d-2)\bigg{(}53d^{8}$
$\displaystyle-670d^{7}+3202d^{6}-6822d^{5}+4889d^{4}+2872d^{3}-2444d^{2}-4656d+3600\bigg{)}^{1/2}\Bigg{\\}}^{1/2},$
$\displaystyle t_{\mu_{\rm min(-)}}(d)$ $\displaystyle=$ $\displaystyle
2\bigg{(}(2d-3)(2d-5)(2d-7)\bigg{)}^{1/2}.$
For example, the minimum eigenvalue $\mu_{\rm min}$ (2.31) for $d=5$ is given
by $\mu_{c}=\mu_{\rm min(-)}\thickapprox 0.837$, which is exactly matched with
that in [34], and $\mu_{\rm min(-)}\thickapprox 1.22$ for $d=6$, and $\mu_{\rm
min(-)}\thickapprox 1.58$ for $d=7$.
When the scalar field squared mass $m^{2}$ is bigger than the Breitenlohner-
Freedman bound squared mass $m_{\rm BF}^{2}=-(d-1)^{2}/4$, the ${\cal O}_{+}$
is normalizable. Furthermore, since it is possible that the analysis in
previous case is applied to any $m^{2}$ in the range $m_{\rm BF}^{2}<m^{2}<0$,
the chemical potential $\mu_{c}$ is investigated for more general squared mass
$m^{2}$. We now deal with operator of the dimension $\lambda_{+}=d/2$ before
operators of general dimensions.
In the same way in previous case $\mu_{\rm min(-)}$, taking $m^{2}=d(d-2)/4$,
the dimension of operator $\lambda_{+}$ (2.16) reduces to $\lambda_{+}=d/2$.
Then the minimum eigenvalue $\mu_{\rm min(+)}$ is obtained as
$\displaystyle\mu_{\rm min(+)}=\frac{s_{\mu_{\rm min(+)}}(d)}{t_{\mu_{\rm
min(+)}}(d)},$ (2.33)
with
$\displaystyle s_{\mu_{\rm min(+)}}(d)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}11d^{5}-33d^{4}-13d^{3}+94d^{2}-60d-d\bigg{(}53d^{8}-266d^{7}-114d^{6}$
$\displaystyle+2558d^{5}-3451d^{4}-4192d^{3}+13804d^{2}-12000d^{1}+3600\bigg{)}^{1/2}\Bigg{\\}}^{1/2},$
$\displaystyle t_{\mu_{\rm min(+)}}(d)$ $\displaystyle=$ $\displaystyle
2\bigg{(}(2d-1)(2d-3)(2d-5)\bigg{)}^{1/2}.$
The critical value $\mu_{c}=\mu_{\rm min(+)}\thickapprox 1.890$ for $d=5$ is
absolute agreement with the numerical result in [34], and $\mu_{\rm
min(+)}\thickapprox 2.205$ for $d=6$, and $\mu_{\rm min(+)}\thickapprox 2.531$
for $d=7$.
After taking the dimension of operator ${\lambda_{-}}=(d-2)/2$ and
$\lambda_{+}=d/2$, we obtain the critical chemical potential $\mu_{c}$ as the
total dimension $d=5$ to $d=21$, and so it is linearly proportional to $d$. We
plot these results in Figure 1.
Figure 1: The critical chemical potential $\mu_{c}$ is plotted as the total
dimension $d=5$ to $d=21$ where red is the dimension of operator
${\lambda_{-}}=(d-2)/2$ and blue $\lambda_{+}=d/2$.
Considering the operators of more general dimensions, the square of chemical
potential is obtained as
$\displaystyle\mu^{2}=\frac{s_{\mu_{m^{2}}}(d)}{t_{\mu_{m^{2}}}(d)},$ (2.35)
with
$\displaystyle s_{\mu_{m^{2}}}(d)$ $\displaystyle=$
$\displaystyle\left(\frac{2m^{2}+(d-1)\sqrt{(d-1)^{2}+4m^{2}}+(d-2)(d-7)}{\sqrt{(d-1)^{2}+4m^{2}}+2(d-1)}\right.$
$\displaystyle\hskip
8.5359pt\left.+\frac{8}{\sqrt{(d-1)^{2}+4m^{2}}+d-1}\right)a^{2}$
$\displaystyle+2\left(\frac{2m^{2}+(d-1)\sqrt{(d-1)^{2}+4m^{2}}+(d-1)^{2}}{\sqrt{(d-1)^{2}+4m^{2}}+2(d-2)}\right)a$
$\displaystyle+\frac{2m^{2}+2(d-1)\sqrt{(d-1)^{2}+m^{2}}+(d-1)^{2}}{\sqrt{(d-1)^{2}+2m^{2}}+2(d-3)},$
$\displaystyle t_{\mu_{m^{2}}}(d)$ $\displaystyle=$
$\displaystyle\frac{2a^{2}}{\sqrt{(d-1)^{2}+4m^{2}}+d+1}+\frac{4a}{\sqrt{(d-1)^{2}+4m^{2}}+d-1}$
$\displaystyle+\frac{1}{\sqrt{(d-1)^{2}+4m^{2}}+d-3}.$
In spite of getting the explicit form of the critical potential $\mu_{c}$, the
result is not shown in this article since it is rather lengthy, so we attempt
to show the result for $d=7$ instead. $\mu^{2}$ for $d=7$ yields
$\displaystyle\mu^{2}=\frac{s_{\mu_{m^{2}}}(7)}{t_{\mu_{m^{2}}}(7)},$ (2.37)
with
$\displaystyle s_{\mu_{m^{2}}}(7)$ $\displaystyle=$
$\displaystyle\bigg{\\{}18m^{4}+786m^{2}+6396\left(m^{4}+146m^{2}+2124\right)\sqrt{m^{2}+9}\bigg{\\}}a^{2}$
$\displaystyle-2\bigg{\\{}19m^{4}+855m^{2}+6804\left(m^{4}+159m^{2}+2268\right)\sqrt{m^{2}+9}\bigg{\\}}a$
$\displaystyle+20m^{4}+954m^{2}+7776\left(m^{4}+174m^{2}+2592\right)\sqrt{m^{2}+9},$
$\displaystyle t_{\mu_{m^{2}}}(7)$ $\displaystyle=$
$\displaystyle\bigg{(}m^{2}+15+5\sqrt{m^{2}+9}\bigg{)}a^{2}-2\bigg{(}m^{2}+17+6\sqrt{m^{2}+9}\bigg{)}a$
$\displaystyle+m^{2}+21+7\sqrt{m^{2}+9},$
which leads to the minimum eigenvalue $\mu_{\rm min(+)}$
$\displaystyle\mu_{\rm min(+)}=\frac{s_{\mu_{\rm min(+)}}(7)}{t_{\mu_{\rm
min(+)}}(7)},$ (2.40)
with
$\displaystyle s_{\mu_{\rm min(+)}}(7)$ $\displaystyle=$
$\displaystyle\Bigg{[}m^{8}-22m^{6}-513m^{4}+(8m^{6}-422m^{4})\sqrt{m^{2}+9}$
$\displaystyle+m^{2}\bigg{\\{}15198+6126\sqrt{m^{2}+9}+30\bigg{(}5m^{8}+2907m^{6}+200595m^{4}$
$\displaystyle+3778092m^{2}+\sqrt{m^{2}+9}(178m^{6}+28296m^{4}+882612m^{2}+6781536)$
$\displaystyle+20344608\bigg{)}^{1/2}-2\sqrt{m^{2}+9}\bigg{(}5m^{8}+2907m^{6}+200595m^{4}+3778092m^{2}$
$\displaystyle+\sqrt{m^{2}+9}(178m^{6}+28296m^{4}+882612m^{2}+6781536)+20344608\bigg{)}^{1/2}\bigg{\\}}$
$\displaystyle-2\bigg{\\{}37692+12564\sqrt{m^{2}+9}-255\bigg{(}5m^{8}+2907m^{6}+200595m^{4}$
$\displaystyle+3778092m^{2}+\sqrt{m^{2}+9}(178m^{6}+28296m^{4}+882612m^{2}+6781536)$
$\displaystyle+20344608\bigg{)}^{1/2}+83\sqrt{m^{2}+9}\bigg{(}5m^{8}+2907m^{6}+200595m^{4}+3778092m^{2}$
$\displaystyle+\sqrt{m^{2}+9}(178m^{6}+28296m^{4}+882612m^{2}+6781536)+20344608\bigg{)}^{1/2}\bigg{\\}}\Bigg{]}^{1/2},$
$\displaystyle t_{\mu_{\rm min(+)}}(7)$ $\displaystyle=$
$\displaystyle\sqrt{(m^{2}-7)(m^{2}-16)(m^{2}-27)}.$ (2.42)
We plot the function (2.37) in Figure 2. (a) for $-9<m^{2}<0$, which indicates
that there is always the minimum value of chemical potential squared for
various $a$’s and $m^{2}$’s when $a\rightarrow 0$. As squared mass $m^{2}$
increases up to the Breitenlohner-Freedman bound squared mass $m_{\rm
BF}^{2}$, the critical chemical potential $\mu_{c}$ increases (see in Figure
2. (b)).
(a) (b)
Figure 2: (a) The square of chemical potential $\mu^{2}$ is plotted as the
constant $a$ and the square of mass $m^{2}$ for $d=7$. (b) A plot of the
function $\mu_{c}(m^{2})$ for $d=7$. $\mu_{c}$ has 2.531 when $m^{2}=-35/4$.
When $\mu$ is very closely located near $\mu_{c}$, we have
$\displaystyle\phi^{{}^{\prime\prime}}+\left\\{\frac{f^{{}^{\prime}}}{f}-\left(\frac{d-4}{2n-1}-2\right)\frac{1}{z}\right\\}\phi^{{}^{\prime}}=\frac{<{\cal
O}_{\pm}>^{2}z^{-4n+2\lambda_{\pm}}F^{2}(z)}{\alpha
n(2n-1)(-1)^{3n+1}2^{n}}\frac{\phi}{(\phi^{{}^{\prime}})^{2(n-1)}f},$ (2.43)
by plugging Eq. (2.21) into Eq. (2.8). In such a limit, we may take $\phi(z)$
as
$\displaystyle\phi(z)=\mu_{c}+<{\cal O}_{\pm}>\chi(z),$ (2.44)
where the boundary condition near the tip imposes
$\displaystyle\chi(z)|_{z\rightarrow 1}=0.$ (2.45)
Substituting in Eq. (2.43), we obtain
$\displaystyle\chi^{{}^{\prime\prime}}-\frac{\bigg{(}2d(n-1)-2n+5\bigg{)}z^{d-1}+d-4}{(2n-1)(z-z^{d})}\chi^{{}^{\prime}}=\frac{<{\cal
O}_{\pm}>^{3-2n}z^{-4n+2\lambda_{\pm}}F^{2}(z)}{\alpha
n(2n-1)(-1)^{3n+1}2^{n}f}\frac{\mu_{c}}{(\chi^{{}^{\prime}})^{2(n-1)}},$
(2.46)
which for $n=1$ reduces to
$\displaystyle\frac{d}{dz}\left[T_{1}(z)\chi^{{}^{\prime}}\right]=-\frac{<{\cal
O}_{\pm}>\mu_{c}F^{2}(z)}{2\alpha}\frac{z^{2+2\lambda_{\pm}}}{z^{d}}$ (2.47)
by introducing the function $T_{1}(z)$
$\displaystyle T_{1}(z)=\frac{z^{d-1}-1}{z^{d-4}}.$ (2.48)
Considering the operator of dimension $\lambda_{-}=(d-2)/2$ and taking
$\alpha=1/4$, the above Eq. (2.47) is obtained as
$\displaystyle\frac{d}{dz}\left[\frac{z^{d-1}-1}{z^{d-4}}\chi^{{}^{\prime}}\right]=-2<{\cal
O}_{-}>\mu_{c}F^{2}(z),$ (2.49)
from which it follows that, integrating both sides,
$\displaystyle\left.\frac{z^{d-1}-1}{z^{d-4}}\chi^{{}^{\prime}}\right|_{0}^{1}$
$\displaystyle=$
$\displaystyle\left.\frac{\chi^{{}^{\prime}}}{z^{d-4}}\right|_{z\rightarrow
0}=-2<{\cal
O}_{-}>\mu_{c}\left.z\left(\frac{a^{2}z^{4}}{5}-\frac{2az^{2}}{3}+1\right)\right|_{0}^{1}$
$\displaystyle=$ $\displaystyle-2<{\cal
O}_{-}>\mu_{c}\left(\frac{a^{2}}{5}-\frac{2a}{3}+1\right).$
$\phi(z)$ near $z=0$ is asymptotically given as
$\displaystyle\phi(z)|_{z\rightarrow 0}\thicksim\mu-\rho
z^{2}\thickapprox\mu_{c}+<O_{-}>\bigg{(}\chi(0)+\chi^{{}^{\prime}}(0)z+\frac{1}{2}\chi^{{}^{\prime\prime}}(0)z^{2}+{\cal
O}(z^{3})\bigg{)},$ (2.51)
which leads to
$\displaystyle\mu-\mu_{c}=<{\cal O}_{-}>\chi(0),$ (2.52)
by comparing the coefficients of zeroth order in $z$ in both sides, and from
first order we can read
$\displaystyle\chi^{{}^{\prime}}(0)=0.$ (2.53)
After imposing two boundary conditions (2.45) and (2.53), Eq. (2.51) $\chi(z)$
for $d=7$ is explicitly obtained as
$\displaystyle\chi(z)=\begin{array}[]{cl}&\frac{<{\cal
O}_{-}>\mu_{c}}{90}\bigg{[}-\frac{36}{5}a^{2}(z^{5}-1)+120a(z-1)-\frac{3}{2}(3a^{2}-10a+15)\ln(z^{4}+1)\\\
&\hskip
49.50795pt+2(3a^{2}-10a+15)\ln(z^{3}+1)+15\sqrt{2}a\ln(z^{2}-\sqrt{2}z+1)\\\
&\hskip
49.50795pt-15\sqrt{2}a\ln(z^{2}+\sqrt{2}z+1)+4\sqrt{3}(3a^{2}-10a+15)\tan^{-1}(\frac{2z-1}{\sqrt{3}}z)\\\
&\hskip
49.50795pt+15(\sqrt{2}-1)\tan^{-1}(\sqrt{2}z+1)-15(\sqrt{2}+1)\tan^{-1}(\sqrt{2}z-1)\\\
&\hskip
49.50795pt+3\bigg{\\{}3(\sqrt{2}z+1)a^{2}-10a\bigg{\\}}-3\bigg{\\{}3(\sqrt{2}z-1)a^{2}+10a\bigg{\\}}\\\
&\hskip
49.50795pt-\frac{1}{2}(3a^{2}-10a+15)\ln(2)+30\sqrt{2}a\coth^{-1}(\sqrt{2})\\\
&\hskip
49.50795pt-\frac{1}{12}\bigg{\\{}3(9-18\sqrt{2}+8\sqrt{3})a^{2}-10(9+8\sqrt{2})a+15(9-18\sqrt{2}+8\sqrt{3})\bigg{\\}}\pi\bigg{]}\end{array}.$
(2.61)
Thus, from Eq. (2.52) we get the qualitative relation between the condensation
value $<{\cal O}_{-}>$ and the chemical potential difference ($\mu-\mu_{c}$)
for arbitrary dimension $d$
$\displaystyle<{\cal O}_{-}>\thicksim\gamma_{-}\sqrt{\mu-\mu_{c}},$ (2.62)
and comparing the coefficients of the $z^{2}$ term in (2.51), we read the
linear relation between the charge density $\rho$ and ($\mu-\mu_{c}$)
$\displaystyle\rho\thicksim\delta_{-}(\mu-\mu_{c}),$ (2.63)
where $\gamma_{-}$ and $\delta_{-}$ are positive constants. For example
$\gamma_{-}$ and $\delta_{-}$ for $d=5$, $d=6$, and $d=7$ are given as
$\displaystyle\gamma_{-}=\left\\{\begin{array}[]{cl}&1.940~{}~{}~{}~{}~{}{\rm
for}~{}d=5\\\ &1.987~{}~{}~{}~{}~{}{\rm for}~{}d=6\\\
&2.042~{}~{}~{}~{}~{}{\rm for}~{}d=7\end{array}\right.,\hskip
49.50795pt\delta_{-}=\left\\{\begin{array}[]{cl}&2.700~{}~{}~{}~{}~{}{\rm
for}~{}d=5\\\ &4.050~{}~{}~{}~{}~{}{\rm for}~{}d=6\\\
&5.399~{}~{}~{}~{}~{}{\rm for}~{}d=7.\end{array}\right.$ (2.70)
Taking $\lambda_{+}=d/2$, the Eq. (2.47) is
$\displaystyle\frac{d}{dz}\left[\frac{z^{d-1}-1}{z^{d-4}}\chi^{{}^{\prime}}\right]=-2<{\cal
O}_{+}>\mu_{c}F^{2}(z)z^{2},$ (2.71)
which leads to
$\displaystyle\left.\frac{\chi^{{}^{\prime}}}{z^{d-4}}\right|_{z\rightarrow
0}=-2<{\cal
O}_{+}>\mu_{c}\left(\frac{a^{2}}{7}-\frac{2a}{5}+\frac{1}{3}\right).$ (2.72)
From following the preceding steps, we obtain
$\displaystyle<{\cal O}_{+}>\thicksim\gamma_{+}\sqrt{\mu-\mu_{c}}\,,\hskip
28.45274pt\rho\thicksim\delta_{+}(\mu-\mu_{c}),$ (2.73)
where $\gamma_{+}$ and $\delta_{+}$ are positive constants. For example
$\gamma_{+}$ and $\delta_{+}$ for $d=5$, $d=6$, and $d=7$ are given as
$\displaystyle\gamma_{+}=\left\\{\begin{array}[]{cl}&1.801~{}~{}~{}~{}~{}{\rm
for}~{}d=5\\\ &2.099~{}~{}~{}~{}~{}{\rm for}~{}d=6\\\
&2.316~{}~{}~{}~{}~{}{\rm for}~{}d=7\end{array}\right.,\hskip
49.50795pt\delta_{+}=\left\\{\begin{array}[]{cl}&1.329~{}~{}~{}~{}~{}{\rm
for}~{}d=5\\\ &1.994~{}~{}~{}~{}~{}{\rm for}~{}d=6\\\
&2.659~{}~{}~{}~{}~{}{\rm for}~{}d=7.\end{array}\right.$ (2.80)
As Figure 3 shows, supposing the total dimension $d$ more increases than
$d=5$, the coefficient $\gamma_{+}$ in Eq. (2.73) is bigger than the
coefficient $\gamma_{-}$ Eq. (2.62), and $\delta_{\pm}$ increase linearly as
$d$ grows up.
(a) (b)
Figure 3: (a) The coefficient $\gamma_{\pm}$ in Eqs. (2.62) and (2.73) is
plotted as the total dimension $d=5$ to $d=21$. (b) The coefficient
$\delta_{\pm}$ is plotted as $d=5$ to $d=21$. Here, red is the dimension of
operator ${\lambda_{-}}=(d-2)/2$ and blue $\lambda_{+}=d/2$.
We now come back to any power of PM field $n$, and the Eq. (2.46) leads to
$\displaystyle\frac{d}{dz}\left[T_{n}(z)(\chi^{{}^{\prime}})^{2n-1}\right]=-\frac{<{\cal
O}_{\pm}>^{3-2n}z^{-2n+2\lambda_{\pm}}(z^{d}-1)^{2n-2}F^{2}(z)}{\alpha
n(-1)^{3n+1}2^{n}}\mu_{c},$ (2.81)
with
$\displaystyle T_{n}(z)=\frac{(z^{d-1}-1)^{2n-1}}{z^{d-4}}.$ (2.82)
Then the condensation value $<{\cal O}_{\pm}>$ and the charge density $\rho$
are qualitatively
$\displaystyle<{\cal
O}_{\pm}>\thicksim\xi_{\pm}(\mu-\mu_{c})^{\frac{1}{4-2n}}\,,\hskip
28.45274pt\rho\thicksim\zeta_{\pm}(\mu-\mu_{c})^{\frac{1}{2-n}},$ (2.83)
where $\xi_{\pm}$ and $\zeta_{\pm}$ are positive constants. It implies that
the critical exponent of condensation operator can be changed into $1/(4-2n)$
for various $n$ unlike that of Maxwell field.
## 3 Conclusion
Previous work on the analytical behavior of the holographic superconductors in
five-dimensional AdS soliton spacetime [34] have found that the critical
exponent of condensation operator is 1/2, and the charge density is linearly
depending on the chemical potential. We also get the same results in higher-
dimensional cases. However, the critical exponent of condensation operator is
changed into $1/(4-2n)$ in the context of the Einstein-Power-Maxwell field
theory, and the charge density is proportional to the chemical potential to
the power of $1/(2-n)$. In addition, since analytical calculations in Eqs.
(2.62) and (2.73) indicate AdS soliton background is unstable below the
threshold value $\mu_{c}$ but are stable above this value, they may play the
role of higher-dimensional insulator and superconductor in the dual field
theory, respectively, as in the case of $5$-dimensional AdS soliton [33, 34,
35].
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|
arxiv-papers
| 2012-02-23T10:39:43 |
2024-09-04T02:49:27.746338
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chong Oh Lee",
"submitter": "Chongoh Lee",
"url": "https://arxiv.org/abs/1202.5146"
}
|
1202.5195
|
# On the injection of helicity by shearing motion of fluxes in relation to
Flares and CMEs
P. Vemareddy1, A. Ambastha1, R. A. Maurya2 and J. Chae2 1Udaipur Solar
Observatory, Physical Research Laboratory, Udaipur-313 001,India. 2Astronomy
Program, Department of Physics and Astronomy, Seoul National University, Seoul
151-747, Korea vema@prl.res.in, ambastha@prl.res.in, ramajor@astro.snu.ac.kr,
jcchae@snu.ac.kr
###### Abstract
An investigation of helicity injection by photospheric shear motions is
carried out for two active regions(ARs), NOAA 11158 and 11166, using line-of-
sight magnetic field observations obtained from the Helioseismic and Magnetic
Imager on-board Solar Dynamics Observatory. We derived the horizontal
velocities in the active regions from the Differential Affine Velocity
Estimator(DAVE) technique. Persistent strong shear motions at the maximum
velocities in the range of 0.6–0.9km/s along the magnetic polarity inversion
line and outward flows from the peripheral regions of the sunspots were
observed in the two active regions. The helicities injected in NOAA 11158 and
11166 during their six days’ evolution period were estimated as $14.16\times
10^{42}$Mx2 and $9.5\times 10^{42}$Mx2, respectively. The estimated injection
rates decreased up to 13% by increasing the time interval between the
magnetograms from 12 min to 36 min, and increased up to 9% by decreasing the
DAVE window size from $21\times 18$ to $9\times 6$ pixel2, resulting in 10%
variation in the accumulated helicity. In both ARs, the flare prone regions
(R2) had inhomogeneous helicity flux distribution with mixed helicities of
both signs and that of CME prone regions had almost homogeneous distribution
of helicity flux dominated by single sign. The temporal profiles of helicity
injection showed impulsive variations during some flares/CMEs due to negative
helicity injection into the dominant region of positive helicity flux. A
quantitative analysis reveals a marginally significant association of helicity
flux with CMEs but not flares in AR 11158, while for the AR 11166, we found
marginally significant association of helicity flux with flares but not CMEs,
providing evidences of the role of helicity injection at localized sites of
the events. These short-term variations of helicity flux are further discussed
in view of possible flare-related effects. This study suggests that flux
motions and spatial distribution of helicity injection are important to
understand the complex nature of magnetic flux system of the active region
leading to conditions favorable for eruptive events.
Sun: activity — Sun: flares — Sun: magnetic fields— Sun:Coronal Mass
ejections— Sun: helicity injection
## 1 Introduction
Magnetic helicity is an important topological property of solar active regions
(ARs) and is a measure of twist and writhe of the field lines (Berger & Field,
1984; Finn & Antonsen, 1985). It is gauge invariant for a closed volume of
space. The Sun’s outer atmosphere is dominated by magnetic field at all
scales. Dynamic phenomena, such as, energetic flares and coronal mass
ejections (CMEs) occur due to the loss of equilibrium during the evolution of
magnetic fields in solar ARs. Magnetic helicity has become an important
physical parameter in the context of solar transient phenomena. It is one of
the few global quantities which is conserved even in resistive magneto-
hydrodynamics on a timescale less than that of the global diffusion. There
exists no absolute measure of helicity within a sub-volume of space if that
sub-volume is not bounded by a magnetic surface. However, a topologically
meaningful and gauge invariant relative helicity for such volumes can be
defined.
There are several methods for estimating helicity in solar ARs. By the force-
free field assumption of coronal magnetic field, we have:
$\nabla\cdot\mathbf{B}=\alpha\mathbf{B}$ (1)
where $\alpha$ is the force-free parameter, also known as helicity or twist
parameter. Assuming $\alpha$ to be constant for the whole AR, we can fit
observed vector magnetograms to deduce the value of $\alpha$ (Pevtsov et al.,
1995; Hagyard & Pevtsov, 1999; Tiwari et al., 2009). Latitudinal variation of
helicity of photospheric magnetic fields, and statistical significance of the
observed temporal variations of the ARs’ hemispheric helicity rule, as
measured by the latitudinal gradient of the best-fit linear force-free
parameter $\alpha$, etc., have been discussed by Pevtsov et al. (2008).
The Poynting-like theorem for helicity in an open volume as derived by Berger
& Field (1984) is given by:
$\frac{dH}{dt}=\oint 2(\mathbf{B_{t}}\cdot\mathbf{A_{p}})v_{z}ds-\oint
2(\mathbf{A_{p}}\cdot\mathbf{v})B_{z}ds$ (2)
where $\mathbf{A_{p}}$ is the vector potential of the potential magnetic
field, $\mathbf{B_{p}}$, which is uniquely specified by the observed flux
distribution on the surface(x-y plane) as
$\nabla\times\mathbf{A_{p}}\cdot\hat{z}=B_{z};\hskip
8.5359pt\nabla\cdot\mathbf{A_{p}}=0;\hskip
8.5359pt\mathbf{A_{p}}\cdot\hat{z}=0$ (3)
where $\hat{z}$ refers to unit vector along vertical direction of Cartesian-
geometry. Equation 2 shows that the helicity of magnetic fields in an open
volume may change by the passage of helical field lines through the surface
(first term) and/or by photospheric footpoint motions of the field lines
(second term). The temporal evolution of magnetic helicity flux across the
photosphere characterizes the injection of magnetic helicity from the sub-
photospheric layers into the solar atmosphere, horizontal flux motions, and
the changes in the coronal magnetic field configurations related to eruptive
events, such as the CMEs, propagating into the interplanetary medium.
During the past years, several attempts have been made to estimate magnetic
helicity from suitable solar observations. Chae (2001) developed a method for
determining the helicity flux (the second term in Equation 2) passing through
the photosphere. They used a time series of photospheric line-of-sight (LOS)
magnetograms to determine horizontal velocities by local correlation tracking
(LCT) technique (November & Simon, 1988). Using this method, vector potential
$\mathbf{A_{p}}$ was constructed by using photospheric LOS field (as an
approximation to $B_{z}$ field) as boundary conditions with Coulomb gauge in
terms of Fourier Transform (FT) as:
$\displaystyle A_{\rm p,x}$ $\displaystyle=$ $\displaystyle
FT^{-1}\left[\frac{jk_{y}}{k_{x}^{2}+k_{y}^{2}}FT\left(B_{\rm
z}\right)\right]$ $\displaystyle A_{\rm p,y}$ $\displaystyle=$ $\displaystyle
FT^{-1}\left[\frac{-jk_{x}}{k_{x}^{2}+k_{y}^{2}}FT\left(B_{\rm
z}\right)\right]$
where $k_{x}$, $k_{y}$ are spatial frequencies in the x, y directions,
respectively. Later, this method was applied to many ARs by several authors
(Chae et al., 2001; Moon et al., 2002; Nindos et al., 2003; Chae et al.,
2004). However, Pariat et al. (2005) showed that this method of calculation
introduced artificial polarities of both signs in the helicity flux density
maps with many flow patterns. Therefore, they suggested to use relative
velocities for calculating the helicity injection rate:
$\frac{dH}{dt}=\frac{-1}{2\pi}\int_{S}\int_{S^{\prime}}\frac{[(\mathbf{x}-\mathbf{x^{\prime}})\times(\mathbf{u}-\mathbf{u^{\prime}})]_{n}}{|\mathbf{x}-\mathbf{x^{\prime}}|^{2}}B^{\prime}_{z}(\mathbf{x^{\prime}})B_{z}(\mathbf{x})dS^{\prime}dS$
(4)
where $\mathbf{u}$ is the foot-point velocity at the position vector
$\mathbf{x}$, and $B_{z}$ is the vertical component of the observed magnetic
field. This equation shows that the helicity injection rate can be understood
as the summation of relative rotation rates of all the pairs of elementary
fluxes weighted with their magnetic flux.
Furthermore, Schuck (2005) has shown that the LCT method is inconsistent with
the magnetic induction equation, which governs the temporal evolution of the
photospheric magnetic fields. Tracking methods have serious practical
limitations that might result in the failure of detecting significant shear
velocity fields and hence in the underestimate of the amount of helicity
injected by such velocity fields. Démoulin & Berger (2003) reported that the
magnetic energy and helicity fluxes should be computed only from the
horizontal motions deduced by tracking the photospheric cross-section of
magnetic flux tubes. These authors contend that the apparent horizontal
motions include the effect of both the emergence and the shearing motions.
They analyzed the observational difficulties involved in deriving such fluxes
and in particular, the limitations of the correlation tracking methods. One of
the main limitations in the previous studies has been the coarse spatial
resolution of the available observations which limits the deduced velocities
to the velocity corresponding to the group motion of an unresolved bunch of
thin flux tubes covered by a pixel. Also, tracking motions faced difficulties
in the areas lacking sufficient contrast, such as in the sunspot umbrae.
Several alternative, improved methods have been developed for inferring plasma
velocities consistent with the induction equation at the photospheric level,
based on the LOS, as well as, vector magnetograms. The Induction method (IM;
Kusano et al. 2002), induction local correlation tracking(ILCT; Welsch et al.
2004), minimum energy fit (MEF; Longcope 2004), differential affine velocity
estimator (DAVE; Schuck 2005, 2006) and differential affine velocity estimator
for vector magnetograms (DAVE4VM; Schuck 2008) have been developed for the
determination of horizontal component of motion. In contrast, the normal
component of velocity can be determined from the doppler observations of ARs
located near the disk center. DAVE4VM method requires vector magnetograms. The
performance of different techniques has been examined in Welsch et al. (2007)
which showed that the MEF, DAVE, FLCT, IM, and ILCT algorithms performed
comparably. Furthermore, they reported that while the DAVE estimated the
magnitude and direction of velocities slightly more accurately than the other
methods, MEF’s estimates of the fluxes of magnetic energy and helicity were
more accurate.
Time series data of photospheric magnetograms have been extensively used to
derive magnetic helicity and its evolution in order to examine its role in the
level of transient activity of the ARs. Moon et al. (2002) reported impulsive
variations of helicity during some M and X-class flares. In a survey, LaBonte
et al. (2007) revealed that X-flaring ARs have a higher net helicity change
with peak helicity rate $>6\times 10^{36}$Mx2s-1 with weak hemispheric
preference. Park et al. (2010b) have also studied the solar flare productivity
in relation to the helicity injection using a large sample of 378 active
regions. Using SOHO-MDI magnetograms, they reported variation of helicity
injection rates and a significant helicity accumulation of $(3-45\times
10^{42})$ Mx2 over several days around the time of flares above M5.0. Most of
the previous studies that used data from Michelson Doppler Imager (MDI)
onboard SOHO had the time resolution of 96 minutes. The rather coarse time
resolution between two consecutive observations has been a matter of concern
in the above calculations because the contribution from the motion of short
lived magnetic features in small intervals is difficult to be accounted
suitably (e.g., Chae et al. 2004). This underlines the need for observations
of magnetic fields with higher temporal resolution.
The above mentioned issues can now be addressed with the availability of a
better cadence of 12 minutes by the recently launched Helioseismic and
Magnetic Imager (HMI) onboard Solar Dynamics Observatory (SDO). Our main
objective in the present work is to reinvestigate the role of helicity
injection in relation with flares and CMEs using the high-resolution data
obtained from SDO-HMI. We intend to utilize this opportunity to revisit some
of the previous studies involving computations of helicity rate for two ARs,
NOAA 11158 and NOAA 11166, that appeared during February and March 2011,
respectively, in the ascending phase of the current Solar Cycle 24.
Using the high quality HMI data obtained for the two ARs, we intend to examine
whether the variations as reported by Moon et al. (2002) and Park et al.
(2010a) for energetic flares occurred also during the flares of lower
magnitude. It is of particular interest to investigate if such changes were
associated with the CMEs as well. For our analysis, we use DAVE technique for
retrieving horizontal foot-point velocity from the LOS magnetograms.
Thereafter, using Equation 4 we determine helicity injection rates and the
accumulated helicity in the two ARs due to foot-point shearing motions during
their disk transit. It has been inferred from the previous studies that most
of the helicity injection corresponds to magnetic flux emergence in the
ARs(Jeong & Chae, 2007). We, therefore, attempt to interpret the variations
found in these physical parameters in relation to the occurrence of flares and
CMEs. In particular, we investigate whether the transport of magnetic helicity
plays a role in solar eruptions.
We organize this paper as follows. Description of the data used in this study
and the procedures of the data processing are given in Section 2. Results
obtained for the two selected ARs are presented in Section 3 and the following
discussions in Section 4. The summary of the work presented in this article is
given in Section 5.
## 2 Data and Method of Analysis
For our study, we have used high resolution LOS magnetograms at a cadence of
12 minutes obtained from the Helioseismic Magnetic Imager (HMI; Schou et al.
2012) on board Solar Dynamic Observatory (SDO). HMI observes the full solar
disk in the Fe i 6173Å spectral line with a spatial resolution of 1 arc-
second. HMI provides four main types of data: dopplergrams (maps of solar
surface LOS velocity), continuum filtergrams (broad-wavelength maps of the
solar photosphere), LOS and vector magnetograms (maps of the photospheric
magnetic field).
NOAA 11158 (19∘S) and 11166 (10∘N) appeared on the disk during February 11-20,
2011 and March 03-16 2011 respectively. These ARs were very active, and
produced some intense X-class flares associated with CMEs in addition to many
M- and C- class flares during their disk transits. From the AIA observations,
intermittent mass expulsions were seen, many of which turned into large, fast
moving CMEs as further confirmed by STEREO. Table 1 gives a list of
flares111Obtained from the website www.solarmonitor.org (as recorded by GOES)
and CMEs222by scrutinizing AIA 304Å quicklook images mirrored at
http://jsoc.stanford.edu/data/aia/images/2011/ and further confirmed by the
timings from http://spaceweather.gmu.edu/seeds/.
Table 1: List of Flares and CMEs
AR | Date | Flares | CMEs
---|---|---|---
(NOAA) | dd/mm/yyyy | magnitude(time UT) | (time UT)
11158 | 11/02/2011 | No flares | No CMEs
| 12/02/2011 | No flares | No CMEs
| 13/02/2011 | C1.1(12:29),C4.7(13:44),M6.6(17:28) | 21:30,23:30
| 14/02/2011 | C1.6(02:35),C8.3(04:29),C7.1(06:51) | 02:40,07:00,12:50,17:30,19:20
| | C1.8(08:39),C1.7(11:51),C9.4(12:41) |
| | C7.0(13:47),M2.2(17:20)*,C6.6(19:23) |
| | C1.2(23:14), C2.7(23:40) |
| 15/02/2011 | X2.2(01:44),C2.7(00:31) | 00:40,02:00*,03:00,04:30,05:00
| | C4.8(04:27),C4.8(14:32),C1.7(18:07) | 09:00
| | C6.6(19:30),C1.3(22:49) |
| 16/02/2011 | C2.0(00:58),C2.2(01:56),C5.9(05:40) | 14:35
| | C2.2(06:18),C9.9(09:02),C3.2(10:25) |
| | C1.0(11:58),M1.0(01:32),M1.1(07:35) |
| | M1.6(14:19),C7.7(15:27),C1.3(19:29) |
| | C1.1(20:11),C4.2(21:06),C2.8(23:02) |
11166 | 06/03/2011 | C5.1(11:56),C3.9(15:21) | 02:00,15:20
| 07/03/2011 | M1.9(13:45) | 14:25,22:10
| 08/03/2011 | C7.7(23:10) | 14:30,19:00
| 09/03/2011 | C9.4(08:23),M1.7(10:35),M1.7(13:17) | 06:40,10:40,21:45
| | C9.4*(22:03),X1.5(23:13) |
| 10/03/2011 | C2.9(03:50),C6.2(07:03),C4.2(13:19) |
| | C4.7(13:42),C2.0(14:21) | 04:50,07:10
| | C4.0(19:00),M1.1(22:34) |
| 11/03/2011 | C1.4(00:29),C1.1(01:46),C2.8(02:24) | 00:50
| | C5.5(04:15),C4.3(07:22),C1.1(08:13) |
| | C2.0(11:10),C3.6(11:43),C1.1(16:04) |
| | C1.0(22:20),C1.0(22:50) |
Note: All flares(CMEs) associated to source region R2(R1) of respective ARs
except those marked by *
Magnetograms obtained at different times were aligned by the method of Chae et
al. (2004). In this method, an image of the AR taken at the central meridian
is considered as the reference image. All other images, in time accounted for
differential rotation (Howard et al., 1990) along with the latitudinal
difference of center of reference image from the ephemeris information, were
remapped on to the disk center. This method is intended to reduce errors due
to geometrical foreshortening and the AR is transformed to the disk center.
Since at disk center, normal and vertical components of magnetic fields are
same, the difference between the normal and LOS component was corrected by
cosine of the distance of the AR center from the disk center by assuming the
horizontal field contribution for the transformation to be negligible
(Venkatakrishnan & Gary, 1989).
We followed the transits of the two selected ARs from east to west on the
solar disk. In order to have negligible errors in geometric correction, we
restricted ourselves to a region within $\pm$40∘ longitude from the central
meridian. With this constraint, we confined our study of the temporal
evolution of the ARs to six days’s period around their central meridian
passage.
We derived the horizontal velocities of foot-points on the photosphere by
using DAVE technique (Schuck, 2006). The DAVE technique is essentially a local
optical flow method that determines the magnetic footpoint velocities within
the windowed region. Further, it adopts an affine velocity profile specifying
velocity field in the windowed region about a point and constrains that
profile to satisfy the induction equation. Any tracking method depends on two
parameters, viz., the window size and the time interval. For a given time
interval $\Delta$t, the window should be large enough so that tracked features
remain confined within the window. Also, it should be small enough to be
consistent with an affine velocity profile. Schuck (2008) presented a way to
select an optimal window objectively, using the degree of consistency between
change in the observed magnetic field ($\Delta{\rm B}/\Delta t$) and the
expected magnetic field change based on the flow estimated with several trial
windows. They found the best performance of this method at approximately a
square window of pixels. Since the ARs evolved rapidly, we chose a window size
of 21$\times$18 pixel2 after a careful verification of the physical flux
motions and directions of estimated flows. The dependence of helicity
injection rate on window size and time difference between the tracked maps
using this method were investigated. Moreover, as the HMI magnetic field
measurement precision is 10G (Schou et al., 2012), we have set this as the
threshold to avoid errors while retrieving velocities. Further details of this
method are given in a recent work of Tian et al. (2011).
Computation of the helicity rate using the method (direct integration)
proposed by Pariat et al. (2005) at each pixel of the AR map (cf., Equation 4)
is a tedious, time consuming process. However, we chose to use this method for
reducing the effect of fake polarities of helicity flux. Restricting the
calculations at pixels with magnetic field above the threshold ($\geq$10G)
helps to reduce the computation time typically by 15-25%. Parallelization in
integrand computation further reduces the time approximately by a factor of
the number of processors used. The same equation as rewritten by Chae (2007)
to suit the convolution algorithm by Fourier transform is faster than the
direct integration method. The intrinsic problem of Fourier transform with
periodicity could be overcome by padding the array corresponding to the data
points with rows and columns of zeros to get results as obtained by direct
integration method. In this study, we have implemented the former approach
(direct integration) to get sufficiently accurate results.
## 3 Evolution of Magnetic Flux and Helicity
The evolution of observed magnetic flux and the computed helicity rates are
presented in the following for the two selected ARs NOAA 11158 and NOAA 11166
with the methods and procedures explained in Section 2.
### 3.1 AR NOAA 11158
This AR appeared as small pores at the heliographic location E33S19 on 2011
February 11 as seen in the full disk HMI photoheliograms. Thereafter, it grew
very rapidly during the next two days as the small pores merged and formed
bigger sunspots. It was a newly emerging region which developed to a large AR
having $\beta\gamma\delta$ magnetic complexity during its rapid evolution. It
consisted of four large regions of opposite polarities in quadrupolar
configuration. Figure 1(top row) shows the evolution of NOAA 11158 during 2011
February 13-15 in HMI intensity maps. The prominent positive polarity sunspots
of the AR are labeled as SP1, SP2, SP3 and the negative polarity spots as SN1,
SN2, SN3 for identification. LOS contours are overlaid on the intensity image
showing the respective polarity distribution.
The spatial evolution of the AR shows a large shearing motion of SP2 that
rotated around SN2 about its umbral axis during 2011 February 13-15. It then
detached and moved towards SP3 along with small patches of both polarities
appearing and disappearing over short periods of time. This motion appeared to
have created a twist in magnetic fieldlines connecting these spots. A careful
examination of the animation made from magnetograms and intensity maps
revealed a significant counter clock-wise (CCW) rotation of SN1 during the
same period, while a small positive polarity region SP1 located to the north
of SN1 rotated in the counter clock-wise direction along with a proper motion
away from SN1. The rotations of SN1 and SP1 increased the twist of the field
lines, and the magnetic non-potentiality of the sigmoid structure (Canfield et
al., 1999). Several mass expulsions were launched intermittently from this
region, as seen from the quick look images in AIA. These turned into CMEs as
confirmed by STEREO observations.
In order to quantitatively analyze the magnetic complexity or twist
contributed by the observed shearing motions of the magnetic foot points, we
computed the helicity injection rates using the temporal sequence of
magnetograms of the AR. Figure 1(bottom row) shows the computed helicity flux
density maps corresponding to the HMI continuum intensity images (top row).
The dark (white) patches in the right panel represent negative (positive)
helicity flux density according to the usual convention. Contours of LOS
magnetic field at [-150, 150]G levels are overlaid for a better visualization
of helicity flux density with respect to the magnetic polarity. Evidently,
negative polarity region of SN1 injected negative (dark) helicity during 2011
February 14-15 which is also consistent with its physical CCW rotation. In
contrast, SP2, SN2 and SN3 injected positive (white) helicity along with
negative (dark) helicity in some small patches. We expect that the nature of
motions in these areas could have influenced the helicity pattern there.
The photospheric maps of helicity flux (and its injection rate) provides
spatial information about the basic properties of a link between the activity
and its sub-photospheric roots as reflected by the flux emergence process. In
a sample of four active regions, Jeong & Chae (2007) found that helicity was
mostly injected while fluxes emerged in the AR, suggesting it to be the major
source of helicity injection. The flux cancellation process, on the other
hand, resulted in a loss of coronal magnetic helicity, or inverse helicity
injection. We thus infer that the AR possessed two main sites, of unstable
energy storage systems marked by the rectangular boxes R1 and R2 in Figure 1.
These sites had distinctly different injection of helicity flux density
corresponding to the flux (or foot-point) motions, polarities and activity.
In order to show the transient activity of the AR as it evolved, we have
plotted the disk integrated GOES soft X-ray flux (1-8Å channel) during
February 11–17 in Figure 2(top) where the start times of flares of NOAA 11158
are marked by arrows. After its birth, the AR gradually evolved during 2011
February 11–13 as evident from the monotonic increase of fluxes in both
polarities corresponding to $3\times 10^{21}$ Mx (Figure 2, middle). Then
followed a rapid phase of flux emergence (of $9\times 10^{21}$ Mx) during
February 13–14 after which it reached a plateau. Also plotted is the flux
imbalance, i.e, the ratio of the net flux and absolute total flux in the AR.
The dominance of negative flux during February 13–15, and thereafter of the
positive flux, is evident. Flux variations occurred in the range of
(9.5–12.5)$\times 10^{21}$ Mx with the imbalance within $10\%$ over six days.
A significant flux decrease in both polarities by $\sim 1\times 10^{21}$Mx
occurred till the time of the X2.2 flare. We shall discuss more about flux
changes during X-flares in Section 4. The unusual rotating sunspots along with
the increased fluxes indicated emergence of highly twisted fluxes from the
sub-photospheric region (Leka et al., 1996), and not resulting from the
surface flows alone. Most of the flare and CME activity of this AR occurred
only after February 13/12:00UT, indicating that the rapid flux emergence could
have played important role in triggering the transients.
In Figure 2(bottom), we have plotted the time profile of helicity injection
rate($dH/dt$), which is the summation of helicity flux density over the AR .
Also plotted is the accumulated helicity, i.e, the integrated helicity change
rate over time ($\Delta H=\int\frac{dH}{dt}\Delta t$). The total accumulated
helicity is estimated as 14.16$\times 10^{42}$Mx2 during the six day period of
2011 February 11–16, with the peak helicity rate of 31.54$\times
10^{40}$Mx2h-1. The occurrence times of the CMEs associated with the AR are
marked by arrows in this panel for reference. An impulsive variation of
helicity injection rate due to injection of negative helicity is discernible
during the X2.2 flare and the concomitant CME. The helicity injection rate
decreased during the period February 14/11:00–February 15/13:00 UT, and
increased thereafter till February 16 along with fluctuations in the range
2–4$\times 10^{40}$Mx2h-1. We notice a large dip of helicity injection around
X2.2 flare with associated CMEs. We have smoothed the original time profile at
12 minute interval by a box car window of five data points (i.e., 1 hour).
Similar sudden dips in injection rates during other events can be further
analyzed for examining their association.
Figure 3 shows transverse velocities in the rectangular sub-regions R1 (top
row) and R2 (bottom row) of NOAA 11158 overlaid on the corresponding maps of
helicity flux density during three flare events. Also overlaid are the
contours of the LOS magnetic flux at $\pm 150$G levels. Maximum rms velocities
in the range of 0.6–0.9 $\mbox{km s}^{-1}$ were found over the observed period
in the AR. Spiral or vortex like velocity patterns are obviously related to
the counter rotation of SN1 in Figure 3(b–c). A notable observation is that
the sub-region R1 possessed negative helicity flux density distribution which
is consistent with the chirality associated with the physically observed
counter rotation of SN1 whereas R2 possessed mixed helicity flux dominated by
positive helicity flux distribution. Because of the continued shearing motions
at the interface of SP2 and SN2, the flow field vectors almost aligned with
the polarity inversion line (PIL) as seen in panels (d–f). Interaction of
fluxes with this shear motion can squeeze and converge the flux in both SP2
and SN2. We hypothesize that the field lines were stressed and twisted by this
motion leading to the storage of free energy adequate to account for the
release in the energetic X2.2 flare of February 15/01:44UT. As almost all
flares (except M2.2 at 14/17:20) occurred in R2, we examined the spatial
distribution of helicity flux before and after the flare events to know
whether any sudden changes are found related to the occurrence of flare.
During some events, we noticed negative patch of helicity flux in the regions
of positive helicity flux. Especially, in the panels (e–g), a negative
helicity flux distribution near the PIL during M6.6, C7.0, and X2.2 flares can
be observed. There may be some concern about these flare-related changes, as
it is known that during the impulsive phase of large flares, the spectral line
profile itself may undergo some change affecting the magnetic (and velocity)
field measurements.
Most of these flares occurred in R2 while the mass expulsions(or CMEs) were
associated to R1. In order to relate helicity rate changes to these events,
therefore, we have computed and plotted the total injected quantities for R1
and R2 in Figure 4(a-b). Injection of helicity in a region of dominant
opposite sign can be understood as a sudden dip in the time profile plot. Of
course, the corresponding spatial information is lost in the averaged
quantity. The advantage of using localized analysis of selected sub-areas in
the ARs is that it reduces complex variations occurring over a much larger
area of the entire AR while showing only the variations occurring in the
areas-of-interest. It also enhances the dips corresponding to the identified
events (marked by the arrows). However, it is important to identify the
location of individual event in order to correctly attribute a particular
change of helicity rate to it. NOAA 11158 was essentially a positive helicity
injecting region, while its sub-region R1 had a negative injection rate and
accumulated quantity due to the presence of rotational motion. We expect that
as the sunspots SN1 and SP1 rotated, the injection rate increased to a maximum
of $-16\times 10^{40}$Mx2h-1 on February 14/18:00UT. A total helicity
accumulation of $-5.60\times 10^{42}$Mx2 occurred during the six day period in
this region. Noticeably, a steep accumulation occurred during Feb 14–15 along
with many observed mass expulsions shown by arrows. This could be interpreted
as shedding of excess helicity from the corona in the form of eruptive events.
The steep accumulation of helicity by monotonic injection rate, therefore, is
suggested to be a cause of expulsions. Accumulated helicity amounting to
$14.44\times 10^{42}$Mx2 in sub-region R2 with steep accumulation observed
from February 13 onwards, could be mostly associated with the observed large
shear motion of SP2.
For a quantitative study of the association of short term variations in
helicity rate to the flaring or CME, the following analysis is carried out.
The absolute time difference of the helicity flux ($|\Delta(dH/dt)|$, having
units same as dH/dt) averaged over start and stop times of GOES flares above
C2.0 is computed. This is compared to that of randomly selected but equal
length time intervals containing no flares. A significantly higher mean of
$|\Delta(dH/dt)|$ during flares compared to quiet times would indicate a
robust association between flaring and helicity fluxes. A similar analysis is
undertaken for time windows around CMEs to look for a CME-helicity flux
association. We assume that there is no time lag between flaring and helicity
flux signal while carrying out this analysis. We first interpolated the signal
at 1 min interval from 12 min interval to get values as required by the GOES
flare times, then it was smoothed to a boxcar width of 30 minutes. Within
start and stop times of flares, the averaged value of absolute variation was
computed to compare with that calculated during randomly selected, constant
interval(30 min) quiet times.
The time difference of helicity rate in R1-R2 is shown in Figure 4(c-d) with
CMEs and flares marked by arrows. Large amplitude variations are discernible
during M6.6, X2.2 and the CME at 12:30UT indicating some association, but
similar variations are present around the mean position even in quiet times.
From the above described analysis, we found a significantly higher mean during
CME’s ($0.054\pm 0.007$) compared to quiet times($0.032\pm 0.008$). The
difference in CME versus quiet time helicity fluctuations are marginally
statistically significant, at better than one-sigma. Similarly, a mean of
$0.044\pm 0.004$($0.049\pm 0.008$) during flare (quiet) times indicate poor or
no association of flaring to helicity flux variations. The same analysis for
the helicity flux over the entire AR improved the association (in terms of
mean absolute helicity variation) slightly for CMEs but worsened it for
flaring. We shall further discuss these helicity variations during flare/CMEs
in view of the involved flare-related effects in Section 3.3.
### 3.2 AR NOAA 11166
AR NOAA 11166 appeared on the east limb of solar disk on 2011 March 03 at the
location N10E64. We monitored its activity during the period of 2011 March
6–11 in which it produced a large X1.5 flare, two M-class flares and several
C-class flares, some of which were also associated with plasmoid ejections or
CMEs. Table 1 lists the flares and CMEs of this AR. Daily evolution of the AR
in the period of March 8-11, 2011 is shown in Figure 5(top row).
The major sunspots of the AR are labeled as SP1, SP2, SN1 and SN2. The
identification of SP2 was somewhat unclear before March 10 as several small
umbrae were spread over its location. They moved and coalesced to form SP2.
Polarities of the respective sunspots are identified by the overlaid LOS
magnetic field contours. This AR also consisted of a complex magnetic
configuration with two positive (SP1, SP2) and two negative (SN1, SN2)
polarity sunspots located within the surrounding diffused fluxes. Emerging and
moving flux regions, FP3 and FN2, were identified in the course of the
evolution in the sunspot periphery (March 11/22:00UT panel), having opposite
sign to that of their native sunspots. However, there were no intrinsic
rotating sunspots or flux patches as observed in the case of AR NOAA 11158.
We computed the helicity flux density for AR NOAA 11166 during its evolution
in the period 2011 March 6–11. The corresponding maps for three successive
days are plotted in Figure 5(bottom row). Locations of helicity flux density
of mixed sign were distributed all over the AR through out the evolution
period. The peripheral sites of the sunspots exhibited helicity flux density
predominantly of negative sign. However, patches of negative helicity flux
were also observed embedded in the positive helicity flux site of the flare
(March 09/23:00UT panel). For further close examination, we consider two sub-
areas R1 and R2, as marked by the boxes in this panel.
The disk integrated GOES soft X-ray flux (1-8Å channel) during 2011 March 6-11
is plotted in Figure 6(top). The arrows in this panel indicate the start time
of flares in NOAA 11166. During the disk transit of the AR, fluxes of both
polarities increased corresponding to $5\times 10^{21}$Mx, with the imbalance
varying below 6% (Figure 6, middle). As observed for NOAA 11158, a rapid flux
emergence occurred in this AR too during March 7–9. Thereafter, only small
variations associated with local cancellations/emergence of about $\sim
1\times 10^{21}$Mx took place pertaining to the gradual evolution of the AR.
Positive flux dominated in the AR during March 7-11, and then a near balance
was established. It is worth noticing that magnetic fluxes in both polarities
decreased by $\sim 0.9\times 10^{21}$Mx while evolution of fluxes leading to
the occurrence of a CME following the X1.5 flare. However, it is not clear
whether this decrease in flux six hours before the flare/CME has some role in
these events. But, the flux imbalance, increasing prior to the flare, reduced
significantly after the flare consistent with observations reported by Wang &
Liu (2010). Most of the flares and CME activity of this AR occurred only after
March 8, suggesting that the rapid emergence of fluxes could be an important
factor for triggering of these transients.
Temporal evolution of helicity injection rate and the accumulated helicity for
NOAA 11166 are shown in Figure 6(bottom) with arrows marking the times of the
CMEs. A five magnetogram boxcar was used to smooth the profile to reduce
fluctuations in the profile. As expected, these parameters increased in the
first phase corresponding to the flux emergence, in agreement with Jeong &
Chae (2007) that helicity is mostly injected while the fluxes emerged. Total
helicity accumulated during the six days’ period of the AR’s evolution was
estimated to $\sim 9.5\times 10^{42}$Mx2. The maximum helicity injection
occurred during 2011 March 8 at the rate of $30\times 10^{40}$Mx2h-1.
Thereafter, it reduced gradually to the minimum rate at $-10\times
10^{40}$Mx${}^{2}h^{-1}$ on 2011 March 10. The coronal helicity of the AR is
likely to be positive as a result of this positive helicity injection.
Horizontal, or transverse, velocity vectors corresponding to the tracked flux
motions are plotted in Figure 7 separately for R1 (top row) and R2(bottom
row). The rms velocities of flux motions are found to have the maximum values
in the range 0.5–0.9 $\mbox{km s}^{-1}$. Strong moat flows were
systematically dominant in both regions from the peripheral regions of
sunspots in addition to the shear flows. Persistent strong shear motions due
to the merging SP2 group were identified in R2. These flows appear to collide
head on with those from SP1 resulting in the flux submergence/cancellation.
Flux emergence was also identified from the diverging flow field observed in
animated flows from R1. From this region, flux moved towards R2 as the AR
evolved. Such motions appear to be associated with injection of negative
helicity into a region with predominantly positive flux, increasing the
complexity of the magnetic flux system as shown in panels (d)–(f) of R2.
Further, these negative helicity injections often coincided with some observed
events, such as the three of them shown in this plot. For the X1.5 flare the
distribution of helicity flux is shown in panel (e) on March 09/23:36UT.
The injection rates and accumulated helicities deduced from sub-regions R1 and
R2 are plotted in Figure 8(a–b). Also the contribution of each signed helicity
flux in the net helicity flux is plotted separately. The time profile of R1
shows it to have positive helicity injection with a steep increasing phase
during March 7–9 at a peak rate of $27\times 10^{40}$Mx2h-1. Thereafter,
gradual decrease in the rate of injection is evident from the plot. As
mentioned earlier, R1 was a site of emerging flux that resulted in
contributing to accumulation of helicity amounting to $11\times 10^{42}$Mx2.
While R2 exhibited mixed sign injection rates during its evolution. As in the
previous AR, continuous injection of dominant positive helicity from R1 is
suggested to be the cause of observed mass expulsions, whereas the injection
from R2 is of mixed signs suggested to result in flares. An enhanced peak of
helicity rate was seen around the time of the X1.5 flare in R2 of AR 11166
that was not obvious in Figure 6(bottom panel) since we reduced fluctuations
occurring over entire AR by selecting small area. After this event, the
negative injection rate increasingly dominated on March 10, turning the net
injection of the entire AR negative. The implication of this transition of
injection rate from positive to negative sign over a day is not clear in the
observed events shown by the arrows.
The time variation of helicity flux in both R1 and R2 are plotted in Figure
8(c–d) along with the arrows pointing start times of CMEs and flares in the
AR. Some of the large amplitude variations of helicity flux about the mean
position appear to be related to these events. As in the previous AR, we have
analyzed the association of flare/CMEs that originated from the sub-regions R1
and R2 of this AR with the respective helicity flux. The calculated mean of
variation in helicity flux ($|\Delta(dH/dt)|$) during flaring ($0.099\pm
0.020$) is marginally statistically different at about two-sigma level over
that during quiet times ($0.057\pm 0.007$), reflecting a robust association of
flaring and helicity fluxes. The mean of $|\Delta(dH/dt)|$ obtained in quiet
times do not have any information or bias of flaring or CME, therefore higher
mean during the flare/CMEs implies some impact of helicity flux variations in
them. A similar analysis undertaken for CMEs also showed the similar
association( during CMEs of $0.052\pm 0.006$ dominated over quiet times of
$0.047\pm 0.006$, but not statistically significant difference). However, the
association strengthened for flaring and weakened for CMEs when the helicity
flux over the entire AR was considered in the analysis.
### 3.3 Flare-related effects on Helicity flux
It is well known that the photospheric magnetic (and Doppler) field
measurements are affected by flares. During an energetic flare, the profile of
spectral line used for the measurement was reported to change from absorption
to emission, resulting in a change of sign in the deduced magnetic polarity
(Qiu & Gary, 2003, and references therein). This abnormal polarity reversal
was observed to last for about a few minutes during the impulsive phase of the
flare (typically 3-4 minutes). Similar abnormal, transient changes have also
been reported for some other large, white light flares (Maurya & Ambastha,
2009; Maurya et al., 2012). The change in the line profile may arise due to
both thermal effects and non-thermal excitation and ionization by the
penetrating electron jets produced during the large flares. We term these as
flare-related transient changes, considered to be artifacts as they do not
correspond to real magnetic field changes.
There is increasing evidence that flares may change the magnetic field more
significantly on a persistent and permanent manner (Sudol & Harvey, 2005;
Petrie & Sudol, 2010; Wang & Liu, 2010). The persistence of the observed field
changes implies that they are not artifacts of changes in the photospheric
plasma parameters during the flare, and the temporal and spatial coincidences
between flare emission and the field changes suggest the link of the field
changes to the flare. We term these as permanent flare-related changes. With
these known transient and permanent flare-related effects on magnetic fields,
it would not be clear, particularly during the impulsive phase of the flare,
if the change in helicity flux can be interpreted as genuine transport of
helicity across the photosphere.
In addition, an implicit assumption made in our approach of calculating
helicity injection is the ideal evolution of photospheric magnetic fields in
the induction equation used to derive velocities of flux motions. Moreover,
the same assumption is involved in the derivation of helicity injection from
the relative helicity formula (Berger & Field, 1984; Finn & Antonsen, 1985).
This assumption is valid and reasonable outside the flaring time intervals (at
least during permanent changes of fields) as the typical observed photospheric
velocities are far less than the Alfven velocities. In the real conditions of
rapid, transient changes in photospheric magnetic fields spanning impulsive
period of the flare, the assumption of ideal magnetic evolution may not be
applicable. Therefore, there is theoretical uncertainty regarding the
interpretation of helicity fluxes during flares.
In order to inspect these aspects in the signal of the helicity change rate,
we procured 45s cadence magnetograms for some selected flare events and
averaged them to 3min cadence after processing as the previous data set. A
mosaic of distribution of helicity flux around the X2.2 flare is shown in
Figure 9. During the impulsive period (01:48-02:02UT) of this flare, negative
helicity flux is distributed about the PIL which we believe to be due to the
transient flare-related effect. The magnetic (and Doppler) transients and
locations of spectral line reversal associated with this flare are already
reported by Maurya et al. (2012), which are spatially and temporally
consistent with this negative helicity flux distribution. Therefore, the
observed negative helicity flux distribution in the dominant positive site can
be attributed to the transient flare-effect, and is likely to be artifact,
i.e., not a true transfer of helicity.
Similar mosaics of helicity flux distribution maps were made and examined for
other events. The computed magnetic and helicity fluxes are plotted with time
in Figure 10. The flare start time is shown in vertical dotted line labeled
with magnitude of the flare. It should be noted that we have not applied any
smoothing to the computed helicity rate signal in these panels. Magnetic
fluxes of both signs decreased abruptly with a dip during the impulsive period
following with injection of negative helicity flux in the dominant positive
helicity flux, during the M6.6, X2.2 flare events. Magnetic field measurements
could also be underestimated by 18-25% due to enhanced core emission of
spectral line by the heating of the impulsive flare (Abramenko & Baranovsky,
2004) as a result of which the integrated flux profile could show such a dip
during peak phase of the flare. Interpretation of flux annihilation through
reconnection during this peak phase might be ambiguous due to this fact,
although it could be a possible consideration. In the post-flare phase, fluxes
increased in both polarities as field lines reorganized as a post-reconnection
process. This falls under the “permanent” real change related to the flare.
For smaller magnitude flares, transient effects may be absent or not be
prominent in the impulsive phase. Therefore the measurements of magnetic
fields and the computed helicity rate signal are not expected to be affected
during the flare. Hence, they may indicate true transfer of helicity flux,
except for the theoretical uncertainties as mentioned above. In the case of
the 14 February/13:47UT (C7.0) flare, shown in panels (b1)-(b2), indeed the
variation of helicity signal occurs without the variations in magnetic fluxes
associated to the flare-related effects. This may be an example of true
transfer of helicity of the flux system, but with the theoretical uncertainty
in our approach.
There are no significant variations in magnetic and helicity fluxes
corresponding to the 09 March/09:23UT (C9.4), and 10:35UT (M1.7) flares. Large
amplitude fluctuations in both sign of helicity signals during the CME just
before the 09 March/22:03UT (C9.4) flare are apparent in panels (e1)-(e2). We
speculate that these fluctuations subsequently led to the initiation of the
prominent CME that followed the 09 March/23:13UT (X1.5) flare an hour later.
Similarly, the transient flare effects might be responsible for the abrupt
changes in magnetic fluxes resulting in variations of helicity injection
signal during the X1.5 flare (panels (e1)-(e2)). During the 10 March/13:19UT
(C4.2), 13:42UT (C4.7) flares, the transfer of helicity flux from positive to
negative, negative to positive sign is clear from the panels (f1)-(f2),
respectively. These flares are of small magnitude, with no obvious flare-
related artifacts. Therefore, the observed helicity flux changes are expected
to be true (with the implicit theoretical uncertainty in the approach). A
point to be noted is that all large flares (M and X-class) may be involved
with transient flare effects. Therefore, it is better to look for helicity
variations in small flares where magnetic fields are expected to be less
affected, making it easier to examine the possible role of transfer of
helicity flux. Thus, we consider the 14 February/13:47UT (C7.0), 10
March/13:19UT (C4.2) and 13:42UT (C4.7) flares to be the best examples here,
supporting the true transfer of helicity. It is not clear that whether the
helicity transfer in these cases is related to permanent flare-effects.
At present, it is difficult to say much about the physical significance of
these variations over the AR in the corona, i.e., at the primary sites of the
flares. It would be particularly interesting to study the physical
significance of such injection along with the information of coronal
connectivities (e.g., Chae et al. 2010) as suggested by Pariat et al. (2005)
for understanding the possible role of transfer of helicity flux during the
flares/CMEs.
### 3.4 Dependence of Helicity Injection Rate on the DAVE Parameters
Computation of helicity injection rate involves the measurement of magnetic
field and the inferred horizontal velocities. Apart from the errors in the
measurements, the computations involving the DAVE method for deriving
velocities depend on two main parameters viz., the time interval between two
successive magnetic maps, $\Delta$t, and the DAVE window size. For obtaining
optimized results, horizontal displacements of features during the time
interval $\Delta$t should be large enough to be well determined by DAVE. Also,
these displacements should be smaller than the selected window size. To check
our results for consistency, we carried out the DAVE calculations using the
time intervals $\Delta$t = 12, 24 and 36 minutes, while keeping the window
size fixed at $21\times 18$ pixels. Then, calculations were carried out for
different window sizes, viz., 21$\times$18, 15$\times$12, 9$\times$6 while
keeping $\Delta$t fixed at 36 minutes. Furthermore, to avoid the effect
arising from noise, we used a threshold of magnetic field at 10G, which is the
HMI precision. As the HMI provides 12 minute averaged data products, we
averaged them corresponding to our calculations at 24 (2 maps) and 36 (3 maps)
minutes.
The dependence of helicity injection rates on time interval $\Delta$t is shown
in Figure 11(top row) for NOAA 11158. The scattered data are fitted by
straight line in the least square sense. Due to the large volume of data, this
computation is tedious and time consuming. Therefore, results are shown here
only for NOAA 11158, but, we expect they are also valid for other ARs observed
by the HMI. There is an additional issue of unequally spaced data points to be
addressed in case, for example, we intend to plot the results for $\Delta$t=36
with $\Delta$t = 24 minutes. For such cases, we used a cubic spline
interpolation (cf., Press et al. 1992), to get corresponding abscissa values
for the ordinate points or vice-versa. Essentially, this algorithm employs
cubic polynomial between each pair of data points with the constraint that the
second and first derivatives of that polynomial are same at the end points so
that the resulting values are smooth. Table 2 lists the minimum and maximum
values of helicity injection rates (dH/dt, in units of $10^{40}$Mx2h-1) and
the accumulated helicity ($\Delta H$, in units of $10^{42}$Mx2) for the
computational runs carried out with various DAVE parameters as mentioned
above.
Table 2: Helicity injection rates and Accumulated helicities at different DAVE
parameters
DAVE parameters | AR 11158
---|---
$\Delta$t | Window size | dH/dt | | $\Delta H$
min | pixel2 | min | max |
12 | 21x18 | -18.98 | 31.54 | 14.16
24 | 21x18 | -7.48 | 27.27 | 13.09
36 | 21x18 | -1.06 | 22.52 | 12.96
36 | 21x18 | -1.06 | 22.52 | 12.96
36 | 15x12 | -1.06 | 25.02 | 13.51
36 | 9x6 | -1.28 | 26.8 | 14.22
Units of dH/dt are $10^{40}$Mx2h-1 and $\Delta H$ are $10^{42}$Mx2
It can be observed from the scatter plots that the helicity rates decreased
slightly as the time interval $\Delta$t is increased from 12 min to 36 min.
The fitted straight line deviates at a slope of 0.87 and 0.91 corresponding to
$\Delta t=12$ versus 24 and $\Delta t=24$ versus 36 min indicating that
helicity injection decreases by 13% and 9% respectively. This implies that
short-lived features and their dynamics have considerable contribution to
helicity rates. The helicity rates at intervals of 36min are lower by a factor
of 21% than that at 12 min with worst correlation coefficient of 0.79. These
effects in turn reflected in the variation of accumulated helicity by 9%. This
implies that averaging in time between 12-36 min has significant effect on
injected helicity rates up to 13% corresponding to 9% of variation in
accumulated helicity.
The dependence of helicity injection rate on window size by keeping the time
interval $\Delta$t fixed at 36 minutes is shown in Figure 11(bottom row). The
slopes of 1.09 and 1.05 for the DAVE windows $21\times 18$ versus $15\times
12$ and $15\times 12$ versus $9\times 6$ respectively, show increasing trend
of helicity rates with decreasing window size. Indeed, a scalable factor of
14% reduction of helicity rate is evident for windows $21\times 18$ versus
$9\times 6$. Accumulated helicity also showed this increased trend with
decreased window size. A total variation of $10\%$ is found, however, with the
same trend of helicity injection rate profiles which is discernible in
correlation coefficient with the plots. A maximum velocity of 1 km-s-1 during
the time interval of 12 min corresponds to a plasma displacement of an arc-
sec. Hence, for the window size of 4.5″$\times$3″(9$\times$6 pixel2), the
issue of features overflowing out of the window should not pose problem.
These results are consistent with those reported by Chae et al. (2004, their
Figure 7). They deduced and compared velocity and helicity rates by
combinations of time difference between magnetograms and LCT window size.
Their rms velocity values varied up to 0.6km/s at time interval of 5min. They
found that smaller values of LCT parameters result in larger amplitude
fluctuations of the rate of helicity, with variation within 10%. We, in our
computations, found maximum rms velocities for 12min, 24min and 36min in the
AR as 0.95, 0.85 and 0.8km/s respectively. However for the window sizes
$21\times 18$, $15\times 12$ and $9\times 6$, we obtained the rms velocities
as 0.8, 0.9 and 1.5km/s respectively. These are higher by a factor of 2
compared to their values probably due to the higher resolution and sensitivity
of HMI as against the coarser spatial resolution of MDI of 1.98″/pixel.
Nevertheless, the variation in accumulated helicity found in our analysis is
within $10\%$; consistent with their result.
We thus, found the measured helicity injection rate to depend on the time
interval between the two successive magnetograms, i.e., the observational
cadence. The selected window size also influenced the measured quantities. Our
analysis suggests that it is better to use images averaged over up to 24
minutes with relatively small DAVE window size subjected to the overflow
condition as mentioned above. These are important considerations to derive
reasonable and meaningful results in addition to optimizing the computations
involving large data-sets.
## 4 Discussions
Free energy storage and release are some of the most important problems in the
eruption physics of the Sun. There are essentially two effects that can supply
magnetic free energy and helicity from below the solar surface to the corona.
Flux emergence is the process in which vertical motions carry magnetic fluxes
through the photosphere. If the sub-surface fluxes emerging through the
photosphere are already twisted, then it will contribute to the injection of
helicity (cf., the 1st term in Equation 2). Computation of this term requires
the knowledge of the vertical component of velocity and the horizontal or
transverse component of magnetic field. Flux motions in the form of rotation
or proper motions are another process that may efficiently supply helicity
injection (cf., the 2nd term in Equation 2). The helicity injected by solar
differential rotation is rather small, less than 10% of that contributed by
the flux motions (Chae et al., 2004; Démoulin et al., 2002), and has only a
much longer term effect on helicity accumulation (DeVore, 2000).
Magnetic helicity is a physical quantity having a positive or negative sign,
representing a right-handed or left-handed linkage of magnetic fluxes,
respectively. This means that if positive and negative helicities co-exist in
a single domain, magnetic reconnection can cancel magnetic helicity by merging
magnetic flux systems of opposite helicities. Helicity densities are not
gauge-invariant. It is only area-integrated relative helicity flux that is
gauge-invariant. In order to define true helicity flux density, the coronal
linkage needs to be provided (Pariat et al., 2005), so the helicity flux
density inferred from tracking will not be precisely accurate. Our
computations of magnetic helicity injection in both ARs revealed that the
distribution of helicity flux is highly complicated in time and space. Even
the sign of helicity flux often changed within the AR.
It has been suggested earlier by several workers that magnetic helicity must
play an important role in flares as a substantial amount of helicity
accumulation is found before many events (Kusano et al. 1995; Kusano &
Nishikawa 1996; Kusano et al. 2002). However, the correlation between various
magnetic field parameters and the flare index of an AR is not high
irrespective of the method used. This is an intrinsic problem for flare
forecasting as the occurrence of a flare depends not only on the amount of
magnetic energy stored in an AR, but also on how it is triggered. Thus, it
appears that helicity accumulation might be a necessary, but insufficient
condition for the flares requiring a trigger even if a magnetic system has
enough non-potentiality. For instance, Kusano et al. (2003) suggested that
coexistence of positive and negative helicities may be important for the onset
of flares.
Careful three-dimensional simulations have been carried out by Linton et al.
(2001) to explore the physics of flux tube interaction for the co-helicity
(same sign) or counter-helicity (opposite sign). According to them, counter-
helicity presented the most energetic type of slingshot interaction in which
flux is annihilated and twist is canceled. In contrast, co-helicity exhibited
very little interaction, and the flux tubes bounced off resulting in
negligible magnetic energy release.
Magnetic helicity in the solar corona is closely related to the photospheric
magnetic shear, which is usually defined as the extent of alignment of the
transverse component of magnetic field along the neutral or polarity inversion
line (PIL)(Ambastha et al., 1993). Based on this idea, Kusano et al. (2004)
performed a numerical simulation by applying a slow footpoint motion. This
motion can reverse the preloaded magnetic shear at the PIL resulting in a
large scale eruption of the magnetic arcade through a series of two different
kinds of magnetic reconnections. They proposed a model for solar flares in
which magnetic reconnection converts oppositely sheared field into shear-free
fields.
We interpret our observations according to the above observational and
simulation aspects as follows. We have found flux interactions during the
X-class flares and associated CMEs as seen in Figure 3 in the form of
continued shearing motion of SP2 around SN2 in AR 11158. Similar motions are
also associated with SP2 in AR 11166. In both ARs cases, the flare prone
regions (R2) had inhomogeneous the helicity flux distribution with mixed
helicities of both signs. Correspondingly, sudden impulsive peaks appeared in
the profiles of helicity injection due to the injection of negative signed
helicity during some flare events. These were also spatially correlated with
the observed flares. Opposite helicity flux tubes can interact easily leading
to reconnection, thereby unleashing explosive release of magnetic energy.
Impulsive variations of the magnetic helicity injection rate associated with
eruptive X- and M- class flares accompanied with CMEs were reported also by
Moon et al. (2002). Recently, Park et al. (2010a) conjectured that the
occurrence of the X3.4 flare on 2006 December 13 was involved with the
positive helicity injection into an existing system of negative helicity.
Further, a solar eruption triggered by the interaction of two opposite-
helicity flux systems (Chandra et al., 2010; Romano et al., 2011), and
occurrence of flares in relation to spatial distribution of helicity flux
density (Romano & Zuccarello, 2011) were reported. The main drawback of these
findings is that the time span between two magnetograms is more than the
duration of the flare($\geq 96$m), so the time rate of helicity could not be
easily resolved at the onset time of the flare. Therefore, our results appear
to be consistent with the reports of opposite helicity flux tubes reconnecting
to trigger transient events.
However, it should be cautioned that we have not found such variations of
helicity flux clearly in all flare/CME events. From a quantitative analysis,
we found poor association of difference in helicity rate during flares to that
of quiet times in AR NOAA 11158. This indicates such variations are not
prominent or present during all flares. Moreover, statistically significant
association of such impulsive variations was found during CMEs compared to
quiet times. There are many possible reasons for this poor association; one of
them is time duration of helicity flux change. We first interpolated the
signal at 1 min interval from 12 min interval to get values as required by the
GOES flare times. Then, it was smoothed to a boxcar-averaging window of 30
minutes to reduce fluctuations arising due to interpolation. Within start and
stop times of flares, the averaged values of absolute variation were computed.
Here, averaging might have diluted the original helicity variation, so
comparison with the helicity variation during quiet times might not be valid.
In any case, there is no better way to find appreciable variation in the
helicity flux over background fluctuations to incorporate into the correlation
analysis, unless individual events are monitored manually to get variation
timings. Despite these difficulties, statistically significant association of
helicity flux is found during flares, but dominant association that is not
statistically significant during CMEs in the AR 11166 by following the same
approach.
Further, there are concerns about the flare-related effects on magnetic field
measurements resulting in misleading interpretation of helicity flux transfer,
in addition to the theoretical uncertainty with the assumption of ideal
magnetic field evolution in the approach. We therefore investigated this issue
using 3 min interval time sequence magnetograms. We found transient flare
effects resulting in spurious negative helicity flux distribution during the
X2.2, M6.6, and X1.5 flare events. Also, we indeed observed the true transfer
of helicity flux with variations of opposite sign helicity without such flare-
related effects in small flares such as the C7.0 on 14 February, C4.2 at
13:19UT, C4.7 at 13:42UT on 10 March. The important point to note is that we
found statistically significant association of helicity flux variations with
flares/CMEs in above cases of ARs at zero time lags. Also these variations are
clear during the flare events (see Figure 10) and not before their
commencement. Therefore, it is difficult to suggest that these variations
triggered the flares. A study with the information of fieldline connectivity
from coronal observations may be expected to reveal the physical significance
of the role of helicity transfer during these events.
Our computed helicity rates involving photospheric flux motions include the
flux emergence term as explained by Démoulin & Berger (2003). By a simple
geometrical argument, horizontal foot-point velocity ($\mathbf{u}$, here the
DAVE velocity) can be written in terms of horizontal and vertical plasma
velocities, $\mathbf{v}_{h}$, $v_{n}$, respectively:
$\mathbf{u}=\mathbf{{v}_{h}}-\frac{v_{n}}{B_{n}}\mathbf{B_{h}}.$ (5)
From this relation, it is not possible to infer as to which term, viz., the
flux emergence or flux motions, governs the level of activity of the ARs. To
resolve this difficulty, we have plotted the integrated absolute flux and
accumulated helicity computed over the ARs, as shown in Figure 12.
Evidently, the accumulated helicity increased monotonically with the emergence
of magnetic flux in the AR in its first phase (marked by the vertical dashed
line for NOAA 11158). After this phase followed the next, the active phase,
where an appreciable increase of helicity occurred with only small variation
in the flux, i.e., where little emergence of fluxes occurred. This rapid
increase in helicity in the second phase could be interpreted as the dominant
contribution of the flux motions. Intermittent mass expulsions in the form of
CMEs transferred away the excess helicity. The extent of this transfer,
however, is not clear from this plot, although one can make plausible
conclusions from the timings of the flares and CMEs. The X-class flares with
associated CMEs in both ARs occurred at a slowing phase of helicity
accumulation by negative helicity injection. These facts add to the cases as
reported by Park et al. (2010b).
Moreover, it can be inferred for AR 11158, that less than 25% of the total
helicity flux accumulated with the emergence of the first 75% of the magnetic
flux. Most of the helicity flux (from about $3-13\times 10^{42}$Mx2) was
accompanied by very little flux emergence (about $3\times 10^{21}$Mx out of
the $30\times 10^{21}$Mx). Therefore, more than 75% of the helicity flux came
with only 10% of the total magnetic flux. Similarly, the first 60%
($19.5-28.0\times 10^{21}$Mx) of total magnetic flux was associated to less
than 30% ($3\times 10^{42}$Mx2 of $9.5\times 10^{42}$Mx2) of the total
helicity flux in AR 11166. This implies that more than 70% of total helicity
flux was accompanied with less than 40% of total magnetic flux. These two
cases are thus contrary to the findings of Jeong & Chae (2007) stating that
most of the helicity flux occurs during flux emergence. Our study suggests
that flux emergence may not always play a major role in accumulating helicity
flux. It is also evident that although flux emergence is necessary but
horizontal motions also played crucial and dominant role over emergence term
in increasing the complexity of magnetic structures contributing to the
helicity flux. Therefore, we suggest that the horizontal flux motions
contributed further, in addition to the emergence term, in creating more
complex magnetic structures that caused the observed eruptive phenomena.
## 5 Summary
We have studied the evolution of magnetic fluxes, horizontal flux motions,
helicity injection and their relationship with the eruptive transient events
in two recent flare (CME) productive ARs, NOAA 11158 and NOAA 11166 of 2011
February and March, respectively. We have used high resolution, high cadence
data provided by SDO-HMI for these ARs which were in their emerging and active
phases. The emerging AR consisted of rotating sunspots with increasing flux
indicating emergence of twisted flux from the sub-photospheric layers. This
indicated the transfer of twist or helicity injection through the photosphere
to the outer atmosphere.
We suggest that strong shear motions that include rotational and proper
motions played significant role in most of the events in addition to the flux
emergence. Such motions are crucial in twisting or shearing the magnetic field
lines and for further flux interactions. AR NOAA 11158 consisted of a CME-
prone site of rotating main sunspot along with emerging flux of opposite sign
and moving magnetic feature. It also had a flare-prone site consisting of
self-rotating sunspot(SP2) moving about a sunspot of opposite sign(SN2),
leading to flux interaction. These motions are likely to form the sigmoidal
structures, which are unstable, and more likely to produce eruptive events. A
huge expulsion as CME on 2011 February 14/17:30UT occurred in the former site
and a white light, energetic X2.2 flare on 2011 February 15/01:44UT occurred
in the later site. The other case, AR NOAA 11166 was already in its active
phase with further increasing content of flux as it evolved. Group motions of
diffused fluxes merging to form a bigger sunspot manifested major shear
motions in addition to outward flows from sunspot. A large CME on 2011 March
09/21:45UT, followed by an X1.5 flare, was one of the major events in this AR.
AR NOAA 11158 injected $14.16\times 10^{42}$Mx2 while AR NOAA 11166 injected
$9.5\times 10^{42}$Mx2 helicity during the six days’ period of their
evolution. These are consistent with the previously reported order of helicity
accumulation (e.g., Park et al., 2010b). It appears that due to the presence
of rotational motions, the former AR accumulated larger amount of helicity
accounting for its greater activity in the form of flares and CMEs. It is also
evident that flux emergence is necessary and their motions are crucial in
additionally accounting for the accumulated amount of helicity to the
emergence term. In both ARs, X-class flares with associated CMEs were observed
in the decreasing phase of helicity accumulation by the injection of opposite
helicity.
Apart from the instrumental and computational errors, the estimation of
helicity injection rates are also affected by the choice of DAVE parameters
used to track the motion of the fluxes. Helicity injection rates are found to
decrease up to 13% by increasing the time interval between magnetograms from
12 to 36 min whereas an increasing trend upto 9% resulted by decreasing the
window size from $21\times 18$ to $9\times 6$ pixel2, with a total variation
of 10% in the deduced value of accumulated helicity.
The time profile of helicity rate exhibited sudden sharp variations during
some flare events due to injection of opposite helicity flux into the existing
system of helicity flux. In both ARs, the flare prone regions (R2) had
inhomogeneous helicity flux distribution with mixed helicities of both signs
and that of CME prone regions had almost homogeneous distribution of helicity
flux dominated by single sign. A quantitative analysis was carried out to show
the association of these variations to the timings of flares/CMEs. For the AR
11158, we find a marginally significant association of helicity flux with CMEs
but not flares, while for the AR 11166, we find marginally significant
association of helicity flux with flares but not CMEs. Moreover, these
variations of helicity flux may not reflect true transfer; there exists flare-
related transient effects and theoretical uncertainties resulting to these
variations. We believe the helicity transfer in the cases of C7.0 on 14
February, C4.2 at 13:19UT, C4.7 at 13:42UT on 10 March to be true, without
flare-related transient effect but with theoretical uncertainty in the
approach.
Therefore, to further strengthen the above evidences of true helicity
transfer, it would be worthwhile to scrutinize more flare/CMEs cases using 3
min cadence magnetic observations, over a period of a day or so. This will
enable one to find detectable changes in helicity flux signal during smaller
magnitude flares with less transient-flare effects. Interpreting the physical
significance of such variations using the information of coronal
connectivities will be another important aspect to add further to the present
knowledge of helicity physics. Our study reveals that the spatial information
of helicity injection is a key factor to understand its role in the
flares/CMEs.
The data have been used here courtesy of NASA/SDO and HMI science team. We
thank Dr. Etienne Pariat for checking our helicity program with comments and
suggestions. The author expresses his gratitude to Prof. P. Venkatakrishnan
for some useful discussions on the concept of helicity. We thank an anonymous
referee for carefully reading the manuscript and making valuable comments
which led to improved clarity and readability of the manuscript. We thank
Mr.Jigar Raval and Mr.Anish Parwage for their help in running program on one
of the nodes of 3TFLOP HPC cluster at PRL computer center.
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Figure 1: (_Top row_) The daily HMI continuum intensity maps of AR NOAA 11158,
and (_Bottom row_) the corresponding helicity flux density maps (scaled to
$\pm 0.05\times 10^{20}$Mx2cm-2s-1 and also in subsequent plots) computed from
Equation 4. The field of view is $275\times 200$ arcsec2. The overlaid gray
and black contours correspond to LOS magnetic fields at [-150,150]G levels,
respectively. Rectangular boxes in intensity image of 2011 February 14 mark
the selected sub-areas R1 and R2 in which velocity flows are shown in the
subsequent figures. Figure 2: _Top_ : Solar disk integrated GOES Soft X-ray
flux during February 11-16, 2011. The arrows on top panel indicate the start
times of flares in AR NOAA 11158. _Middle_ : Time profiles of the magnetic
fluxes and flux imbalance in the AR. _Bottom_ : The computed helicity rates
integrated over the whole AR. Arrows in this panel indicate the onset time of
CMEs that were launched from this AR.
Figure 3: Transverse velocity field vectors as inferred from DAVE technique
superposed on helicity flux density maps with the LOS magnetic field contours
for the rectangular regions of Figure 1 – R1 (_Top row_) and R2 (_Bottom
row_). Spiral or vortex like velocity patterns in sunspot penumbra in (b-c)
are due to umbral rotation of sunspot SN1. Sites of negative helicity
injection are seen around the magnetic polarity inversion line in (d)-(f) at
the peak times of the flares noted in each panel. Figure 4: Temporal evolution
of helicity rate and accumulated helicity integrated over (a) R1 and (b) R2.
The time difference of helicity rate($\Delta(dH/dt)$) in (c) for region R1
with arrows marking CME timings, (d) for region R2 with pointed flares
originated from this AR. Figure 5: (_Top row_) The daily HMI continuum
intensity maps of AR NOAA 11166, and (_Bottom row_) the corresponding helicity
flux density maps computed from Equation 4. The field of view is $350\times
200$ arcsec2. The overlaid gray and black contours correspond to LOS magnetic
fields at [-150,150]G levels, respectively. Rectangular boxes in intensity
image of March 9 mark the selected sub-areas in which velocity flows are shown
in the next figure. Emerging fluxes from sunspot periphery are indicated as
FN2 and FP3 on March 11/22:00UT Figure 6: Same as Figure 2 but for AR NOAA
11166.
Figure 7: Transverse velocity field vectors in the rectangular region R1 (_Top
row_) and R2 (_Bottom row_) of Figure 5 overlaid on the helicity flux density
maps with iso-contours of LOS magnetic field during flare events. Figure 8:
Same as Figure 4 but for AR NOAA 11166. Figure 9: Mosaic of injection of
helicity flux distribution around the time of X2.2 flare in AR 11158 with iso-
contour of LOS positive(negative) flux in black(white). Intense negative
helicity flux about the PIL during peak time(01:48–02:00UT) of the flare is
evident possibly due to flare-related transient effect on the magnetic field
measurements during the impulsive period.
Figure 10: Temporal profiles of magnetic and helicity fluxes during some
selected flare events in both ARs. Vertical dashed lines indicate onset time
of flares as labeled in each panel. See text for more details.
Figure 11: Dependence of helicity injection rate (in units of 1040 Mx2h-1) for
AR NOAA 11158 on (_Top row_) the time interval $\Delta$t(minutes), and
(_bottom row_) the window size(pixel2). The solid line represents the straight
line fit to the scattered data points whereas the dotted line indicates
slope=1 line for reference. Correlation coefficient and slope of the fitting
are noted in each panel. Figure 12: Plot of accumulated helicity with total
absolute flux computed for NOAA 11158(_Left_) and NOAA 11166(_Right_). The
flare/CME events are labeled and shown by circles in each panel.
|
arxiv-papers
| 2012-02-23T14:47:40 |
2024-09-04T02:49:27.755016
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Vemareddy, A. Ambastha, R. A. Maurya and J. Chae",
"submitter": "Vema Reddy Panditi",
"url": "https://arxiv.org/abs/1202.5195"
}
|
1202.5229
|
# Forward and Adjoint Sensitivity Computation of Chaotic Dynamical Systems
Qiqi Wang Department of Aeronautics and Astronautics, MIT, 77 Mass Ave,
Cambridge, MA 02139, USA Corresponding author. qiqi@mit.edu
###### Abstract
This paper describes a forward algorithm and an adjoint algorithm for
computing sensitivity derivatives in chaotic dynamical systems, such as the
Lorenz attractor. The algorithms compute the derivative of long time averaged
“statistical” quantities to infinitesimal perturbations of the system
parameters. The algorithms are demonstrated on the Lorenz attractor. We show
that sensitivity derivatives of statistical quantities can be accurately
estimated using a single, short trajectory (over a time interval of 20) on the
Lorenz attractor.
###### keywords:
Sensitivity analysis, linear response, adjoint equation, unsteady adjoint,
chaos, statistical average, Lyapunov exponent, Lyapunov covariant vector,
Lorenz attractor.
††journal: Journal of Computational Physics
,
## 1 Introduction
Computational methods for sensitivity analysis is a powerful tool in modern
computational science and engineering. These methods calculate the derivatives
of output quantities with respect to input parameters in computational
simulations. There are two types of algorithms for computing sensitivity
derivatives: the forward algorithms and the adjoint algorithms. The forward
algorithms are more efficient for computing sensitivity derivatives of many
output quantities to a few input parameters; the adjoint algorithms are more
efficient for computing sensitivity derivatives of a few output quantities to
many input parameters. Key application of computational methods for
sensitivity analysis include aerodynamic shape optimization [3], adaptive grid
refinement [9], and data assimilation for weather forecasting [8].
In simulations of chaotic dynamical systems, such as turbulent flows and the
climate system, many output quantities of interest are “statistical averages”.
Denote the state of the dynamical system as $x(t)$; for a function of the
state $J(x)$, the corresponding statistical quantity $\langle J\rangle$ is
defined as an average of $J(x(t))$ over an infinitely long time interval:
$\langle
J\rangle=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}J(x(t))\,dt\;,$ (1)
For ergodic dynamical systems, a statistical average only depends on the
governing dynamical system, and does not depend on the particular choice of
trajectory $x(t)$.
Many statistical averages, such as the mean aerodynamic forces in turbulent
flow simulations, and the mean global temperature in climate simulations, are
of great scientific and engineering interest. Computing sensitivities of these
statistical quantities to input parameters can be useful in many applications.
The differentiability of these statistical averages to parameters of interest
as been established through the recent developments in the Linear Response
Theory for dissipative chaos [6][7]. A class of chaotic dynamical systems,
known as “quasi-hyperbolic” systems, has been proven to have statistical
quantities that are differentiable with respect to small perturbations. These
systems include the Lorenz attractor, and possibly many systems of engineering
interest, such as turbulent flows.
Despite recent advances both in Linear Response Theory [7] and in numerical
methods for sensitivity computation of unsteady systems [10] [4], sensitivity
computation of statistical quantities in chaotic dynamical systems remains
difficult. A major challenge in computing sensitivities in chaotic dynamical
systems is their sensitivity to the initial condition, commonly known as the
“butterfly effect”. The linearized equations, used both in forward and adjoint
sensitivity computations, give rise to solutions that blow up exponentially.
When a statistical quantity is approximated by a finite time average, the
computed sensitivity derivative of the finite time average diverges to
infinity, instead of converging to the sensitivity derivative of the
statistical quantity [5]. Existing methods for computing correct sensitivity
derivatives of statistical quantities usually involve averaging over a large
number of ensemble calculations [5] [1]. The resulting high computation cost
makes these methods not attractive in many applications.
This paper outlines a computational method for efficiently estimating the
sensitivity derivative of time averaged statistical quantities, relying on a
single trajectory over a small time interval. The key idea of our method,
inversion of the “shadow” operator, is already used as a tool for proving
structural stability of strange attractors [6]. The key strategy of our
method, divide and conquer of the shadow operator, is inspired by recent
advances in numerical computation of the Lyapunov covariant vectors [2][11].
In the rest of this paper, Section 2 describes the “shadow” operator, on which
our method is based. Section 3 derives the sensitivity analysis algorithm by
inverting the shadow operator. Section 4 introduces a fix to the singularity
of the shadow operator. Section 5 summarizes the forward sensitivity analysis
algorithm. Section 6 derives the corresponding adjoint version of the
sensitivity analysis algorithm. Section 7 demonstrates both the forward and
adjoint algorithms on the Lorenz attractor. Section 8 concludes this paper.
The paper uses the following mathematical notation: Vector fields in the state
space (e.g. $f(x)$, $\phi_{i}(x)$) are column vectors; gradient of scalar
fields (e.g. $\frac{\partial a_{i}^{x}}{\partial x}$) are row vectors;
gradient of vector fields (e.g. $\frac{\partial f}{\partial x}$) are matrices
with each row being a dimension of $f$, and each column being a dimension of
$x$. The ($\cdot$) sign is used to identify matrix-vector products or vector-
vector inner products. For a trajectory $x(t)$ satisfying $\frac{dx}{dt}=f(x)$
and a scalar or vector field $a(x)$ in the state space, we often use
$\frac{da}{dt}$ to denote $\frac{da(x(t))}{dt}$. The chain rule
$\frac{da}{dt}=\frac{da}{dx}\cdot\frac{dx}{dt}=\frac{da}{dx}\cdot f$ is often
used without explanation.
## 2 The “Shadow Operator”
For a smooth, uniformly bounded $n$ dimensional vector field $\delta x(x)$,
defined on the $n$ dimensional state space of $x$. The following transform
defines a slightly “distorted” coordinates of the state space:
$x^{\prime}(x)=x+\epsilon\,\delta x(x)$ (2)
where $\epsilon$ is a small real number. Note that for an infinitesimal
$\epsilon$, the following relation holds:
$x^{\prime}(x)-x=\epsilon\,\delta x(x)=\epsilon\,\delta
x(x^{\prime})+O(\epsilon^{2})$ (3)
We call the transform from $x$ to $x^{\prime}$ as a “shadow coordinate
transform”. In particular, consider a trajectory $x(t)$ and the corresponding
transformed trajectory $x^{\prime}(t)=x^{\prime}(x(t))$. For a small
$\epsilon$, the transformed trajectory $x^{\prime}(t)$ would “shadow” the
original trajectory $x(t)$, i.e., it stays uniformly close to $x(t)$ forever.
Figure 1 shows an example of a trajectory and its shadow.
Figure 1: A trajectory of the Lorenz attractor under a shadow coordinate
transform. The black trajectory shows $x(t)$, and the red trajectory shows
$x^{\prime}(t)$. The perturbation $\epsilon\,\delta x$ shown corresponds to an
infinitesimal change in the parameter $r$, and is explained in detail in
Section 7.
Now consider a trajectory $x(t)$ satisfying an ordinary differential equation
$\dot{x}=f(x)\;,$ (4)
with a smooth vector field $f(x)$ as a function of $x$. The same trajectory in
the transformed “shadow” coordinates $x^{\prime}(t)$ do not satisfy the same
differential equation. Instead, from Equation (3), we obtain
$\begin{split}\dot{x^{\prime}}&=f(x)+\epsilon\,\frac{\partial\delta
x}{\partial x}\cdot f(x)\\\ &=f(x^{\prime})-\epsilon\,\frac{\partial
f}{\partial x}\cdot\delta x(x^{\prime})+\epsilon\,\frac{\partial\delta
x}{\partial x}\cdot f(x^{\prime})+O(\epsilon^{2})\end{split}$ (5)
In other words, the shadow trajectory $x^{\prime}(t)$ satisfies a slightly
perturbed equation
$\dot{x^{\prime}}=f(x^{\prime})+\epsilon\,\delta
f(x^{\prime})+O(\epsilon^{2})$ (6)
where the perturbation $\delta f$ is
$\begin{split}\delta f(x)&=-\frac{\partial f}{\partial x}\cdot\delta
x(x)+\frac{\partial\delta x}{\partial x}\cdot f(x)\\\ &=-\frac{\partial
f}{\partial x}\cdot\delta x(x)+\frac{d\delta x}{dt}\\\ :&=(S_{f}\delta
x)(x)\end{split}$ (7)
For a given differential equation $\dot{x}=f(x)$, Equation (7) defines a
linear operator $S_{f}:\delta x\Rightarrow\delta f$. We call $S_{f}$ the
“Shadow Operator” of $f$. For any smooth vector field $\delta x(x)$ that
defines a slightly distorted “shadow” coordinate system in the state space,
$S_{f}$ determines a unique smooth vector field $\delta f(x)$ that defines a
perturbation to the differential equation. Any trajectory of the original
differential equation would satisfy the perturbed equation in the distorted
coordinates.
Given an ergodic dynamical system $\dot{x}=f(x)$, and a pair $(\delta x,\delta
f)$ that satisfies $\delta f=S_{f}\delta x$, $\delta x$ determines the
sensitivity of statistical quantities of the dynamical system to an
infinitesimal perturbation $\epsilon\delta f$. Let $J(x)$ be a smooth scalar
function of the state, consider the statistical average $\langle J\rangle$ as
defined in Equation (1). The sensitivity derivative of $\langle J\rangle$ to
the infinitesimal perturbation $\epsilon\,\delta f$ is by definition
$\frac{d\langle J\rangle}{d\epsilon}=\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon}\left(\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}J(x^{\prime}(t))\,dt-\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}J(x(t))\,dt\right)$
(8)
where by the ergodicity assumption, the statistical average of the perturbed
system can be computed by averaging over $x^{\prime}(t)$, which satisfies the
perturbed governing differential equation. Continuing from Equation (8),
$\begin{split}\frac{d\langle J\rangle}{d\epsilon}&=\lim_{\epsilon\rightarrow
0}\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\frac{J(x^{\prime}(t))-J(x(t))}{\epsilon}\>dt\\\
&=\lim_{T\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\frac{1}{T}\int_{0}^{T}\frac{J(x^{\prime}(t))-J(x(t))}{\epsilon}\>dt\\\
&=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\frac{\partial J}{\partial
x}\cdot\delta x\>dt=\left\langle\frac{\partial J}{\partial x}\cdot\delta
x\right\rangle\;.\end{split}$ (9)
Equation (9) represents the sensitivity derivative of a statistical quantity
$\langle J\rangle$ to the size of a perturbation $\epsilon\delta f$. There are
two subtle points here:
* •
The two limits $\lim_{\epsilon\rightarrow 0}$ and $\lim_{T\rightarrow\infty}$
can commute with each other for the following reason: The two trajectories
$x^{\prime}(t)$ and $x(t)$ stay uniformly close to each other forever;
therefore,
$\frac{J(x^{\prime}(t))-J(x(t))}{\epsilon}\overset{\epsilon\rightarrow
0}{\longrightarrow}\frac{\partial J}{\partial x}\cdot\delta x$ (10)
uniformly for all $t$. Consequently,
$\frac{1}{T}\int_{0}^{T}\frac{J(x^{\prime}(t))-J(x(t))}{\epsilon}\>dt\overset{\epsilon\rightarrow
0}{\longrightarrow}\frac{1}{T}\int_{0}^{T}\frac{\partial J}{\partial
x}\cdot\delta x\;dt$ (11)
uniformly for all $T$. Thus the two limits commute.
* •
The two trajectories $x^{\prime}(t)$ and $x(t)$ start at two specially
positioned pair of initial conditions $x^{\prime}(0)=x(0)+\epsilon\,\delta
x(x(0))$. Almost any other pair of initial conditions (e.g.
$x^{\prime}(0)=x(0)$) would make the two trajectories diverge as a result of
the “butterfly effect”. They would not stay uniformly close to each other, and
the limits $\lim_{\epsilon\rightarrow 0}$ and $\lim_{T\rightarrow\infty}$
would not commute.
Equation (9) represents the sensitivity derivative of the statistical quantity
$\langle J\rangle$ to the infinitesimal perturbation $\epsilon\,\delta f$ as
another statistical quantity $\langle\frac{\partial J}{\partial x}\cdot\delta
x\rangle$. We can compute it by averaging $\frac{\partial J}{\partial
x}\cdot\delta x$ over a sufficiently long trajectory, provided that $\delta
x=S^{-1}\delta f$ is known along the trajectory. The next section describes
how to numerically compute $\delta x=S^{-1}\delta f$ for a given $\delta f$.
## 3 Inverting the Shadow Operator
Perturbations to input parameters can often be represented as perturbations to
the dynamics. Consider a differential equation
$\dot{x}=f(x,s_{1},s_{2},\ldots,s_{m})$ parameterized by $m$ input variables,
an infinitesimal perturbation in a input parameter $s_{j}\rightarrow
s_{j}+\epsilon$ can be represented as a perturbation to the dynamics
$\epsilon\,\delta f=\epsilon\,\frac{df}{ds_{j}}$.
Equation (9) defines the sensitivity derivative of the statistical quantity
$\langle J\rangle$ to an infinitesimal perturbation $\epsilon\,\delta f$,
provided that a $\delta x$ can be found satisfying $\delta f=S_{f}\delta x$,
where $S_{f}$ is the shadow operator. To compute the sensitivity by evaluating
Equation (9), one must first numerically invert $S_{f}$ for a given $\delta f$
to find the corresponding $\delta x$.
The key ingredient of numerical inversion of $S_{f}$ is the Lyapunov spectrum
decomposition. This decomposition can be efficiently computed numerically [11]
[2]. In particular, we focus on the case when the system $\dot{x}=f(x)$ has
distinct Lyapunov exponents. Denote the Lyapunov covariant vectors as
$\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{n}(x)$. Each $\phi_{i}$ is a vector
field in the state space satisfying
$\frac{d}{dt}\phi_{i}(x(t))=\frac{\partial f}{\partial
x}\cdot\phi_{i}(x(t))-\lambda_{i}\phi_{i}(x(t))$ (12)
where $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ are the Lyapunov exponents
in decreasing order.
The Lyapunov spectrum decomposition enables a divide and conquer strategy for
computing $\delta x=S_{f}^{-1}\delta f$. For any $\delta f(x)$ and every point
$x$ on the attractor, both $\delta x(x)$ and $\delta f(x)$ can be decomposed
into the Lyapunov covariant vector directions almost everywhere, i.e.
$\delta x(x)=\sum_{i=1}^{n}a^{x}_{i}(x)\,\phi_{i}(x)\;,$ (13) $\delta
f(x)=\sum_{i=1}^{n}a^{f}_{i}(x)\,\phi_{i}(x)\;,$ (14)
where $a^{x}_{i}$ and $a^{f}_{i}$ are scalar fields in the state space. From
the form of $S_{f}$ in Equation (7), we obtain
$\begin{split}S_{f}(a^{x}_{i}\phi_{i})=&-\frac{\partial f}{\partial
x}\cdot(a^{x}_{i}(x)\phi_{i}(x))+\frac{d}{dt}(a^{x}_{i}(x)\,\phi_{i}(x))\\\
=&-a^{x}_{i}(x)\>\frac{\partial f}{\partial
x}\cdot\phi_{i}(x)+\frac{d\,a^{x}_{i}(x)}{dt}\,\phi_{i}(x)+a^{x}_{i}(x)\>\frac{d\,\phi_{i}(x)}{dt}\;.\end{split}$
(15)
By substituting Equation (12) into the last term of Equation (15), we obtain
$S_{f}(a^{x}_{i}\phi_{i})=\left(\frac{da^{x}_{i}(x)}{dt}-\lambda_{i}\,a^{x}_{i}(x)\right)\,\phi_{i}(x)\;,$
(16)
By combining Equation (16) with Equations (13), (14) and the linear relation
$\delta f=S_{f}\delta x$, we finally obtain
$\delta
f=\sum_{i=1}^{n}S_{f}(a^{x}_{i}\phi_{i})=\sum_{i=1}^{n}\;\underbrace{\left(\frac{da^{x}_{i}}{dt}-\lambda_{i}\,a^{x}_{i}\right)}_{\displaystyle
a^{f}_{i}}\,\phi_{i}\;,$ (17)
Equations (16) and (17) are useful for two reasons:
1. 1.
They indicate that the Shadow Operator $S_{f}$, applied to a scalar field
$a^{x}_{i}(x)$ multiple of $\phi_{i}(x)$, generates another scalar field
$a^{f}_{i}(x)$ multiple of the same vector field $\phi_{i}(x)$. Therefore, one
can compute $S_{f}^{-1}\delta f$ by first decomposing $\delta f$ as in
Equation (14) to obtain the $a^{f}_{i}$. If each $a_{i}^{x}$ can be calculated
from the corresponding $a_{i}^{f}$, then $\delta x$ can be computed with
Equation (13), completing the inversion.
2. 2.
It defines a scalar ordinary differential equation that governs the relation
between $a^{x}_{i}$ and $a^{f}_{i}$ along a trajectory $x(t)$:
$\frac{da^{x}_{i}(x)}{dt}=a^{f}_{i}(x)+\lambda_{i}\,a^{x}_{i}(x)$ (18)
This equation can be used to obtain $a^{x}_{i}$ from $a^{f}_{i}$ along a
trajectory, thereby filling the gap in the inversion procedure of $S_{f}$
outlined above. For each positive Lyapunov exponent $\lambda_{i}$, one can
integrate the ordinary differential equation
$\frac{d\check{a}^{x}_{i}}{dt}=\check{a}^{f}_{i}+\lambda_{i}\,\check{a}^{x}_{i}$
(19)
backwards in time from an arbitrary terminal condition, and the difference
between $\check{a}^{x}_{i}(t)$ and the desired $a^{x}_{i}(x)$ will decrease
exponentially. For each negative Lyapunov exponent $\lambda_{i}$, Equation
(19) can be integrated forward in time from an arbitrary initial condition,
and $\check{a}^{x}_{i}(t)$ will converge exponentially to the desired
$a^{x}_{i}(x)$. For a zero Lyapunov exponent $\lambda_{i}=0$, Section 4
introduces a solution.
## 4 Time Dilation and Compression
There is a fundamental problem in the inversion method derived in Section 3:
$S_{f}$ is not invertible for certain $\delta f$. This can be shown with the
following analysis: Any continuous time dynamical system with a non-trivial
attractor must have a zero Lyapunov exponent $\lambda_{n_{0}}=0$. The
corresponding Lyapunov covariant vector is $\phi_{n_{0}}(x)=f(x)$. This can be
verified by substituting $\lambda_{i}=0$ and $\phi_{i}=f$ into Equation (12).
For this $i=n_{0}$, Equations (19) becomes
$a^{f}_{n_{0}}(x)=\frac{da^{x}_{n_{0}}(x)}{dt}$ (20)
By taking an infinitely long time average on both sides of Equation (20), we
obtain
$\left\langle
a^{f}_{n_{0}}(x)\right\rangle=\lim_{T\rightarrow\infty}\frac{a^{x}_{n_{0}}(x(T))-a^{x}_{n_{0}}(x(0))}{T}=0\;,$
(21)
Equation (21) implies that for any $\delta f=S_{f}\delta x$, the $i=n_{0}$
component of its Lyapunov decomposition (as in Equation (14)) must satisfy
$\langle a^{f}_{n_{0}}(x)\rangle=0$. Any $\delta f$ that do not satisfy this
linear relation, e.g. $\delta f\equiv f$, would not be in the range space of
$S_{f}$. Thus the corresponding $\delta x=S_{f}^{-1}\delta f$ does not exist.
Our solution to the problem is complementing $S_{f}$ with a “global time
dilation and compression” constant $\eta$, whose effect produces a $\delta f$
that is outside the range space of $S_{f}$. We call $\eta$ a time dilation
constant for short. The combined effect of a time dilation constant and a
shadow transform could produce all smooth perturbations $\delta f$.
The idea comes from the fact that for a constant $\eta$, the time dilated or
compressed system $\dot{x}=(1+\epsilon\,\eta)f(x)$ has exactly the same
statistics $\langle J\rangle$, as defined in Equation (1), as the original
system $\dot{x}=f(x)$. Therefore, the perturbation in any $\langle J\rangle$
due to any $\epsilon\,\delta f$ is equal to the perturbation in $\langle
J\rangle$ due to $\epsilon\,(\eta f(x)+\delta f(x))$. Therefore, the
sensitivity derivative to $\delta f$ can be computed if we can find a $\delta
x$ that satisfies $S_{f}\delta x=\eta f(x)+\delta f(x)$ for some $\eta$.
We use the “free” constant $\eta$ to put $\eta f(x)+\delta f(x)$ into the
range space of $S_{f}$. By substituting $\eta f(x)+\delta f(x)$ into the
constraint Equation (21) that identifies the range space of $S_{f}$, the
appropriate $\eta$ must satisfy the following equation
$\eta+\langle a^{f}_{n_{0}}\rangle=0\;,$ (22)
which we use to numerically compute $\eta$.
Once the appropriate time dilation constant $\eta$ is computed, $\eta
f(x)+\delta f(x)$ is in the range space of $S_{f}$. We use the procedure in
Section 3 to compute $\delta x=S_{f}^{-1}(\eta f+\delta f)$, then use Equation
(9) to compute the desired sensitivity derivative $d\langle
J\rangle/d\epsilon$. The addition of $\eta f$ to $\delta f$ affects Equation
(19) only for $i=n_{0}$, making it
$\frac{da^{x}_{n_{0}}(x)}{dt}=a^{f}_{n_{0}}(x)+\eta\;.$ (23)
Equation (23) indicates that $a^{x}_{n_{0}}$ can be computed by integrating
the right hand side along the trajectory.
The solution to Equation (23) admits an arbitrary additive constant. The
effect of this arbitrary constant is the following: By substituting Equations
(13) into Equation (9), the contribution from the $i=n_{0}$ term of $\delta x$
to $d\langle J\rangle/d\epsilon$ is
$\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}a^{f}_{n_{0}}\frac{dJ}{dt}\,dt$
(24)
Therefore, any constant addition to $a^{f}_{n_{0}}$ vanishes as
$T\rightarrow\infty$. Computationally, however, Equation (9) must be
approximated by a finite time average. We find it beneficial to adjust the
level of $a^{f}_{n_{0}}$ to approximately $\langle a^{f}_{n_{0}}\rangle=0$, in
order to control the error due to finite time averaging.
## 5 The Forward Sensitivity Analysis Algorithms
For a given $\dot{x}=f(x)$, $\delta f$ and $J(x)$, the mathematical
developments in Sections 3 and 4 are summarized into Algorithm 1 for computing
the sensitivity derivative $d\delta\langle J\rangle/d\epsilon$ as in Equation
(9).
Algorithm 1 The Forward Sensitivity Analysis Algorithm
1. 1.
Choose a “spin-up buffer time” $T_{B}$, and an “statistical averaging time”
$T_{A}$. $T_{B}$ should be much longer than $1/|\lambda_{i}|$ for all nonzero
Lyapunov exponent $\lambda_{i}$, so that the solutions of Equation (19) can
reach $a_{i}^{x}$ over a time span of $T_{B}$. $T_{A}$ should be much longer
than the decorrelation time of the dynamics, so that one can accurately
approximate a statistical quantity by averaging over $[0,T_{A}]$.
2. 2.
Obtain an initial condition on the attractor at $t=-T_{B}$, e.g., by solving
$\dot{x}=f(x)$ for a sufficiently long time span, starting from an arbitrary
initial condition.
3. 3.
Solve $\dot{x}=f(x)$ to obtain a trajectory $x(t),t\in[-T_{B},T_{A}+T_{B}]$;
compute the Lyapunov exponents $\lambda_{i}$ and the Lyapunov covariant
vectors $\phi_{i}(x(t))$ along the trajectory, e.g., using algorithms in [11]
and [2].
4. 4.
Perform the Lyapunov spectrum decomposition of $\delta f(x)$ along the
trajectory $x(t)$ to obtain $a^{f}_{i}(x),i=1,\ldots,n$ as in Equation (14).
5. 5.
Compute the global time dilation constant $\eta$ using Equation (22).
6. 6.
Solve the differential equations (19) to obtain $a^{x}_{i}$ over the time
interval $[0,T_{A}]$. The equations with positive $\lambda_{i}$ are solved
backward in time from $t=T_{A}+T_{B}$ to $t=0$; the ones with negative
$\lambda_{i}$ are solved forward in time from $t=-T_{B}$ to $t=T_{A}$. For
$\lambda_{n_{0}}=0$, Equation (23) is integrated, and the mean of
$a^{x}_{n_{0}}$ is set to zero.
7. 7.
Compute $\delta x$ along the trajectory $x(t),t\in[0,T_{A}]$ with Equation
(13).
8. 8.
Compute $d\langle J\rangle/d\epsilon$ using Equation (1) by averaging over the
time interval $[0,T_{A}]$.
The preparation phase of the algorithm (Steps 1-3) computes a trajectory and
the Lyapunov spectrum decomposition along the trajectory. The algorithm then
starts by decomposing $\delta f$ (Step 4), followed by computing $\delta x$
(Steps 5-7), and finally computing $d\langle J\rangle/d\epsilon$ (Step 8). The
sensitivity derivative of many different statistical quantities $\langle
J_{1}\rangle,\langle J_{2}\rangle,\ldots$ to a single $\delta f$ can be
computed by only repeating the last step of the algorithm. Therefore, this is
a “forward” algorithm in the sense that it efficiently computes sensitivity of
multiple output quantities to a single input parameter (the size of
perturbation $\epsilon\,\delta f$). We will see that this is in sharp contrast
to the “adjoint” algorithm described in Section 6, which efficiently computes
the sensitivity derivative of one output statistical quantity $\langle
J\rangle$ to many perturbations $\delta f_{1},\delta f_{2},\ldots$.
It is worth noting that the $\delta x$ computed using Algorithm 1 satisfies
the forward tangent equation
$\dot{\delta x}=\frac{\partial f}{\partial x}\cdot\delta x+\eta\,f+\delta f$
(25)
This can be verified by taking derivative of Equation (13), substituting
Equations (19) and (23), then using Equation (14). However, $\delta x$ must
satisfy both an initial condition and a terminal condition, making it
difficult to solve with conventional time integration methods. In fact,
Algorithm 1 is equivalent to splitting $\delta x$ into stable, neutral and
unstable components, corresponding to positive, zero and negative Lyapunov
exponents; then solving Equation (25) separately for each component in
different time directions. This alternative version of the forward sensitivity
computation algorithm could be useful for large systems to avoid computation
of all the Lyapunov covariant vectors.
## 6 The Adjoint Sensitivity Analysis Algorithm
The adjoint algorithm starts by trying to find an adjoint vector field
$\hat{f}(x)$, such that the sensitivity derivative of the given statistical
quantity $\langle J\rangle$ to any infinitesimal perturbation
$\epsilon\,\delta f$ can be represented as
$\frac{d\langle J\rangle}{\epsilon}=\left\langle\hat{f\,}^{T}\cdot\delta
f\right\rangle$ (26)
Both $\hat{f}$ in Equation (26) and $\frac{\partial J}{\partial x}$ in
Equation (9) can be decomposed into linear combinations of the _adjoint
Lyapunov covariant vectors_ almost everywhere on the attractor:
$\hat{f}(x)=\sum_{i=1}^{n}{\hat{a}}^{f}_{i}(x)\,\psi_{i}(x)\;,$ (27)
$\frac{\partial J}{\partial
x}^{T}=\sum_{i=1}^{n}{\hat{a}}^{x}_{i}(x)\,\psi_{i}(x)\;,$ (28)
where the adjoint Lyapunov covariant vectors $\psi_{i}$ satisfy
$-\frac{d}{dt}\psi_{i}(x(t))=\frac{\partial f}{\partial
x}^{T}\cdot\psi_{i}(x(t))-\lambda_{i}\psi_{i}(x(t))$ (29)
With proper normalization, the (primal) Lyapunov covariant vectors $\phi_{i}$
and the adjoint Lyapunov covariant vectors $\psi_{i}$ have the following
conjugate relation:
$\psi_{i}(x)^{T}\cdot\phi_{j}(x)\equiv\begin{cases}0\;,&i\neq j\\\
1\;,&i=j\end{cases}$ (30)
i.e., the $n\times n$ matrix formed by all the $\phi_{i}$ and the $n\times n$
matrix formed by all the $\psi_{i}$ are the transposed inverse of each other
at every point $x$ in the state space.
By substituting Equations (13) and (28) into Equation (9), and using the
conjugate relation in Equation (30), we obtain
$\frac{d\langle
J\rangle}{d\epsilon}=\sum_{i=1}^{n}\left\langle{\hat{a}}_{i}^{x}a_{i}^{x}\right\rangle$
(31)
Similarly, by combining Equations (26), (14), (27) and (30), it can be shown
that $\hat{f}$ satisfies Equation (26) if and only if
$\frac{d\langle
J\rangle}{d\epsilon}=\sum_{i=1}^{n}\left\langle{\hat{a}}_{i}^{f}a_{i}^{f}\right\rangle$
(32)
Comparing Equations (31) and (32) leads to the following conclusion: Equation
(26) can be satisfied by finding ${\hat{a}}_{i}^{f}$ that satisfy
$\left\langle{\hat{a}}_{i}^{f}a_{i}^{f}\right\rangle=\left\langle{\hat{a}}_{i}^{x}a_{i}^{x}\right\rangle\;,\quad
i=1,\ldots,n$ (33)
The ${\hat{a}}_{i}^{f}$ that satisfies Equation (33) can be found using the
relation between $a_{i}^{f}$ and $a_{i}^{x}$ in Equation (18). By multiplying
$\hat{a}_{i}^{f}$ on both sides of Equation (18) and integrate by parts in
time, we obtain
$\frac{1}{T}\int_{0}^{T}{\hat{a}}_{i}^{f}a_{i}^{f}dt=\left.\frac{{\hat{a}}_{i}^{f}a_{i}^{x}}{T}\right|_{0}^{T}-\frac{1}{T}\int_{0}^{T}\left(\frac{d{\hat{a}}_{i}^{f}}{dt}+\lambda_{i}\,{\hat{a}}_{i}^{f}\right)a^{x}_{i}\;dt$
(34)
for $i\neq n_{0}$. Through apply the same technique to Equation (23), we
obtain for $i=n_{0}$
$\frac{1}{T}\int_{0}^{T}{\hat{a}}_{n_{0}}^{f}a_{n_{0}}^{f}dt=\left.\frac{{\hat{a}}_{n_{0}}^{f}a_{n_{0}}^{x}}{T}\right|_{0}^{T}-\frac{1}{T}\int_{0}^{T}\frac{d{\hat{a}}_{n_{0}}^{f}}{dt}\,a^{x}_{n_{0}}dt+\frac{1}{T}\int_{0}^{T}\eta\,\hat{a}^{f}_{n_{0}}dt$
(35)
If we set $\hat{a}_{i}^{f}$ to satisfy the following relations
$\displaystyle-\frac{d{\hat{a}}_{i}^{f}(x)}{dt}$
$\displaystyle=\hat{a}_{i}^{x}(x)+\lambda_{i}\,{\hat{a}}_{i}^{f}(x)\;,$
$\displaystyle i\neq n_{0}\;,$ (36)
$\displaystyle-\frac{d{\hat{a}}_{i}^{f}(x)}{dt}$
$\displaystyle=\hat{a}_{i}^{x}(x)\;,\quad\langle\hat{a}_{i}^{f}\rangle=0\;,\quad$
$\displaystyle i=n_{0}\;,$
then Equations (34) and (35) become
$\displaystyle\frac{1}{T}\int_{0}^{T}{\hat{a}}_{i}^{f}a_{i}^{f}dt$
$\displaystyle=\left.\frac{{\hat{a}}_{i}^{f}a_{i}^{x}}{T}\right|_{0}^{T}+\frac{1}{T}\int_{0}^{T}\hat{a}_{i}^{x}a_{i}^{x}\;dt\;,\quad$
$\displaystyle i\neq n_{0}$ (37)
$\displaystyle\frac{1}{T}\int_{0}^{T}{\hat{a}}_{i}^{f}a_{i}^{f}dt$
$\displaystyle=\left.\frac{{\hat{a}}_{i}^{f}a_{i}^{x}}{T}\right|_{0}^{T}+\frac{1}{T}\int_{0}^{T}\hat{a}_{i}^{x}a_{i}^{x}\;dt+\eta\left(\frac{1}{T}\int_{0}^{T}\hat{a}^{f}_{n_{0}}\,dt-\langle\hat{a}^{f}_{n_{0}}\rangle\right),$
$\displaystyle i=n_{0}$
As $T\rightarrow\infty$, both equations reduces to Equation (33).
In summary, if the scalar fields $\hat{a}_{i}^{f}$ satisfy Equation (36), then
they also satisfy Equation (37) and thus Equation (33); as a result, the
$\hat{f}$ formed by these $\hat{a}^{f}$ through Equation (27) satisfies
Equation (26), thus is the desired adjoint vector field.
For each $i\neq n_{0}$, the scalar field $\hat{a}_{i}^{f}$ satisfying Equation
(36) can be computed by solving an ordinary differential equations
$-\frac{d\check{\hat{a}}_{i}^{f}}{dt}=\check{\hat{a}}_{i}^{x}+\lambda_{i}\,\check{\hat{a}}_{i}^{f}\;.$
(38)
Contrary to computation of $a_{i}^{x}$ through solving Equation (19), the time
integration should be forward in time for positive $\lambda_{i}$, and backward
in time for negative $\lambda_{i}$, in order for the difference between
$\check{\hat{a}}_{i}^{f}(t)$ and ${\hat{a}}_{i}^{f}(x(t))$ to diminish
exponentially.
The $i=n_{0}$ equation in Equation (36) can be directly integrated to obtain
${\hat{a}}_{n_{0}}^{f}(x)$. The equation is well defined because the right
hand side is mean zero:
$\frac{1}{T}\int_{0}^{T}\hat{a}_{n_{0}}^{f}(x(t))\,dt=\frac{1}{T}\int_{0}^{T}\frac{\partial
J}{\partial
x}\cdot\phi_{n_{0}}\,dt=\frac{1}{T}\int_{0}^{T}\frac{dJ}{dt}\,dt\overset{T\rightarrow\infty}{\longrightarrow}0\;.$
(39)
Therefore, the integral of ${\hat{a}}_{n_{0}}^{x}(x)$ over time, subtracted by
its mean, is the solution ${\hat{a}}_{n_{0}}^{f}(x)$ to the $i=n_{0}$ case of
Equation (36).
Algorithm 2 The Adjoint Sensitivity Analysis Algorithm
1. 1.
Choose a “spin-up buffer time” $T_{B}$, and an “statistical averaging time”
$T_{A}$. $T_{B}$ should be much longer than $1/|\lambda_{i}|$ for all nonzero
Lyapunov exponent $\lambda_{i}$, so that the solutions of Equation (19) can
reach $a_{i}^{x}$ over a time span of $T_{B}$. $T_{A}$ should be much longer
than the decorrelation time of the dynamics, so that one can accurately
approximate a statistical quantity by averaging over $[0,T_{A}]$.
2. 2.
Obtain an initial condition on the attractor at $t=-T_{B}$, e.g., by solving
$\dot{x}=f(x)$ for a sufficiently long time span, starting from an arbitrary
initial condition.
3. 3.
Solve $\dot{x}=f(x)$ to obtain a trajectory $x(t),t\in[-T_{B},T_{A}+T_{B}]$;
compute the Lyapunov exponents $\lambda_{i}$ and the Lyapunov covariant
vectors $\phi_{i}(x(t))$ along the trajectory, e.g., using algorithms in [11]
and [2].
4. 4.
Perform the Lyapunov spectrum decomposition of $(\partial J/\partial x)^{T}$
along the trajectory $x(t)$ to obtain $\hat{a}^{x}_{i}(x(t)),i=1,\ldots,n$ as
in Equation (28).
5. 5.
Solve the differential equations (38) to obtain $\hat{a}_{i}^{f}(x(t))$ over
the time interval $[0,T_{A}]$. The equations with negative $\lambda_{i}$ are
solved backward in time from $t=T_{A}+T_{B}$ to $t=0$; the ones with positive
$\lambda_{i}$ are solved forward in time from $t=-T_{B}$ to $t=T_{A}$. For
$i=n_{0}$, the scalar $-a^{x}_{n_{0}}$ is integrated along the trajectory; the
mean of the integral is subtracted from the integral itself to obtain
$\hat{a}^{f}_{n_{0}}$.
6. 6.
Compute $\hat{f}$ along the trajectory $x(t),t\in[0,T_{A}]$ with Equation
(27).
7. 7.
Compute $d\langle J\rangle/d\epsilon$ using Equation (26) by averaging over
the time interval $[0,T_{A}]$.
The above analysis summarizes to Algorithm 2 for computing the sensitivity
derivative derivative of the statistical average $\langle J\rangle$ to an
infinitesimal perturbations $\epsilon\,\delta f$. The preparation phase of the
algorithm (Steps 1-3) is exactly the same as in Algorithm 1. These steps
compute a trajectory $x(t)$ and the Lyapunov spectrum decomposition along the
trajectory. The adjoint algorithm then starts by decomposing the derivative
vector $(\partial J/\partial x)^{T}$ (Step 4), followed by computing the
adjoint vector $\delta f$ (Steps 5-6), and finally computing $d\langle
J\rangle/d\epsilon$ for a particular $\delta f$. Note that the sensitivity of
the same $\langle J\rangle$ to many different perturbations $\delta
f_{1},\delta f_{2},\ldots$ can be computed by repeating only the last step of
the algorithm. Therefore, this is an “adjoint” algorithm, in the sense that it
efficiently computes the sensitivity derivatives of a single output quantity
to many input perturbation.
It is worth noting that $\hat{f}$ computed using Algorithm 2 satisfies the
adjoint equation
$-\dot{\hat{f}}=\frac{\partial f}{\partial x}^{T}\cdot\hat{f}-\frac{\partial
J}{\partial x}$ (40)
This can be verified by taking derivative of Equation (27), substituting
Equation (36), then using Equation (28). However, $\hat{f}$ must satisfy both
an initial condition and a terminal condition, making it difficult to solve
with conventional time integration methods. In fact, Algorithm 2 is equivalent
to splitting $\hat{f}$ into stable, neutral and unstable components,
corresponding to positive, zero and negative Lyapunov exponents; then solving
Equation (40) separately for each component in different time directions. This
alternative version of the adjoint sensitivity computation algorithm could be
useful for large systems, to avoid computation of all the Lyapunov covariant
vectors.
## 7 An Example: the Lorenz Attractor
We consider the Lorenz attractor $\dot{x}=f(x)$, where
$x=(x_{1},x_{2},x_{3})^{T}$, and
$f(x)=\left(\begin{array}[]{c}\sigma(x_{2}-x_{1})\\\ x_{1}(r-x_{3})-x_{2}\\\
x_{1}x_{2}-\beta x_{3}\end{array}\right)$ (41)
The “classic” parameter values $\sigma=10$, $r=28$, $\beta=8/3$ are used. Both
the forward sensitivity analysis algorithm (Algorithm 1) and the adjoint
sensitivity analysis algorithm (Algorithm 2) are performed on this system.
We want to demonstrate the computational efficiency of our algorithm;
therefore, we choose a relatively short statistical averaging interval of
$T_{A}=10$, and a spin up buffer period of $T_{B}=5$. Only a single trajectory
of length $T_{A}+2T_{B}$ on the attractor is required in our algorithm.
Considering that the oscillation period of the Lorenz attractor is around $1$,
the combined trajectory length of $20$ is a reasonable time integration length
for most simulations of chaotic dynamical systems. In our example, we start
the time integration from $t=-10$ at $x=(-8.67139571762,4.98065219709,25)$,
and integrate the equation to $t=-5$, to ensure that the entire trajectory
from $-T_{B}$ to $T_{A}+T_{B}$ is roughly on the attractor. The rest of the
discussion in this section are focused on the trajectory $x(t)$ for
$t\in[-T_{B},T_{A}+T_{B}]$.
### 7.1 Lyapunov covariant vectors
The Lyapunov covariant vectors are computed in Step 3 of both Algorithm 1 and
Algorithm 2, over the time interval $[-T_{B},T_{A}+T_{B}]$. These vectors,
along with the trajectory $x(t)$, are shown in Figure 2.
(a) The state vector $x$
(b) First Lyapunov covariant vector $\phi_{1}$
(c) Second Lyapunov covariant vector $\phi_{2}$
(d) Third Lyapunov covariant vector $\phi_{3}$
Figure 2: The Lyapunov covariant vectors of the Lorenz attractor along the
trajectory $x(t)$ for $t\in[0,10]$. The x-axes are $t$; the blue, green and
red lines correspond to the $x_{1},x_{2}$ and $x_{3}$ coordinates in the state
space, respectively.
The three dimensional Lorenz attractor has three pairs of Lyapunov exponents
and Lyapunov covariant vectors. $\lambda_{1}$ is the only positive Lyapunov
exponent, and $\phi_{1}$ is computed by integrating the tangent linear
equation
$\dot{\tilde{x}}=\frac{\partial f}{\partial x}\cdot\tilde{x}$ (42)
forward in time from an arbitrary initial condition at $t=-T_{B}$. The first
Lyapunov exponent is estimated to be $\lambda_{1}\approx 0.95$ through a
linear regression of $\tilde{x}$ in the log space. The first Lyapunov vector
is then obtained as $\phi_{1}=\tilde{x}\,e^{-\lambda_{1}t}$.
$\lambda_{2}=0$ is the vanishing Lyapunov exponent; therefore,
$\phi_{2}=\theta\,f(x)$, where $\theta=1/\sqrt{\langle\|f\|_{2}^{2}\rangle}$
is a normalizing constant that make the mean magnitude of $\phi_{2}$ equal to
1.
The third Lyapunov exponent $\lambda_{3}$ is negative. So $\phi_{3}$ is
computed by integrating the tangent linear equation (42) backwards in time
from an arbitrary initial condition at $t=T_{A}+T_{B}$. The third Lyapunov
exponent is estimated to be $\lambda_{3}\approx-14.6$ through a linear
regression of the backward solution $\tilde{x}$ in the log space. The third
Lyapunov vector is then obtained as $\phi_{3}=\tilde{x}\,e^{-\lambda_{3}t}$.
### 7.2 Forward Sensitivity Analysis
We demonstrate our forward sensitivity analysis algorithm by computing the
sensitivity derivative of three statistical quantities $\langle
x_{1}^{2}\rangle$, $\langle x_{2}^{2}\rangle$ and, $\langle x_{3}\rangle$ to a
small perturbation in the system parameter $r$ in the Lorenz attractor
Equation (41). The infinitesimal perturbation $r\rightarrow r+\epsilon$ is
equivalent to the perturbation
$\epsilon\,\delta f=\epsilon\,\frac{\partial f}{\partial
r}=\epsilon\,(0,x_{1},0)^{T}\;.$ (43)
(a) $\delta f=\dfrac{df}{dr}$
(b) $a^{f}_{i},i=1,2,3$ for the $\delta f$
Figure 3: Lyapunov vector decomposition of $\delta f$. The x-axes are $t$; the
blue, green and red lines on the left are the first, second and third
component of $\delta f$ as defined in Equation (43); the blue, green and red
lines on the right are $a^{f}_{1}$, $a^{f}_{2}$ and $a^{f}_{3}$ in the
decomposition of $\delta f$ (Equation (14)), respectively.
The forcing term defined in Equation (43) is plotted in Figure 3a. Figure 3b
plots the decomposition coefficients $a_{i}^{f}$, computed by solving a
$3\times 3$ linear system defined in Equation (14) at every point on the
trajectory.
(a) $a^{x}_{i},i=1,2,3$ for the $\delta f$
(b) $\delta x=\sum_{i=1}^{3}a^{x}_{i}\,\phi_{i}$
Figure 4: Inversion of $S_{f}$ for $\delta x=S_{f}^{-1}\delta f$. The x-axes
are $t$; the blue, green and red lines on the left are $a^{x}_{1}$,
$a^{x}_{2}$ and $a^{x}_{3}$, respectively; the blue, green and red lines on
the right are the first, second and third component of $\delta x$, computed
via Equation (13).
For each $a^{f}_{i}$ obtained through the decomposition, Equation (19) or (23)
is solved to obtain $a^{x}_{i}$. For $i=1$, Equation (19) is solved backwards
in time from $t=T_{A}+T_{B}$ to $t=0$. For $i=n_{0}=2$, the time compression
constant is estimated to be $\eta\approx-2.78$, and Equation (23) is
integrated to obtain $a^{x}_{2}$. For $i=3$, Equation (19) is solved forward
in time from $t=-T_{B}$ to $t=T_{A}$.
The resulting values of $a^{x}_{i},i=1,2,3$ are plotted in Figure 4a. These
values are then substituted into Equation (13) to obtain $\delta x$, as
plotted in Figure 4b. The “shadow” trajectory defined as
$x^{\prime}=x+\epsilon\delta x$ is also plotted in Figure 1 as the red lines,
for an $\epsilon=1/3$. This $\delta x=S_{f}^{-1}\delta f$ is approximately the
shadow coordinate perturbation “induced” by a $1/3$ increase in the input
parameter $r$, a.k.a. the Rayleigh number in the Lorenz attractor.
The last step of the forward sensitivity analysis algorithm is computing the
sensitivity derivatives of the output statistical quantities using Equation
(9). We found that using a windowed time averaging [4] yields more accurate
sensitivities. Here our estimates over the time interval $[0,T_{A}]$ are
$\frac{d\langle x_{1}^{2}\rangle}{dr}\approx 2.64\;,\quad\frac{d\langle
x_{2}^{2}\rangle}{dr}\approx 3.99\;,\quad\frac{d\langle
x_{3}\rangle}{dr}\approx 1.01$ (44)
These sensitivity values compare well to results obtained through finite
difference, as shown in Section 7.4.
### 7.3 Adjoint Sensitivity Analysis
We demonstrate our adjoint sensitivity analysis algorithm by computing the
sensitivity derivatives of the statistical quantity $\langle x_{3}\rangle$ to
small perturbations in the three system parameters $s$, $r$ and $b$ in the
Lorenz attractor Equation (41).
(a) $\dfrac{\partial J}{\partial x}$ for $J=x_{3}$
(b) $\hat{a}^{x}_{i},i=1,2,3$ for the $\dfrac{\partial J}{\partial x}$
Figure 5: Adjoint Lyapunov vector decomposition of $\partial J/\partial x$.
The x-axes are $t$; the blue, green and red lines on the left are the first,
second and third component of $\partial J/\partial x$; the blue, green and red
lines on the right are $\hat{a}^{x}_{1}$, $\hat{a}^{x}_{2}$ and
$\hat{a}^{x}_{3}$ in the decomposition of $\partial J/\partial x$ (Equation
(28)), respectively.
The first three steps of Algorithm 2 is the same as in Algorithm 1, and has
been demonstrated in Section 7.1. Step 4 involves decomposing $(\partial
J/\partial x)^{T}$ into three adjoint Lyapunov covariant vectors. In our case,
$J(x)=x_{3}$, therefore $\partial J/\partial x\equiv(0,0,1)$, as plotted in
Figure 5a. The adjoint Lyapunov covariant vectors $\psi_{i}$ can be computed
using Equation (30) by inverting the $3\times 3$ matrix formed by the (primal)
Lyapunov covariant vectors $\phi_{i}$ at every point on the trajectory. The
coefficients $\hat{a}^{x}_{i},i=1,2,3$ can then be computed by solving
Equation (28). These scalar quantities along the trajectory are plotted in
Figure 5b for $t\in[0,T_{A}]$.
(a) $\hat{a}^{f}_{i},i=1,2,3$ solved using Equation (38)
(b) $\hat{f}=\sum_{i=1}^{3}\hat{a}^{f}_{i}\,\psi_{i}$
Figure 6: Computation of the adjoint solution $\hat{f}$ for the Lorenz
attractor. The x-axes are $t$; the blue, green and red lines on the left are
$\hat{a}^{f}_{1}$, $\hat{a}^{f}_{2}$ and $\hat{a}^{f}_{3}$, respectively; the
blue, green and red lines on the right are the first, second and third
component of $\hat{f}$, computed via Equation (27). Figure 7: The adjoint
sensitivity derivative $\hat{f}$ as in Equation (26), represented by arrows on
the trajectory.
Once we obtain $\hat{a}_{i}^{x}$, $\hat{a}_{i}^{f}$ can be computed by solving
Equation (38). The solution is plotted in Figure 6a. Equation (27) can then be
used to combine the $\hat{a}_{i}^{f}$ into the adjoint vector $\hat{f}$. The
computed $\hat{f}$ along the trajectory is plotted both in Figure 6b as a
function of $t$, and also in Figure 7 as arrows on the trajectory in the state
space.
The last step of the adjoint sensitivity analysis algorithm is computing the
sensitivity derivatives of $\langle J\rangle$ to the perturbations $\delta
f_{s}=\frac{df}{ds}$, $\delta f_{r}=\frac{df}{dr}$ and $\delta
f_{b}=\frac{df}{db}$ using Equation (26). Here our estimates over the time
interval $[0,T_{A}]$ are computed as
$\frac{d\langle x_{3}\rangle}{ds}\approx 0.21\;,\quad\frac{d\langle
x_{3}\rangle}{dr}\approx 0.97\;,\quad\frac{d\langle
x_{3}\rangle}{db}\approx-1.74$ (45)
Note that $\frac{d\langle x_{3}\rangle}{dr}$ estimated using adjoint method
differs from the same value estimated using forward method (44). This
discrepancy can be caused by the different numerical treatments to the time
dilation term in the two methods. The forward method numerically estimates the
time dilation constant $\eta$ through Equation (22); while the adjoint method
sets the mean of $\hat{a}_{i}^{f}$ to zero (36), so that the computation is
independent to the value of $\eta$. This difference could cause apparent
discrepancy in the estimated sensitivity derivatives.
The next section compares these sensitivity estimates, together with the
sensitivity estimates computed in Section 7.2, to a finite difference study.
### 7.4 Comparison with the finite difference method
To reduce the noise in the computed statistical quantities in the finite
difference study, a very long time integration length of $T=100,000$ is used
for each simulation. Despite this long time averaging, the quantities computed
contain statistical noise of the order $0.01$. The noise limits the step size
of the finite difference sensitivity study. Fortunately all the output
statistical quantities seem fairly linear with respect to the input
parameters, and a moderately large step size of the order $0.1$ can be used.
To further reduce the effect of statistical noise, we perform linear
regressions through $10$ simulations of the Lorenz attractor, with $r$ equally
spaced between $27.9$ and $28.1$. The total time integration length (excluding
spin up time) is $1,000,000$. The resulting computation cost is in sharp
contrast to our method, which involves a trajectory of only length $20$.
Similar analysis is performed for the parameters $s$ and $b$, where 10 values
of $s$ equally spaced between $9.8$ and $10.2$ are used, and 10 values of $b$
equally spaced between $8/3-0.02$ and $8/3+0.02$ are used. The slopes
estimated from the linear regressions, together with $3\sigma$ confidence
intervals (where $\sigma$ is the standard error of the linear regression) is
listed below:
$\begin{split}&\frac{d\langle x_{1}^{2}\rangle}{dr}=2.70\pm
0.10\;,\quad\frac{d\langle x_{2}^{2}\rangle}{dr}=3.87\pm
0.18\;,\quad\frac{d\langle x_{3}\rangle}{dr}=1.01\pm 0.04\\\ &\frac{d\langle
x_{3}\rangle}{ds}=0.16\pm 0.02\;,\quad\frac{d\langle
x_{3}\rangle}{db}=-1.68\pm 0.15\;.\end{split}$ (46)
(a) $\dfrac{\partial\langle x_{1}^{2}\rangle}{\partial r}$
(b) $\dfrac{\partial\langle x_{2}^{2}\rangle}{\partial r}$
(c) $\dfrac{\partial\langle x_{3}\rangle}{\partial r}$
Figure 8: Histogram of sensitivities computed using Algorithm 1 (forward
sensitivity analysis) starting from 200 random initial conditions.
$T_{A}=10,T_{B}=5$. The red region identifies the $3\sigma$ confidence
interval estimated using finite difference regression.
(a) $\dfrac{\partial\langle x_{3}\rangle}{\partial s}$
(b) $\dfrac{\partial\langle x_{3}\rangle}{\partial r}$
(c) $\dfrac{\partial\langle x_{3}\rangle}{\partial b}$
Figure 9: Histogram of sensitivities computed using Algorithm 2 (adjoint
sensitivity analysis) starting from 200 random initial conditions.
$T_{A}=10,T_{B}=5$. The red region identifies the $3\sigma$ confidence
interval estimated using finite difference regression.
To further assess the accuracy of our algorithm, which involves finite time
approximations to Equations (9) and (26), we repeated both Algorithm 1 and
Algorithm 2 for 200 times, starting from random initial conditions at $T=-10$.
We keep the statistical averaging time $T_{A}=10$ and the spin up buffer time
$T_{B}=5$. The resulting histogram of sensitivities computed with Algorithm 1
is shown in Figure 8; the histogram of sensitivities computed with Algorithm 2
is shown in Figure 9. The finite difference estimates are also indicated in
these plots.
We observe that our algorithms compute accurate sensitivities most of the
time. However, some of the computed sensitivities seems to have heavy tails in
their distribution. This may be due to behavior of the Lorenz attractor near
the unstable fixed point $(0,0,0)$. Similar heavy tailed distribution has been
observed in other studies of the Lorenz attractor [1]. They found that certain
quantities computed on Lorenz attractor can have unbounded second moment. This
could be the case in our sensitivity estimates. Despite this minor drawback,
the sensitivities computed using our algorithm have good quality. Our
algorithms are much more efficient than existing sensitivity computation
methods using ensemble averages.
## 8 Conclusion
This paper derived a forward algorithm and an adjoint algorithm for computing
sensitivity derivatives in chaotic dynamical systems. Both algorithms
efficiently compute the derivative of statistical quantities $\langle
J\rangle$ to infinitesimal perturbations $\epsilon\,\delta f$ to the dynamics.
The forward algorithm starts from a given perturbation $\delta f$, and
computes a perturbed “shadow” coordinate system $\delta x$, e.g. as shown in
Figure 1. The sensitivity derivatives of multiple statistical quantities to
the given $\delta f$ can be computed from $\delta x$. The adjoint algorithm
starts from a statistical quantity $\langle J\rangle$, and computes an adjoint
vector $\hat{f}$, e.g. as shown in Figure 7. The sensitivity derivative of the
given $\langle J\rangle$ to multiple input perturbations can be computed from
$\hat{f}$.
We demonstrated both the forward and adjoint algorithms on the Lorenz
attractor at standard parameter values. The forward sensitivity analysis
algorithm is used to simultaneously compute $\frac{\partial\langle
x_{1}^{2}\rangle}{\partial r}$, $\frac{\partial\langle
x_{2}^{2}\rangle}{\partial r}$, and $\frac{\partial\langle
x_{3}\rangle}{\partial r}$; the adjoint sensitivity analysis algorithm is used
to simultaneously compute $\frac{\partial\langle x_{3}\rangle}{\partial s}$,
$\frac{\partial\langle x_{3}\rangle}{\partial r}$, and $\frac{\partial\langle
x_{3}\rangle}{\partial b}$. We show that using a single trajectory of length
about $20$, both algorithms can efficiently compute accurate estimates of all
the sensitivity derivatives.
## References
* [1] G. Eyink, T. Haine, and D. Lea. Ruelle’s linear response formula, ensemble adjoint schemes and Lévy flights. Nonlinearity, 17:1867–1889, 2004.
* [2] F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi. Characterizing dynamics with covariant Lyapunov vectors. Physical Review Letters, 99:130601, Sep 2007.
* [3] A. Jameson. Aerodynamic design via control theory. Journal of Scientific Computing, 3:233–260, 1988.
* [4] J. Krakos, Q. Wang, S. Hall, and D. Darmofal. Sensitivity analysis of limit cycle oscillations. Journal of Computational Physics, (0):–, 2012.
* [5] D. Lea, M. Allen, and T. Haine. Sensitivity analysis of the climate of a chaotic system. Tellus, 52A:523–532, 2000.
* [6] D. Ruelle. Differentiation of SRB states. Communications in Mathematical Physics, 187:227–241, 1997.
* [7] D. Ruelle. A review of linear response theory for general differentiable dynamical systems. Nonlinearity, 22(4):855, 2009.
* [8] J.-N. Thepaut and P. Courtier. Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model. Quarterly Journal of the Royal Meteorological Society, 117(502):1225–1254, 1991.
* [9] D. Venditti and D. Darmofal. Grid adaptation for functional outputs: Application to two-dimensional inviscid flow. Journal of Computational Physics, 176:40–69, 2002.
* [10] Q. Wang, P. Moin, and G. Iaccarino. Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation. SIAM Journal on Scientific Computing, 31(4):2549–2567, 2009.
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|
arxiv-papers
| 2012-02-23T16:44:51 |
2024-09-04T02:49:27.769004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qiqi Wang",
"submitter": "Qiqi Wang",
"url": "https://arxiv.org/abs/1202.5229"
}
|
1202.5252
|
# Theory of ZT enhancement in nanocomposite materials.
Paul M. Haney1 1Center for Nanoscale Science and Technology, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899-6202, USA
###### Abstract
The effect of interface scattering on the performance of disordered,
nanocomposite thermoelectric materials is studied theoretically using
effective medium theory and direct numerics. The interfacial electronic and
phonon scattering properties which lead to an enhancement of the
thermoelectric figure of merit $ZT$ are described. Generally, $ZT$ enhancement
requires the interfacial electrical conductance to be within a range of
values, and the thermal phonon conductance to be below a critical value. For
the systems considered, these requirements on interface scattering for $ZT$
enhancement are expressed in terms of the bulk properties of the high-$ZT$
material, and the ratio of the constituent bulk $Z$ values.
## I introduction
There has been considerable recent interest in utilizing nanostructure to
enhance thermoelectric performance review1 . A good thermoelectric has
scattering mechanisms for phonons and electrons with different features:
electron scattering should be strongly energy-dependent, while phonon
scattering should simply be strong. Nanostructured materials may provide a
route to meeting both requirements review2 . Nanostructure can change a
material’s basic electronic properties; for example, the inclusion of
localized impurity states can enhance peaks in the density of states heremans
, leading to stronger energy-dependence of scattering. Alternatively,
nanostructure on a length scale greater than the mean free path does not
change the constituent materials’ basic electronic properties, but scattering
at the interface between material phases changes the bulk composite
properties. A mismatch in material density or sound speed generally decreases
the phonon conductivity through interface scattering, and some interfaces
provide a potential (e.g. a Schottky barrier) which serves as an effective
energy filter, transmitting higher energy electrons, while blocking lower
energy electrons martin . The effect of nanostructuring on the thermoelectric
figure of merit $ZT$ was systematically studied in Ref. (poudeu, ), where $ZT$
enhancement was observed for a range of nanocomposite mixing. Previous
theoretical works have analyzed in detail the electron leonard ; popescu and
phonon scattering chen at specific interfaces. Interfaces that scatter
electrons and phonons as described above may increase $ZT$, and a more
quantitative and general description of the required interfacial properties
for $ZT$ enhancement in composite materials is provided here.
In this work, I employ a linear response model of transport to study
disordered, two-component materials - including the effects of interface
scattering - using effective medium theory and direct numerics. I describe the
specific electronic and phonon scattering properties which lead to $ZT$
enhancement of the composite. The requirements for $ZT$ enhancement are
expressed in terms of the bulk properties of the high $ZT$ constituent, and
the ratio of the constituent bulk $Z$ values. Analysis of these requirements
demonstrates the challenges with the nanostructuring approach for $ZT$
enhancement, but should facilitate an efficient search for materials that
provide higher efficiency.
## II Model
The starting point is the linear response description of transport for the
electrical current $j$ and thermal current $j_{Q}$ footnote :
$\displaystyle j$ $\displaystyle=$ $\displaystyle-\sigma\nabla V+\sigma
S\nabla T~{},$ $\displaystyle j_{Q}$ $\displaystyle=$
$\displaystyle-\left(\kappa^{e}+\kappa^{\gamma}\right)\nabla T+\sigma ST\nabla
V~{},$ (1) $\displaystyle\nabla\cdot j=0;~{}~{}~{}\nabla\cdot j_{Q}=0,$ (2)
where $\sigma$ is the local electrical conductivity,
$\kappa^{e}~{}\left(\kappa^{\gamma}\right)$ is the electron (phonon)
contribution to the total local thermal conductivity $\kappa$
($\kappa=\kappa^{e}+\kappa^{\gamma}$) (all thermal conductivities evaluated
for zero electric field), $S$ is the thermopower, $V$ is the electrostatic
potential, and $T$ is the temperature. I assume that $\sigma$ and $\kappa^{e}$
obey the Wiedamann-Franz law: $\kappa_{e}=\sigma L_{0}T$, where $L_{0}$ is the
Lorenz number. Eq. (1) is valid for length scales greater than a mean free
path, which for relevant materials is on the order of 10 nm.
The figure of merit $ZT$ is:
$\displaystyle ZT$ $\displaystyle=$ $\displaystyle\frac{S^{2}\sigma
T}{\kappa-S^{2}\sigma T}=\frac{N}{1-N+K}~{}.$ (3)
where $K=\left(\kappa^{\gamma}/\kappa^{e}\right)$ and $N=S^{2}\sigma
T/\kappa^{e}$. $N$ can be rewritten in terms of the thermopower only, using
the Wiedamann-Franz law: $N=S^{2}/L_{0}$. $N$ is constrained by the second law
of thermodynamics to be less than 1. Equivalently, $S$ is always less than
$\sqrt{L_{0}}\equiv S_{\rm max}$. An ideal thermoelectric has electronic
properties such that $N\rightarrow 1$, and low phonon thermal conductivity
such that $K\rightarrow 0$.
To study the thermoelectric properties of nanocomposites, I solve Eqs. (1-2)
directly for an ensemble of randomly disordered configurations in 3-d. Fig.
(1) shows a schematic of the method. I use a random site approach in which
sites are randomly assigned as material 1 with probability $c$, or material 2
with probability $(1-c)$. The link between two sites represents a resistor (or
conductance), whose value is set by the adjacent site types. Fig. (1) shows
the conductance values for the three possible cases, along with the associated
probability for each case. In the table, $\sigma_{1}$ ($\sigma_{2}$) is the
bulk conductivity for material 1 (material 2), and
$\sigma_{12}=\sigma_{1}\sigma_{2}/\left(\sigma_{1}+\sigma_{2}\right)$.
$\sigma_{\rm int}$ is the interface conductance, and $\Delta x$ is the grain
size of the material. In the absence of interface scattering, $\Delta x$
factors out of the problem and is not important. In the presence of interface
scattering, $\Delta x$ is a key parameter: a small $\Delta x$ implies a higher
interface density, and a more significant effect of the interface scattering.
It’s important to note that this theory applies only to materials for which
$\Delta x$ is greater than the mean free path. Finally, I note that this
scheme is not unique; Appendix B discusses more complicated schemes, and shows
comparisons between different schemes. The advantage of the simple approach
described here is that it captures the physics of the systems studied well,
and is amenable to analytic treatment with effective medium theory.
Numerically, the system is discretized into $20^{3}$ sites (the results do not
change appreciably when going to $30^{3}$ sites), and the ensemble size is
such that the statistical error of the effective transport parameters is
converged (this typically requires about 30 configurations). The error bars on
the plots of numerical results indicate the statistical uncertainty (one
standard deviation).
Figure 1: (a) depicts a typical random site configuration, where the links
between sites are set by the adjacent site types. (b) shows the values of
conductance for each link type, along with the probability for each link type.
As discussed in the text, $\Delta x$ is the grain size, and $\sigma_{12}$ is
the series conductivity of $\sigma_{1}$ and $\sigma_{2}$.
The transport properties of a multi-component, disordered system can be
approximated with effective medium theory (EMT). As shown in Ref. (cohen, ),
the effective medium electrical conductivity $\sigma$, total thermal
conductivity $\kappa$, and thermopower $S$ satisfy:
$\displaystyle\sum_{i}P_{i}\left(\frac{\sigma_{i}-\sigma}{\sigma_{i}+2\sigma}\right)=\sum_{i}P_{i}\left(\frac{\kappa_{i}-\kappa}{\kappa_{i}+2\kappa}\right)$
$\displaystyle=0~{},$ (4)
$\displaystyle S=$ $\displaystyle
3\kappa\sigma\left(\sum_{i}P_{i}\frac{\sigma_{i}S_{i}}{\left(\kappa_{i}+2\kappa\right)\left(\sigma_{i}+2\sigma\right)}\right)\times$
(5)
$\displaystyle\left(\sum_{i}P_{i}\left[\frac{\sigma_{i}\kappa+\sigma\kappa_{i}+2\sigma\kappa-\sigma_{i}\kappa_{i}}{\left(\kappa_{i}+2\kappa\right)\left(\sigma_{i}+2\sigma\right)}\right]\right)^{-1},$
where $i$ labels the link type, and $P_{i}$ is the probability of a link with
transport parameter values $\sigma_{i},~{}\kappa_{i},~{}S_{i}$.
Each material type (bulk 1, bulk 2, and interface) is described by three
parameters: $\left(\sigma,\kappa^{\gamma},S\right)$, so that 9 material
parameters (plus the concentration $c$) describe a specific two-component
system. This parameter space is too large to describe in its entirety. To make
progress, I generally present results for fixed bulk properties, fixed
interfacial thermopower, and vary the interfacial electrical and phonon
thermal conductivities.
Appendix A discusses the scaling of Eqs. (1) to dimensionless form. The
transport coefficients $\left(\sigma,\kappa^{\gamma},S\right)$ end up being
scaled by those of material 1 (so that
$\sigma_{2}\rightarrow\left(\sigma_{2}/\sigma_{1}\right)$; the interface
values also have a value of $\Delta x$ present in their dimensionless form:
$\sigma_{\rm int}\rightarrow\left(\sigma_{\rm int}/\sigma_{1}\right)\Delta
x$). For ease of presentation, I omit this explicit scaling in most plots; the
axis label $\bar{\sigma}_{\rm int}$ refers to $\left(\sigma_{\rm
int}/\sigma_{1}\right)\Delta x$ , and the label $K_{\rm int}$ refers to
$\left(\kappa_{\rm int}^{\gamma}/\kappa_{1}^{e}\right)\Delta x$.
## III Results
### III.1 single interface
To illustrate the qualitative effect of interface scattering on thermoelectric
performance - and the conditions under which $ZT$ is enhanced - it suffices to
consider the simplest possible system: 1-d transport in a bilayer. This maps
onto a 3-resistor-in-series problem.
Fig. (2) illustrates the role of interface scattering in increasing $ZT$. The
solid red lines denote paths for heat current (top red line for phonons,
bottom red line for electrons), the green dashed line for thermoelectric
charge current. The cylinder size represents the magnitude of the conductance
for a specific transport path. Interface scattering can increase $ZT$ in two
ways: 1. by reducing the phonon thermal conductivity, which leads to
$K\rightarrow 0$, or 2. by increasing the thermopower, which leads to
$N\rightarrow 1$. The interface conductances in Fig. (2) improve $ZT$ in both
ways. In the rest of the paper, I focus on the scenario in which $ZT$ is
enhanced via increased phonon scattering. One reason for this is that
enhancement via increase in thermopower is less well described by this
numerical model. See Appendix B for further discussion on this point.
Figure 2: Depiction of the scattering in a simple bilayer. The phonon thermal
conductance (red cylinder) is detrimental to thermoelectric performance, while
the thermoelectric conductance (green cylinder) is beneficial. Interface
scattering can improve overall thermoelectric performance by improving either
or both of these transport processes, as shown in the figure. Here $\Delta x$
refers to the layer thickness.
Fig. (3) shows the transport properties of the layer for fixed bulk
properties, and varying the interface electrical conductance and phonon
thermal conductance (the interface thermopower is fixed). The results are
intuitively clear: when the interface electrical conductance is small, it
determines the overall layer conductance; conversely when the interface
electrical conductance is large, the interface is transparent and the overall
conductance is set by the bulk. A similar scenario holds for the thermal
conductance (though now the total thermal conductance depends on both
electrical and phonon components). I’ve assumed the thermopower is high for
all constituents, so that its value is relatively unaffected by the interface.
This leads to a $ZT$ value which is enhanced relative to the bulk for a
certain range of interfacial transport parameters.
Figure 3: Transport parameters of bilayer as $\bar{\sigma}_{\rm int}$ and
$K_{\rm int}$ are varied. (Recall the axes labels omit scaling factors. Their
inclusion is via: $\bar{\sigma}_{\rm int}=\left(\sigma_{\rm
int}/\sigma_{1}\right)\Delta x$ and $K_{\rm int}=\left(\kappa_{\rm
int}^{\gamma}/\kappa_{1}^{e}\right)\Delta x$.) The fixed system parameters
are:
$\sigma_{2}=\sigma_{1},~{}\kappa_{2}^{\gamma}=\kappa_{1}^{\gamma},~{}S_{1}=S_{\rm
max},~{}S_{2}=0.9~{}S_{\rm max},~{}S_{\rm int}=0.9~{}S_{\rm max}$ (so that
$Z_{1}T=0.5,~{}Z_{2}T=0.375$).
The region of $ZT$ enhancement is shown again in Fig. (4), where only values
for which $ZT$ is more than 5% greater than the bulk value are shown. For
disordered materials in 2 and 3 dimensions, the phase space of $ZT$
enhancement is very similar to this 3-resistor case, so it’s worth
investigating this simple example fully.
In the limit of low interface conductance (the lower left-hand portion of Fig.
(4)), the interface properties dominate. $ZT$ of the layer is then
approximately that of the interface, so the contours of Fig. (4) in this
region are simply those of Eq. (3), with $N\rightarrow N_{\rm
int},~{}K\rightarrow K_{\rm int}$. The small $\sigma_{\rm int}$ in this region
implies a small electron thermal conductance $\kappa^{e}_{\rm int}$, via the
Wiedamann-Franz law. A large $ZT$ then requires a very small phonon thermal
conductance $\kappa_{\rm int}^{\gamma}$, making $ZT$ enhancement in this
region difficult to achieve. (Recall that for high thermopower, $ZT$ is set by
the ratio of $\kappa^{e}$ to $\kappa^{\gamma}$, see Eq. (3).) In the opposite
limit of high interface conductance ($\bar{\sigma}_{\rm int}\gg 1$), the
interface is transparent electrically and thermally (thermal transparency
follows from Wiedamann-Franz law: $\sigma_{\rm
int}\rightarrow\infty\Rightarrow\kappa^{e}_{\rm int}\rightarrow\infty$). Here
purely bulk properties are recovered, and $ZT$ is not increased.
The crossover between these limits occurs around $\bar{\sigma}_{\rm int}=1$
(or $\sigma_{\rm int}=\sigma_{1}/\Delta x$), when both interface and bulk
properties are important. This is the region most accessible for $ZT$
enhancement. Not surprisingly, $ZT$ is always increased as the phonon thermal
conductance of the interface is decreased. I label the maximum value of
$K_{\rm int}$ for which there is a $ZT$ enhancement of 5% over the bulk value
as $K_{\rm int}^{\rm max}$. (Recall this parameter in full scaled form is
$K_{\rm int}^{\max}=\left(\kappa^{\gamma}_{\rm
int}/\kappa^{e}_{1}\right)\Delta x$.) This is a key parameter because finding
materials with interface scattering that lies below this value is a primary
challenge for using nanocomposites for $ZT$ enhancement review . I label the
associated electrical conductance $\sigma_{\rm int}^{\rm opt}$ (see Fig. (4)).
The next section is largely devoted to describing how the values of $K_{\rm
int}^{\rm max}$ and $\sigma_{\rm int}^{\rm opt}$ depend on the properties of
the bulk material constituents.
Figure 4: Replot of Fig. (3d): $Z$ of the trilayer normalized by $Z$ of the
high-$ZT$ bulk constituent. Only values for which $ZT$ of the trilayer is 5%
greater than the bulk are shown. I use the parameters $K_{\rm int}^{\rm
max},~{}\sigma_{\rm int}^{\rm opt}$ to characterize the phase space of
interface properties that lead to $ZT$ enhancement.
So far I have fixed the interface thermopower $S_{\rm int}$. To illustrate how
the space of $ZT$ enhancement depends on $S_{\rm int}$, I make some slices
through the full parameter space of the interface, shown in Fig. (5). Not
surprisingly, as the thermopower of the interface decreases, the space of $ZT$
enhancement in $\left(\bar{\sigma}_{\rm int},~{}K_{\rm int}\right)$ gets
smaller (i.e, it’s harder to achieve enhancement when the interfacial
thermopower is weak). In the rest of the paper, I fix $S_{\rm int}=S_{\rm
max}$ (or $N_{\rm int}=1$). It should be kept in mind that an interface with
smaller $S$ will have more stringent requirements on phonon thermal
conductance (i.e a smaller $K_{\rm int}^{\rm max}$) for $ZT$ enhancement.
Figure 5: Region of $ZT$ enhancement for the full parameter space of the
interface. Interfaces with high $N$ (high thermopower) are advantageous for
$ZT$ enhancement. The same bulk parameters are used as in Fig. (3).
### III.2 3-d disordered material
Moving to disordered materials in 3-d introduces a new system parameter - the
concentration of one material with respect to the other. Fig. (6) shows the
bulk transport parameters as a function of concentration calculated
numerically and with effective medium theory. The interface scattering leads
to a decrease in electric and thermal conductivity relative to the bulk values
of the constituents. $ZT$ is enhanced relative to the bulk value for a range
of concentrations, shown in Fig. (6d). Note there is excellent agreement
between effective medium theory and the numeric results; most of the results
presented in the rest of the paper are derived from effective medium theory,
except where explicitly noted.
Figure 6: The transport parameters of a two-component disordered medium as a
function of relative concentration. System parameters are:
$\sigma_{2}=1.1~{}\sigma_{1},~{}\kappa_{1}^{\gamma}=2~{}\kappa_{1}^{e},~{}\kappa_{2}^{\gamma}=2.3~{}\kappa_{1}^{e},S_{1}=S_{\rm
max},S_{2}=0.77~{}S_{\rm max},~{}\sigma_{\rm int}=0.24~{}\sigma_{1}/\Delta
x,~{}\kappa_{\rm int}^{\gamma}=0.48~{}\kappa_{1}^{e}/\Delta x,~{}S_{\rm
int}=0.97~{}S_{\rm max}$. (a) and (b) show a decrease in the conductance due
to interface scattering. (d) shows an enhancement in $ZT$.
The rest of this section describes how the interface properties needed for
$ZT$-enhancement depend on the constituent bulk materials. I show that the
phase space for $ZT$ enhancement essentially depends only on a small number of
parameters of the constituent materials. This is a useful simplification. It
enables a precise answer to the question: “given a high-$ZT$ material with
thermopower $S_{1}$ and phonon thermal conductivity $\kappa_{1}^{\gamma}$, and
another material with $Z$ value $Z_{2}$, what are the interface scattering
properties that are required to observe $ZT$ enhancement of the composite
material?”. I describe the required interface properties using the parameters
$(K_{\rm int}^{\rm max},\sigma_{\rm int}^{\rm opt})$ introduced in the
previous section and in Fig. (4).
For each set of bulk and interface properties, I vary the concentration and
determine the maximum possible $ZT$ \- this maximum value is what is reported
in the following results. In all of these results, I assume that the bulk
thermopower of the high-$ZT$ constituent is large ($S_{1}=S_{\rm max}$), so
that $ZT$ enhancement is a consequence of reducing phonon thermal
conductivity.
Fig. (7) is an illustration of how $K_{\rm int}^{\rm max}$ characterizes the
phase space of $ZT$ enhancement as bulk materials change. Fig. (7a) shows how
the region of enhancement changes as the $ZT$ value of one bulk material
component gets smaller. As one component’s $ZT$ value decreases, it’s more
difficult to achieve $ZT$ enhancement of the composite via interface
scattering. Fig. (7b) shows how this behavior is translated into the parameter
$K_{\rm int}^{\rm max}$. In this example, $ZT$ of material 2 is degraded due
to a higher phonon thermal conductivity of material 2.
Figure 7: (a) shows regions of ZT enhancement with respect to interface
properties for $Z_{1}T=0.5$ ($N_{1}=1,~{}K_{1}=2$), $Z_{2}T$ is decreased by
increasing $K_{2}$, with values (2.67, 4.0, 8.0) ($N_{2}=1$ for all cases).
(b) shows how this phase plot is translated to a plot of $K_{\rm int}^{\rm
max}$ versus $Z_{2}T$.
Fig. (8) shows that for a fixed high-$ZT$ constituent, $K_{\rm int}^{\rm max}$
essentially only depends on $Z_{2}$. In Fig. (8a) I show plots of $K_{\rm
int}^{\rm max}$ as $Z_{2}T$ is degraded in three different ways: with a “bad”
$K_{2}$ (or high phonon thermal conductivity), a “bad” $S_{2}$ (or low
thermopower), and a combination of both. Fig. (8b) shows the same thing for a
different high-$ZT$ material. Fig. (8b) also shows numeric results (with a
“bad $K_{2}$” scenario), which confirm that the effective medium theory and
numerical results are very similar. What’s important is that $K_{\rm int}^{\rm
max}$ is quite insensitive to how the low-$ZT$ material is deficient. The bulk
$ZT$ values of the constituent alone determines the required interfacial
phonon scattering for $ZT$ enhancementfootnote1 .
Figure 8: In (a), $ZT_{1}=1$ ($N_{1}=1,~{}K_{1}=1$), and $ZT_{2}$ is reduced
in three ways: by decreasing $N_{2}$, increasing $K_{2}$, or a combination of
both. (b) shows the same plot, with $ZT_{1}=0.5$ ($N_{1}=1,~{}K_{1}=2$), and
also shows results obtained numerically.
Fig. (9a) shows the result of plotting all the curves of Fig. (8) together,
normalized by their maximum value. Again a remarkable and useful
simplification takes place, where the curves collapse on an approximately
“universal” curve. The vertical spread of this normalized curve shows the
spread in values for the different curves of Fig. (8).
The right hand-side of the normalized curve of Fig. (9a), where $Z_{2}=Z_{1}$,
corresponds to a system with identical bulk phases, with interface scattering
between the identical grains. The value of $K_{\rm int}^{\rm max}$ for such a
system sets the overall normalization for plots like Fig. (8). In Fig. (8b), I
plot the value of this normalization as a function of $K_{\rm bulk}$ and
$N_{\rm bulk}$. The two parts of Fig. (9) enable an estimate for the required
interface phonon thermal conductance for $ZT$ enhancement.
As an example, let the high-$ZT$ constituent have $N_{1}=0.8$, $K_{1}=2$ (this
implies $Z_{1}T=0.36$); this is shown as a white dot in Fig. (9). Fig. (9b)
shows the normalization for the $K_{\rm int}^{\rm max}$ curve is 4. Now let
the low-$ZT$ material have $Z_{2}T=0.27$, so that $Z_{2}/Z_{1}=0.75$. Using
Fig. (9a), I conclude $K_{\rm int}^{\rm max}$ for this material combination is
$0.25\times 4=1$ (in dimensionful terms, $K_{\rm int}^{\rm
max}=\left(\kappa_{\rm int}^{\gamma}/\kappa_{1}^{e}\right)\Delta x=1$). This
means that $ZT$ enhancement requires thermal transport parameters and grain
size such that $\left(\kappa_{\rm int}^{\gamma}/\kappa_{1}^{e}\right)\Delta
x<1$.
Figure 9: (a) shows the range of $K_{\rm int}^{\rm max}$ values as a function
of $Z_{2}/Z_{1}$ (from Fig. 8), when normalized by their maximum value. (b)
shows this overall normalization constant as a function of the thermoelectric
parameters $N$ and $K$ of the high-$ZT$ bulk material. The white dot refers to
an example described in the text.
I next go through a similar description of how $\sigma_{\rm int}^{\rm opt}$
depends on bulk material parameters. Fig. (10) shows $\sigma_{\rm int}^{\rm
opt}$ as a function of the $Z_{2}/Z_{1}$, for two different high-$ZT$
constituents. I again find that, for fixed high-$ZT$ material, $\sigma_{\rm
int}^{\rm opt}$ essentially only depends on $Z_{2}$. Fig. (11a) shows the
result of plotting all the curves of Fig. (10) by their maximum value. Again,
I find that $\sigma_{\rm int}^{\rm opt}$ as a function of $Z_{2}/Z_{1}$ is an
approximately “universal” curve. Fig. (11b) shows the normalization value of
this “universal” curve as a function of $N$ and $K$ of the high-$ZT$
constituent. These plots again enable an estimate of $\sigma_{\rm int}^{\rm
opt}$ in terms of just a few bulk material parameters.
Returning to the example before (where we assumed a high $ZT$ material with
parameters $N_{1}=0.8,~{}K_{1}=2$, and a low $ZT$ material with
$Z_{2}/Z_{1}=0.75$), Fig. (11b) shows the normalization constant of about 5,
which is used with Fig. (11a) to infer $\sigma_{\rm int}^{\rm opt}=5\times
0.25=1.25$. In other words, attaining $ZT$ enhancement through interfacial
scattering is most easily accessible with a combination of electrical
conductivity values and grain size such that $\left(\sigma_{\rm
int}/\sigma_{1}\right)\Delta x=1.25$.
$\sigma_{\rm int}^{\rm opt}$ is an important constraint on the interface; even
if an interface blocks phonons effectively, if it also blocks electrons too
much (i.e has too low $\bar{\sigma}_{\rm int}$), or is transparent to
electrons (too high $\bar{\sigma}_{\rm int}$), then it does not lead to
overall $ZT$ enhancement. The reason for this is the same as in the simple
3-resistor-in-series case, described earlier. Note that the value of the
overall normalization for $\sigma_{\rm int}^{\rm opt}$ is fairly constant over
the range of bulk material parameters. Generally, $ZT$ enhancement requires an
interface conductance on the order of the bulk conductivity divided by the
grain size.
Figure 10: In (a), $ZT_{1}=1$ ($N_{1}=1,~{}K_{1}=1$), and $ZT_{2}$ is reduced
in three ways: by decreasing $N_{2}$, increasing $K_{2}$, or a combination of
both. (b) shows the same plot, with $ZT_{1}=0.5$ ($N_{1}=1,~{}K_{1}=2$), and
also shows results obtained numerically. Figure 11: (a) shows the range of
$\sigma_{\rm int}^{\rm opt}$ values as a function of $Z_{2}/Z_{1}$ (from Fig.
10), when normalized by their maximum value. (b) shows this overall
normalization constant as a function of the thermoelectric parameters $N$ and
$K$ of the high-$ZT$ bulk material.
Figs. (9) and (11) represent the main results of the paper. They provide a
blueprint to choosing material properties such that a two-component composite
results in $ZT$ enhancement. An important aspect of Fig. (9a) is the rapid
decrease of $K_{\rm int}^{\rm max}$ as one of the material’s $ZT$ value
decreases. This means interfacial phonon scattering can most easily enhance
$ZT$ when the two materials have similar $ZT$ values. This poses a key
materials science challenge in pursuing this technique for $ZT$ enhancement:
often materials with similar (high) $ZT$ values have similar density (i.e.
both composed of heavy atoms); however, interfacial phonon scattering is
usually strongest between materials with very dissimilar density and speed of
sound review .
For a rough estimate of required material values, the above analysis shows
$ZT$ enhancement via interface phonon scattering requires material parameters
which satisfy an inequality on the order of $\kappa_{\rm
int}^{\gamma}<\kappa_{\rm bulk}^{e}/\Delta x$. A typical thermoelectric has
$\kappa_{\rm bulk}^{e}=1~{}{\rm W/\left(m\cdot K\right)}$. Assuming a grain
size of $10~{}{\rm nm}$, the interface phonon conductance must be less than
$10^{8}~{}{\rm W/\left(m^{2}\cdot K\right)}$ for $ZT$ enhancement. This value
is certainly attainable for some material combinations review , though
obtaining this value of $\kappa_{\rm int}^{\gamma}$ for two materials with
high $ZT$ values (and low $\kappa_{\rm bulk}^{\gamma}$ values) is likely to be
a challenge.
### III.3 Dimension and concentration dependence
Here I briefly compare the results obtained for the space of $ZT$ enhancement
in 1-d, 2-d, and 3-d. The comparison is shown in Fig. (12). The interface
parameter space for enhancement is very similar in all cases, but that the
enhancement is reduced in higher dimensions. This is because some portion of
interface scattering in higher dimensions occurs in directions orthogonal to
the transport direction. This scattering is not effective in reducing the
phonon thermal conductivity along the overall direction of the temperature
gradient, and therefore does not aid in increasing $ZT$. Also shown in Fig.
(12) is the concentration in 2-d and 3-d for which the maximum $ZT$ occurs.
This value depends on the specific material parameters chosen. For example. if
the two bulk materials are equivalent, the optimum enhancement is always at
$c=0.5$. As the two bulk materials properties deviate, the optimum
concentration moves away from $0.5$ \- it’s more advantageous to have a higher
concentration of the high-$ZT$ material. At the edge of the phase space of
enhancement, the optimum concentration is such that the composite is mostly
high-$ZT$ bulk.
Figure 12: (a-c) show the region of $ZT$ enhancement in 1, 2, and 3 dimensions
(1d refers to the bilayer case). Below the 2-d and 3-d cases, the
concentration with the maximum $ZT$ is shown (concentration refers to
percentage of material 1). Fixed system parameters in all cases are:
$N_{1}=N_{2}=1$, $K_{1}=K_{2}=2$, $S_{1}=S_{\rm max},~{}S_{2}=0.9S_{\rm
max},~{}S_{\rm int}=0.9S_{\rm max}$.
## IV Conclusion
In this work I described the conditions under which the formation of a
nanocomposite material results in enhancement of $ZT$ over the constituent
bulk values. $ZT$ enhancement is the result of electronic and phonon
scattering at the interface between different materials, and occurs over a
range of $\bar{\sigma}_{\rm int}$, and for sufficiently low $K_{\rm int}$.
Using effective medium theory and numerical simulation, I give a prescription
for the required value of interface conductances for $ZT$ enhancement, as a
function of the bulk $N$ and $K$ of the high $ZT$ material, and the ratio of
the bulk $Z$ values. The results presented in the 3-d disordered case are for
$S_{\rm int}=S_{\rm max}$, and therefore represent the most optimistic
requirements on $K_{\rm int}$ and $\bar{\sigma}_{\rm int}$.
I emphasize that this theory applies for composites with phase separation
greater than the mean free path of electrons and phonons. It’s therefore most
applicable to nanostructuring techniques such as ball milling and hot
pressing. These techniques have shown the potential for $ZT$ enhancement
martin2 ; poudel . Although not emphasized in this work, scattering at
interfaces can also improve efficiency via improved energy filtering,
resulting in enhanced power factor. The material constraints to achieve $ZT$
enhancement are obviously challenging, but the precise specification of these
constraints should aid in the search for the best material choices for more
efficient thermoelectrics.
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* (13) One reason for the insensitivity of $K_{\rm int}^{\rm max}$ to how $Z_{2}T$ is decreased is the fact that the absolute differences in parameters along different paths of $Z_{2}T$ degradation are relatively small. For example, for $Z_{2}T=0.375$, the “bad $N_{2}$” case has $\left(\sigma_{2},\kappa_{2},S_{2}\right)=(1,3,0.9)$, while the “bad $K_{2}$” has $\left(\sigma_{2},\kappa_{2},S_{2}\right)=(1,3.6,1)$. Generally $K_{\rm int}^{\rm max}$ is slightly greater for the “bad $K_{2}$” case. This makes sense, as here the deficiency of material 2 (high conduction of phonons) is more directly addressed by the interface scattering.
* (14) U. Sivan and Y. Imry, Phys. Rev. B 33, 551 (1986).
* (15) R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O Quinn, Nature 413, 597 (2001).
* (16) D. G. Cahilla, Wayne K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot. App. Phys. Rev. 93, 793 (2003).
* (17) J. Martin, G. S. Nolas, W. Zhang, and L. Chen, App. Phys. Lett. 90, 222112 (2007).
* (18) B. Poudel, Q. Hao, Y. Ma, Y. C. Lan, A. Minnich, B. Yu, X. Yan, D. Wang, A. Muto, D. Vashaee, X. Y. Chen, X. Y, J. M. Liu, M. S. Dresselhaus, G. Chen, Z. Ren, Z. Science 320, 634 (2008).
## V Appendix
### V.1 Dimensionless variables
To write Eqs. (1) in dimensionless form, I introduce the following variables.
$\displaystyle\overline{x}=\frac{x}{L};$
$\displaystyle\overline{\nabla}=L\nabla;$ (6)
$\displaystyle\overline{T}=\frac{T}{T_{0}};$
$\displaystyle~{}~{}\overline{V}=V\left(\frac{S_{1}\sigma_{1}}{\kappa_{1}^{e}}\right);$
(7) $\displaystyle\overline{j}=j\left(\frac{L}{S_{1}\sigma_{1}T_{0}}\right);$
$\displaystyle~{}~{}\overline{j_{Q}}=j_{Q}\left(\frac{L}{\kappa_{1}^{e}T_{0}}\right),$
(8)
where $L$ is the length of the sample in the transport direction, $T_{0}$ is a
fixed reference temperature. This leads to the dimensionless equations:
$\displaystyle\overline{j}$ $\displaystyle=$
$\displaystyle-\frac{1}{N_{1}}\left(\frac{\sigma_{i}}{\sigma_{1}}\right)\overline{\nabla}~{}\overline{V}+\left(\frac{S_{i}\sigma_{i}}{S_{1}\sigma_{1}}\right)\overline{\nabla}~{}\overline{T}$
$\displaystyle\overline{j_{Q}}$ $\displaystyle=$
$\displaystyle-\left(\frac{\kappa_{i}}{\kappa_{1}}\right)\overline{\nabla}~{}\overline{T}+\left(\frac{S_{i}\sigma_{i}}{S_{1}\sigma_{1}}\right)\overline{T}~{}\overline{\nabla}~{}\overline{V},$
(9)
where $N_{1}=\left(\frac{S_{1}^{2}\sigma_{1}T_{0}}{\kappa_{1}^{e}}\right)$.
The prefactor $1/N_{1}$ of the dimensionless conductivity results in an
“effective” conductivity
$\frac{1}{N_{1}}\left(\frac{\sigma_{i}}{\sigma_{1}}\right)$ that is used when
solving Eqs. (V.1). Extracting an effective conductivity from evaluating the
charge current response to an electric potential requires accounting for
$N_{1}$: $\sigma=N_{1}\left(\frac{j}{\Delta V}\right)$, where $\Delta V$ is
the applied potential difference.
### V.2 Discretization scheme
The inclusion of interface scattering complicates the scheme used to
discretize Eqs. (1-2), which we discuss more fully here. The relevant question
is: given a continuous distribution of material, what discrete set of points
should we choose to represent the potential and temperature fields? The answer
depends on the spatial variation of the fields; to accurately represent the
continuous fields requires a more dense mesh near areas of rapid variation in
potential and temperature. For example, small interface electrical conductance
(compared to the bulk conductivity divided by grain length) implies a sharp
potential drop across an interface. This suggests a discretization scheme as
shown in Fig. (13a). The conductance on the link separating two plaquettes is
set to $\sigma_{\rm int}$ for plaquettes with different identities, and set to
$\infty$ otherwise. I call this discretization scheme the “edge scheme”. In
two dimensions the sampling may be chosen as shown in Fig. (14a).
Figure 13: Two different discretization schemes represented in 1-d. In the
“center scheme” (a), the interface conductance is partially combined with bulk
conductances, and the potential is evaluated at the center of each plaquette.
In the “edge scheme” (b), the conductances are separate and the potential is
evaluated at both edges of the plaquettes. Figure 14: Implementation of (a)
center and (b) edge schemes in 2-d.
In the body of the paper I use a simpler scheme, depicted in Fig. (13b). Here
the fields are evaluated at the center of the plaquette, and the interface
conductance is put in series with the adjacent bulk conductance a priori. I
call this the “center scheme”. This results in a less dense sampling, and is
therefore not as accurate as the edge scheme. However, as mentioned in the
body of the paper, this scheme is easily adopted to effective medium theory,
which is very powerful and much more convenient than direct numerics. To
compare the two schemes, I consider a two-component mixture in two dimensions.
Fig. (15) shows the $ZT$ value of the composite as I vary the interface
electrical conductance $\bar{\sigma}_{\rm int}$ and phonon thermal conductance
$K_{\rm int}$. In this case, I let $S_{1}=S_{2}=S_{\rm max}$, so that $ZT$
enhancement is the result of increased phonon scattering. Both schemes give
similar results, although the edge scheme shows slightly greater $ZT$
enhancement. In the region of $ZT$ enhancement, the interfacial conductance is
not appreciably larger than the bulk, so that the temperature and voltage
drops aren’t strongly localized at the interface. This enables the center
scheme to represent the fields reasonably well. Moreover the enhancement is
due to blocking phonons, or a small $\kappa_{\rm eff}^{\gamma}$. Adding the
large bulk $\kappa^{\gamma}_{\rm bulk}$ with the small $\kappa_{\rm
int}^{\gamma}$ in series a priori results in an effective $\kappa^{\gamma}$
that’s still small. (For conductors in series, the smallest conductance
dominates). I therefore conclude that the approach adopted in the paper works
well to describe $ZT$ enhancement via phonon scattering at the interface.
Figure 15: $ZT$ of the composite versus interface $\bar{\sigma}_{\rm int}$ and
$K_{\rm int}$. The system parameters are: $40{\rm x}40$ plaquettes in 2-d,
$N_{1}=N_{2}=1$, $K_{1}=K_{2}=2$ (so that $Z_{1}T=Z_{2}T=0.5$), $S_{\rm
int}=0.9~{}S_{\rm max}$. Interface scattering of phonons reduce $K$ of the
composite, resulting in an enhancement of $ZT$.
Fig. (16) shows $ZT$ as a function of interface properties for the two schemes
when the bulk thermopower is small ($S_{1}=S_{2}=0.5S_{\rm max}$). The role of
the interface in $ZT$ enhancement is to provide energy filtering of the
electrons, increasing $S$ of the composite. The two schemes’ results are now
rather different - the center scheme underestimates the $ZT$ enhancement by a
notable margin. This is because energy filtering is accomplished with a sharp
temperature drop across the interface, which is not represented in the center
scheme. Moreover, adding the low bulk value of $\left(S\sigma\right)_{\rm
bulk}$ in series with the high interface $\left(S\sigma\right)_{\rm int}$ a
priori leads to a small effective $\left(S\sigma\right)$ (again, when adding
these “conductances” in series, the smallest one dominates); the potential
increase in $S\sigma$ is partially nullified by the model construction.
Figure 16: $ZT$ of the composite versus interface $\bar{\sigma}_{\rm int}$ and
$K_{\rm int}$. The system parameters are: $40{\rm x}40$ plaquettes in 2-d,
$N_{1}=N_{2}=0.5$, $K_{1}=K_{2}=0.5$ (so that $Z_{1}T=Z_{2}T=0.5$), $S_{\rm
int}=0.9~{}S_{\rm max}$. These parameters lead to the same $ZT_{\rm int}$ as
in Fig. (15). Further analysis of the data shows that the center scheme
underestimates the increase in $N$ (equivalently $S$) of the composite,
resulting in a smaller $ZT$ relative to the edge scheme.
|
arxiv-papers
| 2012-02-23T18:15:52 |
2024-09-04T02:49:27.778862
|
{
"license": "Public Domain",
"authors": "Paul M. Haney",
"submitter": "Paul Haney Mr.",
"url": "https://arxiv.org/abs/1202.5252"
}
|
1202.5334
|
# An allocation scheme for estimating the reliability of a parallel-series
system
Z. BENKAMRA$\ {}^{1}$, M. TERBECHE$\ {}^{2}$ and M. TLEMCANI$\ {}^{1,*}$
$\ {}^{1}$ University Mohamed Boudiaf, L.A.A.R, Algeria
$\ {}^{2}$ University of Oran, Algeria
$\ {}^{*}$ mounir.tlemcani@univ-pau.fr (M.Tlemcani)
###### Abstract
We give a hybrid two stage design which can be useful to estimate the
reliability of a parallel–series and/or by duality a series–parallel system,
when the component reliabilities are unknown as well as the total numbers of
units allowed to be tested in each subsystem. When a total sample size is
fixed large, asymptotic optimality is proved systematically and validated via
Monte Carlo simulation.
Keywords. Asymptotic optimality; Hybrid; Reliability; Parallel-series; Two
stage design.
## 1 Introduction
In reliability engineering two crucial objectives are considered: (1) to
maximize an estimate of system reliability and (2) to minimize the variance of
the reliability estimate. Because system designers and users are risk-averse,
they generally prefer the second objective which leads to a system design with
a slightly lower reliability estimate but a lower variance of that estimate ,
(eg, [4]). It provides decision makers efficient rules compared to other
designs which have a higher system reliability estimate, but with a high
variability of that estimate. In the case of parallel–series and/or by duality
series–parallel systems, the variance of the reliability estimate can be
lowered by allocation of a fixed sample size (the number of observations or
units tested in the system), while reliability estimate is obtained by testing
components, see Berry [3]. Allocation schemes for estimation with cost, see
for example [3, 5, 7, 8, 10, 11], lead generally to a discrete optimization
problem which can be solved sequentially using adaptive designs in a fixed or
a Bayesian framework. Based on a decision theoretic approach, the authors seek
to minimize either the variance or the Bayes risk associated to a squared
error loss function. The problem of optimal reliability estimation reduces to
a problem of optimal allocation of the sample sizes between Bernoulli
populations. Such problems can be solved via dynamic programming but this
technique becomes costly and intractable for complex systems. In the case of a
two components series or parallel system, optimal procedures can be obtained
and solved analytically when the coefficients of variation of the associated
Bernoulli populations are known, cf. eg, [6, 8]. Unfortunately, the
coefficients of variation are not known in practice since they depend
themselves on the unknown components reliabilities of the system. In [9], the
author has defined a sequential allocation scheme in the case of a series
system and has shown its first order asymptotic optimality for large sample
sizes with comparison to the balanced scheme. In [1], a reliability sequential
schemes (R-SS) was applied successfully to parallel–series systems, when the
total number of units to be tested in each subsystem was fixed. Recently, in
[2], a two stage design for the same purpose was presented and shown to be
asymptotically optimal when the subsystems sample sizes are fixed and large at
the same order of the total sample size of the system. The problem considered
in this paper is useful to estimate the reliability of a parallel-series
and/or by duality a series-parallel system, when the components reliabilities
are unknown as well as the total numbers of units allowed to be tested in each
subsystem. This work improves the results in [2] by developing a hybrid two
stage design to get a dynamic allocation between the sample sizes allowed for
subsystems and those allowed for their components. For example, consider a
parallel system of four components (1),(2),(3) and (4), with reliabilities
0.05, 0.1, 0.95 and 0.99, respectively, under the constraint that the total
number of observations allowed is $T=100$. Then, the sequential scheme given
in [1] suggests to test, respectively, 10, 10, 28 and 52 units and produces a
variance of the system reliability estimate equal $10^{-7}$, approximately.
This is visibly better, compared to the balanced scheme which takes an
allocation equal 25 in each component and produces a variance ten times
greater then the former. The hybrid sequential scheme proposed in this paper
is a tool to solve the same problem when the components are replaced by
subsystems. More precisely, it combines the schemes developed for parallel
and/or series systems in order to obtain approximately the best allocation at
subsystems level as well as at components level.
In section 2, definitions and preliminary results are presented accompanied by
the proper two stage design for a parallel subsystem just as was defined in
[2] and its asymptotic optimality is proved for a fixed and large sample size.
In section 3, a parallel-series system is considered and it is shown that the
variance of its reliability estimate has a lower bound independent of
allocation. This leads, in section 4, to the main result of this paper which
lies in the hybrid two stage algorithm and its asymptotic optimality for a
fixed and large sample size allowed for the system. In section 5, the results
are validated via Monte Carlo simulation and it is shown that our algorithm
leads asymptotically to the best allocation scheme to reach the lower bound of
the variance of the reliability estimate. The last section is reserved for
conclusion and remarks.
## 2 Preliminary results
Consider a system $S$ of $n$ subsystems $S_{1},S_{2},\ldots,S_{n}$ connected
in series, each subsystem $S_{j}$ contains $n_{j}$ components
$S_{1j},S_{2j},\ldots,S_{n_{j}j}$ connected in parallel. The system should be
referred as parallel-series system. Assume s-independence within and across
populations, then the system reliability is
$R=\prod\limits_{j=1}^{n}R_{j},$ (1)
where
$R_{j}=1-\prod\limits_{i=1}^{n_{j}}\left(1-R_{ij}\right)$
is the reliability of the parallel subsystem $S_{j}$ and $R_{ij}$ the
reliability of component $S_{ij}$. An estimator of $R$ is assumed to be the
product of sample reliabilities
$\hat{R}=\prod\limits_{j=1}^{n}\hat{R}_{j},$
where
$\hat{R}_{j}=1-\prod\limits_{i=1}^{n_{j}}\left(1-\hat{R}_{ij}\right)$
and $\hat{R}_{ij}$ is the sample mean of functioning units in component
$S_{ij}$,
$\hat{R}_{ij}=\frac{\sum\limits_{l=1}^{M_{ij}}X_{ij}^{(l)}}{M_{ij}},$
$\hat{R}_{ij}$ is used to estimate $R_{ij}$ where $M_{ij}$ is the sample size
and $X_{ij}^{(l)}$ is the binary outcome of the unit $l$ in component
$S_{ij}$. It should be pointed that a unit is not necessarily a physical
object in a component, but it represents just a Bernoulli observation of the
functioning/failure state of that component. Hence, for each subsystem
$S_{j}$, one must allocate
$T_{j}=\sum\limits_{i=1}^{n_{j}}M_{ij}$
units such that the estimated reliability of the system is based on a total
sample size
$T=\sum\limits_{j=1}^{n}T_{j}$
As in the series case, with the help of s-independence and the fact that a
sample mean is an unbiased estimator of a Bernoulli parameter, see [1, 2, 4],
the variance of the estimated reliability $\hat{R}$ incurred by any allocation
scheme can be obtained,
$Var\left\\{\hat{R}\right\\}=\prod\limits_{j=1}^{n}\left(Var\left(\hat{R}_{j}\right)+R_{j}^{2}\right)-\prod\limits_{j=1}^{n}R_{j}^{2},$
(2)
where
$Var\left\\{\hat{R}_{j}\right\\}=\left(1-R_{j}\right)^{2}\left[\prod\limits_{i=1}^{n_{j}}\left(1+\frac{c_{ij}^{-2}}{M_{ij}}\right)-1\right]$
(3)
is given as a function of the allocation numbers $M_{ij}$ and the coefficients
of variation of Bernoulli populations
$c_{ij}=\sqrt{1/R_{ij}-1}$
We have found convenient to work with the equivalent expression of (3),
$Var\left\\{\hat{R}_{j}\right\\}=\left(1-R_{j}\right)^{2}\left[\sum\limits_{i=1}^{n_{j}}\frac{c_{ij}^{-2}}{M_{ij}}+F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)\right],$
where
$F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)$
is a sum over all the products of at least two of its arguments.
The problem is to estimate $R$ when components reliabilities are unknowns and
a total number of $T$ units must be tested in the system at components level.
The aim is to minimize the variance of $\hat{R}$. Hence, the problem can be
addressed by developing allocation schemes to select $M_{ij}$, the numbers of
units to be tested in each component $i$ in the subsystem $j$, under the
constraint
$\sum_{j=1}^{n}\sum_{i=1}^{n_{j}}M_{ij}=T,$ (4)
such that the variance of $\hat{R}$ is as small as possible. Reliability
sequential schemes (R-SS) exist for the series, parallel or parallel-series
configurations when the sample sizes $T_{j}$ of the subsystems are fixed.
Therefore, one can fully optimize the variance of $\hat{R}$ just by applying
the (R-SS) to find the best partition $T_{1},T_{2},...,T_{n}$ of $T$.
Unfortunately, a full sequential design can not be used in practice for large
systems since the number of operations will growth dramatically. For this
reason, we reasonably propose a hybrid two stage design which is shown to be
asymptotically optimal when $T$ is large.
### 2.1 Lower bound for the variance of the estimated reliability of the
parallel subsystem $S_{j}$
For the asymptotic optimization of the variance of the estimated
reliabilities, we make use of the well-known Lagrange’s identity which can be
written in the form:
Let $a_{i}>0$, $N_{i}>0,$ for $i=1,...,k$ and $N=N_{1}+\cdots+N_{k}$, then the
following identity holds.
$\sum\limits_{i=1}^{k}\frac{a_{i}}{N_{i}}=N^{-1}\left[\left(\sum\limits_{i=1}^{k}\sqrt{a_{i}}\right)^{2}+\sum\limits_{i=1}^{k-1}\sum\limits_{j=i+1}^{k}\frac{\left(N_{i}\sqrt{a_{j}}-N_{j}\sqrt{a_{i}}\right)^{2}}{N_{i}N_{j}}\right]$
(5)
###### Proposition 1.
Denote by
$Q_{j}=\left(1-R_{j}\right)^{2}T_{j}^{-1}\left(\sum_{i=1}^{n_{j}}c_{ij}^{-1}\right)^{2}$
(6)
then
$Var\left\\{\hat{R}_{j}\right\\}\geq Q_{j}$
###### Proof.
The proof is a direct consequence of the previous identity (5). Indeed
$\displaystyle
Var\left\\{\hat{R}_{j}\right\\}=\left(1-R_{j}\right)^{2}T_{j}^{-1}\left(\sum_{i=1}^{n_{j}}c_{ij}^{-1}\right)^{2}$
$\displaystyle+T_{j}^{-1}\left(1-R_{j}\right)^{2}\sum_{i=1}^{n_{j}-1}\sum_{k=i+1}^{n_{j}}\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}$
$\displaystyle+\left(1-R_{j}\right)^{2}F\left(\frac{c_{1j}^{-2}}{M_{1j}},\frac{c_{2j}^{-2}}{M_{2j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)$
(7)
∎
### 2.2 The two stage design for the parallel subsystem $S_{j}$
Following the expansion (7) and since $F$ contains second order terms (see
later), one gives interest to the numbers $M_{ij}$ which minimize the
expression
$T_{j}^{-1}\sum_{i=1}^{n_{j}-1}\sum_{k=i+1}^{n_{j}}\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}$
Thus $M_{ij}$ must verify for $i=1,...,n_{j}$
$M_{ij}c_{kj}^{-1}=M_{kj}c_{ij}^{-1}$
which implies that
$M_{ij}=T_{j}\frac{c_{ij}^{-1}}{\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}}$ (8)
If one assumes that $T_{j}$ is fixed then a proper two stage scheme can be
used to determine $M_{ij}$, just as was defined in [2], as follows:
Choose $L_{j}$ as a function of $T_{j}$ such that:
1. (i)
$L_{j}$ must be large if $T_{j}$ is large,
2. (ii)
$L_{j}\leq\frac{T_{j}}{n_{j}},$
3. (iii)
$\lim\limits_{T_{j}\rightarrow\infty}\frac{L_{j}}{T_{j}}=0$.
One can take for example $L_{j}=\left[\sqrt{T_{j}}\right]$, where
$\left[.\right]$ denotes the integer part.
Stage 1.
Sample $L_{j}$ units from each component $i$ in the subsystem $j$, estimate
$c_{ij}$ by its maximum likelihood estimator (M.L.E)
$\hat{c}_{ij}=\sqrt{\frac{L_{j}}{\sum\limits_{l=1}^{L_{j}}X_{ij}^{(l)}}-1}$
and define the predictor, according to (8),
$\hat{M}_{ij}=\left[T_{j}\frac{\hat{c}_{ij}^{-1}}{\sum\limits_{k=1}^{n_{j}}\hat{c}_{kj}^{-1}}\right],~{}i=1,\ldots,n_{j}-1$
Stage 2.
Sample $T_{j}-n_{j}L_{j}$ units for which $M_{ij}-L_{j}$ are units from
component $i$ in the subsystem $j$ where $M_{ij}$ is the corrector of
$\hat{M}_{ij}$ defined by
$\displaystyle M_{ij}$ $\displaystyle=$
$\displaystyle\max\left\\{L_{j},\hat{M}_{ij}\right\\},~{}i=1,\ldots,n_{j}-1,$
$\displaystyle M_{n_{j}j}$ $\displaystyle=$ $\displaystyle
T_{j}-\sum\limits_{k=1}^{n_{j}-1}M_{kj}$
###### Theorem 1.
Choosing the $M_{ij}$ according to the previous two stage sampling scheme, one
obtains
$\lim_{T_{j}\rightarrow\infty}T_{j}\left(Var\left\\{\hat{R}_{j}\right\\}-Q_{j}\right)=0$
###### Proof.
From relation (7), one can write
$\displaystyle
T_{j}\left(Var\left\\{\hat{R}_{j}\right\\}-Q_{j}\right)=\left(1-R_{j}\right)^{2}\sum_{i=1}^{n_{j}-1}\sum_{k=i+1}^{n_{j}}\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}$
$\displaystyle+\left(1-R_{j}\right)^{2}T_{j}.F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)$
(9)
When $T_{j}$ is large enough, condition (iii) gives $M_{ij}={\hat{M}_{ij}}$
for $i=1,...,n_{j}-1$. So, the strong law of large numbers with the integer
part properties give, when $T_{j}\rightarrow\infty$,
$\frac{M_{ij}}{M_{kj}}{\rightarrow}\frac{c_{kj}}{c_{ij}},$
for $i=1,...,n_{j}$. Hence,
$\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}=\frac{M_{ij}}{M_{kj}}\left(c_{kj}^{-1}-\frac{M_{kj}}{M_{ij}}c_{ij}^{-1}\right)^{2}{\rightarrow}0,\>as\>T_{j}\rightarrow\infty,$
(10)
and on the other hand
$T_{j}.F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right){\rightarrow}0,\>as\>T_{j}\rightarrow\infty,$
(11)
which achieves the proof. ∎
## 3 Lower bound for the variance of the estimated reliability of the
parallel–series system
We consider now the parallel–series system $S$. From expression (2), one can
write
$Var\left\\{\hat{R}\right\\}=R^{2}\left[\prod\limits_{j=1}^{n}\left(\frac{Var\left(\hat{R}_{j}\right)}{R_{j}^{2}}+1\right)-1\right]$
The following theorem gives a lower bound for the variance of $\hat{R}$.
###### Theorem 2.
Denote by
$Q=T^{-1}R^{2}\left[\sum\limits_{j=1}^{n}\frac{1-R_{j}}{R_{j}}\left(\sum_{i=1}^{n_{j}}c_{ij}^{-1}\right)\right]^{2}$
then
$Var\left\\{\hat{R}\right\\}\geq Q$
###### Proof.
Expanding the right hand side of (2) and using (1), one obtains
$Var\left\\{\hat{R}\right\\}=R^{2}\left[\sum\limits_{j=1}^{n}\frac{Var\left(\hat{R}_{j}\right)}{R_{j}^{2}}+F\left(\frac{Var\left(\hat{R}_{1}\right)}{R_{1}^{2}},...,\frac{Var\left(\hat{R}_{n}\right)}{R_{n}^{2}}\right)\right],$
which gives with the help of Theorem 1
$Var\left\\{\hat{R}\right\\}\geq
R^{2}\sum\limits_{j=1}^{n}\frac{Q_{j}}{R_{j}^{2}}=R^{2}\sum\limits_{j=1}^{n}\frac{\left(\frac{1-R_{j}}{R_{j}}\sum\limits_{i=1}^{n_{j}}c_{ij}^{-1}\right)^{2}}{T_{j}}$
(12)
This last expression has the form
$R^{2}\sum\limits_{j=1}^{n}\frac{a_{j}}{T_{j}}$
which can be expanded, thanks to identity (5), as follows
$\displaystyle
R^{2}T^{-1}\left[\sum\limits_{j=1}^{n}\frac{1-R_{j}}{R_{j}}\left(\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}\right)\right]^{2}$
(13) $\displaystyle+$ $\displaystyle
R^{2}T^{-1}\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}\frac{\left(T_{i}\frac{1-R_{j}}{R_{j}}\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}-T_{j}\frac{1-R_{i}}{R_{i}}\sum\limits_{k=1}^{n_{i}}c_{ki}^{-1}\right)^{2}}{T_{i}T_{j}}$
and as a consequence
$Var\left\\{\hat{R}\right\\}\geq
T^{-1}R^{2}\left[\sum_{j=1}^{n}\frac{1-R_{j}}{R_{j}}\left(\sum_{k=1}^{n_{j}}c_{kj}^{-1}\right)\right]^{2}=Q,$
which achieves the proof. ∎
## 4 The hybrid two stage design for the parallel–series system $S$
Similarly to the case of a subsystem an from expressions (12) and (13), one
gives interest to the numbers $T_{j}$ which minimize the quantity
$\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}\frac{\left(T_{i}\frac{1-R_{j}}{R_{j}}\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}-T_{j}\frac{1-R_{i}}{R_{i}}\sum\limits_{k=1}^{n_{i}}c_{ki}^{-1}\right)^{2}}{T_{i}T_{j}},$
and obtains the asymptotic optimality criteria
$\frac{T_{i}}{T_{j}}=\frac{\frac{1-R_{i}}{R_{i}}\sum\limits_{k=1}^{n_{i}}c_{ki}^{-1}}{\frac{1-R_{j}}{R_{j}}\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}},$
for all $i,j\in\left\\{1,2,...,n\right\\}$, which gives the rule
$T_{j}=T\frac{\frac{1-R_{j}}{R_{j}}\sum\limits_{k=1}^{n_{j}}c_{kj}^{-1}}{\sum\limits_{k=1}^{n}\frac{1-R_{k}}{R_{k}}\sum\limits_{i=1}^{n_{k}}c_{ik}^{-1}}$
(14)
We can now implement a hybrid two stage design for the determination of the
numbers $T_{j}$ as well as $M_{ij}$ as follows:
Stage 1
choose $L=\left[\sqrt{T}\right]$: one applies the two stage scheme given in
subsection 2.1 for each subsystem $S_{j}$ with $T_{j}=L$ and
$L_{j}=\left[\sqrt{T_{j}}\right]$. Next, obtain the predictor, according to
the rule (14),
$\hat{T}_{j}=\left[T\frac{\frac{1-\hat{R}_{j}}{\hat{R}_{j}}\sum\limits_{k=1}^{n_{j}}\hat{c}_{kj}^{-1}}{\sum\limits_{k=1}^{n}\frac{1-\hat{R}_{k}}{\hat{R}_{k}}\sum\limits_{i=1}^{n_{k}}\hat{c}_{ik}^{-1}}\right],~{}j=1,\ldots,n-1.$
Stage 2
define the corrector
$\displaystyle T_{j}$ $\displaystyle=$
$\displaystyle\max\left\\{L,\hat{T}_{j}\right\\},~{}j=1,\ldots,n-1,$
$\displaystyle T_{n}$ $\displaystyle=$ $\displaystyle
T-\sum_{j=1}^{n-1}T_{j},$
and take back the two stage scheme for each subsystem $S_{j}$ to calculate
$M_{ij}$ with the sample size equals $T_{j}$.
Now, the main result of this paper is given by the following theorem.
###### Theorem 3.
Choosing the $T_{j}$ and $M_{ij}$ according to the hybrid two stage design,
one obtains
$\lim_{T\rightarrow\infty}T\left(Var\left\\{\hat{R}\right\\}-Q\right)=0,$
where $Q$ is defined in Theorem 2.
###### Proof.
The relation (9) implies that
$\displaystyle
Var\left\\{\hat{R}_{j}\right\\}=Q_{j}+T_{j}^{-1}\left(1-R_{j}\right)^{2}\sum\limits_{i=1}^{n_{j}-1}\sum\limits_{k=i+1}^{n_{j}}\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}$
$\displaystyle+\left(1-R_{j}\right)^{2}F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)$
As a consequence of the hybrid two stage design and the strong law of large
numbers, $T/T_{j}$ and $T_{j}/M_{ij}$ remain bounded for all $i,j$ as
$T\rightarrow\infty$. It follows that, as $T\rightarrow\infty$,
$F\left(\frac{c_{1j}^{-2}}{M_{1j}},...,\frac{c_{n_{j}j}^{-2}}{M_{n_{j}j}}\right)=o\left(T^{-1}\right),$
and
$T_{j}^{-1}\sum_{i=1}^{n_{j}-1}\sum_{k=i+1}^{n_{j}}\frac{\left(M_{ij}c_{kj}^{-1}-M_{kj}c_{ij}^{-1}\right)^{2}}{M_{ij}M_{kj}}=o\left(T^{-1}\right),$
thanks to (10) and (11). Thus,
$Var\left\\{\hat{R}_{j}\right\\}=Q_{j}+o\left(T^{-1}\right),\>as\>T\rightarrow\infty,$
which implies that
$\displaystyle\prod\limits_{j=1}^{n}\left(\frac{Var\left\\{\hat{R}_{j}\right\\}}{R_{j}^{2}}+1\right)$
$\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{n}\left(\frac{Q_{j}}{R_{j}^{2}}+1+o\left(T^{-1}\right)\right)$
$\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{n}\left(\frac{Q_{j}}{R_{j}^{2}}+1\right)+o\left(T^{-1}\right)$
As a consequence,
$\lim\limits_{T\rightarrow\infty}T\left(Var\left\\{\hat{R}\right\\}-Q\right)=R^{2}\lim\limits_{T\rightarrow\infty}T.\left[\prod\limits_{j=1}^{n}\left(\frac{Q_{j}}{R_{j}^{2}}+1\right)-1-Q\right]$
Now, expanding the product within the limit and applying identity (5), after
having replaced $Q_{j}$ by its expression (6), one obtains
$\displaystyle\prod\limits_{j=1}^{n}\left(\frac{Q_{j}}{R_{j}^{2}}+1\right)-1$
$\displaystyle=$ $\displaystyle
R^{2}\left[\sum\limits_{j=1}^{n}\frac{Q_{j}}{R_{j}^{2}}+F\left(\frac{Q_{1}}{R_{1}^{2}},...,\frac{Q_{n}}{R_{n}^{2}}\right)\right]$
$\displaystyle=$ $\displaystyle Q+R^{2}\left(A+B\right),$
where
$\displaystyle A$ $\displaystyle=$ $\displaystyle
T^{-1}\sum\limits_{i=1}^{n-1}\sum\limits_{k=i+1}^{n}\frac{\left(T_{i}\left(\frac{1-R_{k}}{R_{k}}\right)\left(\sum\limits_{l=1}^{n_{k}}c_{lk}^{-1}\right)-T_{k}\left(\frac{1-R_{i}}{R_{i}}\right)\left(\sum\limits_{l=1}^{n_{k}}c_{li}^{-1}\right)\right)^{2}}{T_{i}T_{k}}$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
F\left(\frac{Q_{1}}{R_{1}^{2}},...,\frac{Q_{n}}{R_{n}^{2}}\right)$
Once more, the hybrid two stage allocation scheme and the strong law of large
numbers provide
$\lim_{T\rightarrow\infty}T.A=0$
and
$\lim_{T\rightarrow\infty}T.B=0,$
which achieves the proof. ∎
## 5 Monte Carlo simulation
Let us remark first that the lower bound $Q$ is a first order approximation of
the optimal variance of the reliability estimate under the constraint (4) when
$T$ is large.
In the first experiment, we will validate the fact that the hybrid scheme
provides the best allocation at system level. As in Figure 1, we consider a
simple parallel-series system of two subsystems each one, with varying
reliabilities and a fixed sample size $T=20$. For each situation A,B,C and D
and for each partition sample size $\left\\{T_{1},T-T_{1}\right\\}$ where
$T_{1}$ varies from $\left[\sqrt{T}\right]$ to $T-\left[\sqrt{T}\right]$, by
step one , we have applied the proper two stage design for each parallel
subsystem and reported in a bar diagram $Var\left(\hat{R}\right)$ as a
function of $T_{1}$, see Figure 3. On the other hand, in Table 1, we have
reported the expected value of $T_{1}=M_{11}+M_{21}$ given by the hybrid two
stage design. As expected, our scheme gives the best allocation for each
situation.
The second experiment deals with a non trivial parallel-series system just as
in [2], where subsystems are composed, respectively, of 2,3,4 and 5
components, see Figure 2. The partition total numbers $T_{j}$ to test in each
subsystem are evaluated systematically by the hybrid two stage design while
their sum $T$ is incremented from 100 to 10000 by step of 100. Figure 4 shows
the rate of the excess of variance $T\left(Var\left(\hat{R}\right)-Q\right)$
at logarithmic scale as a function of the sample size $T$. The asymptotic
optimality of the hybrid scheme is validated.
Figure 1: A simple parallel-series system of two subsystems with two components each one Figure 2: A non trivial parallel-series system. Figure 3: Bar diagram $Var\left(\hat{R}\right)$ as a function of $T_{1}$ for each case A,B,C and D : $\hat{}$ shows the minimum of $Var\left(\hat{R}\right)$ Figure 4: Asymptotic optimality of the hybrid two stage design : the speed of the excess of variance $T\left(Var\left(\hat{R}\right)-Q\right)$ at logarithmic scale as a function of the sample size $T$ System | $R_{11}$ | $R_{21}$ | $R_{12}$ | $R_{22}$ | $E(T_{1})$
---|---|---|---|---|---
A | $0.1$ | $0.11$ | $0.9$ | $0.99$ | $16$
B | $0.5$ | $0.55$ | $0.51$ | $0.6$ | $11$
C | $0.9$ | $0.99$ | $0.1$ | $0.11$ | $4$
D | $0.2$ | $0.4$ | $0.6$ | $0.3$ | $12$
Table 1: Expected value of $T_{1}=M_{11}+M_{21}$ given by the hybrid two stage
design
## 6 Conclusion
The proof of the first order asymptotic optimality for the proper two stage
design for a parallel subsystem as well as for the hybrid two stage design for
the full system has been obtained mainly through the following steps
* •
an adequate writing of the variance of the reliability estimate,
* •
a lower bound for this variance, independent of allocation,
* •
the allocation defined by the hybrid sampling scheme and the strong law of
large numbers.
With a straightforward but tedious adaptation, the above study can be namely
extended to deal with complex systems involving a multi-criteria optimization
problem under a set of constraints such as risk, system weight, cost,
performance and others, in a fixed or in a Bayesian framework.
## Acknowledgments
This work is supported with grants by the national research project (PNR) and
the L.A.A.R laboratory of the department of physics in the university Mohamed
Boudiaf of Oran.
## References
* [1] Z. Benkamra, M. Terbeche, M. Tlemcani, Procédures d’échantillonnage efficaces. Estimation de la fiabilité des systèmes séries/parallèles, Revue ARIMA, 13 (2010) 119–133.
* [2] Z. Benkamra, M. Terbeche, M. Tlemcani, Tow stage design for estimating the reliability of series/parallel systems, Mathematics and Computer in Simulation 81 (2011) 2062–2072.
* [3] D. A. Berry, Optimal sampling schemes for estimating System reliability by testing components –1 : fixed sample sizes. Journal of the American Statistical Association 69(346) (1974) 485–491.
* [4] D. W. Coit, System-reliability confidence-intervals for complex-systems with estimated component-reliability, IEEE Transactions on Reliability, 46 (4) (1997) 487–493.
* [5] M. Djerdjour, K. Rekab, A sampling scheme for reliability estimation, Southwest Journal of Pure and Applied Mathematics, electronic(2) (2002) 1–5.
* [6] J. P. Hardwick, Q.F. Stout, Optimal allocation for estimating the mean of a bivariate polynomial, Sequential Analysis, 15 (1996) 71–90.
* [7] J. P. Hardwick, Q.F. Stout, Optimal few-stage designs, Journal of Statistical Planning and Inference 104 (2002) 121-145.
* [8] C.F. Page, Allocation proportional to coefficients of variation when estimating the product of parameters, Journal of the American Statistical Association 85 (412) (1990) 1134–1139.
* [9] K. Rekab, A sampling scheme for estimating the reliability of a series system, IEEE Trans. Reliability 42(2) (1993), pp. 287–290.
* [10] M. Terbeche, O. Broderick, Two stage design for estimation of mean difference in the exponential family, Advances and Applications in Statistics 5(3) (2005) 325–339.
* [11] M. Woodroofe, J. Hardwick, Sequential allocation for an estimation problem with ethical costs, Annals of Statistics 18(3) (1990) 1358–1377.
|
arxiv-papers
| 2012-02-23T21:52:57 |
2024-09-04T02:49:27.792839
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zohra Benkamra, Mekki Terbeche, Mounir Tlemcani",
"submitter": "Tlemcani Mounir",
"url": "https://arxiv.org/abs/1202.5334"
}
|
1202.5417
|
# On logically-geometric types of algebras
###### Abstract
The connection between classical model theoretical types (MT-types) and
logically-geometrical types (LG-types) introduced by B. Plotkin is considered.
It is proved that MT-types of two $n$-tuples in two universal algebras
coincide if and only if their LG-types coincide. An algebra $H$ is called
logically perfect if for every two $n$-tuples in $H$ whose types coincide, one
can be sent to another by means of an automorphism of this algebra. Some
sufficient condition for logically perfectness of free finitely generated
algebras is given which helps to prove that finitely generated free Abelian
groups, finitely generated free nilpotent groups and finitely generated free
semigroups are logically perfect. It is proved that if two Abelian groups have
the same type and one of them is finitely generated and free then these groups
are isomorphic.
G.Zhitomirski
Department of Mathematics, Bar-Ilan University,
52900, Ramat Gan, Israel
E-mail address:zhitomg@012.net.il
## 1 Introduction
The ideas suggested and developed by B. Plotkin in the field of algebraic
logic seem to be very interesting and efficient. It turns out that the
geometrical notions and the geometrical intuition can be successfully applied
in studying algebras from arbitrary varieties. Such approach leads to so
called universal algebraic geometry and multi-sorted logical geometry.
The sketch of the ideas of universal algebraic geometry, problems and results
can be found for example in [6], [5], [7], [10], [11]. The notions of logical
geometry and obtained facts are presented in [12],[8], [9].
The purpose of this paper is to consider only one but important notion of
model theory, namely, the notion of type. The model theoretic notion of type
is well known [1]. Such a type is denoted in the paper by MT-type. MT-type is
related to one-sorted logics. On the other hand, the ideas of universal
logical geometry give rise to logically-geometric types (LG-types). This
notion is related to multi-sorted logic [12],[9]. Some of the problems
discussed in the literature are the following ones: how are connected
algebraically two $n$-tuples in an algebra whose types coincide and what we
can say about two algebras whose types coincide.
Let $\Theta$ be a variety of universal algebras of some signature and $W(X)$
denote the free $\Theta$-algebra over a set $X=\\{x_{1},x_{2},\dots,x_{n}\\}$.
In the universal algebraic geometry, the set $A^{n}$ of all $n$-tuples in a
$\Theta$-algebra $A$ is replaced by the set $\hom(W(X),A)$ which is called an
$n$-dimensional affine space and whose elements are called points. Since a
point $\mu\in\hom(W(X),A)$ is a map we can speak about its kernel. Along with
this usual kernel, so called logical kernel of $\mu$ is defined. The notion of
logical kernel of a point leads to the notion of LG-type of an algebra. All
notions mentioned above are defined in Section 2.
Although the two kinds of types mentioned above are related to different
languages we show that MT-types of two $n$-tuples coincide if and only if the
logical kernels of the corresponding points coincide (Theorem 3.1).
Then in Section 4 we consider so called logically perfect algebras. An algebra
$H$ is said to be logically perfect if for every its two $n$-tuples whose
types coincide there exists an automorphism of $H$ which sends one of these
tuples to another. A sufficient condition for logically perfectness of free
finitely generated algebras is given. The main result in this section is
Theorem 4.5.
The last Section 5 is devoted to algebras having the same type (isotyped
algebras). It is proved (Theorem 5.3) that if two Abelian groups have the same
type and one of them is finitely generated and free then these groups are
isomorphic.
The obtained results solve some problems set in [9].
Acknowledgments The author is pleased to thank B. Plotkin for useful
discussions and interesting suggestions.
## 2 Preliminaries
Throughout this paper, $\Theta$ is a variety of universal algebras of some
signature which determines the corresponding first-order language $L$ with
equality ”$\equiv$” and the infinite set $X^{0}=\\{x_{1},x_{2},\dots\\}$ of
variables. Let $W(X)$ denote the free $\Theta$-algebra generated by $X\subset
X^{0}$. We consider finite subsets $X\subset X^{0}$ only and follow the
conception suggested by B. Plotkin (see for example [6], [9], [8], [12]).
Let $\mathbb{M}$ be a $\Theta$-algebra with the domain $M$. Every $n$-tuple
$\bar{a}=(a_{1},\dots,a_{n})$ of elements of $M$ determines a homomorphism
$\mu:W(X)\to\mathbb{M}$ where $X=\\{x_{1},\dots,x_{n}\\}$, viz
$\mu(x_{i})=a_{i}$ for $i=1,\dots,n$. And vice versa, every such homomorphism
determines an $n$-tuple in $M$. Thus the set $M^{n}$ can be identified with
$\hom(W(X),\mathbb{M})$ which is called an affine space and whose elements are
called points. Considering the tuples in $M$ as points in the corresponding
affine space gives us new interesting opportunities.
First of all, the kernel of a point $\mu$ appears:
$Ker\mu=\\{(w,w^{\prime})|\mu(w)=\mu(w^{\prime})\\}$. It is useful to consider
equalities $w\equiv w^{\prime}$ instead of corresponding pairs in $W(X)$. Such
an approach leads to connections between sets of points and systems of
identities, that is, to something like to algebraic geometry for an universal
algebra. For details see papers cited above. In the present paper, we focus on
the notion of so called logical kernel of a point $\mu$: $LKer\mu$. We recall
the definition according to [9].
Let $\Gamma$ denote the set of all finite subsets of $X^{0}$. For every
$X\in\Gamma$, consider the signature $L_{X}=\\{\vee,\wedge,\neg,\exists x,x\in
X,M_{X},\\}$, where $M_{X}$ is the set of all equalities $w\equiv w^{\prime}$,
where $w,w^{\prime}\in W(X)$. By adding for every $X\in\Gamma$ symbols
$s=s^{XY}:W(X)\to W(Y)$, we obtain multi-sorted signature $L_{\Theta}$ . The
corresponding multi-sorted language is defined by induction on length and sort
of formulas.
###### Definition 2.1.
1\. Each equality $w\equiv w^{\prime}$ is a formula of the length zero and
sort $X$ if $w\equiv w^{\prime}\in M_{X}$.
2\. Let $u$ be a formula of the length $n$ and the sort $X$. Then the formulas
$\neg u$ and $\exists xu$ are the formulas of the same sort $X$ and the length
$(n+1)$.
3\. For the given $s:W(X)\to W(Y)$ we have the formula $s_{*}u$ with the
length $(n+1)$ and the sort $Y$.
4\. Let $u_{1}$ and $u_{2}$ be formulas of the same sort $X$ and the length
$n_{1}$ and $n_{2}$ accordingly. Then the formulas $(u_{1}\vee u_{2})$ and
$(u_{1}\wedge u_{2})$ have the length $(n_{1}+n_{2}+1)$ and the sort $X$.
The set of all formulas of the sort $X$ will be denote by $\Phi(X)$.
The value $Val^{X}_{H}(u)$ of a formula $u\in\Phi(X)$ in a $\Theta$-algebra
$H$ is defined according to the construction. Elements of $Val^{X}_{H}(u)$ are
points $\mu:W(X)\to H$.
###### Definition 2.2.
(1). $Val^{X}_{H}(w\equiv w^{\prime})=\\{\mu\mid\mu(w)=\mu(w^{\prime})\\}$.
(2). If $v=\exists xu$ and $u\in\Phi(X)$, then $\mu\in Val^{X}_{H}(v)$ if and
only if there exists a point $\nu:W(X)\to H$ such that $\nu$ coincides with
$\mu$ for all $y\in X$ besides $x$ and $\nu\in Val^{X}_{H}(u)$.
(3). If $u_{1},u_{2}\in\Phi(X)$ then $Val^{X}_{H}(u_{1}\vee
u_{2})=Val^{X}_{H}(u_{1})\cup Val^{X}_{H}(u_{2})$, $Val^{X}_{H}(u_{1}\wedge
u_{2})=Val^{X}_{H}(u_{1})\cap Val^{X}_{H}(u_{2})$.
(4). $Val^{X}_{H}(\neg u)=\hom(W(X),H)\setminus Val^{X}_{H}(u)$.
(5). Let $s:W(X)\to W(Y)$ be a homomorphism, $v\in\Phi(X)$ and $u=s_{*}v$.
Then $\mu\in Val^{Y}_{H}(u)$ if and only if $\mu\circ s\in Val^{X}_{H}(v)$.
###### Definition 2.3.
A formula $u\in\Phi(X)$ belongs to the logical kernel $LKer(\mu)$ of a point
$\mu:W(X)\to H$ if and only if $\mu\in Val^{X}_{H}(u)$.
The set $LKer(\mu)$ of formulas from $\Phi(X)$ is called logically-geometric
$X$-type of the point $\mu$ ($X$-$LG$-type).
###### Definition 2.4 ([12]).
The set $T$ of formulas from $\Phi(X)$ is called $X$-$LG$-type of the algebra
$H$, if there is a point $\mu:W(X)\to H$ such that $T=LKer(\mu)$.
Algebras $H_{1}$ and $H_{2}$ in $\Theta$ are called $LG$-isotyped, if for any
finite $X$, every $X$-type of the algebra $H_{1}$ is an $X$-type of the
algebra $H_{2}$ and vice versa.
###### Definition 2.5.
An algebra $H$ is called logically perfect if for every two points $\mu$ and
$\nu$ in $H$ having the same $X$-$LG$-type (that is, $LKer(\mu)=LKer(\nu)$)
there exists an automorphism $\varphi$ of $H$ such that $\mu=\varphi\circ\nu$,
that is, $\varphi$ transports $n$-tuple $(\nu(x_{1}),\dots,\nu(x_{n}))$ to
$n$-tuple $(\mu(x_{1}),\dots,\mu(x_{n}))$.
Now we recall the model-theoretical notion of type of an $n$-tuple $\bar{a}$.
###### Definition 2.6.
The type $tp^{\mathbb{M}}(\bar{a})$ consists of all formulas
$u(x_{1},\dots,x_{n})\in L$ with free variables $x_{1},\dots,x_{n}$ (all other
variables in this formula are bounded) such that $\mathbb{M}\models
u(a_{1},\dots,a_{n})$, that is, $u(x_{1},\dots,x_{n})$ is true under
interpretation which assigns $a_{i}$ to $x_{i}$.
Such a type will be called $MT$-type. It is worth to mention that we do not
consider types depending of parameters (the more general definition can be
found in [2]). The problem arises how two tuples are algebraically connected
if their $MT$-types coincide.
Two kinds of types defined above (MT- and LG-type) are sets of formulas in
different languages. We will prove below that the two points $\mu$ and $\nu$
have the same $X$-$LG$-type if and only if the $n$-tuples
$(\nu(x_{1}),\dots,\nu(x_{n}))$ and $(\mu(x_{1}),\dots,\mu(x_{n}))$ have the
same $MT$-type.
## 3 Relations between logical-geometrical types and model-theoretical types
###### Theorem 3.1.
Let $H_{1}$ and $H_{2}$ be $\Theta$-algebras. Let
$\bar{a}=(a_{1},\dots,a_{n})$ and $\bar{b}=(b_{1},\dots,b_{n})$ be $n$-tuples
in $H_{1}$ and $H_{2}$ respectively. Consider two corresponding points
$\nu:W(X)\to H_{1}$ and $\mu:W(X)\to H_{2}$ where $X=\\{x_{1},\dots,x_{n}\\}$,
$\nu(x_{i})=a_{i}$ and $\mu(x_{i})=b_{i}$, $i=1,\dots,n$. Then
$LKer(\nu)=LKer(\mu)$ if and only if
$tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$.
###### Proof.
We will prove this statement by several steps.
###### Lemma 3.2.
$LKer(\nu)=LKer(\mu)\Rightarrow tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$.
###### Proof.
Let $LKer(\nu)$=$LKer(\mu)$. Let $u\in tp^{H_{1}}(\bar{a})$. Under Definition
2.6, we have that $u=u(x_{1},\dots,x_{n},\;y_{1},\dots,y_{m})$ with listed
variables, where $x_{1},\dots,x_{n}$ have free occurrences only, all
$y_{1},\dots,y_{m}$ are bounded, and $H_{1}\models u(a_{1},\dots,a_{n})$.
On the other hand, according to Definition 2.1, $u\in\Phi(X\cup Y)$, where
$Y=\\{y_{1},\dots,y_{m}\\}$. Therefore for every homomorphism $\gamma:W(X\cup
Y)\to H_{1}$ such that $\gamma(x_{i})=a_{i}$,$i=1,\dots,n$ (values of
$\gamma(y_{j})$ do not influence), we have $\gamma\in Val^{X\cup
Y}_{H_{1}}(u)$ (see definitions (1)-(4) from 2.2). Consider arbitrary
homomorphism $s:W(X\cup Y)\to W(X)$ such that $s(x_{i})=x_{i}$, $i=1,\dots,n$
and construct the formula $v=s_{*}u\in\Phi(X)$. Since $\nu\circ
s(x_{i})=a_{i}$, $i=1,\dots,n$, we have $\nu\circ s\in Val^{X\cup
Y}_{H_{1}}(u)$. Under definition 2.2 (5), we obtain that $\nu\in
Val^{X}_{H_{1}}(v)$ and therefore $v\in LKer\nu$.
Since $LKer(\nu)$=$LKer(\mu)$, we have $v\in LKer\mu$, that is, $\mu\in
Val^{X}_{H_{2}}(v)$ which implies that $\mu\circ s\in Val^{X\cup
Y}_{H_{2}}(u)$. Let $\delta:W(X\cup Y)\to H_{2}$ be an an arbitrary
homomorphism such that $\delta(x_{i})=b_{i}$ for all $i=1,\dots,n$. Since
$\mu\circ s:W(X\cup Y)\to H_{2}$, $\mu\circ s(x_{i})=b_{i}$ for all
$i=1,\dots,n$ and the variables from $Y$ are bounded in $u$ , we obtain that
the values of the formula $u$ under interpretations $\delta$ and $\mu\circ s$
coincide. Therefore $H_{2}\models u(b_{1},\dots,b_{n})$, that is, $u\in
tp^{H_{2}}(\bar{b})$. Consequently $tp^{H_{1}}(\bar{a})\subseteq
tp^{H_{2}}(\bar{b})$. The inverse inclusion is also true by symmetry. ∎
Now we assign to every formula $u\in\Phi(X),\;X\in\Gamma,$ a formula
$\tilde{u}$ in the one-sorted first order language, that is, a formula which
does not contain symbols $s_{*}$. Let $\tilde{X}^{0}$ be a copy of $X^{0}$
such that to every variable $x\in X^{0}$ the variable
$\tilde{x}\in\tilde{X}^{0}$ corresponds one to one. Consider the first-order
language $L$ associated with the variety $\Theta$ with set
$X^{0}\cup\tilde{X}^{0}$ of variables using variables from $X^{0}$ for free
variables and variables from $\tilde{X}^{0}$ for bounded ones only.
We construct the formula $\tilde{u}$ for every formula
$u\in\Phi(X),\;X\in\Gamma$, inductively.
1\. If $u$ is $w\equiv w^{\prime}$ then $\tilde{u}=u$.
2\. If $u$ is $\neg v,(u_{1}\vee u_{2})$ or $(u_{1}\wedge u_{2})$ then
$\tilde{u}=\neg\tilde{v},\;(\tilde{u_{1}}\vee\tilde{u_{2}})$ or
$(\tilde{u_{1}}\wedge\tilde{u_{2}})$ respectively.
3\. If $u=\exists xv$ and $x\in X$ then
$\tilde{u}=\exists\tilde{x}\tilde{v}|^{x}_{\tilde{x}}$, where
$\tilde{v}|^{x}_{\tilde{x}}$ denotes the formula in $L$ which is obtained by
replacing of all occurrences of the variable $x$ in $\tilde{v}$ by
$\tilde{x}$.
4\. Let $Y=\\{y_{1},\dots,y_{m}\\}\in\Gamma$ and $s:W(Y)\to W(X)$ be a
homomorphism, $v\in\Phi(Y)$ and $u=s_{*}v$. Then
$\tilde{u}=\tilde{v}|^{y_{1}}_{s(y_{1})},\dots,^{y_{m}}_{s(y_{m})}$. Notice
that all occurrences of elements from $X$ and $Y$ in $\tilde{v}$ can be free
only.
###### Lemma 3.3.
For every point $\mu:W(X)\to H$ and every $u\in\Phi(X)$
$u\in LKer(\mu)\Leftrightarrow\tilde{u}\in tp^{H}(\bar{a}),$
where $\bar{a}=(\mu(x_{1}),\dots,\mu(x_{n}))$, $X=\\{x_{1},\dots,x_{n}\\}$.
###### Proof.
We will prove this statement by induction according to the construction of
formulas of sort $X$.
1\. Let $u$ be $w\equiv w^{\prime}$. Under definition, $u\in LKer(\mu)$ means
that $\mu(w)=\mu(w^{\prime})$. In the considered case, $\tilde{u}=u$ and we
obtain that $u\in LKer(\mu)$ is equal to
$H\models\tilde{u}(a_{1},\dots,a_{n})$, that is, to $\tilde{u}\in
tp^{H}(\bar{a})$.
2\. For $u=\neg v,(u_{1}\vee u_{2}$) or $(u_{1}\wedge u_{2})$ our statement is
obviously true.
3\. Let $u=\exists xv$ , where $x\in X$. Assume that our statement is true for
$v$. The fact $u\in LKer(\mu)$ means that there exists a point $\nu:W(X)\to H$
which coincides with $\mu$ for all $y\in X$ besides $x$ and such that $\nu\in
Val^{X}_{H}(v)$. Under assumption, $\nu\in Val^{X}_{H}(v)$ is equal to
$\tilde{v}\in tp^{H}(\bar{b})$ where $\bar{b}=(\nu(x_{1}),\dots,\nu(x_{n}))$.
Since $\tilde{u}=\exists\tilde{x}\tilde{v}|^{x}_{\tilde{x}}$, we obtain that
$u\in LKer(\mu)$ is equal to $\tilde{u}\in tp^{H}(\bar{a})$ where
$\bar{a}=(\mu(x_{1}),\dots,\mu(x_{n}))$. Notice that $\tilde{u}$ does not
contain $x$.
4\. Let $Y=\\{y_{1},\dots,y_{m}\\}$, $s:W(Y)\to W(X)$ be a homomorphism,
$v\in\Phi(Y)$, and $u=s_{*}v$. Assume that our statement is true for $v$. This
means that $v\in LKer(\mu\circ s)$ is equal to $\tilde{v}\in tp^{H}(\bar{b})$,
where $\bar{b}=\mu\circ s(\bar{y})$. Further, $v\in LKer(\mu\circ s)$ is equal
to $u\in LKer(\mu)$ and $\tilde{v}\in tp^{H}(\bar{b})$ is equal to
$\tilde{u}\in tp^{H}(\bar{a})$ because
$\tilde{u}=\tilde{v}|^{y_{1}}_{s(y_{1})},\dots,^{y_{m}}_{s(y_{m})}$ according
to the definition, and hence $H\models\tilde{u}(a_{1},\dots,a_{n})$ is the
same that $H\models\tilde{v}(b_{1},\dots,b_{m})$. Thus our statement is true
for $u$ too.
∎
###### Lemma 3.4.
$tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})\Rightarrow LKer(\nu)=LKer(\mu)$
###### Proof.
Let $tp^{H_{1}}(\bar{a})=tp^{H_{2}}(\bar{b})$. Let $u\in\Phi(X)$ and $u\in
LKer\nu$. Then according to Lemma 3.3, $\tilde{u}\in tp^{H_{1}}(\bar{a})$.
Consequently $\tilde{u}\in tp^{H_{2}}(\bar{b})$ and therefore $u\in LKer\mu$
according to the same Lemma. ∎
In virtue of Lemmas 3.2 and 3.4, Theorem 3.1 is proved. ∎
## 4 Logically perfect algebras
The purpose of this section is to present some results concerning logically
perfect algebras. Some authors call an algebra $H$ homogeneous if every
automorphism between two finitely generated subalgebras of $H$ can be extended
to an automorphism of $H$. It is easy to see that every homogeneous algebra is
logically perfect [8].
It is obvious that every finite dimensional linear space $V$ is a homogeneous
algebra, and therefore $V$ is logically perfect. On the other hand, it is easy
to see that free finitely generated semigroups and free finitely generated
Abelian groups are not homogeneous, nevertheless we will show below that all
of them are logically perfect. Thus the homogeneity is not a necessary
condition for an algebra to be logically perfect. There is a logical condition
equivalent to homogeneity obtained by the author. This condition is cited in
[8] and called there strictly logically perfectness. The following
generalization of homogeneity will be useful.
###### Definition 4.1.
An algebra $H$ is called weakly homogeneous if for every isomorphism
$\varphi:A\to B$ between two its finitely generated subalgebras $A$ and $B$,
the following condition is satisfied: if $\varphi$ itself and its inverse map
$\varphi^{-1}:B\to A$ both can be extended to endomorphisms of $H$ then
$\varphi$ can be extended to an automorphism of $H$.
###### Theorem 4.2.
Every weakly homogeneous finitely generated free algebra is logically perfect.
###### Proof.
Let $H$ be weakly homogeneous and $e_{1},...,e_{n}$ be free generators of $H$.
Let $X=\\{x_{1},...,x_{k}\\}$. Consider two points $\nu,\mu:W(X)\to H$ and
suppose that $LKer\nu=LKer\mu$. Let $\nu(x_{i})=a_{i}$ and $\mu(x_{i})=b_{i}$
for all $i=1,...,k$. Take $Y=\\{y_{1},...,y_{n}\\}$, such that $X\bigcap
Y=\emptyset$, and define a homomorphism $\gamma:W(Y)\to H$ by the values:
$\gamma(y_{i})=e_{i},\;i=1,...,n$. Let $w_{1},...,w_{k}\in W(Y)$ be any $k$
words such that $a_{i}=\gamma(w_{i}),\;i=1,...,k$.
Consider a formula $u$ of sort $X$ of the kind $u=s_{*}(v)$ where
$v=(\exists y_{1})...(\exists y_{n})(x_{1}\equiv w_{1}\wedge...\wedge
x_{k}\equiv w_{k})$
and $s:W(X\bigcup Y)\to W(X)$ defined by
$s(x_{i})=x_{i},s(y_{1})=...=s(y_{n})=x_{1}$.
It is obvious that $\nu\in Val_{H}(u)$. Thus under assumption, $\mu\in
Val_{H}(u)$ and therefore $\mu\circ s\in Val_{H}(v)$. The last one means that
there exists a homomorphism $\delta:W(Y)\to H$ such that
$b_{i}=\delta(w_{i}),\;i=1,...,k$. Define an endomorphism $\sigma$ of $H$
setting $\sigma(e_{i})=\delta(y_{i}),\;i=1,...,n$, that is,
$\sigma\circ\gamma=\delta$. We have
$\sigma(a_{i})=\sigma(\gamma(w_{i}))=\delta(w_{i})=b_{i}$ for $i=1,...,k$.
Hence $\sigma$ determines a homomorphism $\varphi$ of the subalgebra $A$
generated by $a_{1},...,a_{k}$ on the subalgebra $B$ generated by
$b_{1},...,b_{k}$.
Similarly, we can define an endomorphism $\tau$ of $H$ such that
$\tau(b_{i})=a_{i}$ for $i=1,...k$. Consequently
$\sigma\circ\tau(b_{i})=b_{i}$ and $\tau\circ\sigma(a_{i})=a_{i}$ which means
that the restriction $\varphi$ of $\sigma$ to $A$ is an isomorphism of $A$ on
$B$ and $\varphi^{-1}$ is a restriction of $\tau$. Since $H$ is weakly
homogeneous, $\varphi$ can be extended up to automorphism $\tilde{\varphi}$ of
$H$ for which we have $\tilde{\varphi}\circ\nu=\mu$. ∎
###### Lemma 4.3.
Finitely generated free Abelian groups and finitely generated free nilpotent
groups are weakly homogeneous.
###### Proof.
1. We start with considering Abelian groups. Let $G$ and $F$ be free Abelian groups of the same rank $n$. Let $A$ and $B$ be two subgroups of $G$ and $F$ respectively which are isomorphic by means of an isomorphism $\varphi:A\to B$. We will prove that if $\varphi$ and $\varphi^{-1}$ both can be extended up to homomorphisms $\sigma:G\to F$ and $\tau:F\to G$ respectively, then $\varphi$ can be extended up to an isomorphism of $G$ onto $F$.
It is known ([4], Theorem 3.5) that there exists a base $g_{1},...,g_{n}$ of
$G$ and a base $a_{1},...,a_{k}$ of $A$ such that $a_{i}=p_{i}g_{i}$ for
$1\leq i\leq k$ where $p_{1},...,p_{k}$ are integers and every $p_{i+1}$ is
divisible by $p_{i}$ for $1\leq i\leq k-1$ . Exactly in the same way, there
exists a base $f_{1},...,f_{n}$ of $F$ and a base $b_{1},...,b_{k}$ of $B$
such that $b_{i}=q_{i}f_{i}$ for $1\leq i\leq k$ and every integer $q_{i+1}$
is divisible by the integer $q_{i}$ for $1\leq i\leq k-1$.
Let $\sigma(g_{i})=\sum_{j=1}^{n}s^{j}_{i}f_{j}$ and
$\tau(f_{i})=\sum_{j=1}^{n}t^{j}_{i}g_{j}$. We obtain two integer matrices of
order $n$: $S=||s^{j}_{i}||$ and $T=||t^{j}_{i}||$. Since $\varphi:A\to B$ is
an isomorphism, $\varphi$ provides an invertible integer matrix
$||a^{j}_{i}||$ of order $k$, where
$\varphi(a_{i})=\sum_{j=1}^{k}a^{j}_{i}b_{j}$. Let $||b^{j}_{i}||$ be its
inverse matrix: $\varphi^{-1}(b_{i})=\sum_{j=1}^{k}b^{j}_{i}a_{j}$.
Since $\sigma(a_{i})=\varphi(a_{i})$, we obtain
$p_{i}\sigma(g_{i})=\sum_{j=1}^{k}a^{j}_{i}b_{j}=\sum_{j=1}^{k}a^{j}_{i}q_{j}f_{j}$
for $1\leq i\leq k$. Thus for all $1\leq i\leq k$ we have
$p_{i}\sum_{j=1}^{n}s^{j}_{i}f_{j}=\sum_{j=1}^{k}a^{j}_{i}q_{j}f_{j}$. This
implies that $p_{i}s^{j}_{i}=a^{j}_{i}q_{j}$ for $1\leq i,j\leq k$ and
$p_{i}s^{j}_{i}=0$ for $1\leq i\leq k$, $k+1\leq j\leq n$. In view of the
definitions of $p_{i},q_{i}$, we have $p_{1}=q_{1}$ and $s^{j}_{i}=0$ for all
$1\leq i\leq k$ and $k+1\leq j\leq n$.
By duality, we obtain $q_{i}t^{j}_{i}=b^{j}_{i}p_{j}$ for $1\leq i,j\leq k$
and $t^{j}_{i}=0$ for all $1\leq i\leq k$ and $k+1\leq j\leq n$. Therefore we
obtain for all $1\leq i,j\leq k$ :
$\sum_{l=1}^{k}s^{j}_{l}t^{l}_{i}=\sum_{l=1}^{k}\frac{a^{j}_{l}q_{j}}{p_{l}}\frac{b^{l}_{i}p_{l}}{q_{i}}=\sum_{l=1}^{k}\frac{q_{j}}{q_{i}}a^{j}_{l}{b^{l}_{i}}=\begin{cases}1,&\text{if
$i=j$;}\\\ 0,&\text{if $i\not=j$.}\end{cases}$ (1)
Consider the left corner $k$-th minor $M$ of the matrix $S$, that is, the
determinant of the matrix $||s^{j}_{i}||_{1\leq i,j\leq k}$. According to (1)
$M=1$ or $M=-1$. Define map $\tilde{\varphi}:G\to F$ setting
$\tilde{\varphi}(g_{i})=\begin{cases}\sigma(g_{i}),&\text{if $i\leq k$;}\\\
f_{i},&\text{if $k+1\leq i\leq n$.}\end{cases}$ (2)
The matrix $V$ of this map is
$V=\left(\begin{matrix}s^{1}_{1}&...&s^{1}_{k}&0&...&0\\\
...&...&...&0&...&0\\\ s^{k}_{1}&...&s^{k}_{k}&0&...&0\\\ 0&...&0&1&...&0\\\
0&...&0&0&1...&0\\\ 0&...&0&0&...&1\end{matrix}\right)$
We see that $DetV=M=\pm 1$ and therefore $\tilde{\varphi}$ is an isomorphism.
By construction,
$\tilde{\varphi}(a_{i})=m_{i}\tilde{\varphi}(g_{i})=m_{i}\sigma(g_{i})=\sigma(a_{i})=\varphi(a_{i})$
for all $i\leq k$. Consequently $\tilde{\varphi}$ extends $\varphi$.
2. Now let $H$ be a finitely generated free nilpotent group of class $c>1$ and rank $n$. Let $A$ and $B$ be two subgroups of $H$ which are isomorphic by means of an isomorphism $\varphi:A\to B$. Let $\varphi$ and $\varphi^{-1}$ both can be extended up to endomorphisms $\sigma$ and $\tau$ of $H$ respectively.
The quotient group $G=H/H^{\prime}$ is a free Abelian group of the same rank
$n$. Let $\eta:H\to G$ be the corresponding epimorphism. Then
$\bar{A}=\eta(A)$ and $\bar{B}=\eta(B)$ are isomorphic subgroups of $G$ under
isomorphism $\bar{\varphi}=\eta\circ\varphi\circ\eta^{-1}$. This isomorphism
is contained in the endomorphism $\bar{\sigma}=\eta\circ\sigma\circ\eta^{-1}$
and the inverse isomorphism $\bar{\varphi}^{-1}$ is contained in the
endomorphism $\bar{\tau}=\eta\circ\tau\circ\eta^{-1}$. Thus we can apply the
fact proved above in the point 1, that is, $\bar{\varphi}$ can be extended up
to automorphism $\bar{\Phi}$ of $G$.
Consider this extension in details. A base $g_{1},...,g_{n}$ of Abelian group
$G$ and a base $\bar{a}_{1},...,\bar{a}_{k}$ of its subgroup $\bar{A}$ are
chosen such that $\bar{a}_{i}=g_{i}^{p_{i}}$ for $1\leq i\leq k$ (now we use
the multiplicative notation). The automorphism $\bar{\Phi}$ of $G$ extending
$\bar{\varphi}$ is constructed in such a way that
$\bar{\Phi}(g_{i})=\bar{\sigma}(g_{i})$ for $i\leq k$. The elements
$f_{i}=\bar{\Phi}(g_{i})$ for $i=1,\dots,n$ form a base of $G$ in which first
$k$ elements are equal to corresponding $\bar{\sigma}(g_{i})$.
It is known from the theory of nilpotent groups (see for example [3]) that a
system $h_{1},\dots,h_{n}$ of elements of $H$ is a system of free generators
of some free nilpotent subgroup of the same class if and only if the the
system $\eta(h_{1}),\dots,\eta(h_{n})$ is linear independent in
$G=H/H^{\prime}$. So if $\eta(h_{1}),\dots,\eta(h_{n})$ is a base of $G$ then
$h_{1},\dots,h_{n}$ is a base of a free nilpotent subgroup $H_{0}$ of $G$.
Since $\eta(H_{0})=G$, we have $H_{0}H^{\prime}=H$. The last one implies that
$H_{0}=H$. We obtain that if $\eta(h_{1}),\dots,\eta(h_{n})$ is a base of $G$
then $h_{1},\dots,h_{n}$ is a base of $H$. Below we apply this property of
finitely generated free nilpotent groups.
There exist bases $h_{1},\dots,h_{n}$ and $u_{1},\dots,u_{n}$ of $H$ such that
$\eta(h_{i})=g_{i}$ and $\eta(u_{i})=f_{i}$ for $1\leq i\leq n$. Of course we
can chose $u_{i}=\sigma(h_{i})$ for $1\leq i\leq k$ because
$\eta(\sigma(h_{i}))=\bar{\sigma}(g_{i})=f_{i}$ for $1\leq i\leq k$.
Now we define an automorphism $\Phi$ of $H$ setting $\Phi(h_{i})=u_{i}$ for
$1\leq i\leq n$. On the other hand, elements $h_{i}^{p_{i}}$ $1\leq i\leq k$
form a base of the free nilpotent subgroup $AH^{\prime}$ because
$\eta(h_{i}^{p_{i}})=g_{i}^{p_{i}}=\bar{a_{i}}$. We have
$\Phi(h_{i}^{p_{i}})=(\Phi(h_{i}))^{p_{i}}=u_{i}^{p_{i}}=(\sigma(h_{i}))^{p_{i}}=\sigma(h_{i}^{p_{i}})$.
Thus $\Phi$ coincides with $\sigma$ on the subgroup $AH^{\prime}$. Since
$\sigma$ contains $\varphi$ which is defined on $A\subset AH^{\prime}$, $\Phi$
is an extension of $\varphi$. ∎
###### Lemma 4.4.
Every finitely generated free semigroup is weakly homogeneous.
###### Proof.
Let $S$ be a free semigroup with the set $X=\\{x_{1},\dots,,x_{k}\\}$ of free
generators . Let $\varphi:A\to B$ be an automorphism between two subsemigroups
$A$ and $B$ of $S$, where $A$ and $B$ are generated by elements
$a_{1},\dots,a_{n}$ and $b_{1},\dots,b_{n}$ respectively. We may assume that
$\varphi(a_{i})=b_{i}$ for $1\leq i\leq n$.
Suppose that there exist two endomorphisms $\sigma$ and $\tau$ first of which
extends $\varphi$ and the second one extends $\varphi^{-1}$. Thus
$\sigma(a_{i})=b_{i}$ and $\tau(b_{i})=a_{i}$ Denote by $|w|$ the length of
the word $w$ in alphabet $X$. Since $|\sigma(w)|\geq|w|$ and
$|\tau(w)|\geq|w|$ for every $w\in S$, we obtain that $|a_{i}|=|b_{i}|$. Let
$y_{1},\dots,y_{p}$ be the list of all variables from $X$ which occur in
$a_{1},\dots,a_{n}$ and $z_{1},\dots,z_{q}$ be the analogical list of all
variables which occur in $b_{1},\dots,b_{n}$. It is obvious that
$|\sigma(y_{i})|=1$ for all $1\leq i\leq p$ and $|\tau(z_{i})|=1$ for all
$1\leq i\leq q$. Therefore we have that
$\sigma(y_{i})\in\\{z_{1},\dots,z_{q}\\}$ and
$\tau(z_{i})\in\\{y_{1},\dots,y_{p}\\}$.
Since $\tau(\sigma(a_{i}))=a_{i}$ and $\sigma(\tau(b_{i}))=b_{i}$ for $1\leq
i\leq n$, we have that the restrictions of $\sigma$ and $\tau$ to variables
$y_{1},\dots,y_{p}$ and $z_{1},\dots,z_{q}$ respectively are mutually inverse
maps. Thus $p=q$ and $\sigma$ and $\tau$ induce two mutually inverse partial
one-to-one transformations of $X$. Let $\alpha$ be a bijection of
$X\setminus\\{y_{1},\dots,y_{p}\\}$ on $X\setminus\\{z_{1},\dots,z_{p}\\}$.
Setting $\tilde{\varphi}(y_{i})=\sigma(y_{i})$ for $1\leq i\leq p$ and
$\tilde{\varphi}(x)=\alpha(x)$ for all other variables from $X$, we obtain the
automorphism $\tilde{\varphi}$ of $S$ which extends $\varphi$. ∎
Lemmas 4.3, 4.4 and 4.2 give us the following result:
###### Theorem 4.5.
Finitely generated free Abelian groups, finitely generated free nilpotent
groups of any class and finitely generated semigroups are logically perfect.
The method which has been used to prove the theorem above can not be applied
to non-Abelian finitely generated free groups.
###### Proposition 4.6.
Free groups of rank 2 are not weakly homogeneous.
###### Proof.
Consider the free group $\mathbb{F}_{2}$ of rank 2 free generated by
$x_{1},x_{2}$. Let $a=x_{1}^{2}x_{2}x_{1}^{-1}x_{2}$ and $b=x_{1}x_{2}$.
Define endomorphisms $\sigma$ and $\tau$ setting
$\sigma(x_{1})=x_{1}x_{2},\;\sigma(x_{2})=1$ and
$\tau(x_{1})=x_{1}^{2}x_{2},\;\tau(x_{2})=x_{1}^{-1}x_{2}$. We see that
$\sigma(a)=b$ and $\tau(b)=a$. Thus $\sigma$ induces an isomorphism of
$\varphi:\langle a\rangle\to\langle b\rangle$ and $\tau$ induces the inverse
isomorphism $\varphi^{-1}$.
Suppose that there exists an automorphism $\tilde{\varphi}$ of
$\mathbb{F}_{2}$ which sends $a$ to $b$. Let $\tilde{\varphi}(x_{1})=w_{1}$ ,
$\tilde{\varphi}(x_{2})=w_{2}$, where $w_{1}$,$w_{2}$ are words in symbols
$x_{1},x_{2}$. Thus we have a relation in our free group: $x_{1}x_{2}\equiv
w_{1}^{2}w_{2}w_{1}^{-1}w_{2}$. (*)
This relation must be an identity in the group variety. Let $l_{1},l_{2}$ be
the sums of all exponents of $x_{1},x_{2}$ incoming in $w_{1}$ and
$m_{1},m_{2}$ the sums of all exponents of $x_{1},x_{2}$ incoming in $w_{2}$
respectively. It is obvious that $l_{1}+2m_{1}=l_{2}+2m_{2}=1$. Thus
$l_{1},l_{2}$ must be odd numbers.
Consider the group $S_{3}$ of all permutations of the set $\\{1,2,3\\}$ . This
group is a homomorphic image of $\mathbb{F}_{2}$ under the map $\gamma$ which
maps $x_{1}$ to $(213)$ and $x_{2}$ to $(132)$. Since
$\gamma(x_{1}^{2})=\gamma(x_{2}^{2})=(123),\;\gamma(x_{1}x_{2})=(312),\;\gamma(x_{2}x_{1})=(231),\;\gamma(x_{1}x_{2}x_{1})=\gamma(x_{2}x_{1}x_{2})=(321),\;\gamma((x_{1}x_{2})^{2})=\gamma(x_{2}x_{1}),\;\gamma((x_{2}x_{1})^{2})=\gamma(x_{1}x_{2})$,
we obtain that the following equalities are satisfied in $S_{3}$: $w_{1}\equiv
x_{1}x_{2}$ or $w_{1}\equiv x_{2}x_{1}$. For $w_{2}$ we have variants:
$w_{2}\equiv 1,x_{1},x_{2},x_{1}x_{2},x_{2}x_{1},x_{1}x_{2}x_{1}$. Since
$w_{1},w_{2}$ generate $\mathbb{F}_{2}$, their images generate $S_{3}$.
Therefore we have only three variants for $w_{2}$: $w_{2}\equiv
x_{1},x_{2},x_{1}x_{2}x_{1}$. Directly calculations show that in all mentioned
cases $\gamma(w_{1}^{2}w_{2}w_{1}^{-1}w_{2})=(123)$ which contradicts to the
identity (*).
Consequently there is no automorphism of $\mathbb{F}_{2}$ sending $a$ to $b$.
∎
Nevertheless all free finitely generated non-Abelian free groups are logically
perfect. This fact is proved in [2] in view of Theorem 3.1.
## 5 Isotyped algebras
We consider the following problem: in what cases isotyped algebras are
necessarily isomorphic. At first, we generalize the result obtained in [12],
Theorem 3.11.
###### Theorem 5.1.
If two algebras $H_{1}$ and $H_{2}$ from the same variety $\Theta$ are
isotyped then for every finitely generated subalgebra $A$ of $H_{1}$ there
exists a subalgebra $B$ of $H_{2}$ isomorphic to $A$, and if $A$ is a proper
subalgebra then $B$ can be chosen as a proper subalgebra too.
###### Proof.
Let $H_{1}$ and $H_{2}$ be isotyped $\Theta$-algebras. Let $A=\langle
a_{1},\dots,a_{n}\rangle$ where $a_{1},\dots,a_{n}$ are different elements in
$H_{1}$. Consider the free $\Theta$-algebra $W(X)$, where
$X=\\{x_{1},\dots,x_{n}\\}$. Let $\nu\in\hom(W(X),H_{1})$ defined by
$\nu(x_{i})=a_{i}$ for $1\leqq i\leqq n$. Since $H_{1}$ and $H_{2}$ are
isotyped there exists a point $\mu\in\hom(W(X),H_{2})$ such that
$LKer\nu=LKer\mu$. We obtain a subalgebra
$B=\langle\mu(a_{1}),\dots,\mu(a_{n})\rangle$ of $H_{2}$ and $B=\mu(W(X))$.
Since $Ker\nu=Ker\mu$, algebras $A$ and $B$ are isomorphic.
Let now $A$ be a proper subalgebra of $H_{1}$ and let $a_{n+1}\in
H_{1}\setminus A$. Add to $X$ a new variable $x_{n+1}\not\in X$ and consider a
new point $\nu:W(X\cup\\{x_{n+1}\\})\to H_{1}$ setting $\nu(x_{i})=a_{i}$ for
all $1\leqq i\leqq n+1$. For every $w\in W(X)$ consider the following formula
$v_{w}\in\Phi(X\cup\\{x_{n+1}\\})$:
$v_{w}=\neg(x_{n+1}\equiv w).$
Under condition that $H_{1}$ and $H_{2}$ are isotyped, there exists a point
$\mu\in\hom(W(X\cup\\{x_{n+1}\\}),H_{2})$ such that $LKer\nu=LKer\mu$. Since
$LKer\nu\cap M_{X}=LKer\mu\cap M_{X}$, the subalgebra $B$ generated by
$\mu(x_{1}),\dots,\mu(x_{n})$ is isomorphic to $A$. On the other hand, it is
obvious that $v_{w}\in LKer\nu$ and hence $v_{w}\in LKer\mu$ for every $w\in
W(X)$. The last one means that $\mu(x_{n+1})$ does not belong to $B$, that is,
$B$ is a proper subalgebra of $H_{2}$.
∎
###### Corollary 5.2.
Let a finitely generated algebra $H$ contain no proper subalgebra isomorphic
to $H$. Then every algebra $G$ isotyped to $H$ is isomorphic to $H$.
###### Proof.
Let $H$ and $G$ be isotyped algebras. Since $H$ is finitely generated, there
exists a subalgebra $B$ of $G$ isomorphic to $H$. If $B$ is a proper
subalgebra of $G$ then $H$ contains a proper subalgebra $A$ which is
isomorphic to $B$ and therefore $A$ is isomorphic to $H$ but this is
impossible according to the hypotheses. Thus $B=G$. ∎
We can apply this result to finitely dimensional linear spaces but it is not
the case for finitely generated free Abelian groups. However the next result
can be obtained using Theorem 5.1.
###### Theorem 5.3.
If two Abelian groups are isotyped and one of them is free and finitely
generated then they are isomorphic.
###### Proof.
Let $H$ and $G$ be isotyped Abelian groups and $H$ be free of rank $n$. Then
every finitely generated subgroup of $G$ is isomorphic to a subgroup of $H$.
Therefore every finitely generated subgroup of $G$ is free of a rank $k\leq
n$. This means that every $n+1$ elements of $G$ are linearly dependent. On the
other hand, $H$ is isomorphic to a subgroup $B$ of $G$. Let
$g_{1},\dots,g_{n}$ is a base of $B$. These elements form a maximal linearly
independent system in $G$. We obtain that rank of $G$ is equal to $n$.
It remains to show that $G$ is finitely generated. Let $h_{1},\dots,h_{n}$ be
a base of $H$. Consider the following countable set of formulas
$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})$, indexed by $n$-tuples
$(q_{1},\dots,q_{n})$ of integers, which not all are equal to zero and
formulas $v_{(q_{1},\dots,q_{n},q)}(x_{1},\dots,x_{n})$, indexed by
$n+1$-tuples $(q_{1},\dots,q_{n},q)$ of integers , where $q\not=0$ :
$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})=q_{1}x_{1}+q_{2}x_{2}+\dots+q_{n}x_{n}\not\equiv
0,$ $v_{(q_{1},\dots,q_{n},q)}(x_{1},\dots,x_{n})=\forall
y(q_{1}x_{1}+q_{2}x_{2}+\dots+q_{n}x_{n}+qy\equiv 0\\\
\Longrightarrow\bigvee_{|k_{i}|\leq|\frac{q_{i}}{q}|,i=1,\dots n}y\equiv
k_{1}x_{1}+\dots+k_{n}x_{n}).$
Every such formula is satisfied in $H$ by the tuple
$\bar{h}=(h_{1},\dots,h_{n})$. Indeed, for the formulas
$u_{(q_{1},\dots,q_{n})}(x_{1},\dots,x_{n})$ this statement is obvious.
Consider the formulas $v_{(q_{1},\dots,q_{n},q)}$. Suppose that for an element
$h\in H$ we have $q_{1}h_{1}+q_{2}h_{2}+\dots+q_{n}h_{n}+qh=0$ for some
integers $(q_{1},\dots,q_{n},q)$ and $q\not=0$. Since $(h_{1},\dots,h_{n})$ is
a base, $h=k_{1}h_{1}+\dots+k_{n}h_{n}$ for some integers
$k_{i},\;i=1,\dots,n$. It obvious that $k_{i}=-\frac{q_{i}}{q}$. Thus all
considered formulas belong to $tp^{H}(\bar{h})$.
Since $H$ and $G$ are isotyped, all formulas $u_{(q_{1},\dots,q_{n})}$ and
$v_{(q_{1},\dots,q_{n},q)}$ belong to $tp^{G}(\bar{g})$ for some $n$-tuple
$\bar{g}=(g_{1},\dots,g_{n})$ in $G$. First of all this means that elements
$g_{1},\dots,g_{n}$ are linearly independent. Let $g$ be an arbitrary element
in $G$. Since rank of $G$ is $n$, the elements $g_{1},\dots,g_{n},g$ are
linearly dependent, that is, $q_{1}g_{1}+\dots+q_{n}g_{n}+qg=0$ for some
integers $(q_{1},\dots,q_{n},q)$, which not all are equal to zero. Taking into
account that the first $n$ elements are linearly independent, we conclude that
$q\not=0$. Since $v_{(q_{1},\dots,q_{n},q)}(g_{1},\dots,g_{n})$ is valid in
$G$, we obtain that
$\bigvee_{|k_{i}|\leq|\frac{q_{i}}{q}|,i=1,\dots
n}g=k_{1}g_{1}+\dots+k_{n}g_{n}.$
This means that $g=k_{1}g_{1}+\dots+k_{n}g_{n}$ for some integers
$k_{1},\dots,k_{n}$.
Consequently $G$ it is generated by $g_{1},\dots,g_{n}$, and therefore $G$ is
isomorphic to $H$. ∎
Conjecture. It seems to be probable that analogous result takes place for
nilpotent groups too.
Remark B. Plotkin writes [8] that Z. Sela has proved a similar fact for free
non-commutative groups (unpublished).
## References
* [1] C. C. Chang, H. J. Keisler: _Model Theory_ , North-Holland Publishing Company (1973).
* [2] Chloe Perin and Rizos Sklinos: _Homogeneity in the free group_ , Preprint (2005).ArXiv: math.GR/1003.4095v1
* [3] A.G. Kurosh: _Theory of Groups_ , ”Nauka” (1967)
* [4] W. Magnus, A. Karrass, D. Solitar: _Combinatorial Group Theory_ , IP (1966)
* [5] B. Plotkin: _Seven lectures on the universal algebraic geometry_ , Preprint,(2002), Arxiv:math, GM/0204245, 87pp.
* [6] B. Plotkin: _Algebraic geometry in First Order Logic_ , Sovremennaja Matematika and Applications 22 (2004), p. 16–62. Journal of Math. Sciences, 137, n.5, (2006), p. 5049– 5097. http:// arxiv.org/ abs/ math GM/0312485.
* [7] B. Plotkin: _Some results and problems related to universal algebraic geometry,_ International Journal of Algebra and Computation, 17(5/6), (2007), p. 1133–1164.
* [8] B. Plotkin: _Isotyped algebras._ Arxiv: math.LO/0812.3298v2 (2009). Submitted.
* [9] B. Plotkin, E. Aladova, E. Plotkin: _Algebraic logic and logically-geometric types in varieties of algebras_ , Preprint (2011). ArXiv:math.LO/1108.0573v1
* [10] B. Plotkin, G. Zhitomirski: _Automorphisms of categories of free algebras of some varieties_ , J. Algebra, 306, (2006), no. 2, p. 344 -367.
* [11] B. Plotkin, G. Zhitomirski: _On automorphisms of categories of universal algebras_ , Internat. J. Algebra Comput. 17, (2007), no. 5-6, p. 1115–1132.
* [12] B. Plotkin, G. Zhitomirski: _Some logical invariants of algebras and logical relations between algebras_ , Algebra and Analysis, 19:5, (2007), p. 214–245, St. Peterburg Math. J., 19:5, (2008), p. 859–879.
|
arxiv-papers
| 2012-02-24T11:08:49 |
2024-09-04T02:49:27.802255
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Grigori Zhitomirski",
"submitter": "Grigori Zhitomirski",
"url": "https://arxiv.org/abs/1202.5417"
}
|
1202.5462
|
# Propagation of Vortex Electron Wave Functions in a Magnetic Field
Gregg M. Gallatin
National Institute of Standards and Technology
Center for Nanoscale Science and Technology
Gaithersburg, MD 20899-6203
gregg.gallatin@nist.gov Ben McMorran
Physics Department, University of Oregon, Eugene, OR 97403-1274
###### Abstract
The physics of coherent beams of photons carrying axial orbital angular
momentum (OAM) is well understood and such beams, sometimes known as vortex
beams, have found applications in optics and microscopy. Recently electron
beams carrying very large values of axial OAM have been generated. In the
absence of coupling to an external electromagnetic field the propagation of
such vortex electron beams is virtually identical mathematically to that of
vortex photon beams propagating in a medium with a homogeneous index of
refraction. But when coupled to an external electromagnetic field the
propagation of vortex electron beams is distinctly different from photons.
Here we use the exact path integral solution to Schrodingers equation to
examine the time evolution of an electron wave function carrying axial OAM.
Interestingly we find that the nonzero OAM wave function can be obtained from
the zero OAM wave function, in the case considered here, simply by multipling
it by an appropriate time and position dependent prefactor. Hence adding OAM
and propagating can in this case be replaced by first propagating then adding
OAM. Also, the results shown provide an explicit illustration of the fact that
the gyromagnetic ratio for OAM is unity. We also propose a novel version of
the Bohm-Aharonov effect using vortex electron beams.
## 1 Introduction
Coherent beams of photons carrying axial orbital angular momentum (OAM),
sometimes referred to as vortex beams, are well understood.[1][2][3] and have
various uses in optics and microscopy.[4][5][6][7] Recently electron beams
carrying very high amounts of axial OAM have been generated[8] and the
properties of such beams have been studied.[9][11] Mathematically the
propagation of a vortex photon beam in a medium with a homogeneous index of
refraction is virtually identical to that of a freely propagating vortex
electron beam. This is obviously not the case when the electrons are
propagating in an external electromagnetic field. Here we use the exact path
integral solution to examine how an electron wave function carrying axial OAM
evolves in time. We find that the propagation of a wave function carrying
nonzero axial OAM is equivalent to the the propagation of a zero OAM wave
function multiplied by an appropriate position and time dependent prefactor.
Also, the results provide an explicit illustration of the fact the the (non-
radiatively corrected) gyromagnetic ratio for OAM is unity as it must be.[11]
We will see that from a practical point of view this means that the OAM vector
rotates at half the rate of that the electron circulates in a magnetic field,
i.e., at half the cyclotron or Landau frequency
The paper is organized as follows Section 2 briefly reviews the derivation of
the gyromagnetic ratios for orbital and spin angular momentum from the Dirac
equation Section 3 discusses the path integral solution for the (non-
relativistic) propagation of the electron wave function in a magnetic field.
Section 4 uses the path integral solution to study how a vortex electron beam,
actually a wave packet, evolves in a magnetic and shows explicitly that the
gyromagnetic ratio for OAM is unity.
## 2 Dirac to Schrodinger
For completeness we provide a brief review of the derivation of the
Schrodinger equation from the Dirac equation which shows explicitly that the
(non-radiatively corrected) gyromagnetic ratio for orbital angular momentum is
unity.[10]
The Dirac equation in SI units is
$\left(i\gamma^{\mu}D_{\mu}-mc\right)\psi_{D}\left(\vec{x},t\right)=0$ (1)
where $\psi_{D}$ is a four-component Dirac spinor and
$D_{\mu}=\hbar\partial_{\mu}-ieA_{\mu}.$Here $A_{\mu}$ is the four-vector
potential and $e$ is the electron charge. The indices $\mu,\nu,\cdots$ take
the values 0,1,2,3 which correspond to the $t,x,y,z$ directions, respectively
$x_{0}=ct,x_{1}=x,x_{2}=y,x_{3}=z$. The Einstein summation convention wherein
repeated indices are summed over their appropriate range is used throughout,
e.g., $u_{\mu}v^{\mu}\equiv\sum_{\mu=0}^{3}u_{\mu}v^{\mu}.$
Multiplying Eq (1) by $\left(i\gamma^{\mu}D_{\mu}+mc\right),$ and using
$\displaystyle\gamma^{\mu}\gamma^{\nu}D_{\mu}D_{\nu}$
$\displaystyle=D^{\mu}D_{\mu}-i\sigma^{\mu\nu}\frac{1}{2}\left[D_{\mu},D_{\nu}\right]$
$\displaystyle=D^{\mu}D_{\mu}-\frac{1}{2}e\hbar\sigma^{\mu\nu}F_{\mu\nu}$ (2)
which follows from $\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}$
where $\gamma^{\mu}$ are the gamma matrices, $\eta^{\mu\nu}$is the Minkowski
metric,
$\sigma^{\mu\nu}=\left(i/2\right)\left[\gamma^{\mu},\gamma^{\nu}\right]$ and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
tensor we get[10]
$\left(D^{\mu}D_{\mu}-\frac{1}{2}e\hbar\sigma^{\mu\nu}F_{\mu\nu}+m^{2}c^{2}\right)\psi_{D}\left(\vec{x},t\right)=0$
(3)
Consider a constant magnetic field $B$ pointing the in the $z$ direction.
Using gauge invariance we can write
$A_{0}=0,~{}A_{1}=-\frac{1}{2}Bx_{2}~{},A_{2}=\frac{1}{2}Bx_{1},~{}A_{3}=0$ or
equivalently
$A_{i}=-\epsilon_{ij3}\frac{B}{2}x_{j}=-\frac{B}{2}\epsilon_{ij}x_{j}$. Here
$\epsilon_{ijk}$ and $\epsilon_{ij}s$are the totally antisymmetric Levi-Civita
tensors. $\epsilon_{ijk}$ is $+1\left(-1\right)$ when $i,j,k$ is an even(odd)
permutation of $1,2,3$ and is zero otherwise and $\epsilon_{ij}$ is
$+1\left(-1\right)$ for $i,j=1,2\left(2,1\right)$ and is zero otherwise[10]
Note that $\partial_{i}A_{i}=0$. We now have
$F_{12}=-F_{21}=\partial_{1}A_{2}-\partial_{2}A_{1}=B.$ Working in the so
called ”weak field limit”, i.e. dropping the $\vec{A}^{2}$ term, gives
$\left(\hbar^{2}\left(\frac{1}{c^{2}}\partial_{t}^{~{}2}-\partial_{i}^{~{}2}\right)+ie\hbar
B\left(x_{1}\partial_{2}-x_{2}\partial_{1}\right)-e\hbar\sigma^{12}B+m^{2}c^{2}\right)\psi_{D}\left(\vec{x},t\right)=0$
(4)
In the Dirac basis
$\sigma^{ij}=\epsilon_{ijk}\begin{bmatrix}\sigma^{k}&0\\\
0&\sigma^{k}\end{bmatrix}$ (5)
where the $\sigma^{k}$ are the Pauli matrices.[10] In terms of two-component
spinors $\phi$ and $\chi,$ $\psi_{D}=\begin{bmatrix}\phi\\\ \chi\end{bmatrix}$
and for a slowly moving electron (in the Dirac basis) we can set $\chi=0$ and
so finally
$\left(\hbar^{2}\left(\frac{1}{c^{2}}\partial_{t}^{~{}2}-\partial_{i}^{~{}2}\right)-eBL_{3}-e2BS_{3}+m^{2}c^{2}\right)\phi\left(\vec{x},t\right)=0$
(6)
Here $L_{3}=-i\hbar\left(x_{1}\partial_{2}-x_{2}\partial_{1}\right)$ is the
orbital angular momentum and $S_{3}=\frac{\hbar}{2}\sigma^{3}$ is the spin
angular momentum, both in the $z$ direction. More generally[10] we can write
$\left(\hbar^{2}\left(\frac{1}{c^{2}}\partial_{t}^{~{}2}-\partial_{i}^{~{}2}\right)-e\vec{B}\cdot\left(\vec{L}+2\vec{S}\right)+m^{2}c^{2}\right)\phi\left(\vec{x},t\right)=0$
(7)
for a constant $\vec{B}$ field. Thus we see that the OAM, $\vec{L},$ couples
to the magnetic field as $\vec{B}\cdot\vec{L}$ whereas the spin angular
momentum, $\vec{S},$ couples as $2\vec{B}\cdot\vec{S}$ and so the (non-
radiatively corrected) gyromagnetic ratio for orbital angular momentum
$g_{L}=1$ whereas for spin angular momentum $g_{S}=2.$ This difference has the
effect that electron helicity, i.e., the spin projected in the direction of
propagation, remains tangent to the trajectory, i.e, it rotates at the same
rate that the electron circulates in a magnetic field. We will see below that
because $g_{L}=1$ this is not the case for electron beams carrying axial OAM.
Note that the values of $g_{L}$ and $g_{S}$ are a property of the Hamiltonian
and not of the wave function. The vortex wave function studied below, which
carries nonzero axial OAM, still couples to the magnetic field with a $g_{L}$
value of unity.
## 3 Path Integral Solution for Propagation in a Magnetic Field
We are interested in OAM and not spin and so we will drop the spin term in (7)
and let $\phi\left(\vec{x},t\right)$ be a single component wave function. To
reduce to the nonrelativistic case substitute
$\phi\left(\vec{x},t\right)=e^{-imc^{2}t/\hbar}\psi\left(\vec{x},t\right)$ (8)
with $\psi\left(\vec{x},t\right)$ slowly varying compared to
$\exp\left[-imc^{2}t/\hbar\right]$ into (7) and dropping the
$\partial_{t}^{~{}2}\psi$ term we get the standard Schrodinger equation
$\left(i\hbar\partial_{t}+\frac{\hbar^{2}}{2m}\vec{\partial}^{2}+e\vec{B}\cdot\vec{L}\right)\psi\left(\vec{x},t\right)=0$
(9)
with $\vec{L}=-i\hbar\varepsilon_{ijk}\hat{x}_{i}x_{j}\partial_{k}$ where
$\hat{x}_{i}$ is the unit vector in the $i$ direction.
Because (9) is linear and first order in the time derivative the solution can
be written in the form
$\psi\left(\vec{x},t\right)=\int
d^{3}x^{\prime}K\left(\vec{x},t,\vec{x}^{\prime},t^{\prime}\right)\psi\left(\vec{x}^{\prime},t^{\prime}\right)$
(10)
where $K\left(\vec{x},t,\vec{x}^{\prime},t^{\prime}\right)$ is called the
”propagator” and the integral is nominally over all space. The fact that (9)
is first order in time allows the propagator to be written as a path
integral[10][12][13], i.e.,
$K\left(\vec{x},t,\vec{x}^{\prime},t^{\prime}\right)=\int\limits_{\left(\vec{x}^{\prime},t^{\prime}\right)}^{\left(\vec{x},t\right)}\delta\vec{x}\left(t\right)\exp\left[\frac{i}{\hbar}\int_{t_{a}}^{t_{b}}dt\mathcal{L}\left(\vec{x}\left(t\right),\partial_{t}\vec{x}\left(t\right),t\right)\right]$
(11)
Here
$\mathcal{L}\left(\vec{x}\left(t\right),\partial_{t}\vec{x}\left(t\right),t\right)$
is the classical Lagrangian corresponding to the quantum Hamiltonian, and the
integral is over all paths or trajectories which go from $\vec{x}^{\prime}$ at
time $t^{\prime}$ to $\vec{x}$ at time $t.$ The Lagrangian corresponding to
(9) has the form
$\mathcal{L}\left(\vec{x}\left(t\right),\partial_{t}\vec{x}\left(t\right),t\right)=\frac{1}{2}m\left(\partial_{t}\vec{x}\left(t\right)\right)^{2}-e\vec{A}\left(\vec{x}\left(t\right),t\right)\cdot\partial_{t}\vec{x}\left(t\right)$
(12)
where $\vec{A}$ is the vector potential with the magnetic field
$\vec{B}=\vec{\partial}\times\vec{A}.$ Using the form for $\vec{A}$ given
above we get, for a constant magnetic field in the $z$ direction,
$\mathcal{L}\left(\vec{x}\left(t\right),\partial_{t}\vec{x}\left(t\right)\right)=\frac{m}{2}\left(\partial_{t}\vec{x}\left(t\right)\right)^{2}+\frac{eB}{2}\epsilon_{ij}x_{i}\partial_{t}x_{j}\left(t\right)$
(13)
It should be noted that the Lagrangian in (12) and (13) is the full
Lagrangian, not the weak field approximation . This can be seen simply by
calculating the corresponding classical Hamiltonian which yields
$H=\left(\vec{p}-e\vec{A}\right)^{2}/2m$.with
$\vec{p}=m\partial_{t}x\left(t\right).$
The solution for the propagator with this Lagrangian is
straightforward[12][13], indeed it’s given as a problem in Feynman and Hibbs
book.[14] Transform to a rotating frame in the $xy$ or $1,2$ plane by writing
$x_{i}=\exp\left[\frac{eBt}{2m}\epsilon\right]_{ij}X_{j}\ \ \ \Rightarrow\ \ \
\
\binom{x_{1}}{x_{2}}=\begin{pmatrix}\cos\left[\frac{eBt}{2m}\right]&\sin\left[\frac{eBt}{2m}\right]\\\
-\sin\left[\frac{eBt}{2m}\right]&\cos\left[\frac{eBt}{2m}\right]\end{pmatrix}\binom{X_{1}}{X_{2}}$
(14)
In terms of the new variables the Lagrangian corresponds to free propagation
in the $z$ direction and a harmonic oscillator in the $X_{i},$ $i=1,2$
directions with radian frequency $eB/2m.$ The path integral solutions for free
propagation and for a harmonic oscillator are well known[12][13]. Using these
results and transforming back to the non-rotating coordinates we get
$K\left(\vec{x},t,\vec{x}^{\prime},t^{\prime}\right)=\left(\frac{m}{2\pi
i\hbar
T}\right)^{3/2}\frac{\frac{\omega}{2}T}{\sin\left[\frac{\omega}{2}T\right]}\exp\left[\frac{i}{2\hbar}\left(\begin{array}[c]{c}\frac{m\left(z-z^{\prime}\right)^{2}}{T}+\frac{m\omega}{2}\cot\left[\frac{\omega}{2}T\right]\left(x_{i}-x_{i}^{\prime}\right)^{2}\\\
+m\omega\epsilon_{ij}x_{i}x_{j}^{\prime}\end{array}\right)\right]$ (15)
with
$\omega=\frac{eB}{m}$ (16)
which is the standard cyclotron frequency[13] and $T\equiv t-t^{\prime}.$ In
(15) the combination $\omega T$ always occurs divided by 2 and so we should
expect various aspects of the wave function to evolve at half the rate at
which the electron circulates in the magnetic field.
Note that in the limit as $\omega\rightarrow 0$ the propagator in (15) reduces
to the free propagator
$K_{free}\left(\vec{r}-\vec{r}^{\prime},t-t^{\prime}\right)=\left(\frac{m}{2\pi
i\hbar\left(t-t^{\prime}\right)}\right)^{3/2}\exp\left[\frac{im}{2\hbar}\frac{\left(x_{i}-x_{i}^{\prime}\right)^{2}}{t-t^{\prime}}\right]$
(17)
which is explicitly space and time translation invariant as it should be.
## 4 Evolution of a Gaussian wave function with and without OAM
The propagator given in (15) is Gaussian in form and so if we choose a
Gaussian for the wave function at $t^{\prime}=0$ it will remain Gaussian.
Also, in this case the integral in (10) can be evaluated analytically.
First consider propagation perpendicular to the magnetic field. In this case
let the initial normalized wave function be a Gaussian centered at the origin
and propagating in the $x_{2}=y$ direction
$\psi_{0}\left(\vec{r},0\right)=\frac{1}{\sqrt{\pi\sigma^{2}\sqrt{\pi
L^{2}}}}\exp\left[-\frac{x^{2}+z^{2}}{2\sigma^{2}}-\frac{y^{2}}{2L^{2}}+\frac{i}{\hbar}py\right]$
(18)
where we have switched from the $x_{i}$ notation to the more convenient at
this stage $x,y,z$ notation with $\vec{r}=x\hat{x}+y\hat{y}+z\hat{z}$. This
wave function is roughly $\sigma$ in width in the $x$ and $z$ directions and
has length $L$ in the $y$ direction. If we specify the values of $\omega$ and
the radius $R$ of the classical orbit of the electron then $p=m\omega R.$ If
we take $\sigma$ and $L$ to be much larger than the nominal de Broglie
wavelength of $2\pi\hbar/p$ then we expect mininal ”diffraction” effects to
occur during propagation and as shown explicitly below this is exactly the
case. This initial wave function has zero OAM about it’s direction of
propagation, the $y$ direction, since
$L_{y}\psi_{0}\left(\vec{r},0\right)=i\hbar\left(x\partial_{z}-z\partial_{x}\right)\psi_{0}\left(\vec{r},0\right)=0$
(19)
To generate axial OAM the so called ladder operator approach[15] is used.
Consider an operator $\mathbf{A}$ with eigenstate $\left|a\right\rangle$ such
that $\mathbf{A}\left|a\right\rangle=a\left|a\right\rangle.$ We now want to
generate a state $\left|a+1\right\rangle$ such that
$\mathbf{A}\left|a+1\right\rangle=\left(a+1\right)\left|a+1\right\rangle.$ To
do this we only need to find an operator $\mathbf{B}$ such that
$\left[\mathbf{A},\mathbf{B}\right]=\mathbf{B}$ since then
$\mathbf{AB}\left|a\right\rangle=\mathbf{B}\left|a\right\rangle+\mathbf{BA}\left|a\right\rangle=\left(a+1\right)\mathbf{B}\left|a\right\rangle$
and so the state $\mathbf{B}\left|a\right\rangle=\left|a+1\right\rangle,$ up
to normalization and phase factors. Noting that
$\left[L_{y}/\hbar,\left(\partial_{x}-i\partial_{z}\right)\right]=\left[i\left(x\partial_{z}-z\partial_{x}\right),\left(\partial_{x}-i\partial_{z}\right)\right]=\left(\partial_{x}-i\partial_{z}\right)$
(20)
it follows that a state with 1 unit of axial OAM,
$\psi_{1}\left(\vec{r},0\right),$ is given (up to normalization and phase
factors) by
$\psi_{1}\left(\vec{r},0\right)=\left(\partial_{x}-i\partial_{z}\right)\psi_{0}\left(\vec{r},0\right)=\frac{1}{\sigma^{2}}\left(-x+iz\right)\psi_{0}\left(\vec{r},0\right)=\frac{1}{\sigma^{2}}\rho
e^{i\theta}\psi_{0}\left(\vec{r},0\right)$ (21)
Here $\rho=\sqrt{x^{2}+z^{2}}$ and $\theta$ increases in the counterclockwise
direction when looking in the $-y$ direction and is measured from the $-x$
axis. Using the fact that
$i\left(x\partial_{z}-z\partial_{x}\right)=-i\partial_{\theta}$ we immediately
see that $L_{y}\psi_{1}=\hbar\psi_{1}.$and so $\psi_{1}$ carries one unit of
axial OAM. The factor of $\rho,$ which appears automatically, is necessary
since at $\rho=0$ (= the $y$ axis in this case) the phase
$\exp\left[i\theta\right]$ is not defined and the wave function must vanish
there.
Substituting $\psi_{0}\left(\vec{r},0\right)$ into (10) and using (15) gives
$\displaystyle\psi_{0}\left(\vec{r},t\right)$ $\displaystyle=N\int
d^{3}r^{\prime}\exp\left[\begin{array}[c]{c}\begin{array}[c]{c}\frac{im}{2\hbar
t}\left(z-z^{\prime}\right)^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}\right)\\\
+\frac{im\omega}{2\hbar}\left(xy^{\prime}-yx^{\prime}\right)\end{array}\\\
-\frac{1}{2\sigma^{2}}\left(x^{\prime 2}+z^{\prime
2}\right)-\frac{1}{2L^{2}}y^{\prime 2}+\frac{im\omega
R}{\hbar}y^{\prime}\end{array}\right]$ (26)
$\displaystyle=N\exp\left[\frac{im}{2\hbar
t}z^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(x^{2}+y^{2}\right)\right]$ $\displaystyle\times\int
d^{3}r^{\prime}\exp\left[\alpha_{x}x^{\prime}+\alpha_{y}y^{\prime}+\alpha_{z}z^{\prime}-\frac{1}{2\beta_{x}}x^{\prime
2}-\frac{1}{2\beta_{y}}y^{\prime 2}-\frac{1}{2\beta_{z}}z^{\prime 2}\right]$
$\displaystyle=N\exp\left[\frac{im}{2\hbar
t}z^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(x^{2}+y^{2}\right)\right]$
$\displaystyle\times\sqrt{\left(2\pi\right)^{3}\beta_{x}\beta_{y}\beta_{z}}\exp\left[\frac{1}{2}\beta_{x}\alpha_{x}^{~{}2}+\frac{1}{2}\beta_{y}\alpha_{y}^{~{}2}+\frac{1}{2}\beta_{z}\alpha_{z}^{~{}2}\right]$
(27)
where
$\displaystyle N$ $\displaystyle=\left(\frac{m}{2\pi i\hbar
t}\right)^{3/2}\frac{\frac{\omega t}{2}}{\sin\left[\frac{\omega
t}{2}\right]}\frac{1}{\sqrt{\pi\sigma^{2}\sqrt{\pi L^{2}}}}$
$\displaystyle\alpha_{x}$
$\displaystyle=-\frac{im\omega}{2\hbar}\cot\left[\frac{\omega
t}{2}\right]x-\frac{im\omega}{2\hbar}y$ $\displaystyle\alpha_{y}$
$\displaystyle=-\frac{im\omega}{2\hbar}\cot\left[\frac{\omega
t}{2}\right]y+\frac{im\omega}{2\hbar}x+\frac{im\omega R}{\hbar}$
$\displaystyle\alpha_{z}$ $\displaystyle=-\frac{im}{\hbar t}z$ (28)
$\displaystyle\beta_{x}$
$\displaystyle=\left(\frac{1}{\sigma^{2}}-\frac{im\omega}{2\hbar}\cot\left[\frac{\omega
t}{2}\right]\right)^{-1}$ $\displaystyle\beta_{y}$
$\displaystyle=\left(\frac{1}{L^{2}}-\frac{im\omega}{2\hbar}\cot\left[\frac{\omega
t}{2}\right]\right)^{-1}$ $\displaystyle\beta_{z}$
$\displaystyle=\left(\frac{1}{\sigma^{2}}-\frac{im}{\hbar t}\right)$
To propagate $\psi_{1}$ we can write
$\displaystyle\psi_{1}\left(\vec{r},t\right)$ $\displaystyle=N\int
d^{3}r^{\prime}K\left(\vec{r},t,\vec{r}^{\prime},0\right)\left(\partial_{x^{\prime}}-i\partial_{z/}\right)\psi_{0}\left(\vec{r}^{\prime},0\right)$
$\displaystyle=\frac{N}{\sigma^{2}}\int
d^{3}r^{\prime}K\left(\vec{r},t,\vec{r}^{\prime},0\right)\left(-x^{\prime}+iz^{\prime}\right)\psi_{0}\left(\vec{r}^{\prime},0\right)$
$\displaystyle=\frac{N}{\sigma^{2}}\left.\partial_{\lambda}\int
d^{3}r^{\prime}K\left(\vec{r},t,\vec{r}^{\prime},0\right)\exp\left[\lambda\left(-x^{\prime}+iz^{\prime}\right)\right]\psi_{0}\left(\vec{r}^{\prime},0\right)\right|_{\lambda=0}$
(29)
The integral is still Gaussian and can be evaluated as above by letting
$\alpha_{x}\rightarrow\alpha_{x}-\lambda$ and
$\alpha_{z}\rightarrow\alpha_{z}+i\lambda$ in (27). Taking the derivative with
respect to $\lambda$ and setting $\lambda=0$ then yields
$\displaystyle\psi_{1}\left(\vec{r},t\right)$
$\displaystyle=\frac{N}{\sigma^{2}}\exp\left[\frac{im}{2\hbar
t}z^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(x^{2}+y^{2}\right)\right]$
$\displaystyle\times\sqrt{\left(2\pi\right)^{3}\beta_{x}\beta_{y}\beta_{z}}\left(-\beta_{x}\alpha_{x}+i\beta_{z}\alpha_{z}\right)\exp\left[\frac{1}{2}\beta_{x}\alpha_{x}^{~{}2}+\frac{1}{2}\beta_{y}\alpha_{y}^{~{}2}+\frac{1}{2}\beta_{z}\alpha_{z}^{~{}2}\right]$
$\displaystyle=\left(-\beta_{x}\alpha_{x}+i\beta_{z}\alpha_{z}\right)\frac{1}{\sigma^{2}}\psi_{0}\left(\vec{r},t\right)$
(30)
with $\alpha_{x},\beta_{x},\ldots$the same as in (28).
Even though both these analytic solutions can be manipulated into somewhat
more convenient forms, this is not very illuminating and so we will simply
plot these solutions for a set of conditions which nicely illlustrate the
relevant aspects of their time evolution. On the other hand it is worthwhile
to examine the factor $\left(-\beta_{x}\alpha_{x}+i\beta_{z}\alpha_{z}\right)$
to get a better understanding of how it evolves and controls the orientation
of the OAM. Substituting from above we find, after some algebra,
$f\left(\vec{r},t\right)\equiv-\beta_{x}\alpha_{x}+i\beta_{z}\alpha_{z}=\frac{\cos\left[\frac{\omega
t}{2}\right]x+\sin\left[\frac{\omega
t}{2}\right]y}{\left(\sin\left[\frac{\omega
t}{2}\right]\frac{2\hbar}{im\omega\sigma^{2}}-\cos\left[\frac{\omega
t}{2}\right]\right)}+i\frac{z}{\left(1-\frac{\hbar t}{im\sigma^{2}}\right)}$
(31)
We see that $f\left(\vec{r},0\right)=-x+iz$ at $t=0,$ as it should, and that
it rotates in time in the $xy$ plane at a radian frequency of $\omega/2,$ The
origin of this factor obvious. In operator notation, ignoring the
$1/\sigma^{2}$, (21) becomes
$\left|\psi_{1}\right\rangle=\left(-\mathbf{X}+i\mathbf{Z}\right)\left|\psi_{0}\right\rangle$
(32)
The time evolution is given by
$\displaystyle e^{-i\mathbf{H}t/\hbar}\left|\psi_{1}\right\rangle$
$\displaystyle=e^{-i\mathbf{H}t/\hbar}\left(-\mathbf{X}+i\mathbf{Z}\right)\left|\psi_{0}\right\rangle$
$\displaystyle=\left(e^{-i\mathbf{H}t/\hbar}\left(-\mathbf{X}+i\mathbf{Z}\right)e^{+i\mathbf{H}t/\hbar}\right)e^{-i\mathbf{H}t/\hbar}\left|\psi_{0}\right\rangle$
$\displaystyle=f\left(\overset{\rightarrow}{\mathbf{R}},t\right)e^{-i\mathbf{H}t/\hbar}\left|\psi_{0}\right\rangle$
(33)
where
$\mathbf{H=}\left(\overset{\rightarrow}{\mathbf{P}}-e\vec{A}\left(\overset{\rightarrow}{\mathbf{R}}\right)\right)^{2}/2m$
is the quantum Hamiltonian corresponding to the Lagrangian (13). Note this is
the full Hamiltonian, not the weak field approximation.
The position of the node of $\psi_{1}\left(\vec{r},t\right)$ follows from the
solution to $f\left(\vec{r},t\right)=0.$ At $t=0$ this is the $y$ axis as
shown above. For arbitrary $t$ we have the solution
$\displaystyle y$ $\displaystyle=-\cot\left[\frac{\omega t}{2}\right]x$
$\displaystyle z$ $\displaystyle=0$ (34)
This solution is illustrated in Figure 1 for several values of $t$. This
”nodal line” rotates only by $\pi$ during one full period, $\tau=2\pi/\omega,$
of the electron cyclotron orbit and since this factor is the origin of the OAM
carried by $\psi_{1}$ this shows explicity that the OAM rotates at half the
cyclotron frequency, i.e., $g_{L}=1.$ This also shows that the OAM is axially
oriented only at times $t=n\tau,$ with $n=0,1,2,\cdots$, and its direction
switches between being parallel and antiparallel to the direction of
propagation at each of these times.
Figure 1: The graph shows the nodal lines (red) at different positions in the
electron orbit. The OAM lies along the nodal lines and thus rotates at half
the cyclotron frequency $\omega=eB/m.$
Note that $\psi_{0}\left(\vec{r},t\right)$ and
$\psi_{1}\left(\vec{r},t\right)$ are not simply propagating Gaussian envelope
functions multiplied by a propagating plane wave factor of the form
$\exp\left[i\vec{p}\cdot\vec{r}/\hbar-iEt/\hbar\right]$ with
$\left|\vec{p}\right|$ constant (but rotating at radian frequency $\omega)$
and $E=\left|\vec{p}\right|^{2}/2m$. For both wave functions the de Broglie
wavelength varies in time. This is to be expected since the coupling to the
vector potential contributes an extra phase to the wave function of the form
$-i/\hbar\int_{0}^{t}dt\vec{A}\left(\vec{r}\right)\cdot\partial_{t}\vec{r}\left(t\right)$
which varies with position in generally an nonlinear fashion . Figures 2 and 3
show slices of the modulus squared and the real parts of $\psi_{0}$ and
$\psi_{1}$ in the $xy$ plane at different positions in the electron orbit. The
values chosen for $\sigma,L,\omega$ and $R$ are such that the size of the wave
packet at $t=0$, $L$ in the $y$ direction and $\sigma$ in the $x$ direction
are both much larger than the wavelength (so that diffraction effects are
minimal) and $R$ is much larger than $L$. The actual ratios used for the plots
are $R=10^{3}L,~{}L=10\sigma$ and $\sigma\simeq 10^{5}2\pi\hbar/m\omega$ hence
the spatial range of the $\operatorname{Re}\left[\psi_{0}\right]$ and
$\operatorname{Re}\left[\psi_{1}\right]$ plots is about 5 orders of magnitude
smaller than for the $\left|\psi_{0}\right|^{2}$ and
$\left|\psi_{1}^{2}\right|$ plots so that the phase variation is visible. In
Figure 2 we see that the long axis of the wave function tracks the nodal line
and the spatial extent of the wave function varies with period $\tau$ and thus
the length and width return, up to diffraction effects to their initial values
at every $t=\tau,~{}2\tau,~{}3\tau,\cdots.$ This periodic variation in the
spatial extent of the wave function can be traced back to the fact that in the
rotating frame the Lagrangian is that of a harmonic oscillator.The free
propagation part of the Langrangian, $m\left(\partial_{t}x\right)^{2}/2$ cause
the wave function to expand or diffract as it propagates. The harmonic
oscillator part, $m\omega^{2}\vec{x}^{2}/2$ causes the wave function to
contract and unless these two effects are precisely balanced the wave function
will oscillate in size This is exactly analogous to the propagation of a
paraxial Gaussian optical beam.centered on the $z$ axis and propagating in the
$z$ direction in a medium with an index of refraction of the form
$n\left(x,y\right)=n_{0}-c\left(x^{2}+y^{2}\right)$, i.e, a harmonic
osciallator potential. In the paraxial approximation the propagator for the
photon beam has the same Gaussian form as the propagator for the harmonic
oscillator. The quadratic variation of the index of refraction will case the
beam to focus or shrink in size as it propagates whereas diffraction effects
cause the beam to expand as it propagates. If the beam is large, so that the
focusing effect dominates, then the beam will shrink in size as it propagates.
Eventually it reaches a size where the diffraction effect dominates and it
begins to expand. This process repeats itself causing the beam to oscillate in
size with a fixed period along its length.[16] These oscillations can be
prevented if the size of the beam is fine tuned so that the diffraction and
focusing effects exactly cancel out.[16] Figure 3 shows the propagation of the
wave function $\psi_{1}$ carrying a single unit of OAM. The node in the center
of the wave function maintains its alignment on the nodal line during each
cycle. The spiral form the phase of $\psi_{1}$ is apparent in the
$\operatorname{Re}\left[\psi_{1}\right]$ plots. Clearly the OAM is rotating at
half the cyclotron frequency $\omega$.
Figure 2: Slices in the $xy$ plane of $\left|\psi_{0}\right|^{2}$ and
$\operatorname{Re}\left[\psi_{0}\right]$ at different positions around the
cyclotron orbit where $\psi_{0}$ is a Gaussian wavepacket carrying 0 axial
orbital angular momentum(OAM). The values chosen for the width $\sigma$ and
length $L$ of the wavepacket, the cyclotron frequency $\omega=eB/m,$ and the
radius of the cycloctron orbit $R$ are such that the size of the wave packet
at $t=0$ ($L$ in the $y$ direction and $\sigma$ in the $x$ direction) are much
larger than the wavelength so that diffraction effects are minimal. All the
plots are the same fixed spatial scale with that of the
$\operatorname{Re}\left[\psi_{0}\right]$ plots being about 5 orders of
magnitude smaller than the $\left|\psi_{0}\right|^{2}$ plots so that the phase
of the wavepacket is visible. At $t=0.5\tau$ the wavepacket would be too small
to be seen at this fixed spatial scale and so it is shown at times $t=0.4\tau$
and $t=0.6\tau$ instead. Figure 3: Slices in the $xy$ plane of
$\left|\psi_{1}\right|^{2}$ and $\operatorname{Re}\left[\psi_{1}\right]$ at
different positions around the cyclotron orbit where $\psi_{1}$ is a Gaussian
wavepacket carrying 1 unit axial orbital angular momentum(OAM) oriented in the
$y$ direction at $t=0$. The values chosen for the width $\sigma$ and length
$L$ of the wavepacket, the cyclotron frequency $\omega=eB/m,$ and the radius
of the cycloctron orbit $R$ are the same as in Figure 2, i.e., they are such
that the size of the wave packet at $t=0$ ($L$ in the $y$ direction and
$\sigma$ in the $x$ direction) are much larger than the wavelength so that
diffraction effects are minimal. All the plots are the same fixed spatial
scale with that of the $\operatorname{Re}\left[\psi_{1}\right]$ plots being
about 5 orders of magnitude smaller than the $\left|\psi_{1}\right|^{2}$ plots
so that the phase of the wavepacket is visible. At $t=0.5\tau$ the wavepacket
would be too small to be seen at this fixed spatial scale and so it is shown
at times $t=0.4\tau$ and $t=0.6\tau$ instead.
Now consider propagation parallel to the magetic field. In this case we let
$\psi_{0}\left(\vec{r},0\right)=\frac{1}{\sqrt{\pi\sigma^{2}\sqrt{\pi
L^{2}}}}\exp\left[-\frac{x^{2}+y^{2}}{2\sigma^{2}}-\frac{z^{2}}{2L^{2}}+\frac{i}{\hbar}pz\right]$
(35)
and
$\displaystyle\psi_{0}\left(\vec{r},t\right)$ $\displaystyle=N\int
d^{3}r^{\prime}\exp\left[\begin{array}[c]{c}\begin{array}[c]{c}\frac{im}{2\hbar
t}\left(z-z^{\prime}\right)^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}\right)\\\
+\frac{im\omega}{2\hbar}\left(xy^{\prime}-yx^{\prime}\right)\end{array}\\\
-\frac{1}{2\sigma^{2}}\left(x^{\prime 2}+y^{\prime
2}\right)-\frac{1}{2L^{2}}z^{\prime
2}+\frac{ip}{\hbar}z^{\prime}\end{array}\right]$ (40)
$\displaystyle=N\exp\left[\frac{im}{2\hbar
t}z^{2}+\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]\left(x^{2}+y^{2}\right)\right]$ $\displaystyle\times\int
d^{3}r^{\prime}\exp\left[\alpha_{x}x^{\prime}+\alpha_{y}y^{\prime}+\alpha_{z}z^{\prime}-\frac{1}{2\beta_{\rho}}\left(x^{\prime
2}+y^{\prime 2}\right)-\frac{1}{2\beta_{z}}z^{\prime 2}\right]$
$\displaystyle=N\sqrt{\left(2\pi\right)^{3}\beta_{\rho}^{~{}2}\beta_{z}}$
$\displaystyle\times\exp\left[\begin{array}[c]{c}\left(\frac{im\omega}{4\hbar}\cot\left[\frac{\omega
t}{2}\right]-\frac{1}{2}\beta_{\rho}\left(\frac{m\omega}{2\hbar\sin\left[\frac{\omega
t}{2}\right]}\right)^{2}\right)\left(x^{2}+y^{2}\right)\\\
-\beta_{z}\left(\frac{m}{\hbar
t}\right)^{2}\left(z-\frac{p}{m}t\right)^{2}+\frac{im}{2\hbar
t}z^{2}\end{array}\right]$ (43)
where $N$ is the same as in (28) but now
$\displaystyle\beta_{\rho}$
$\displaystyle=\left(\frac{1}{\sigma^{2}}-\frac{im\omega}{2\hbar}\cot\left[\frac{\omega
t}{2}\right]\right)^{-1}$ $\displaystyle\beta_{z}$
$\displaystyle=\left(\frac{1}{L^{2}}-\frac{im}{\hbar t}\right)$ (44)
Because $\psi\left(\vec{r},t\right)$ depends on $x$ and $y$ only in the
combination $\rho^{2}=x^{2}+y^{2}$ it follows that the initial Gaussian wave
function chosen here does not pick up angular momentum as it propagates along
the magnetic field. In fact for propagation parallel to the magnetic field the
axial OAM of an eigenstate of $\mathbf{L}_{z}$ is conserved. This follows
directly from
$\left[\mathbf{L}_{z}\mathbf{,H}\right]=0$ (45)
where again
$\mathbf{H=}\left(\overset{\rightarrow}{\mathbf{P}}-e\vec{A}\left(\overset{\rightarrow}{\mathbf{R}}\right)\right)^{2}/2m$
and
$\mathbf{A}_{i}\mathbf{=-}\frac{B}{2}\epsilon_{ij}\mathbf{X}_{j}\mathbf{.}$
Indeed it can be shown that
$\mathbf{H}=\frac{1}{2m}\overset{\rightarrow}{\mathbf{P}}^{2}-\frac{eB}{2m}\mathbf{L}_{z}+\frac{e^{2}B^{2}}{2m}\left(\mathbf{X}^{2}+\mathbf{Y}^{2}\right)$
which obviously yields (45).
## 5 Conclusion
Using the exact path integral solution for the propagator in a constant
magnetic field we have derived the evolution of a Gaussian wave function and
shown explicitly that the (non-radiatively corrected) gyromagnetic ratio
$g_{L}$ for OAM is unity. This must be the case since $g_{L}$ is a property of
the Hamiltonian and not of the wave function.
The results presented above a novel version of the Aharonov-Bohm effect.[17]
Consider a long thin solenoid aligned along the $z$ axis. Outside the solenoid
(far from the ends) $\vec{A}$ varies as $1/\rho=1/\sqrt{x^{2}+y^{2}}$ and so
$\vec{B}$ is zero outside. Inside the solenoid $\vec{A}$ varies as $\rho$ and
so $\vec{B}$ is constant and nonzero. A Gaussian wave function like those
considered above carrying nozero OAM that propagates along the $z$ axis has a
node on the $z$ axis. In fact wave functions carrying large values of OAM have
a very large region around the $z$ axis where the wave function is effectively
zero.[8] As in the standard Aharonov-Bohm experiment[17] this is a case where
there is no overlap between the wave function and the magnetic field. The wave
function only overlaps with the magnetic vector potential. Hence the presence
of the solenoid will cause a change in how the wave function propagates
relative to the no solenoid case. This effect will be predominantly a change
in the focus position of the wave function. Experimental verification of this
would provide yet another example of the fact $A_{\mu}$ is the fundamental
quantity and not $\vec{E}$ and $\vec{B}.$
## References
* [1] Mark R. Dennis, Kevin O’Holleran, Miles J. Padgett, ”Singular Optics: Optical Vortices and Polarization Singularities”, Chapter 5, Progress in Optics, vol. 53, 293-363, Elsevier (2009).
* [2] Miles Padgett, Johannes Courtial and Les Allen, ”Light’s Angular Momentum”, Physics Today, May 2004, p 35.
* [3] U. D. Jentschura and B. G. Serbo, ”Generation of High-Energy Photons with Large Orbital Angular Momentum by Compton Backscattering”, Phys. Rev. Letts. 106, 013001 (2011).
* [4] Sri Rama Prasanna Pavani and Rafael Peistun, ”High-efficiency rotating point spread functions”, Opt. Exp. 16, 3484 (2008).
* [5] Gabriel Molina-Terriza, Juan P. Torres, and Lluis Torner, ”Twisted Photons”, Nat. Phys. 3, p. 305 (2007).
* [6] Sri Rama Prasanna Pavani, Michael A. Thompson, Julie S. Biteen, Samuel J. Lord, Na Liu, Robert J. Twieg, Rafael Piestun and W. E. Moerner, ”Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function”, PNAS 106, p. 2995 (2009).
* [7] Michael A. Thompson, Matthew D. Lew, Majid Badieirostami and W. E. Moerner, ”Localizing and Tracking Single Nanoscale Emitters in Three Dimensions with High Spatiotemporal Resolution Using a Double-Helix Point Spread Function”, Nano Lett. 10, p. 211 (2010).
* [8] Benjamin J. McMorran, Amit Agrawal, Ian M. Anderson, Andrew A. Herzing, Henri J. Lezec, Jabez J. McClelland, and John Unguris, ”Electron Vortex Beams with High Quanta of Orbital Angular Momentum”, Science 331, p 192 (2011).
* [9] J. Verbeek, H. Tian, and P. Schattschneider, ”Production and application of electron vortex beams”, Nature 467, p. 301 (2010).
* [10] A. Zee, Quantum Field Theory in a Nutshell, Chapter III.6, 2nd ed., Princeton University Press (2010).
* [11] Konstantin Yu. Bliokh, Mark R. Dennis and Franco Nori, ”Relativistic Electron Vortex Beams: Angular Momentum and Spin-Orbit Interaction”, Phys. Rev. Lett. 107, 174802 (2011).
* [12] Richard P. Feynman, Albert R. Hibbs, and Daniel F. Styer, Quantum Mechanics and Path Integrals: Emended Edition, Dover Publications (2010).
* [13] Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, Chapter 2.18, World Scientific Publishing Company (2009).
* [14] Richard P. Feynman, Albert R. Hibbs, and Daniel F. Styer, Quantum Mechanics and Path Integrals: Emended Edition, Problem 3-10, Dover Publications (2010).
* [15] see for example, J. J. Sakurai and Jim J. Napolitano, Modern Quantum Mechanics, 2nd edition, Addison Wesley (2010).
* [16] see for example, Amnon Yariv and Pochi Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Chapter 2, Wiley-Interscience (2002).
* [17] see for example, A. Zee, Quantum Field Theory in a Nutshell, Chapter IV.4, 2nd ed., Princeton University Press (2010).
|
arxiv-papers
| 2012-02-24T14:42:47 |
2024-09-04T02:49:27.812172
|
{
"license": "Public Domain",
"authors": "Gregg M. Gallatin and Ben McMorran",
"submitter": "Gregg Gallatin",
"url": "https://arxiv.org/abs/1202.5462"
}
|
1202.5579
|
# The suppression of magnetism and the development of superconductivity within
the collapsed tetragonal phase of Ca0.67Sr0.33Fe2As2 at high pressure
J. R. Jeffries Condensed Matter and Materials Division, Lawrence Livermore
National Laboratory, Livermore, CA 94550, USA N. P. Butch Condensed Matter
and Materials Division, Lawrence Livermore National Laboratory, Livermore, CA
94550, USA K. Kirshenbaum Center for Nanophysics and Advanced Materials,
Department of Physics, University of Maryland, College Park, MD 20742, USA S.
R. Saha Center for Nanophysics and Advanced Materials, Department of Physics,
University of Maryland, College Park, MD 20742, USA S. T. Weir Condensed
Matter and Materials Division, Lawrence Livermore National Laboratory,
Livermore, CA 94550, USA Y. K. Vohra Department of Physics, University of
Alabama at Birmingham, Birmingham, Alabama 35294, USA J. Paglione Center for
Nanophysics and Advanced Materials, Department of Physics, University of
Maryland, College Park, MD 20742, USA
###### Abstract
Structural and electronic characterization of (Ca0.67Sr0.33)Fe2As2 has been
performed as a function of pressure up to 12 GPa using conventional and
designer diamond anvil cells. The compound (Ca0.67Sr0.33)Fe2As2 behaves
intermediate between its end members—CaFe2As2 and SrFe2As$2$—displaying a
suppression of magnetism and the onset of superconductivity. Like other
members of the AEFe2As2 family, (Ca0.67Sr0.33)Fe2As2 undergoes a pressure-
induced isostructural volume collapse, which we associate with the development
of As-As bonding across the mirror plane of the structure. This collapsed
tetragonal phase abruptly cuts off the magnetic state, giving rise to
superconductivity with a maximum $T_{c}$=22.2 K. The maximum $T_{c}$ of the
superconducting phase is not strongly correlated with any structural
parameter, but its proximity to the abrupt suppression of magnetism as well as
the volume collapse transition suggests that magnetic interactions and
structural inhomogeneity may play a role in its development. The pressure-
dependent evolution of the ordered states and crystal structures in
(Ca,Sr)Fe2As2 provides an avenue to understand the generic behavior of the
other members of the AEFe2As2 family.
superconductivity, magnetism, x-ray diffraction, pressure
###### pacs:
74.62.Fj, 74.70.Xa, 75.50.Ee, 61.50.Ks
## I Introduction
Since the first reports of superconductivity with a critical temperature
$T_{c}$=26 K in fluorine-doped LaFeAsO,Kamihara2008 researchers have rapidly
expanded the number of Fe-based superconductors,Ishida2009 raised the $T_{c}$
to about 55 K,Ren2008 and identified five different, but related, crystal
structures in which these Fe-based superconductors crystallize.Paglione2010 ;
Johnston2010 Like the cuprate superconductors,Lee2006 the different Fe-based
superconductors display many common themes in both the electronic and
structural properties: the presence of corrugated Fe-pnictogen or Fe-chalcogen
layers within a tetragonal unit cell, and the occurrence of antiferromagnetic
order in the undoped or ambient-pressure compounds.Lumsden2010 The ubiquity
of these common elements makes these systems fertile playgrounds to explore
the interplay between magnetism, structure, and superconductivity.
One of the archetypal Fe-based superconductor structures is the “122”
structure: AEFe2X2 (ThCr2Si2-type), with AE an alkaline earth element (Ca, Sr,
Ba) an alkali metal (K, Rb, Cs) or Eu and X a pnictogen element.Paglione2010 ;
Johnston2010 ; Lumsden2010 ; Gooch2010 ; Jeevan2008 Variants of the 122
structure have been widely studied owing to the availability of a wide range
of chemical substitutions on different crystallographic sites (e.g., Co for
Fe, K for Ba, P for As, etc.) as well as their tendency to form macroscopic,
high-purity crystals. The parent compounds within the 122 systems are
paramagnetic metals at room temperature, but at low temperatures each member
of the 122 family exhibits a concomitant structural and magnetic transition.
Despite their different structural/magnetic transition temperatures—spanning a
range greater than 100 K—and chemical compositions, each of the 122 parent
compounds displays the same low-temperature structural and magnetic phases.
The tetragonal I4/mmm space group stable at room temperature undergoes a
distortion that leads to a low-temperature orthorhombic (Fmmm space group)
crystal structure, where the basal plane of the orthorhombic unit cell is
rotated by 45∘ with respect to that of the tetragonal unit cell.Rotter2008 ;
Huang2008 At ambient pressure or without doping, spin-density-wave (SDW),
antiferromagnetic (AFM) order occurs simultaneous with the tetragonal-
orthorhombic structural transition. The AFM state is characterized by a (101)
wavevector (note: the magnetic and orthorhombic unit cells are identical),
yielding Fe moments directed along the orthorhombic $a$-axis that are
antiferromagnetically arranged along $a$ and $c$ (between Fe layers) and
ferromagnetically coupled along $b$. In contrast to the wide range of AFM
ordering temperatures ($T_{N}$), the ordered moment within the 122 family
varies only slightly between 0.80 and 1.01 $\mu_{B}$.Goldman2008 ; Kaneko2008
; Su2009
With applied pressure or doping, both the structural and AFM transitions are
generally suppressed. In the case of Co or K doping in BaFe2As2, the nominally
concomitant structural and magnetic transitions separate from one another,
with the structural transition preceding the magnetic transition upon
cooling.Ni2008 ; Chu2009 ; Pratt2009 ; Urbano2010 Near the suppression of the
AFM state, with either doping or pressure, superconductivity arises with
critical temperatures ranging from roughly 9-47 K. Han2009 ; Saha2012 A
notable exception to this general observation is the lack of superconductivity
in CaFe2As2 under highly hydrostatic pressure conditions.Yu2009 In addition
to superconductivity, an isostructural volume collapse to a collapsed
tetragonal phase is seen as a function of pressure as well as for a small
minority of chemical substitutions.Saha2012 ; Goldman2009 ; Uhoya2010a ;
Uhoya2010b ; Uhoya2011 ; Mittal2011 ; Danura2011 The proximity of the
superconducting state with both structural and magnetic instabilities has
prompted suggestions that the maximum $T_{c}$ in the 122 family of compounds
could be controlled by structural parameters, magnetic interactions, or
both.Ishida2009 ; Paglione2010 ; Johnston2010 ; Lumsden2010 ; Yildrim2009
Each of these factors could have ramifications on the pairing symmetry of the
superconducting state itself,Mazin2009 and, as such, exploring the
relationships between superconductivity, magnetism, and structural
instabilities is an important component of understanding the unconventional,
high-temperature superconductivity seen in the ferropnictide compounds.
In this article we report a pressure-dependent structural and electrical
transport study of (Ca0.67Sr0.33)Fe2As2. The isoelectronic substitution of Sr
for Ca in this pseudobinary alloy expands the ambient-pressure lattice volume
and rapidly increases $T_{N}$ close to that of SrFe2As2 with with the addition
of approximately 30% Sr.Kirshenbaum2012 The effects of the larger volume and
the enhanced $T_{N}$ are to expand the phase space occupied by the AFM state
to higher temperatures and higher pressures relative to that of pure CaFe2As2,
thus pushing the destruction of magnetism to higher pressures and allowing for
a larger region of study under pressure.
## II Experimental Details
Single crystals of (Ca0.67Sr0.33)Fe2As2 were synthesized with a flux-growth
technique previously described.Saha2009 The samples were verified with x-ray
diffraction to crystallize in the I4/mmm ThCr2Si2-type crystal structure with
ambient-pressure lattice constants $a$=3.907 Å and $c$=11.988 Å.
Pressure-dependent electrical transport measurements were performed using two
pressure cells: (i) a hydrostatic clamp cell employing n-pentane:isoamyl
alcohol as a pressure-transmitting medium was used up to 1.1 GPa; and (ii) a
designer diamond anvil cell (DAC) loaded with quasihydrostatic solid steatite
as a pressure-transmitting medium was used for pressures above 1.76 GPa. The
designer DAC was composed of a 300-$\mu$m culet, 8-probe designer diamond
anvilWeir2000 ; Patterson2000 ; Jackson2006 paired with a matching standard
diamond anvil. In order to facilitate electrical contact with the sample,
tungsten contact pads were lithographically deposited onto the microprobes
exposed at the culet of the designer diamond anvil. A non-magnetic MP35N
gasket was pre-indented to a thickness of 40 $\mu$m and a 130-$\mu$m hole was
drilled in the center of the indentation by means of an electric discharge
machine (EDM). A small, thin crystallite (approximately 70 $\times$ 70
$\times$ x 20 $\mu$m) was placed on the culet of the designer diamond anvil in
contact with the tungsten contact pads. The pressure was calibrated using the
shift in the R1 fluorescence line of ruby.Mao1986 ; Vos1991 The ruby R2
fluorescence line remained distinguishable from the R1 line to pressures in
excess of 7 GPa, implying a nearly hydrostatic environment below 7 GPa.
Temperature-dependent, electrical resistance measurements were performed in a
commercial cryostat.
For x-ray diffraction measurements, the DAC was composed of a pair of opposed
diamond anvils with 700-$\mu$m culets and a nickel gasket. The gasket was pre-
indented to a thickness of 65 $\mu$m and a 250-$\mu$m hole was drilled in the
center of the indentation with an EDM. The (Ca0.67Sr0.33)Fe2As2 crystals were
crushed in a mortar and pestle and loaded into the sample space along with a
few small ruby chips for initial pressure calibration and fine Cu powder (3-6
$\mu$m, Alfa Aesar) for in situ x-ray pressure calibration. A 4:1
methanol:ethanol mixture served as the pressure-transmitting medium.
Room-temperature, angle-dispersive x-ray diffraction (ADXD) experiments were
performed at the HPCAT beamline 16 BM-D of the Advanced Photon Source at
Argonne National Laboratory. A 5x10 $\mu$m, 29.2 keV (${\lambda}_{inc}$=0.4246
Å) incident x-ray beam, calibrated with CeO2, was used. The diffracted x-rays
were detected with a Mar345 image plate; exposure times ranged from 300-600
seconds. 2D diffraction patterns were collapsed to 1D intensity versus
2$\Theta$ plots using the program FIT2D.Hammersley1996 Pressure-dependent
lattice parameters were extracted by indexing the positions of the Bragg
reflections using the EXPGUI/GSAS package.Larson1994 ; Toby2001
## III Results
### III.1 Crystal structure
A typical, powder x-ray diffraction pattern for (Ca0.67Sr0.33)Fe2As2, taken at
a pressure of 1.98 GPa in the DAC, is shown in Fig. 1. The Bragg reflections
corresponding to the tetragonal I4/mmm structure of (Ca0.67Sr0.33)Fe2As2 are
indicated by the red tickmarks below the data (thin, black crosses), for which
the background has been subtracted.The green stars represent the positions of
the Bragg peaks of the Cu pressure marker. The diffraction pattern is well
described, as indicated by the absence of additional peaks in the pattern, by
including only a combination of (Ca0.67Sr0.33)Fe2As2 and Cu. The
(Ca0.67Sr0.33)Fe2As2 specimen displays a preferred orientation, with the
crystallites of the powder tending to form small platelets aligned with the
$c$-axis parallel to surface of the culet of the diamond anvil (i.e., parallel
to the incident x-ray beam). This preferred orientation results in a relative
decrease in the intensity of the (00l) reflections and an increase in the
intensity of the (hk0) reflections. Nonetheless, a full refinement (red line
through data) of the diffraction pattern results in a good fit to the data,
allowing for determination of the lattice parameters as well as the
$z$-coordinate of the As atoms.
Figure 1: (color online) An example x-ray diffraction pattern acquired at 1.98
GPa in a DAC. The refinement runs through the data points as a red line, and
the residual of the refinement is shown below the pattern as a light blue
line. Bragg reflections of the (Ca0.67Sr0.33)Fe2As2 sample are shown as red
tickmarks, while Bragg peaks from the Cu pressure marker are indicated by the
green stars.
The structural parameters extracted from refinements of the x-ray diffraction
data under pressure are shown in Fig. 2 up to 12 GPa. Ambient-pressure values
are from [Saha2011, ]. With increasing pressure, the $c$-axis of the
tetragonal unit cell monotonically decreases, but with a steeper slope between
roughly 2 and 6 GPa. The $a$-axis, on the other hand, increases with pressure
within the same 2-6 GPa range, followed by a more conventional compression for
pressures in excess of 6 GPa. The unit cell volume and the $c/a$ ratio (Fig.
2b) naturally reflect the pressure dependences of the lattice parameters, with
both quantities exhibiting an increased slope centered around 4 GPa.
Figure 2: (color online) Structural parameters extracted from refinements of
x-ray diffraction patterns under pressure: (a) tetragonal lattice parameters
$c$ (red circles, left axis) and $a$ (blue squares, right axis), (b) unit cell
volume (green diamonds, left axis) and $c/a$ ratio (orange triangles, right
axis). The inset shows the refined z-coordinate of the As site as a function
of pressure. In all cases, lines are guides to the eye.
These pressure-dependent evolution of the structural parameters shown in Fig.
2 indicate the presence of an isostructural volume collapse, identical to that
seen in the other pure, alkaline-earth 122 compounds: CaFe2As2, SrFe2As2, and
BaFe2As2. The structural parameters all exhibit inflection points near 4 GPa,
providing a consistent estimate for the volume-collapse transition pressure
(vertical, grey bar in Fig. 2) in (Ca0.67Sr0.33)Fe2As2. Above 4 GPa,
(Ca0.67Sr0.33)Fe2As2 is in the collapsed tetragonal phase. The $z$-coordinate
of the As atoms, a free parameter within the ThCr2Si2 structure, also exhibits
an anomaly near the volume-collapse transition as seen in the inset of Fig.
2b. The $z$-coordinate increases slightly at low pressures before exhibiting a
significant increase near 4 GPa. Above 4 GPa, the $z$-coordinate decreases
before increasing and recovering toward the general trend (dashed line) seen
at low pressure, suggesting a correlation between the As atoms and onset of
the collapsed tetragonal phase.
### III.2 Electrical transport
The electrical resistivity $\rho$ as a function of temperature for selected
pressures is presented in Fig. 3. The electrical resistivity data have been
normalized such that the ambient pressure value of $\rho$(300 K) is equal to
one. In the ambient-pressure curve, the concomitant magnetic and structural
transition is evident as a pronounced jump near 200 K. With applied pressure,
$\rho$(300 K) decreases and the magnetic/structural transition is smoothly
suppressed, disappearing between 1.10 and 1.76 GPa. There is no evidence
suggesting a splitting of the structural and magnetic transitions. Within the
magnetic state at 0.87 and 1.10 GPa, there is a rapid reduction in resistivity
just below 20 K; this behavior is reminiscent of the strain-induced
superconductivity observed in pure SrFe2As2 crystals,Saha2009 although a full
resistive transition is lacking here within the magnetic state of
(Ca0.67Sr0.33)Fe2As2.
Figure 3: (color online) Normalized electrical resistivity as a function of
temperature for selected pressures (denoted in GPa unless otherwise
specified). The magnetic/structural transition and the onset of
superconductivity are visible in (a). The downward arrows ($T_{ct}$) indicate
inflection points in the electrical resistivity curves (see text). The
evolution of the superconducting transition with pressure is highlighted in
(b).
At 1.76 GPa, the lowest measured pressure where the signature of the
magnetic/structural transition is no longer visible, the electrical
resistivity displays an inflection point (downward arrows in Fig. 3a) near 80
K and full resistive superconducting transition at $T_{c}$=22.2 K. Higher
pressures reduce $T_{c}$, as clearly seen in Fig. 3b, but substantially
increase the temperature of the inflection point. Previous experiments on
CaFe2As2 revealed a similar occurrence and pressure-dependent behavior of this
inflection point.Torikachvili2008a
The features extracted from the pressure-dependent electrical resistivity
measurements are collected in the phase diagram of Fig. 4. The closed, red
squares represent the onset of magnetic order ($T_{N}$) and its associated
structural transition, while the closed, blue circles reveal the evolution of
$T_{c}$ with pressure. The open, blue circles at 0.87 and 1.10 GPa indicate
incomplete transitions possibly associated with strain-induced filamentary
superconductivity. The inflection point, seen for $P>$1.76 GPa, is shown as
open, green squares. The pressure dependence of the inflection point in the
electrical resistivity intersects with the volume collapse transition pressure
at room temperature (described in III.1), leading to the conclusion that the
inflection point seen in the temperature-dependent electrical resistivity
signifies the onset of the collapsed tetragonal phase, a conclusion consistent
with previous pressure-dependent and Rh-substitution studies of
CaFe2As2Torikachvili2008a ; Torikachvili2008b ; Yu2009 ; Canfield2009 ;
Danura2011 We thus denote this feature (i.e., the inflection point in the
electrical resistivity) as $T_{ct}$. Bulk superconductivity, as inferred from
a complete resistive transition, is seemingly limited to the collapsed
tetragonal phase.
Figure 4: (color online) Phase diagram of (Ca0.67Sr0.33)Fe2As2 showing the
suppression of magnetism ($T_{N}$ \- red squares), the development of
superconductivity ($T_{c}$ \- blue circles), and the progression of the volume
collapse transition ($T_{ct}$ \- green, crossed squares) with pressure. The
room-temperature value of $T_{ct}$ is determined from x-ray diffraction data,
all other data points are from electrical transport measurements. The open,
blue circles at lower pressures represent incomplete superconducting
transitions. Lines and shaded regions are guides to the eye.
## IV Discussion
Given the strong link between the appearances of superconductivity and the
collapsed tetragonal phase in the 122 ferropnictide family of superconductors,
it is naturally important to explore what driving mechanisms or correlations
may be responsible for each phenomena.
### IV.1 Isostructural volume collapse
At room temperature, the isostructural volume collapse in (Ca0.67Sr0.33)Fe2As2
occurs near 4 GPa (III.1); the volume collapse transition shifts to lower
pressures with reduced temperature (III.2). The evolution of the position of
the As atoms within the unit cell (given by the $z$-coordinate in Fig. 2) upon
passing through the volume collapse transition suggests that the As atoms are
involved in this transition.
Figure 5: (color online) Pressure dependence of the As-As distance ($d_{As-
As}$) across the mirror plane of the crystal structure shown in the inset.
Lines through the data are guides to the eye.
Figure 5 shows the interlayer As-As spacing across the mirror plane of the
unit cell, $d_{As-As}$, as a function of pressure at room temperature. From
ambient pressure, the As-As spacing decreases continuously, and nearly
linearly, with applied pressure, reaching a value of $d_{As-As}$=3.06 Å at 3.2
GPa. Between 3.2 and 3.8 GPa, $d_{As-As}$ abruptly decreases to a value of
2.94 Å, a 4% reduction occurring over 0.6 GPa. Further pressure causes a
continuous, monotonic decrease in $d_{As-As}$ up to the highest pressure
measured. The onset of the collapsed tetragonal phase in (Ca0.67Sr0.33)Fe2As2,
therefore, is signified by a collapse in the As-As separation across the
mirror plane of the unit cell. From Fig. 5, the midpoint of this collapse
occurs when $d_{As-As}$=3.0 Å.
The tendency for a collapse across the mirror plane of the ThCr2Si2-type
structure has been discussed previously. Hoffman and Zheng formulated this
collapse for BaMn2P2,Hoffman1985 but the effect can be generalized to other
compounds with this structure. For the purpose of discussion we refer to a
general formula AB2X2. The basic description of the collapse put forth by
Hoffman and Zheng is predicated on the chemistry of the B2X${}_{2}^{-2}$
layer, which yields a schematic density of states (Fig. 6a) with X-X bonding
and anti-bonding $p$-states separated by the $d$-states arising from the B
atom. As the atomic number of B increases within a row of the Periodic Table,
the Fermi level shifts downward, leaving the anti-bonding $p$-states of the
density of states unfilled, resulting in the development of an X-X bond across
the mirror plane of the structure. X-X bonding across the width of the unit
cell does not occur because that dimension is fixed by the $a$ lattice
parameter, which is at least partly set by the size of the A cation and
typically larger than 3.5 Å.Just1996 There is thus a chemical route to
describing the uncollapsed and collapsed tetragonal phases of the
ThCr2Si2-type structure, which sheds light on the mechanism under pressure.
Figure 6: (color online) (a) Schematic density of states of the
B2As${}_{2}^{-2}$ layers for a hypothetical AB2As2 compound. The Fermi level
lowers with increasing d-band occupancy, depleting the As-As anti-bonding
states, creating an As-As mirror plane bond, and collapsing the structure. (b)
Schematic unit cell energy (adapted from [Hoffman1985, ]) versus $d_{As-As}$
for a hypothetical AB2As2 specimen; pressure or d-electron element
substitution should push $d_{As-As}$ leftward, providing a driving force for
the collapsed phase.
Full electronic structure calculations by Hoffman and Zheng indicate that for
BaMn2P2 there is a maximum structural energy (Fig. 6b) when the P-P distance
is about 2.7 Å.Hoffman1985 In BaMn2P2, the P-P bond length is thus shifted
roughly 0.5 Å above the bare P-P bond length.Pyykko2009 If a similar value of
the X-X bond-length shift occurs in (Ca0.67Sr0.33)Fe2As2, then the As-As bond
length would shift from the bare As-As bond length of 2.4 Å to about 2.9 Å, in
excellent agreement with the value $d_{As-As}$=3.0 Å defining the volume
collapse transition pressure.
The onset of such As-As bonding would naturally be directed along the $c$-axis
of the unit cell (inset, Fig. 5), and would tend to pull the previously weakly
connected FeAs cages toward one another, accounting for the contraction of the
crystallographic $c$-axis upon entering the collapsed tetragonal phase. By
conservation, the increase in bonding between mirror plane As atoms would
likely reduce the Fe-As bond strength, which would relax the FeAs cages, alter
the Fe-As bond angles, and increase the $a$-axis of the unit cell as seen
experimentally.
In addition to the structural consequences of the development of this new As-
As bond within the structure, there are likely electronic structure effects.
Band structure calculations conclude that the transition into the collapsed
tetragonal phase results in a downward shift of the bands relative to the
uncollapsed phase and a reduction in the density of states at the Fermi
level.Yildrim2009 ; Goldman2009 Indeed, magnetotransport measurements in
rare-earth doped CaFe2As2 show a dramatic reduction in the magnitude of the
Hall coefficient upon cooling through the volume collapse transition.Saha2012
In addition, first-principles calculations for CaFe2As2 show that the strength
of the Fe-As bonds, the As-As mirror plane bonds, and the Fe spin-state, and
henceforth magnetism, are strongly coupled.Yildrim2009 Thus, the
disappearance of magnetic order with the onset of the collapsed tetragonal
phase may be an unsurprising consequence of the electronic structure mandated
by the collapsed tetragonal phase.
The onset of As-As interlayer bonding, as indicated by the contraction of
$d_{As-As}$, is not unique to (Ca0.67Sr0.33)Fe2As2. In fact, the other pure
alkaline earth 122 ferropnictide superconductors as well as some of their
rare-earth-doped counterpartsSaha2012 display identical behavior in $d_{As-
As}$, albeit at different pressures. Figure 7 shows $d_{As-As}$ as a function
of pressure for members of the (AE)Fe2As2 family. The horizontal line
represents the onset of As-As bonding at $d_{As-As}$=3.0 Å. Each compound has
been shown to undergo the volume collapse transition, and, accordingly, each
displays an abrupt contraction of the As-As separation. As the size of the
alkaline earth element increases, the unit cell volume of the crystal
structure increases, and $d_{As-As}$ increases. A natural consequence of this
unit cell volume expansion is that a larger pressure is required to achieve
sufficient lattice compression to invoke As-As bonding across the mirror plane
of the unit cell, and the volume collapse transition concordantly shifts to
higher pressures with increasing atomic radius of the alkaline earth element
(inset, Fig. 7).
Figure 7: (color online) Comparison of the pressure dependence of $d_{As-As}$
for the alkaline earth (AE)Fe2As2 compounds. The horizontal dashed line
indicates $d_{As-As}=$3.0 Å. Data for CaFe2As2, SrFe2As2, and BaFe2As2 are
from references [Goldman2009, ], [Uhoya2011, ], and [Mittal2011, ],
respectively. Inset: volume collapse transition pressure, $P_{ct}$, as a
function of unit cell volume at $P$=0. Lines through data points are guides to
the eye.
### IV.2 As-Fe-As bond angles
Early studies of the iron-bearing oxypnictide superconductors noted an
empirical correlation between the As-Fe-As bond angles and the maximum
observed $T_{c}$.Ishida2009 ; Lee2008 Within the corrugated FeAs layers of
the crystal structure, there are two As-Fe-As bond angles: the two-fold or
intralayer angle, denoted as $\alpha$; and the four-fold or interlayer angle,
denoted as $\beta$ (see the inset of Fig. 8).Johnston2010 Due to the crystal
structure, these two bond angles move oppositely (as $\alpha$ increases,
$\beta$ decreases), but if the As atoms surrounding the Fe atoms are perfectly
tetrahedrally coordinated, then $\alpha$=$\beta$=109.47∘. In reference
[Lee2008, ], it was found that as the As-Fe-As bond angles of LnFeAsO1-y
approached this ideal tetrahedral angle, $T_{c}$ reached a maximum value near
55 K. Since then, this general empirical relationship has been noted in nearly
all families of ferropnictide superconductors.Paglione2010 ; Johnston2010
However, like many “rules” in condensed-matter physics, there are exceptions,
notably CsFe2As2 with a low $T_{c}$=2.6 K and As-Fe-As bond angles of 109.58∘
and 109.38∘.Gooch2010
Figure 8: (color online) (a) The room-temperature values of the two-fold,
$\alpha$, and four-fold, $\beta$, As-Fe-As bond angles as a function of
pressure. The ideal tetrahedral angle (109.47∘) is marked by the horizontal,
green, dashed line. The inset defines $\alpha$ and $\beta$ within the
corrugated FeAs component of the crystal structure. (b) The pressure
dependence of $T_{c}$ on the same pressure axis; symbols identical to Fig. 4.
Lines are guides to the eye.
Figure 8a displays the pressure dependence of the $\alpha$ and $\beta$ As-Fe-
As bond angles at room temperature. At ambient pressure, the corrugated FeAs
layers exhibit coordination close to the ideal tetrahedral configuration
($\alpha$=109.93∘, $\beta$=109.24∘). Applied pressure drives the structure
away from this ideal tetrahedral coordination, and, above the volume collapse
transition, the bond angles settle into relatively pressure-independent values
significantly disparate from the ideal tetrahedral condition.
The evolution of $T_{c}$ with pressure is reproduced in Fig. 8b for comparison
with the bond-angle evolution. While $T_{c}$ seems to be correlated with the
bond angles, with $T_{c}$ decreasing as the bond angles deviate from that of
the ideal tetrahedron, it should be emphasized that a one-to-one
correspondence is likely too simple of an explanation. The bond angle data in
Fig. 8a was determined at room temperature, while the determination of $T_{c}$
is clearly at low temperature. From the phase diagram in Fig. 4, it can be
seen that the superconducting phase occurs within the collapsed tetragonal
phase. If the relatively constant bond angles seen in the collapsed tetragonal
phase at room temperature are representative of that phase even at low
temperatures, then one might expect that the maximum in $T_{c}$ would occur
with bond angles $\alpha{\approx}$116∘ and $\beta{\approx}$106∘, distinctly
deviating from the ideal tetrahedral angle. Unfortunately, no low-temperature
structural data were acquired in this study, but low-temperature structural
characterization of pure and rare-earth doped CaFe2As2, which still exhibit
superconductivity, indicate that the bond angles tend away from the ideal
tetrahedral angle upon cooling.Kreyssig2008 ; Saha2012 Furthermore, the phase
diagram of Fig. 4 reveals that the superconducting state occurs in proximity
to the destruction of magnetic order, its associated structural transition,
and the occurrence of an isostructural volume collapse, certainly suggesting
that the appearance of superconductivity may be correlated with factors other
than structural parameters at room temperature.
### IV.3 Structural and electronic phase diagrams
The structural and electronic phase diagram of (Ca0.67Sr0.33)Fe2As2
interpolates very well with the phase diagrams of its end member compounds as
well as the related BaFe2As2 compound. These phase diagram are shown together
in Fig. 9, highlighting the qualitative similarities within the AEFe2As2
system. Each compound obeys some general behavioral rules. At ambient-
pressure, each compound undergoes a structural/magnetic transition ($T_{N}$)
at sub-ambient temperatures. $T_{N}$ is suppressed with applied pressure and
abruptly disappears well above $T=0$. Superconductivity develops around this
discontinuous destruction of magnetism and persists as a several-GPa-wide dome
or half-dome in P-T space with a maximum $T_{c}$ occurring close to the
pressure at which $T_{N}$ abruptly vanishes. Each compound exhibits a
pressure-induced volume collapse at room temperature, and the volume collapse
transition $T_{ct}$ occurs with positive slope in P-T space (i.e.,
$dT_{c}/dP>0$) where it has been measured. The pressure axis of Fig. 9 does
not extend far enough to include the volume collapse transition in BaFe2As2,
which, from the data shown in Fig. 7, occurs near 26 GPa.Mittal2011
In CaFe2As2 and (Ca0.67Sr0.33)Fe2As2, measurements of the $T_{ct}(P)$ line
strongly suggest that the volume collapse itself is likely responsible for the
abrupt destruction of magnetic order. This further implies that magnetism is
limited to the orthorhombic phase, and that the collapsed tetragonal phase
does not support magnetic order. While the destruction of magnetism may be
linked to the onset of the collapsed tetragonal phase, whether that
destruction is driven by a reduction in the Fe moments, an altering of some
exchange coupling, or a more subtle change in the electronic structure is an
open question likely requiring both theoretical and experimental input to
reach a conclusion.
Figure 9: (color online) Comparison of the temperature-pressure phase diagrams
for (AE)Fe2As2. Closed circles represent $T_{N}$, open diamonds represent
$T_{c}$, and crossed or diagonal boxes represent $T_{ct}$. The volume collapse
transition as determined by electrical transport and x-ray diffraction are
denoted by ET and XRD, respectively. Electrical transport data for CaFe2As2,
SrFe2As2, and BaFe2As2 are from [Torikachvili2008a, ; Torikachvili2008b, ;
Colombier2009, ]; structural data are from [Goldman2009, ; Uhoya2011, ;
Mittal2011, ]. Lines and shaded regions are guides to the eye.
Unlike the volume collapse transition, the facts about the occurrence of
superconductivity in the AEFe2As2 systems are less clear. With the use of more
hydrostatic pressure conditions, researchers have found that the
superconducting dome, as defined by complete resistive transitions or
susceptibility data, in the parent compounds is generally excluded from the
magnetic, orthorhombic phase.Alireza2009 ; Colombier2009 The results on
(Ca0.67Sr0.33)Fe2As2 are consistent with this finding even with the less
hydrostatic steatite pressure-transmitting medium used in this study. However,
it is imperative to note that a study using He as the pressure-transmitting
medium revealed the absence of superconductivity in both the orthorhombic
(magnetic) and the collapsed tetragonal phases of CaFe2As2.Yu2009 A simple,
quantitative shift in the pressure at which superconductivity appears as a
function of hydrostaticity would not be entirely surprising, but the
qualitative difference (i.e., a complete lack of superconductivity) as a
function of pressure media creates a conundrum regarding the appearance of
superconductivity in the AEFe2As2 systems.
While the generic phase diagrams of chemical substitution and doping look
strikingly similar, there are subtle differences that question the roles of
structural and magnetic instabilities and suggest that a one-to-one
correspondence between pressure and substitution is too simple. Studies of
electron- and hole-doped BaFe2As2 have revealed a splitting of the nominally
coupled paramagnetic-antiferromagnetic and tetragonal-orthorhombic phase
transitions with increasing dopant content.Ni2008 ; Chu2009 ; Pratt2009 ;
Urbano2010 Superconductivity is seen to develop within the antiferromagnetic,
orthorhombic phase with both Co and K substitution in BaFe2As2,Ni2008 but the
optimal $T_{c}$ is achieved within the paramagnetic, uncollapsed tetragonal
phase. Though superconductivity develops within the antiferromagnetic state,
the suppression of that state, at least to some degree, seems to be a
necessary ingredient for superconductivity. Unlike doping of the BaFe2As2 end
member, Rh-doping into CaFe2As2 does not reveal a significant splitting of the
paramagnetic-antiferromagnetic and tetragonal-orthorhombic phase transitions,
but $T_{N}$ is nonetheless suppressed with increasing doping.Danura2011
Superconductivity is seen only in the paramagnetic, uncollapsed tetragonal
phase, and the onset of a doping-induced collapsed tetragonal phase destroys
superconductivity. Recently, a doping study using rare earth (RE) elements in
Ca1-xRExFe2As2 has shown that the components of electron doping and chemical
pressure can be effectively separated, and that each of these components plays
a different role in manifesting superconductivity.Saha2012 Within the
Ca1-xRExFe2As2 series, superconductivity occurs in either the uncollapsed or
collapsed tetragonal phases, but not within the antiferromagnetic,
orthorhombic phase.
Assimilating the doping- and pressure-dependent phase diagrams of the 122
systems in order to better understand the nature of the high-temperature
superconductivity seen therein is a challenging problem. Given the
overwhelming evidence that electron- or hole-doping plays an important role in
the development of superconductivity within the chemically substituted
AEFe2As2 systems and given the occurrence of strain-induced superconductivity
in SrFe2As2,Saha2009 it is tempting to posit that the small superconducting
dome seen under pressure may be a product of some effective doping induced by
non-hydrostatic pressure conditions, possibly resulting not only from the
pressure-transmitting medium but from the volume collapse transition itself.
Alternatively, without high-fidelity, low-temperature structural data under
pressure, it is difficult to exclude structural phase inhomogeneity (e.g.,
coexistence of the collapsed and uncollapsed tetragonal phases) as a possible
cause of the proximity of superconductivity to the observed structural
instabilities. More work demarcating the $T_{ct}(P)$ lines in SrFe2As2 and
BaFe2As2 under pressure may help to illuminate any possible connections
between pressure-induced superconductivity and the isostructural volume
collapse.
## V Conclusions
The compound (Ca0.67Sr0.33)Fe2As2 under pressure behaves intermediate between
its two end member parent compounds. With applied pressure, the concomitant
structural/magnetic transition is suppressed with no evidence favoring a
splitting of the two nominally coupled transitions. The AFM, orthorhombic
phase is abruptly cut off by an isostructural volume collapse resulting in a
collapsed tetragonal phase. The volume collapse transition is driven by the
development of As-As bonding across the mirror plane of the crystal structure,
contracting the $c$-axis of the unit cell and likely affecting the Fe-As
bonding and potentially the magnetic state of the Fe atoms. The collapsed
tetragonal phase supports superconductivity with a maximum $T_{c}$ near 22 K.
There is no obvious structural parameter that defines the magnitude of
$T_{c}$, but the proximity of the superconducting phase to the suppression of
magnetism as well as the onset of the collapsed tetragonal phase suggests that
magnetic interactions and/or structural inhomogeneity may both play a role in
the development of pressure-induced superconductivity in these systems.
## VI Acknowledgments
We are grateful to Z. Jenei and K. Visbeck for assistance with cell
preparations. JRJ and STW are supported by the Science Campaign at Lawrence
Livermore National Laboratory. Portions of this work were performed under LDRD
(11-LW-003). Lawrence Livermore National Laboratory is operated by Lawrence
Livermore National Security, LLC, for the U.S. Department of Energy, National
Nuclear Security Administration under Contract DE-AC52-07NA27344. Portions of
this work were performed at HPCAT (Sector 16), Advanced Photon Source (APS),
Argonne National Laboratory. HPCAT is supported by CIW, CDAC, UNLV and LLNL
through funding from DOE-NNSA, DOE-BES and NSF. Use of the Advanced Photon
Source, an Office of Science User Facility operated for the U.S. Department of
Energy (DOE) Office of Science by Argonne National Laboratory, was supported
by the U.S. DOE under Contract No. DE-AC02-06CH11357. Beamtime was provided
through the Carnegie-DOE Alliance Center (CDAC). This work was partially
supported by AFOSR-MURI Grant No. FA9550-09-1-0603. YKV acknowledges support
from DOE-NNSA Grant No. DE-FG52-10NA29660.
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|
arxiv-papers
| 2012-02-24T23:14:01 |
2024-09-04T02:49:27.826029
|
{
"license": "Public Domain",
"authors": "J. R. Jeffries, N. P. Butch, K. Kirshenbaum, S. R. Saha, S. T. Weir,\n Y. K. Vohra, and J. Paglione",
"submitter": "Jason Jeffries",
"url": "https://arxiv.org/abs/1202.5579"
}
|
1202.5676
|
# sums of two biquadrates and elliptic curves of rank $\geq 4$
F.A. Izadi Mathematics Department Azarbaijan university of Tarbiat Moallem ,
Tabriz, Iran f.izadi@utoronto.ca , F. Khoshnam Mathematics Department
Azarbaijan university of Tarbiat Moallem , Tabriz, Iran khoshnam@azaruniv.edu
and K. Nabardi Mathematics Department Azarbaijan university of Tarbiat
Moallem , Tabriz, Iran nabardi@azaruniv.edu
(Date: Februry 25, 2012.)
###### Abstract.
If an integer $n$ is written as a sum of two biquadrates in two different
ways, then the elliptic curve $y^{2}=x^{3}-nx$ has rank $\geq 3$. If moreover
$n$ is odd and the parity conjecture is true, then it has even rank $\geq 4$.
Finally, some examples of ranks equal to $4$, $5$, $6$, $7$, $8$ and $10$, are
also obtained.
###### Key words and phrases:
elliptic curves, rank, biquadrates, sums of two biquadrates, parity conjecture
###### 2000 Mathematics Subject Classification:
Primary 11G05; Secondary 10B10
## 1\. Introduction
Let $E$ be an elliptic curve over $\mathbb{Q}$ defined by the Weierstrass
equation of the form
(1.1) $E:y^{2}=x^{3}+ax+b\quad\ a,b\in\mathbb{Q}.$
In order the curve $(1.1)$ to be an elliptic curve, it must be smooth. This in
turn is equivalent to requiring that the cubic on the right of Eq. $(1.1)$
have no multiple roots. This holds if and only if the $discriminant$ of
$x^{3}+ax+b$, i.e., $\Delta=-16(4a^{3}+27b^{2})$ is non-zero.
By the Mordell-Weil theorem, the set of rational points on $E$ i.e.,
$E(\mathbb{Q})$ is a finitely generated abelian group, i.e.,
$E(\mathbb{Q})\simeq E(\mathbb{Q})_{{\rm{tors}}}\oplus\mathbb{Z}^{r},$
where $E(\mathbb{Q})_{{\rm{tors}}}$ is a finite group called the torsion group
and $r$ is a non-negative integer called the Mordell-Weil rank of
$E(\mathbb{Q})$. In this paper, we consider the family of elliptic curves
defined by
$E_{n}:y^{2}=x^{3}-nx,$
for positive integers $n$ written as sums of two biquadrates in two different
ways, i.e.,
$n=p^{4}+q^{4}=r^{4}+s^{4},$
where the greatest common divisor of all the numbers $p,q,r,s$ is one. Such a
solution is referred to as a primitive solution. In what follows we deal only
with numbers $n$ having primitive solution. This Diophantine equation was
first proposed by Euler [7] in 1772 and has since aroused the interest of
numerous mathematicians. Among quartic Diophantine equations it has a distinct
feature for its simple structure, the almost perfect symmetry between the
variables and the close relationship with the theory of elliptic functions.
The latter is demonstrated by the fact that this equation is satisfied by the
four elliptic theta functions of Jacobi,
$\vartheta_{1},\vartheta_{2},\vartheta_{3},\vartheta_{4}$, in that order [19].
Here in this note, we show that it also has an obvious relationship with the
theory of elliptic curves. To this end, we need some parametric solutions of
the equation for which we use the one that was constructed by Euler as:
(1.2)
$\left\\{\begin{array}[]{ll}p=a^{7}+a^{5}b^{2}-2a^{3}b^{4}+3a^{2}b^{5}+ab^{6},\\\
q=a^{6}b-3a^{5}b^{2}-2a^{4}b^{3}+a^{2}b^{5}+b^{7},\\\
r=a^{7}+a^{5}b^{2}-2a^{3}b^{4}-3a^{2}b^{5}+ab^{6},\\\
s=a^{6}b+3a^{5}b^{2}-2a^{4}b^{3}+a^{2}b^{5}+b^{7}.\\\ \end{array}\right.$
(See Hardy and Wright [8] page 201, problem no.(13.7.11)). It is easy to see
that the two different integers $n_{1}$ and $n_{2}$ having primitive solutions
are independent modulo $\mathbb{Q}^{\ast 4}.$ For let $n_{1}$ and $n_{2}$ be
two such numbers in which $(p_{1},q_{1},r_{1},s_{1})$ is the solution for
$n_{1}$ and $n_{2}=k^{4}n_{1}$ for non-zero rational number $k$. It follows
that $(kp_{1},kq_{1},kr_{1},ks_{1})$ is a solution for $n_{2}$ which is not
primitive. We see that this condition is sufficient for the curves $E_{n_{1}}$
and $E_{n_{2}}$ to be non-isomorphic over $\mathbb{Q}$ (the dependence modulo
$\mathbb{Q}^{\ast k}$ for $k=0,1,2,3$ expresses one curve as the quartic
twists of the other). However, it is not plain that there are infinitely many
integers having primitive solution. To remedy this difficulty, Choudhry [4]
presented a method of deriving new primitive solutions starting from a given
primitive solution. This makes it possible to construct infinitely many non-
isomorphic elliptic curves using the primitive solutions of the biquadrate
equation. Our main results are the following:
###### Theorem 1.1.
If an integer $n$ is written as a sum of two biquadrates in two different
ways, then the elliptic curve $y^{2}=x^{3}-nx$ has rank $\geq 3$. If moreover
$n$ is odd and the parity conjecture is true, then it has even rank $\geq 4$.
###### Remark 1.2.
Our numerical results suggest that the odd ranks for even numbers should be at
least 5.
## 2\. Previous works
For questions regarding the rank, we assume without loss of generality that
$n\not\equiv 0\pmod{4}$. This follows from the fact that $y^{2}=x^{3}-nx$ is a
$2$-isogenous to $y^{2}=x^{3}+4nx$. These curves form a natural family in the
sense that they all have $j$-invariant $j(E)=1728$ regardless of the different
values or various properties that the integers $n$ may have. There have been a
lot of investigations concerning the distribution of ranks of elliptic curves
in natural families, and it is believed that the vast majority of elliptic
curves $E$ over $\mathbb{Q}$ have rank $\leq 1$. Consequently, the
identification of elliptic curves of rank $\geq 2$ is of great interest.
Special cases of the family of the curves $E_{n}$ and their ranks have been
studied by many authors including Bremner and Cassels [3], Kudo and Motose
[10], Maenishi [11], Ono and Ono [13], Spearman [17, 18], and Hollier,
Spearman and Yang [9]. The general cases were studied by Aguirre, Castaneda,
and Peral [1].
The main purpose of Aguirre et al., [1] was to find the elliptic curves of
high rank in this family without restricting $n$ to have any prescribed
property. They developed an algorithm for general $n$, and used it to find 4
curves of rank 13 and 22 of rank 12.
Breamner and Cassels [3] dealt with the case $n=-p$, where $p\equiv 5\pmod{8}$
and less than $1000.$ The rank is always 1 in accordance with the conjecture
of Selmer and Mordell. For each prime in this range, the authors found the
generator for the free part. In some cases the generators are rather large,
the most startling being that for $p=877$, the $x$ has the value
$x=\left(\frac{612776083187947368101}{7884153586063900210}\right)^{2}.$
Kudo and Motose [10] studied the curve for $n=p$, a Fermat or Mersenne prime
and found ranks of $0$, $1$, and $2$. More precisely,
1. (1)
For a Fermat prime $p=2^{2^{n}}+1,$
$E(Q)\cong\left\\{\begin{array}[]{ll}\mathbb{Z}/2\mathbb{Z}&{\rm for}\ p=3\\\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}&{\rm for}\ p=5\\\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}&{\rm for}\
p>5.\end{array}\right.$
2. (2)
In case $p=2^{q}-1$ is a Mersenne prime where $q$ is a prime,
$E(Q)\cong\left\\{\begin{array}[]{ll}\mathbb{Z}/2\mathbb{Z}&{\rm for}\ p=3\\\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}&{\rm for}\ p>3.\end{array}\right.$
Maenishi [11] investigated the case $n=pq$, where $p,q$ are distinct odd
primes and found a condition that the rank of $E_{pq}$ equals 4. This can be
done by taking natural numbers $A,B,C,D$ and two pairs $p$ and $q$ satisfying
the equations:
$pq=A^{2}+B^{2}=2C^{2}-D^{4}=S^{4}-4t^{4}\qquad(p=s^{2}-2t^{2},q=s^{2}+2t^{2}).$
Then using these equations one can construct 4 independent points on the
corresponding elliptic curve.
In [13] the authors examined the elliptic curves for $n=b^{2}+b$, where $b\neq
0,-1$ is an integer, and show that, subject to the parity conjecture, one can
construct infinitely many curves $E_{b^{2}+b}$ with even rank $\geq 2$. To be
more precise they obtained the followings:
Let $b\neq 0,-1$ be an integer for which $n=b^{2}+b$, is forth power free, and
define $T$ by
$T:={\rm{card}}\\{p\ |\ primes\ 3\leq p\equiv 3\pmod{4},\ p^{2}\parallel
b^{2}+b\\}.$
* 1.
If $b\equiv 1,2\pmod{4}$ and $T$ is odd, then $E(b)$ has even rank $\geq$2.
* 2.
If $b\equiv 7,8,11,12,20,23,24,28,35,39,40,43,51,52,55,56\pmod{64}$ and $T$ is
even, then $E(b)$ has even rank $\geq$2.
* 3.
If $b\equiv 3,14,19,27,36,44,59,60\pmod{64}$ and $T$ is odd, then $E(b)$ has
even rank $\geq$2.
* 4.
In all other cases, $E(b)$ has odd rank.
In two separate papers, Spearman [17], [18] gave the following two results:
(1) If $n=p$ for an odd prime $p$ written as $p=u^{4}+v^{4}$ for some integers
$u$ and $v$, then
$E(\mathbb{Q})={\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}\oplus{\mathbb{Z}}.$
(2) If $n=2p,$ where $2p=(u^{2}+2v^{2})^{4}+(u^{2}-2v^{2})^{4}$ for some
integers $u$ and $v$, then
$E(\mathbb{Q})={\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}\oplus{\mathbb{Z}}\oplus\overline{}{\mathbb{Z}}.$
In recent paper Spearman along with Hollier and Yang [9] assuming the parity
conjecture constructed elliptic curves of the form $E_{-pq}$ with maximal rank
4, here $p\equiv 1\pmod{8}$ and $q$ be an odd prime different from $p$
satisfying
$q=p^{2}+24p+400.$
Finally, Yoshida [20] investigated the case $n=-pq$ for distinct odd primes
$p,q$ and showed that for general such $p,q$ the rank is at most 5 using the
fact that
${\rm{rank}}(E_{n}(\mathbb{Q}))\leq 2\\#\\{l\ {\rm prime;\ divides}\ 2n\\}-1.$
If $p$ is an odd prime, the rank of $E_{p}(\mathbb{Q})$ is much more
restricted, i.e.,
${\rm{rank}}(E_{p}(\mathbb{Q}))\leq\left\\{\begin{array}[]{lll}0&{\rm{if}}&p\equiv
7,11\pmod{16}\\\ 1&{\rm{if}}&p\equiv 3,5,13,15\pmod{16}\\\ 2&{\rm{if}}&p\equiv
1\pmod{8}.\end{array}\right.$
If the Legendre symbol $(q/p)=-1$ and $q-p\equiv\pm 6\pmod{16}$, then
$E_{-pq}(\mathbb{Q})=\\{\mathcal{O},(0,0)\\}\cong\mathbb{Z}/2\mathbb{Z}.$
If $p,q$ are twin prime numbers, then $E_{-pq}(\mathbb{Q})$ has a non-torsion
point $(1,(p+q)/2)$. If $p,q$ be twin primes with $(q/p)=-1,$ then
$E_{pq}(\mathbb{Q})\cong\mathbb{Z}\oplus{\mathbb{Z}}/2\mathbb{Z}.$
Having introduced the previous works, one can easily see that all the elliptic
curves including those in our family share three main properties in common.
They have the same $j$-invariant $j(E)=1728$, have positive rank (except for
the case $p=3$ in the Kudo-Motose [10] paper with rank zero), and have the
torsion group $T=\mathbb{Z}/2\mathbb{Z}$, as we will see in the next section.
In spite of these similarities our family has almost higher ranks among all
the other families and can be taken as an extension of the previous results.
Before we proceed to the proofs, we wish to make the following remarks.
###### Remark 2.1.
Our result for odd $n$ is conditional on the parity conjecture. In [2] the
authors using the previous version of our work proved the following two
results unconditionally.
Theorem 1. The family $y^{2}=x^{3}-nx,$ with $n=p^{4}+q^{4}$ has rank at least
2 over $\mathbb{Q}(p,q).$
Theorem 2. The family $y^{2}=x^{3}-nx,$ in which $n$ given by the Euler
parametrization has rank at least $4$ over $\mathbb{Q}(a)$, where $a$ is the
parameter and $b=1.$
One may prove both results by a very straightforward way. For the first
theorem, we note that, by the same reasons as in [2] not only the point
$Q(p,q)=(-p^{2},pq^{2})$, but also the point $R(p,q)=(-q^{2},qp^{2})$ is on
the curve. Then the specialization by $(p,q)=(2,1)$ gives rise to the points
$Q=(-4,2)$ and $R=(-1,4)$. Therefore by using the Sage software, we see that
the associated height matrix has non-zero determinant $1.8567$ showing that
the points are independent. For the second theorem, we see that the points
$Q_{1}(a)=(-p^{2},pq^{2}),$ $Q_{2}(a)=(-q^{2},qp^{2}),$
$Q_{3}(a)=(-r^{2},rs^{2})$ and $Q_{4}(a)=(-s^{2},sr^{2})$ are on the curve and
the specialized points for $a=2$ gave rise to
$Q_{1}=(-24964,549998),\ Q_{2}=(-3481,-1472876),$ $Q_{3}=(-17956,2370326),\
Q_{4}=(-17689,2388148).$
By using the Sage software we find that the elliptic height matrix associated
to $\\{Q_{1},Q_{2},Q_{3},Q_{4}\\}$ has non-zero determinant $5635.73654$
showing that again the 4 points are independent.
###### Remark 2.2.
We see that the map $(u,v)\rightarrow(-u^{2},uv^{2})$ from the quadric curve:
$u^{4}+v^{4}=n$ to the elliptic curve: $y^{2}=x^{3}-nx$ takes the integral
points of the first to the integrals of the second. Now to find the integral
points of the quadric, it is enough to find the integrals of the elliptic
curve. This might suggests that to find $n$ with more representations as sums
of two biquadrates, the corresponding elliptic curve should have many
independent integral points.
## 3\. Method of Computation
To compute the rank of this family of elliptic curves, a couple of facts are
necessary from the literature. We begin by describing the torsion group of the
family. To this end, let $D\in\mathbb{Z}$ be a fourth-power-free integer, and
let $E_{D}$ be the elliptic curve
$E_{D}:y^{2}=x^{3}+Dx.$
Then we have
$E_{D}(\mathbb{Q})_{{\rm{tors}}}\cong\left\\{\begin{array}[]{ll}\mathbb{Z}/4\mathbb{Z}&{\rm{if}}\
D=4,\\\ &\\\ \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}&{\rm{if}}\ -D\
{{\rm{is\ a\ perfect\ square}}},\\\ &\\\
\mathbb{Z}/2\mathbb{Z}&\mbox{otherwise}.\\\ \end{array}\right.$
See( [15] Proposition 6.1, Ch.X, page 311). Since $n=p^{4}+q^{4}$ is not $-4$
and can not be a square, (see for example [5], proposition 6.5.3, page 391),
we conclude that the family has the torsion group $T=\mathbb{Z}/2\mathbb{Z}.$
The second fact that we need is the parity conjecture which takes the
following explicit form (see Ono and Ono [13]).
Let $r$ be the rank of elliptic curve $E_{n}$, then
$(-1)^{r}=\omega(E_{n})$
where
$\omega(E_{n})={\rm{sgn}}(-n)\cdot\epsilon(n)\cdot\prod_{p^{2}||n}\left(\frac{-1}{p}\right)$
with $p\geq 3$ a prime and
(3.1) $\epsilon(n)=\left\\{\begin{array}[]{ll}-1,&n\equiv
1,3,11,13\pmod{16},\\\ &\\\ 1,&n\equiv 2,5,6,7,9,10,14,15\pmod{16}.\\\
\end{array}\right.$
As we see from the parity conjecture formula, the key problem is to calculate
the product $\prod_{p^{2}||n}\left(\frac{-1}{p}\right).$ For this reason it is
necessary to describe the square factors of the numbers $n$ if there is any.
Before discussing the general case, we look at some examples:
$(p,q,r,s)=(3364,4849,4288,4303)$ with $17^{2}|n$,
$(p,q,r,s)=(17344243,6232390,12055757,16316590)$ with $97^{2}|n$,
$(p,q,r,s)=(9066373,105945266,5839429,105946442)$ with $17^{2}|n$,
$(p,q,r,s)=(160954948,40890495,114698177,149599920)$ with $41^{2}|n$.
These examples show that the prime divisor of the square factor of $n$ are of
the form $p=8k+1$. We will see that this is in fact a general result according
to the following proposition.
###### Proposition 3.1.
Let $n=u^{4}+v^{4}=r^{4}+s^{4}$ be such that $\gcd(u,v,r,s)=1$. If $p|n$ for
an odd prime number $p$, then $p=8k+1$.
###### Proof.
Without loss of generality we can assume that $n$ is not divisible by 4. We
use the following result from Cox [6]. Let $p$ be an odd prime such that
$\gcd(p,m)=1$ and $p|x^{2}+my^{2}$ with $\gcd(x,y)=1$, then
$(\frac{-m}{p})=1$. From one hand for
$n=u^{4}+v^{4}=(u^{2}-v^{2})^{2}+2(uv)^{2}$, we get $(\frac{-2}{p})=1$ which
implies that $p=8k+1$ or $p=8k+3$. On the other hand for
$n=u^{4}+v^{4}=(u^{2}+v^{2})^{2}-2(uv)^{2}$, we get $(\frac{2}{p})=1$ which
implies that $p=8l+1$ or $p=8l+7$. Putting these two results together we get
$p=8k+1$. ∎
###### Remark 3.2.
If $n=p^{2}m$ for an odd prime $p,$ then $p=8k+1$ from which we get
$(\frac{-1}{p})=1$. This last result shows that the square factor of $n$ does
not affect the root number of the corresponding elliptic curves on the parity
conjecture formula.
###### Remark 3.3.
First of all, by the above remark, we have
$\omega(E_{n})={\rm{sgn}}(-n)\cdot\epsilon(n).$
On the other hand, for $n=p^{4}+q^{4}$, we note that
$p^{4}\equiv 0\ {\rm{or}}\ 1\pmod{16},$ $q^{4}\equiv 0\ {\rm{or}}\
1\pmod{16}.$
For odd $n$ we note that
$n\equiv 1\pmod{16}.$
Now the parity conjecture implies that
$\omega(E_{n})={\rm{sgn}}(-n)\cdot\epsilon(n)=(-1)\cdot(-1)=1.$
For even $n$ we have $n\equiv 2\pmod{16}$ and therefore $\omega(E_{n})=-1$ in
this case.
Finally, we need the Silverman-Tate computation formula [16] (Ch.3 §.5, p.83)
to compute the rank of this family. Let $G$ denote the group of rational
points on elliptic curve $E$ in the form $y^{2}=x^{3}+ax^{2}+bx$. Let
$\mathbb{Q}^{\ast}$ be the multiplicative group of non-zero rational numbers
and let $\mathbb{Q}^{\ast 2}$ denote the subgroup of squares of elements of
$\mathbb{Q}^{\ast}$. Define the group homomorphism $\phi$ from $G$ to
${\mathbb{Q}^{\ast}}/\mathbb{Q}^{\ast 2}$ as follows:
$\phi(P)=\left\\{\begin{array}[]{lll}1\pmod{\mathbb{Q}^{\ast
2}}&{\rm{if}}&P=\mathcal{O},\\\ b\pmod{\mathbb{Q}^{\ast
2}}&{\rm{if}}&P=(0,0),\\\ x\pmod{\mathbb{Q}^{\ast 2}}&{\rm{if}}&P=(x,y)\ {\rm
with}\ x\not=0.\end{array}\right.$
Similarly we take the dual curve $y^{2}=x^{3}-2ax^{2}+(a^{2}-4b)x$ and call
its group of rational points $\overline{G}.$ Now the group homomorphism $\psi$
from $\overline{G}$ to $\mathbb{Q}^{\ast}/\mathbb{Q}^{\ast 2}$ defined as
$\psi(Q)=\left\\{\begin{array}[]{llll}1&\pmod{\mathbb{Q}^{\ast
2}}&{\rm{if}}&Q=\mathcal{O},\\\ a^{2}-4b&\pmod{\mathbb{Q}^{\ast
2}}&{\rm{if}}&Q=(0,0),\\\ x&\pmod{\mathbb{Q}^{\ast 2}}&{\rm{if}}&Q=(x,y)\
{\rm{with}}\ x\not=0.\end{array}\right.$
Then the rank $r$ of the elliptic curve $E$ satisfies
(3.2) $2^{r+2}=|\phi(G)||\psi(\overline{G}|.$
## 4\. Proof of Theorem 1.1
The following fact is an important tool in the proof of our main result.
###### Lemma 4.1.
Let
$\displaystyle A$ $\displaystyle=$ $\displaystyle b^{4}+6b^{2}a^{2}+a^{4},$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
b^{8}+2b^{6}a^{2}+11b^{4}a^{4}+2b^{2}a^{6}+a^{8},$ $\displaystyle C$
$\displaystyle=$ $\displaystyle
b^{8}-4b^{6}a^{2}+8b^{4}a^{4}-4b^{2}a^{6}+a^{8},$ $\displaystyle D$
$\displaystyle=$ $\displaystyle b^{8}-b^{4}a^{4}+a^{8}.$
We have the following properties:
* 1.
$B\neq D$.
* 2.
$D$ is non-square.
* 3.
$A\neq C$.
* 4.
$A$ is non-square.
Proof of lemma 4.1 Let $B=D.$ Since $ab\neq 0$, we get
$b^{4}+6a^{2}b^{2}+a^{4}=0,$ which has no nontrivial solution.
For part 2, we consider the diophantine equation
$x^{4}-x^{2}y^{2}+y^{4}=z^{2},$ which has only the trivial solutions
$x^{2}=1,y=0$ and $y^{2}=1,x=0$ (see [12] page 20).
If $C=A,$ then $(a^{2}-b^{2})^{2}+8a^{2}b^{2}=(a^{2}-b^{2})^{4}+2a^{4}b^{4}$.
Setting $u=(a^{2}-b^{2}),v=a^{2}b^{2}$ , we get $2(v-2)^{2}=-u^{4}+u^{2}+8$.
This is an elliptic curve with Weierstrass equation
$y^{2}=x^{3}-x^{2}-129x-127$, and integral points $(-1,0)$, and $(17,48)$.
Similarly, for part 4, we get the diophantine equation
$x^{4}+6x^{2}y^{2}+y^{4}=z^{2}$ which has only the solutions $x^{2}=1,y=0$ and
$y^{2}=1,x=0$ (see [12] page 18).
The following corollary is an immediate consequence of the above lemma.
###### Corollary 4.2.
Let $b_{1}=BD$, $b_{2}=-AC$, $n=-b_{1}b_{2}$, where $A$, $B$, $C$, and $D$ as
in the above lemma, then the elements of the sets
$\\{1,-n,-1,n,b_{1},-b_{1},b_{2},-b_{2}\\}$ and $\\{1,2,n,2n\\}$ are
independent modulo ${\mathbb{Q}}^{\star 2}$.
Proof of Corollary 4.2 Without loss of generality we check only independence
of the positive numbers in both sets. By construction we know that the numbers
$n$, $b_{1}$ and $-b_{2}$ are all non-squares. Let $b_{1}=c_{1}c_{2}^{2}$, and
$-b_{2}=d_{1}d_{2}^{2}$, where $c_{1}\geq 2$, $c_{2}\geq 1$ , $d_{1}\geq 2$,
$d_{2}\geq 1$ and $c_{1}\neq d_{1}$. Since
$n=-b_{1}b_{2}=c_{1}d_{1}(c_{2}d_{2})^{2}=rs^{2}$ where $r>2$,
$r\notin{\mathbb{Q}}^{\ast 2}$, and $s\geq 1$, then we have
$\begin{array}[]{l}\displaystyle\frac{n}{1}=rs^{2}\equiv r\not\equiv
1\pmod{{\mathbb{Q}}^{\ast 2}},\\\ \\\
\displaystyle\frac{n}{b_{1}}=-b_{2}=d_{1}d_{2}^{2}\equiv d_{1}\not\equiv
1\pmod{{\mathbb{Q}}^{\ast 2}},\\\ \\\
\displaystyle\frac{n}{-b_{2}}=b_{1}=c_{1}c_{2}^{2}\equiv c_{1}\not\equiv
1\pmod{{\mathbb{Q}}^{\ast 2}},\\\ \\\
\displaystyle\frac{b_{1}}{-b_{2}}=\left(\frac{c_{1}}{d_{1}}\right)\left(\frac{c_{2}}{d_{2}}\right)^{2}\equiv\displaystyle\frac{c_{1}}{d_{1}}\not\equiv
1\pmod{{\mathbb{Q}}^{\ast 2}},\\\ \\\
\displaystyle\frac{n}{2}=\frac{n}{2}=\frac{r}{2}s^{2}\equiv\frac{r}{2}\not\equiv
1\pmod{{\mathbb{Q}}^{\ast 2}},\\\ \\\ \displaystyle\frac{2n}{1}=2rs^{2}\equiv
2r\not\equiv 1\pmod{{\mathbb{Q}}^{\ast 2}}\\\ \\\
\displaystyle\frac{2n}{n}=2\not\equiv 1\pmod{{\mathbb{Q}}^{\ast
2}}.\end{array}$
Proof of theorem 1.1 First of all, we show that
$\phi(G)\supseteq\\{1,-n,-1,n\\}.$
The first two numbers $1$ and $-n$ are obvious from the definition of the map
$\phi$. For the numbers $-1$ and $n$ we note that if $n=p^{4}+q^{4}$, then the
homogenous equation
$N^{2}=-M^{4}+ne^{4}$
has solution $e=1$, $M=p$, $N=q^{2}$. Similarly for $N^{2}=nM^{4}-e^{4}$ we
have $M=1$, $e=p$, $N=q^{2}$.
Next we know that the Euler parametrization for $n$ is a consequence of the
fact that $n=p^{4}+q^{4}=r^{4}+s^{4}$ for different numbers $p,q,r,s$. This
implies that
$\displaystyle n=$
$\displaystyle(b^{4}+6b^{2}a^{2}+a^{4})(b^{8}+2b^{6}a^{2}+11b^{4}a^{4}+2b^{2}a^{6}+a^{8})$
$\displaystyle(b^{8}-4b^{6}a^{2}+8b^{4}a^{4}-4b^{2}a^{6}+a^{8})(b^{8}-b^{4}a^{4}+a^{8}).$
Let $b_{1}=BD$, $b_{2}=-AC$, $n=-b_{1}b_{2}$ be as in lemma 4.1. By taking
$M=1$, and $e=b,$ we have
$\begin{array}[]{l}b_{1}M^{4}=BD\\\ \\\ b_{2}e^{4}=-b^{4}AC.\\\ \end{array}$
Then adding them up we get
(4.1) $\displaystyle K=b_{1}M^{4}+b_{2}e^{4}$
$\displaystyle=(b^{8}+2b^{6}a^{2}+11b^{4}a^{4}+2b^{2}a^{6}+a^{8})(b^{8}-b^{4}a^{4}+a^{8})$
$\displaystyle\ \
-b^{4}(b^{4}+6b^{2}a^{2}+a^{4})(b^{8}-4b^{6}a^{2}+8b^{4}a^{4}-4b^{2}a^{6}+a^{8})$
Now, using Sage to factor K we get
$K=a^{4}(a^{6}+b^{2}a^{4}+4b^{4}a^{3}-5b^{6})^{2}.$ Consequently,
$N=a^{2}(a^{6}+b^{2}a^{4}+4b^{4}a^{3}-5b^{6})$. Since $\phi(G)$ is a subgroup
of ${{\mathbb{Q}}^{\ast}}/{{\mathbb{Q}}^{\ast 2}},$ we get
(4.2) $\phi(G)\supseteq\\{1,-n,-1,n,b_{1},-b_{1},b_{2},-b_{2}\\}.$
On the other hand, for the curve
$y^{2}=x^{3}+4nx$
we have
(4.3) $\psi(\overline{G})\supseteq\\{1,n,2,2n\\}.$
Again the numbers $1$ and $n$ are immediate consequence of the definition of
the map $\psi$. For the numbers $2$ and $2n$ we note that the homogeneous
equation
$N^{2}=2M^{4}+2ne^{4}$
has the solution $M=p+q$, $e=1$, and $N=2(p^{2}+pq+q^{2})$, where
$n=p^{4}+q^{4}$. From Corollary $(4.2)$, we know that the right hand side of
$(4.2)$, $(4.3)$ are independent modulo ${\mathbb{Q}}^{\ast 2}$. Therefore
from these observations together with Eq. $(3.2)$ we get
$2^{r+2}=|\phi(G)||\psi(\overline{G}|\geq 4\cdot 8=32.$
This implies that $r\geq 3$. But from $\omega(E_{n})=1,$ the rank should be
even. Therefore we see that $r$ is even and $r\geq 4$.
### 4.1. Remark
If n is an even number $n$ written in two different ways as sums of two
biquadrates, then since $\omega(E_{n})=-1$ in this case, the rank is odd and
$r\geq 3$.
## 5\. Numerical Examples
We conclude this paper by providing many examples of ranks $4$, $5$, $6$, $7$,
$8$ and $10$ using sage software [14].
Table 1. Curves with even rank $p$ | $q$ | $n$ | $rank$
---|---|---|---
$114732$ | $15209$ | $173329443404113736737$ | $10$
$3494$ | $1623$ | $155974778565937$ | $8$
$43676$ | $11447$ | $3656080821185585057$ | $8$
$500508$ | $338921$ | $75948917104718865094177$ | $8$
$502$ | $271$ | $68899596497$ | $6$
$292$ | $193$ | $8657437697$ | $6$
$32187$ | $6484$ | $1075069703066384497$ | $4$
$7604$ | $5181$ | $4063780581008977$ | $4$
$133$ | $134$ | $635318657$ | $4$
Table 2. Curves with odd rank $p$ | $q$ | $n$ | $rank$
---|---|---|---
$989727$ | $161299$ | $960213785093149760746642$ | $7$
$129377$ | $20297$ | $280344024498199948322$ | $7$
$103543$ | $47139$ | $119880781585424489842$ | $7$
$119183$ | $49003$ | $207536518650314617202$ | $7$
$3537$ | $661$ | $156700232476402$ | $7$
$266063$ | $72489$ | $5038767537882101285602$ | $5$
$139361$ | $66981$ | $397322481336075317362$ | $5$
$38281$ | $25489$ | $2569595578866824162$ | $5$
## References
* [1] Agirre, J., Castaneda, A. and Parel, J.C. Higher rank elliptic curves with torsion group ${\mathbb{Z}}/{2\mathbb{Z}}$, Mathematic of Computation, vol. 73, No. 245, (2003), 323-331.
* [2] Agirre, J. and Parel, J.C. Elliptic curves and biquadrates, preprint, arXiv: 1203.2576v1.
* [3] Bremner, A. and Cassels, J.W.S. On the equation $Y^{2}=X(X^{2}+p)$, Math Comp., 42(1984), 257- 264.
* [4] Choudhry, A. The diophantine equation $A^{4}+B^{4}=C^{4}+D^{4}$, Indian J. pure appl. Math., 22(1): 9-11, January, 1991.
* [5] Cohen, H. Number theory vol. I: tools and diophantine equations, Springer, New York, 2007.
* [6] Cox, D.A. Primes of the form $x^{2}+ny^{2}$: Fermat, class field theory, and complex multiplication (Pure and Applied Mathematics: a Wiley series of texts, monographs and tracts).
* [7] Euler, L. Novi Comm. Acad. Petrop., v. 17, p. 64.
* [8] Hardy, G.H. and Wright, E.M. An Introduction to the theory of the numbers, 4th edt., Oxford Univ. press.
* [9] Hollies, A.J., Spearman, B.K. and Yang, Q. Elliptic curves $y^{2}=x^{3}+pqx$ with maximal rank, International Mathematical Forum, 5, 2010, No. 23, 1105-1110
* [10] Kudo, T. and Motose, K. On group structure of some special elliptic curves, Math. J. Okayama Univ. 47(2005), 81-84
* [11] Maenishi, M. On the rank of elliptic curves $y^{2}=x^{3}-pqx$, Kumamoto J. Math. 15(2002), 1-5.
* [12] Mordell, L.J., Diophantine equations, volume30, Academic Press Inc., (London)LTD, England, 1969.
* [13] Ono, K. and Ono, T. Quadratic form and elliptic curves III, Proc. Japan Acad. Ser. A Math. Sci. 72(1996), 204-205.
* [14] Sage software, Version 4.3.5, http://sagemath.org .
* [15] Silverman, J.H. The arithmetic of Elliptic curves, Springer, New York, 1986.
* [16] J.H. Silverman, J.H. and Tate, J. Rational points on elliptic curves, Springer, New York, 1985.
* [17] Spearman, B.K, Elliptic curves $y^{2}=x^{3}-px$, Math. J. Okayama Univ. 49(2007), 183-184.
* [18] Spearman, B.K, On the group structure of elliptic curves $y^{2}=x^{3}-2px$, International Journal of Algebra, 1(5) (2007), 247-250.
* [19] Whittaker, E.T. and Watson, G. N. it A course of modern analysis, Cambridge univ. Press, Cambridge, 1927.
* [20] Yoshida, S. On the equation $y^{2}=x^{3}+pqx$, Comment. Math. Univ. St. Paul., 49(2000), 23-42.
|
arxiv-papers
| 2012-02-25T17:41:45 |
2024-09-04T02:49:27.836820
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. A. Izadi, F. Khoshnam, and K. Nabardi",
"submitter": "Farzali Izadi",
"url": "https://arxiv.org/abs/1202.5676"
}
|
1202.5819
|
# On the exponent of spinor groups
Sanghoon Baek Department of Mathematics and Statistics, University of Ottawa,
Canada sbaek@uottawa.ca
## 1\. Introduction
Let $G$ be a split simple simply connected group of rank $n$ over a field $F$.
Fix a maximal split torus $T$ of $G$ and a Borel subgroup $B$ containing $T$.
We denote by $W$ the Weyl group of $G$ with respect to $T$. Let $\Lambda$ be
the weight lattice of $G$ (hence, $T^{*}=\Lambda$).
We denote by $\omega_{1},\cdots,\omega_{n}$ the fundamental weights of
$\Lambda$. We let
$I_{K}:=\operatorname{Ker}(\mathbb{Z}[\Lambda]\to\mathbb{Z})$ and
$I_{CH}:=\operatorname{Ker}(S^{*}(\Lambda)\to\mathbb{Z})$ be the augmentation
ideals, where $\mathbb{Z}[\Lambda]\to\mathbb{Z}$ (respectively,
$S^{*}(\Lambda)\to\mathbb{Z}$) is the map from the group ring
$\mathbb{Z}[\Lambda]$ (respectively, the symmetric algebra) of $\Lambda$ to
the ring of integers by sending $e^{\lambda}$ to $1$ (respectively, any
element of positive degree to $0$).
For any $i\geq 0$, we consider the ring homomorphism
$\phi^{(i)}:\mathbb{Z}[\Lambda]\to\mathbb{Z}[\Lambda]/I_{K}^{i+1}\to
S^{*}(\Lambda)/I_{CH}^{i+1}\to S^{i}(\Lambda),$
where the first and the last maps are projections and the middle map sends
$e^{\sum_{j=1}^{n}a_{j}\omega_{j}}$ to
$\prod_{j=1}^{n}(1-\omega_{j})^{-a_{j}}$. The _$i$ th-exponent of $G$_
(denoted by $\tau_{i}$), as introduced in [1], is the gcd of all nonnegative
integers $N_{i}$ satisfying
$N_{i}\cdot(I_{CH}^{W})^{(i)}\subseteq\phi^{(i)}(I_{K}^{W}),$
where $I_{K}^{W}:=\langle\mathbb{Z}[\Lambda]^{W}\cap I_{K}\rangle$
(respectively, $I_{CH}^{W}:=\langle S^{*}(\Lambda)^{W}\cap I_{CH}\rangle$)
denotes the $W$-invariant augmentation ideal of $\mathbb{Z}[\Lambda]$
(respectively, $S^{*}(\Lambda)$) and $(I_{CH}^{W})^{(i)}=I_{CH}^{W}\cap
S^{i}(\Lambda)$. Informally, these numbers $\tau_{i}$ measure how far is the
ring $S^{*}(\Lambda)^{W}$ from being a polynomial ring in basic invariants.
For any $i\leq 4$, it was shown that the $i$th-exponent of $G$ divides the
Dynkin index in [1] and this integer was used to estimate the torsion of the
Grothendieck gamma filtration and the Chow groups of $E/B$, where $E/B$
denotes the twisted form of the variety of Borel subgroups $G/B$ for a
$G$-torsor $E$.
In this paper, we show that all the remaining exponents of spinor groups
divide the Dynkin index $2$.
### Acknowledgments.
The work has been partially supported from the Fields Institute and from
Zainoulline’s NSERC Discovery grant 385795-2010.
## 2\. Exponent
Let $G$ be $\operatorname{\mathbf{Spin}}_{2n+1}$ ($n\geq 3$) or
$\operatorname{\mathbf{Spin}}_{2n}$ ($n\geq 4$). The fundamental weights are
defined by
$\displaystyle\omega_{1}$
$\displaystyle=e_{1},\omega_{2}=e_{1}+e_{2},\cdots,\omega_{n-1}=e_{1}+\cdots+e_{n-1},\omega_{n}=\frac{e_{1}+\cdots+e_{n}}{2},$
$\displaystyle\omega_{1}$
$\displaystyle=e_{1},\omega_{2}=e_{1}+e_{2},\cdots,\omega_{n-1}=\frac{e_{1}+\cdots+e_{n-1}-e_{n}}{2},\omega_{n}=\frac{e_{1}+\cdots+e_{n}}{2},$
respectively, where the canonical basis of $\mathbb{R}^{n}$ is denoted by
$e_{i}$ ($1\leq i\leq n$).
For $1\leq i\leq n$, let
(1) $q_{2i}:=e_{1}^{2i}+\cdots+e_{n}^{2i}$
be the basic invariants of the group $G$, i.e., be algebraically independent
homogeneous generators of $S^{*}(\Lambda)^{W}$ as a $\mathbb{Q}$-algebra (see
[2, §3.5 and §3.12]), together with
(2) $q^{\prime}_{n}:=e_{1}\cdots e_{n}$
if $G=\operatorname{\mathbf{Spin}}_{2n}$.
For any $\lambda\in\Lambda$, we denote by $W(\lambda)$ the $W$-orbit of
$\lambda$. For any finite set $A$ of weights, we denote $-A$ the set of
opposite weights.
The Weyl groups of $\operatorname{\mathbf{Spin}}_{2n+1}$ and
$\operatorname{\mathbf{Spin}}_{2n}$ are $(\mathbb{Z}/2\mathbb{Z})^{n}\rtimes
S_{n}$ and $(\mathbb{Z}/2\mathbb{Z})^{n-1}\rtimes S_{n}$, respectively. Hence,
by the action of these Weyl groups, one has the following decomposition of
$W$-orbits: if $G=\operatorname{\mathbf{Spin}}_{2n+1}$ (respectively,
$G=\operatorname{\mathbf{Spin}}_{2n}$), then for any $1\leq k\leq n-1$
(respectively, $1\leq k\leq n-2$)
(3) $W(\omega_{k})=W_{+}(\omega_{k})\cup-W_{+}(\omega_{k}),$
where $W_{+}(\omega_{k})=\\{e_{i_{1}}\pm\cdots\pm
e_{i_{k}}\\}_{i_{1}<\cdots<i_{k}}$. If $n$ is even, then the $W$-orbits of the
last two fundamental weights of $\operatorname{\mathbf{Spin}}_{2n}$ are given
by
(4) $W(\omega_{n-1})=W_{+}(\omega_{n-1})\cup-W_{+}(\omega_{n-1})\text{ and
}W(\omega_{n})=W_{+}(\omega_{n})\cup-W_{+}(\omega_{n}),$
where $W_{+}(\omega_{n-1})$ (respectively, $W_{+}(\omega_{n})$) is the subset
of $W(\omega_{n-1})$ (respectively, $W(\omega_{n})$) containing elements of
the positive sign of $e_{1}$.
For any $\lambda=\sum_{j=1}^{n}a_{j}\omega_{j}\in\Lambda$ and any integer
$m\geq 0$, we set $\lambda(m)=\sum_{j=1}^{n}a_{j}\omega_{j}^{m}$. For example,
$\lambda(0)=\sum_{j=1}^{n}a_{j}$ and $\lambda(1)=\lambda$. We shall need the
following lemma:
###### Lemma 2.1.
$(i)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n}$$)$, then for any odd integer $p$, any
nonnegative integers $m_{1},\cdots,m_{p}$ and, any $1\leq k\leq n-1$
$($respectively, any $1\leq k\leq n-2$$)$, we have
$\sum_{\lambda\in W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})=0.$
$(ii)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$ with odd $n$, then for
any even integer $p$ and any nonnegative integers $m_{1},\cdots,m_{p}$, we
have
$\sum_{\lambda\in
W(\omega_{n})}\lambda(m_{1})\cdots\lambda(m_{p})=\sum_{\lambda\in
W(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p}).$
$(iii)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$, then for any odd
integer $p<n$ and any nonnegative integers $m_{1},\cdots,m_{p}$, we have
$\sum_{\lambda\in
W(\omega_{n})}\lambda(m_{1})\cdots\lambda(m_{p})=\sum_{\lambda\in
W(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p})=0.$
###### Proof.
$(i)$ It follows from (3) that
$\displaystyle\sum_{\lambda\in
W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$
$\displaystyle=\sum_{\lambda\in
W_{+}(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})+\sum_{\lambda\in-
W_{+}(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$
$\displaystyle=\sum_{\lambda\in
W_{+}(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})-\sum_{\lambda\in
W_{+}(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$ $\displaystyle=0.$
$(ii)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$ with odd $n$, then we
have $W(\omega_{n})=-W(\omega_{n-1})$. Hence, the result immediately follows
from the assumption that $p$ is even.
$(iii)$ If $n$ is even, then the result follows from (4) by the same argument
as in the proof of $(i)$. In general, note that for any
$\lambda_{i_{1}},\cdots,\lambda_{i_{p}}\in W_{+}(\omega_{1})$ the term
$\lambda_{i_{1}}(m_{1})\cdots\lambda_{i_{p}}(m_{p})/2^{p}$ (respectively,
-$\lambda_{i_{1}}(m_{1})\cdots\lambda_{i_{p}}(m_{p})/2^{p}$) appears $2^{n-2}$
times (respectively, $2^{n-2}$) in both sums in $(iii)$. ∎
Let $p$ be an even integer and $q\geq 2$ an integer. For any nonnegative
integers $m_{1},\cdots,m_{p}$, we define
$\Lambda(p,q)(m_{1},\cdots,m_{p}):=\sum\lambda_{j_{1}}(m_{1})\cdots\lambda_{j_{p}}(m_{p}),$
where the sum ranges over all different
$\lambda_{i_{1}},\cdots,\lambda_{i_{q}}\in W_{+}(\omega_{1})$ and all
$\lambda_{i_{1}},\cdots,\lambda_{i_{p}}\\\
\in\\{\lambda_{i_{1}},\cdots,\lambda_{i_{q}}\\}$ such that the numbers of
$\lambda_{i_{1}},\cdots,\lambda_{i_{q}}$ appearing in
$\lambda_{i_{1}},\cdots,\lambda_{i_{p}}$ are all nonnegative even solutions of
$x_{1}+\cdots+x_{q}=p$. If $p<2q$, then we set
$\Lambda(p,q)(m_{1},\cdots,m_{p})=0$. Given $m_{1},\cdots,m_{p}$, we simply
write $\Lambda(p,q)$ for $\Lambda(p,q)(m_{1},\cdots,m_{p})$.
For instance, $\Lambda(4,2)$ is the sum of
$\lambda_{j_{1}}(m_{1})\lambda_{j_{2}}(m_{2})\lambda_{j_{3}}(m_{3})\lambda_{j_{4}}(m_{4})$
for all $j_{1},j_{2},j_{3},j_{4}\in\\{i,j\\}$ and all $1\leq i\neq j\leq n$
such that two $i$’s and two $j$’s appear in $j_{1},j_{2},j_{3},j_{4}$.
###### Example 2.2.
We observe that
(5)
$(x_{1}+x_{2})(x^{\prime}_{1}+x^{\prime}_{2})+(x_{1}-x_{2})(x^{\prime}_{1}-x^{\prime}_{2})=2(x_{1}x^{\prime}_{1}+x_{2}x^{\prime}_{2}).$
If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ or
$\operatorname{\mathbf{Spin}}_{2n}$, then by (3) and (5) we have
$\sum_{W_{+}(\omega_{2})}\lambda(m_{1})\lambda(m_{2})=2(n-1)\sum_{W_{+}(\omega_{1})}\lambda(m_{1})\lambda(m_{2})$
for any nonnegative integers $m_{1}$ and $m_{2}$ as we have $(n-1)$ choices of
such pairs in the left hand side of (5) from $W_{+}(\omega_{2})$, which
implies that
$\sum_{W(\omega_{2})}\lambda(m_{1})\lambda(m_{2})=2(n-1)\sum_{W(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{2}),$
(cf. [1, Lemma 5.1(ii)]). For any even $p\geq 4$, we apply the same argument
with the expansion of
$(x_{1}+x_{2})\cdots(x_{1}^{(p)}+x_{2}^{(p)})+(x_{1}-x_{2})\cdots(x_{1}^{(p)}-x_{2}^{(p)})$.
Then, we have
$\sum_{W_{+}(\omega_{2})}\lambda(m_{1})\cdots\lambda(m_{p})=2(n-1)\sum_{W_{+}(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p})+2\Lambda(p,2),$
which implies that
$\sum_{W(\omega_{2})}\lambda(m_{1})\cdots\lambda(m_{p})=2(n-1)\sum_{W(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p})+2^{2}\Lambda(p,2).$
We generalize Example 2.2 to any $\omega_{k}$ as follows.
###### Lemma 2.3.
If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n}$$)$, then for any $1\leq k\leq n-1$
$($respectively, $1\leq k\leq n-2$$)$, any even $p$, and any nonnegative
integers $m_{1},\cdots m_{p}$ we have
$\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})=2^{k-1}{{n-1}\choose{k-1}}\sum_{W(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p})+\sum_{j=2}^{k}2^{k}{{n-j}\choose{k-j}}\Lambda(p,j).$
###### Proof.
For any $\lambda\in W(\omega_{1})$, there are $2^{k}{{n-1}\choose{k-1}}$
choices of the element containing $\lambda$ in $W(\omega_{k})$, thus we have
the term
$2^{k-1}{{n-1}\choose{k-1}}\sum_{W(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p})$
in $\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$.
If an element $\lambda\in W(\omega_{1})$ appears odd times in a term
$\lambda_{i_{1}}(m_{1})\cdots\lambda_{i_{p}}(m_{p})$ of
$\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$, where
$\lambda_{i_{1}},\cdots,\lambda_{i_{p}}\in W(\omega_{1})$, then by the action
of Weyl group this term vanishes in
$\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$. Hence, the remaining
terms in $\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$ are a linear
combination of $\Lambda(p,j)$ for all $2\leq j\leq k$ such that $p\geq 2k$. As
each term $\Lambda(p,j)$ appears $2^{k}{{n-j}\choose{k-j}}$ times in
$\sum_{W(\omega_{k})}\lambda(m_{1})\cdots\lambda(m_{p})$, the result follows.
∎
For any $\lambda\in\Lambda$, we denote by $\rho(\lambda)$ the sum of all
elements $e^{\mu}\in\mathbb{Z}[\Lambda]$ over all elements $\mu$ of
$W(\lambda)$. Let $i!\cdot\phi^{(i)}(e^{\lambda})=\lambda^{i}+S_{i}$ for any
$i\geq 1$, where $S_{i}$ is the sum of remaining terms in
$i!\cdot\phi^{(i)}(e^{\lambda})$ and $\lambda=\sum a_{j}\omega_{j}$,
$a_{j}\in\mathbb{Z}$. Hence, for any fundamental weight $\omega_{k}$ we have
(6)
$i!\cdot\phi^{(i)}(\rho(\omega_{k}))=\sum_{W(\omega_{k})}\lambda^{i}+\sum_{W(\omega_{k})}S_{i}.$
We view $i!\cdot\phi^{(i)}(e^{\lambda})$ as a polynomial in variables
$\lambda,\lambda(m_{1}),\cdots,\lambda(m_{j})$ for some nonnegative integers
$m_{1},\cdots,m_{j}$. Let $T_{i}$ be the sum of monomials in $S_{i}$ whose
degrees are even.
If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n}$$)$, then by Lemma 2.1(i) the equation (6)
reduces to
(7)
$i!\cdot\phi^{(i)}(\rho(\omega_{k}))=\sum_{W(\omega_{k})}\lambda^{i}+\sum_{W(\omega_{k})}T_{i}.$
for any $1\leq k\leq n-1$ $($respectively $1\leq k\leq n-2$$)$.
Given $p$ and $q$, we define
$\Omega(p,q):=\sum\Lambda(p,q)(m_{1},\cdots,m_{p}),$
where the sum ranges over all $m_{1},\cdots,m_{p}$ which appear in all
monomials of $T_{i}$.
###### Example 2.4.
$(i)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ or
$\operatorname{\mathbf{Spin}}_{2n}$ and $i=4$, then by (7) and Lemma 2.3 we
have
$\displaystyle 4!\phi^{(4)}(\rho(\omega_{1}))$
$\displaystyle=\sum_{W(\omega_{1})}\lambda^{4}+\sum_{W(\omega_{1})}T_{4},$
$\displaystyle 4!\phi^{(4)}(\rho(\omega_{2}))$
$\displaystyle=\sum_{W(\omega_{2})}\lambda^{4}+\sum_{W(\omega_{2})}T_{4}$
$\displaystyle=\sum_{W(\omega_{2})}\lambda^{4}+2(n-1)\sum_{W(\omega_{1})}T_{4},$
which implies that
$4!(\phi^{(4)}(\rho(\omega_{2}))-2(n-1)\phi^{(4)}(\rho(\omega_{1})))=\sum_{W(\omega_{2})}\lambda^{4}-2(n-1)\sum_{W(\omega_{1})}\lambda^{4}.$
By Lemma 2.3, the right-hand side of the above equation is equal to
$4\Lambda(4,2)=4\cdot\frac{4!}{2!2!}\sum_{i<j}e_{i}^{2}e_{j}^{2}.$
Hence, we have
$\phi^{(4)}(\rho(\omega_{2}))-2(n-1)\phi^{(4)}(\rho(\omega_{1}))=\sum_{i<j}e_{i}^{2}e_{j}^{2}.$
$(ii)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ ($n\geq 4$) or
$\operatorname{\mathbf{Spin}}_{2n}$ ($n\geq 5$) and $i=6$, then by (7) and
Lemma 2.3 we have
$\displaystyle 6!\phi^{(6)}(\rho(\omega_{1}))$
$\displaystyle=\sum_{W(\omega_{1})}\lambda^{6}+\sum_{W(\omega_{1})}T_{6},$
$\displaystyle 6!\phi^{(6)}(\rho(\omega_{2}))$
$\displaystyle=\sum_{W(\omega_{2})}\lambda^{6}+2(n-1)\sum_{W(\omega_{1})}T_{6}+4\Omega(4,2),$
$\displaystyle 6!\phi^{(6)}(\rho(\omega_{3}))$
$\displaystyle=\sum_{W(\omega_{3})}\lambda^{6}+4{{n-1}\choose{2}}\sum_{W(\omega_{1})}T_{6}+8(n-2)\Omega(4,2),$
which implies that
$\phi^{(6)}(\rho(\omega_{3}))-2(n-2)\phi^{(6)}(\rho(\omega_{2}))+2(n-1)(n-2)\phi^{(6)}(\rho(\omega_{1}))=\sum_{i<j<k}e_{i}^{2}e_{j}^{2}e_{k}^{2}.$
###### Lemma 2.5.
$(i)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$, then we have
$\sum_{W(\omega_{n})}\lambda^{n}-\sum_{W(\omega_{n-1})}\lambda^{n}=n!e_{1}\cdots
e_{n}.$
$(ii)$ If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$, then for any $1\leq
p\leq n-1$ and any nonnegative integers $m_{1},\cdots,m_{p}$ we have
$\sum_{W(\omega_{n})}\lambda(m_{1})\cdots\lambda(m_{p})=\sum_{W(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p}).$
###### Proof.
$(i)$ First, assume that $n\geq 4$ is even. We show that
$\sum_{W_{+}(\omega_{n})}\lambda^{n}-\sum_{W_{+}(\omega_{n-1})}\lambda^{n}=(n!/2)e_{1}\cdots
e_{n}.$
As $|W_{+}(\omega_{n})|=|W_{+}(\omega_{n-1})|=2^{n-2}$, we have
$(n!/2^{n})2^{n-2}e_{1}\cdots e_{n}-(-(n!/2^{n})2^{n-2}e_{1}\cdots
e_{n})=(n!/2)e_{1}\cdots e_{n}$
in $\sum_{W(\omega_{n})}\lambda^{n}-\sum_{W(\omega_{n-1})}\lambda^{n}$. If one
of the exponents $i_{1},\cdots,i_{n}$ in $e_{1}^{i_{1}}\cdots e_{n}^{i_{n}}$
(except the case $i_{1}=\cdots=i_{n}=1$) is odd, then from the orbits
$W_{+}(\omega_{n})$ and $W_{+}(\omega_{n-1})$ this monomial vanishes in each
sum of
$\sum_{W_{+}(\omega_{n})}\lambda^{n}-\sum_{W_{+}(\omega_{n-1})}\lambda^{n}$.
Otherwise, the terms
$2^{n-2}\sum_{j=1}^{n}e_{j}^{n},\Lambda(n,2)\cdots,\Lambda(n,n/2)$ with
$m_{1}=\cdots=m_{n}=1$ are in both $\sum_{W_{+}(\omega_{n})}\lambda^{n}$ and
$\sum_{W_{+}(\omega_{n-1})}\lambda^{n}$.
Now, we assume that $n\geq 4$ is odd. As
$|W(\omega_{n})|=|W(\omega_{n-1})|=2^{n-1}$, we have
$(n!/2^{n})2^{n-1}e_{1}\cdots e_{n}-(-(n!/2^{n})2^{n-1}e_{1}\cdots
e_{n})=n!e_{1}\cdots e_{n}$
in $\sum_{W(\omega_{n})}\lambda^{n}-\sum_{W(\omega_{n-1})}\lambda^{n}$. By the
same argument, if one of the exponents $i_{1},\cdots,i_{n}$ in
$e_{1}^{i_{1}}\cdots e_{n}^{i_{n}}$ (except the case $i_{1}=\cdots=i_{n}=1$)
is odd, then this monomial vanishes in each sum of
$\sum_{W(\omega_{n})}\lambda^{n}-\sum_{W(\omega_{n-1})}\lambda^{n}$. This
completes the proof of $(i)$.
$(ii)$ By Lemma 2.1(ii)(iii), it is enough to consider the case where both $n$
and $p$ are even. For any $p$ and any $n\geq p+2$, we have
$2^{n-2}(\sum_{W_{+}(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p}))$ in both
$\sum_{W_{+}(\omega_{n})}\lambda(m_{1})\cdots\lambda(m_{p})$ and
$\sum_{W_{+}(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p})$. By the action
of Weyl group, any term $\lambda_{i_{1}}(m_{1})\cdots\lambda_{i_{p}}(m_{p})$,
where an element $\lambda\in W(\omega_{1})$ appears odd times in either
$\sum_{W_{+}(\omega_{n})}\lambda(m_{1})\cdots\lambda(m_{p})-2^{n-2}(\sum_{W_{+}(\omega_{1})}\lambda(m_{1})\\\
\cdots\lambda(m_{p}))$ or
$\sum_{W_{+}(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p})-2^{n-2}(\sum_{W_{+}(\omega_{1})}\lambda(m_{1})\cdots\lambda(m_{p}))$,
vanishes. As each term of $\Lambda(p,2),\cdots,\Lambda(p,p/2)$ appears in both
$\sum_{W_{+}(\omega_{n})}\lambda(m_{1})\cdots\\\ \lambda(m_{p})$ and
$\sum_{W_{+}(\omega_{n-1})}\lambda(m_{1})\cdots\lambda(m_{p})$, this completes
the proof.
∎
###### Theorem 2.6.
If $G$ is $\operatorname{\mathbf{Spin}}_{2n+1}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n}$$)$, then for any $i\geq 3$ and any
$n\geq[i/2]+1$ $($respectively, $n\geq[i/2]+2$$)$ the exponent $\tau_{i}$
divides the Dynkin index $\tau_{2}=2$.
###### Proof.
As $B_{2}=C_{2}$ and $D_{3}=A_{3}$, we have $1=\tau_{3}\mid 2$ by [1, Theorem
5.4]. If $G$ is $\operatorname{\mathbf{Spin}}_{2n}$ for any $n\geq 4$, then by
Lemma 2.5(i)(ii) we have
$q^{\prime}_{n}=\phi^{(n)}(\rho(\omega_{n}))-\phi^{(n)}(\rho(\omega_{n-1})),$
which implies that the invariant $q^{\prime}_{n}$ is in the ideal generated by
the image of $\phi^{(n)}$. As there are no invariants of odd degree except
$q^{\prime}_{n}$, we have
$\tau_{2i+1}\mid\tau_{2i}$
for all $i\geq 1$. Therefore, it suffices to show that $\tau_{2i}\mid\tau_{2}$
for any $i\geq 2$.
By Lemma 2.3 together with the same argument as in Example 2.4 we have
(8)
$\phi^{(2i)}(\rho(\omega_{i}))+\sum_{j=1}^{i-1}a_{j}\phi^{(2i)}(\rho(\omega_{i-j}))=\sum_{j_{1}<\cdots<j_{i}}e_{j_{1}}^{2}\cdots
e_{j_{i}}^{2},$
where the integers $a_{1},\cdots,a_{i-1}$ satisfy
$\Big{(}\sum_{j=k}^{i-2}2^{j+1}{{n-1-k}\choose{j-k}}a_{j+1}\Big{)}+2^{i}{{n-1-k}\choose{i-1-k}}=0,$
for $0\leq k\leq i-2$. Let $p_{i}$ be the right-hand side of (8). Then this
equation implies that $p_{i}$ is in the image of $\phi^{(2i)}$.
We show that the invariant $q_{2i}$ is in the ideal $\phi^{(2i)}(I_{K}^{W})$
for any $i\geq 2$. We proceed by induction on $i$. As
$q_{2}=\phi^{(2)}(\rho(\omega_{1}))$, the case $i=2$ is obvious. By Newton’s
identities we have
(9) $(-1)^{i-1}q_{2i}=ip_{i}-\sum_{j=1}^{i-1}(-1)^{j-1}p_{i-1-j}q_{2j}$
with $p_{0}=1$. By the induction hypothesis, the sum of (9) is in
$\phi^{(2i)}(I_{K}^{W})$. Hence, $q_{2i}$ is in $\phi^{(2i)}(I_{K}^{W})$. ∎
For any nonnegative integer $n$, we denote by $v_{2}(n)$ the $2$-adic
valuation of $n$. For a smooth projective variety $X$ over $F$, we denote by
$\Gamma^{*}K(X)$ the gamma filtration on the Grothendieck ring $K(X)$. We let
$c_{CH}:S^{*}(\Lambda)\to CH(G/B)$ be the characteristic map.
###### Corollary 2.7.
Let $G$ be $\operatorname{\mathbf{Spin}}_{2n}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n+1}$$)$. If $2^{m(i)}(\ker
c_{CH})^{(i)}\subseteq(I_{CH}^{W})^{(i)}$ for some nonnegative integer $m(i)$,
then for any $i\geq 3$ and any $n\geq[i/2]+2$ $($respectively,
$n\geq[i/2]+1$$)$ the torsion of $\Gamma^{i}K(G/B)/\Gamma^{i+1}K(G/B)$ is
annihilated by $2^{g(i)}$, where $g(i)=1+m(i)+v_{2}((i-1)!)$.
###### Remark 2.8.
It is shown that $m(3)=0$ and $m(4)=1$ in [1, Lemma 6.4].
###### Proof.
The proof of [1, Theorem 6.5] still works with Theorem 2.6. ∎
###### Corollary 2.9.
Let $G$ be $\operatorname{\mathbf{Spin}}_{2n}$ $($respectively,
$\operatorname{\mathbf{Spin}}_{2n+1}$$)$. If $2^{m(i)}(\ker
c_{CH})^{(i)}\subseteq(I_{CH}^{W})^{(i)}$ for some nonnegative integer $m(i)$,
then for any $G$-torsor $E$, any $i\geq 3$ and any $n\geq[i/2]+2$
$($respectively, $n\geq[i/2]+1$$)$ the torsion of $\operatorname{CH}^{i}(E/B)$
is annihilated by $2^{t(i)}$, where $t(i)=1+\sum_{j=3}^{i}g(j)+v_{2}((i-1)!)$.
###### Proof.
By [3, Theorem 2.2(2)], we have
$\Gamma^{i}K(G/B)/\Gamma^{i+1}K(G/B)\simeq\Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B).$
As the torsion of $\operatorname{CH}^{i}(E/B)$ is annihilated by
$(i-1)!\prod_{j=1}^{i}e(\Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B)),$
where $e(\Gamma^{i}K(E/B)/\Gamma^{i+1}K(E/B))$ denotes the finite exponent of
its torsion subgroup (see [1, p.149]), the result follows from Corollary 2.7.
∎
## References
* [1] S. Baek, E. Neher, K. Zainoulline, _Basic polynomial invariants, fundamental representations and the Chern class map_ , Doc. Math. 17 (2012), 135–150.
* [2] J. Humphreys, _Reflection groups and Coxeter groups_. Cambridge studies in Advanced Math. 29, Cambridge Univ. Press (1990).
* [3] I. A. Panin, _On the algebraic K-theory of twisted flag varieties_ , K-Theory 8 (1994), no. 6, 541–585.
|
arxiv-papers
| 2012-02-27T03:18:56 |
2024-09-04T02:49:27.847493
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sanghoon Baek",
"submitter": "Sanghoon Baek",
"url": "https://arxiv.org/abs/1202.5819"
}
|
1202.5843
|
# Elasto-plastic flow of a foam around an obstacle
F. Boulogne Institute of Mathematics and Physics, Aberystwyth University,
SY23 3BZ, UK Univ Paris-Sud, Univ Pierre et Marie Curie-Paris 6, CNRS,
Lab FAST, Bat 502, Campus Univ – F-91405, Orsay, France. S.J. Cox
foams@aber.ac.uk Institute of Mathematics and Physics, Aberystwyth University,
SY23 3BZ, UK
###### Abstract
We simulate quasistatic flows of an ideal two-dimensional monodisperse foam
around different obstacles, both symmetric and asymmetric, in a channel. We
record both pressure and network contributions to the drag and lift forces,
and study them as a function of obstacle geometry. We show that the drag force
increases linearly with the cross section of an obstacles. The lift on an
asymmetric aerofoil-like shape is negative and increases with its arc length,
mainly due to the pressure contribution.
###### pacs:
83.80.Iz,47.57.Bc,47.11.Fg
## I Introduction
Foams are used widely, for example in industries associated with mining, oil
recovery and personal care products Prud’homme and Khan (1996). Their use is
often preferred because of properties such as a high surface area, low density
and a yield stress Weaire and Hutzler (1999); Höhler and Cohen-Addad (2005).
In addition to this evidence of plasticity, a foam’s rheology is dominated by
elasticity at low strains and viscous flow at high strain-rates: they are
elasto-visco-plastic fluids Cantat et al. (2010).
A common probe of foam rheology is a variation of Stokes’ experiment Stokes
(1850) in which an object moves relative to a foam Cox et al. (2000);
Asipauskas et al. (2003); de Bruyn (2004); Dollet et al. (2005a, b); Cantat
and Pitois (2005); Dollet et al. (2005c); Cox et al. (2006); Cantat and Pitois
(2006); Dollet et al. (2006); Dollet and Graner (2007); Raufaste et al.
(2007); Tabuteau et al. (2007); Wyn et al. (2008); Davies and Cox (2009).
Foams have an advantage over many complex fluids in that their local structure
(the bubbles) is observable, thus making them an excellent choice to determine
the mechanisms by which non-Newtonian fluids show different responses to
Newtonian fluids. In addition, a two-dimensional foam is a realizable entity,
for example the Bragg bubble raft Bragg and Nye (1947), with which it is
possible to perform a rheological experiment in which the shape and velocity
of each bubble can be tracked in time. Foams are also amenable to numerical
simulation because of the precise local geometry that is found wherever soap
films meet. Plateau’s laws, which describe how the films meet, are a
consequence of each soap film minimizing its energy, equivalent to surface
area, and it is this that provides the algorithm for the work described here.
For flow to occur in a foam, the bubbles must slide past each other. This
occurs through T1 neighbour switching topological changes Weaire and Rivier
(1984), in which small faces and/or short films disappear and new ones appear.
Sometimes referred to as plastic events, these are a visible indication of
plasticity in a foam, and act to reduce the stress and energy. Numerous
contributions to viscous dissipation occur Buzza et al. (1995), although we
assume that if the flow is slow enough they can all be neglected.
Dollet et al. (2006) measured the drag, lift and torque on an ellipse in a
two-dimensional foam flow in a channel. The lift was maximized when the
ellipse was oriented at an angle of $\pi/4$ to the direction of flow. Dollet
et al. (2005a) found that an aerofoil embedded in a foam flow exhibited a
negative lift, which they attributed to the elasticity of the foam. This
augments the list of well-known non-Newtonian effects that contradict the
sense of what is known for Newtonian fluids.
We present here elasto-plastic simulations, in the so-called quasi-static
limit, for 2D foam flow around an obstacle, and investigate the effect of the
symmetry of the obstacle in determining the magnitude and direction of the
drag and lift. Such simulations allow us to exclude consideration of viscous
effects, and even to separate out pressure and film network contributions to
the forces on an obstacle, both of which are difficult to do in experiment. As
a means of determining drag and lift on an obstacle, they have been validated
against experiments on an ellipse Dollet et al. (2006) by Davies and Cox
(2010).
We consider a range of obstacle shapes, illustrated in figure 1. Since the
Evolver uses a gradient descent method, we are unable to simulate an obstacle
with sharp corners. We therefore round the corners of each obstacle with
segments of a circle to smooth the boundary. The shapes are:
1. (a)
a circle, which provides the standard case with full symmetry. Its cross-
section is $H=2R$.
2. (b)
the union of a square and two semi-circles, which we call a “stadium”,
arranged either vertically or horizontally. The side-length $2R$ of the square
is equal to the diameter of each semi-circle, so that the area is determined
by just one parameter, $R$. The cross-section is $2R$ (horizontal stadium) or
$4R$ (vertical stadium).
3. (c)
a square, with rounded corners. The radius of curvature of the corners is set
to one-eighth of the side-length of the square, $R=L/8$ , so that the area is
again determined by just one parameter, and $H=L$. Also a diamond, which is
the square rotated by $\pi/4$, with $H\approx\sqrt{2}L$.
4. (d)
a symmetric aerofoil, with long axis parallel to the direction of foam flow,
defined by two arcs of circles bounded by two tangential straight lines. Three
parameters are needed: length $L$ (distance between the centres of the
circles), and radii $R_{1}$ (leading edge) and $R_{2}$ (trailing edge). This
shape has up-down symmetry but not fore-aft symmetry, and cross-section
$H=2\max(R_{1},R_{2})$. If $R_{1}=R_{2}$, then this is a “long” horizontal
stadium.
5. (e)
an aerofoil-like shape with up-down asymmetry, in which two circles of equal
radius $R_{2}$ are joined by arcs of radius $R_{1}$ and $R_{1}+2R_{2}$. The
distance between the circles is parametrized by the angle $\theta_{1}$. Its
cross-section is $H=(R_{1}+R_{2})(1-\cos\theta_{1})+2R_{2}$. This
approximation to a standard aerofoil dispenses with the singular point at the
trailing edge.
Figure 1: Pictures of the obstacles, oriented with flow from left to right:
(a) circle, (b) horizontal stadium, (c) square, (d) symmetric aerofoil and (e)
asymmetric aerofoil.
We begin by describing our numerical method (§II). The forces on each obstacle
are given in §III.1; we find that the drag is mainly determined by its maximum
cross-section $H$ perpendicular to the direction of flow and that a
significant lift is found only for the aerofoil without up-down symmetry. The
field of bubble pressure around the obstacle, which is the main contribution
to this lift, is described in §III.2, and we make some concluding remarks in
§IV.
## II Method
We use the Surface Evolver Brakke (1992) in the manner described by Davies and
Cox (2009). We create three foams of around 725 bubbles (in this range the
number of bubbles does not affect the results; data not shown) between
parallel walls with a Voronoi construction Brakke (1986); Wyn et al. (2008).
The channel has unit length and width $W=0.8$. The foams are monodisperse,
with bubble area denoted $A_{b}$ and about 22 bubbles in the cross-section of
the channel. A bubble in the centre of the channel is chosen to represent the
obstacle, and its periphery constrained to the required shape; its area is
then increased until it reaches the desired area ratio $a_{r}=A_{obs}/A_{b}$
and it is then fixed – see figure 2(a). The tension of each film, $\gamma$,
which is twice the air-liquid surface tension and is in effect a line tension,
is taken equal to one, without loss of generality.
The boundary conditions are that of free slip on the boundary of the obstacle
and the channel walls, so that the films meet the boundaries at $90^{\circ}$,
and periodicity in the direction of flow. We checked in a few instances that
changing the boundary condition on the channel walls to non-slip has little
effect on the forces on a small obstacle in the centre of the channel. At each
iteration the foam is pushed with a small area increment $dA=5\times 10^{-4}$
to create a pressure gradient Raufaste et al. (2007). The perimeter is then
evolved towards a local minimum and T1s are performed whenever a film length
shrinks below $l_{c}=1\times 10^{-3}$ (representing a foam with low liquid
fraction, of the order of $10^{-4}$). A simulation runs for 1500 iterations to
ensure that the measurements are made beyond any transient in which the foam
retains a memory of its initial state. Each simulation takes about one week on
a 1.5GHz CPU. The method has been validated against experiment in the case of
an elliptical obstacle Dollet et al. (2006); Davies and Cox (2010).
Figure 2: (a) Sketch of the simulation, in this case for a horizontal stadium.
A 2D foam is created between two fixed walls and caused to flow in the
positive $x$ direction by increasing the area of the region to the left of the
dark line of films joining the two walls. The obstacle is created in the
centre of the channel; each film that touches the obstacle applies an equal
force outward in the direction normal to the obstacle and each bubble applies
a pressure force inward at the middle of the shared boundary. The films bunch
up at the trailing edge of the obstacle and the bubble pressures rise at the
leading edge due to the flow, leading to drag and lift forces on the obstacle.
(b) Example (vertical stadium, area ratio $a_{r}=6$) of the pressure ($F_{P}$)
and network ($F_{T}$) contributions to the drag ($x$) and the lift ($y$) as a
function of iteration number. The drag forces increase linearly before
developing a saw-tooth variation which is linked to a build-up of stress
followed by avalanches of T1s in the foam. The horizontal lines show the
average drag forces. In this case the pressure and network contributions to
the lift are both negligible.
### II.1 Drag and lift
Each film that touches the obstacle applies an outward force with magnitude
equal to the force of surface tension and direction perpendicular to the
obstacle boundary. Their resultant is the network force
$\vec{F}_{T}=\gamma\sum_{i}\vec{n}_{i}$ (1)
where $\vec{n}_{i}$ is the unit outward normal at the vertex $i$ terminating
each film that meets the obstacle. See figure 2(a).
Each bubble that touches the obstacle applies a pressure force inward at the
middle of the shared boundary. Their resultant is the pressure force
$\vec{F}_{P}=-\sum_{j}p_{j}l_{j}\vec{n}_{j}$ (2)
where $p_{j}$ is the pressure of bubble $j$, $l_{j}$ the length of shared
boundary and $\vec{n}_{j}$ the unit outward normal to the obstacle at the
midpoint of the line joining the two ends of the shared boundary.
The drag on an obstacle is the component of the sum of the network and
pressure forces in the direction of motion, $F_{D}=F_{T}^{x}+F_{P}^{x}$. The
lift is the component perpendicular to this, $F_{L}=F_{T}^{y}+F_{P}^{y}$, with
the convention that positive values of lift act in the positive $y$ direction.
All four components are recorded at the end of each iteration, and averaged
above 600 iterations, well beyond any transient. An example is shown in figure
2(b). The standard deviation of the fluctuations in force about this average
are used to give the error bars in the figures below.
## III Results
### III.1 Drag and lift force on an obstacle
The drag and lift oscillate in a saw-tooth fashion (figure 2(b)), caused by
intervals in which the imposed strain is stored elastically followed by
cascades of T1 topological changes. Nonetheless, they have a well-defined
average. We find that for all obstacles with up-down symmetry the average lift
is close to zero.
We vary the area ratio of each obstacle, usually in the range one to ten but
occasionally higher. We normalize the cross-section and length of each
obstacle by the average bubble diameter $d_{b}=\sqrt{4A_{b}/\pi}$ which, since
the walls are far enough away not to have an effect on the drag and lift, is
the significant length-scale here. We choose to plot the resulting drag as a
function of cross-section $H/d_{b}$ (figure 3) since it gives an approximately
linear relationship Raufaste et al. (2007). It is apparent that the drag
increases with obstacle cross-section most quickly for “blunt” objects with a
vertical leading edge (square, vertical stadium). Obstacles with a rounded
leading edge (circle Cox et al. (2006), horizontal stadium) experience lower
drag for given cross-section. In each case, the main contribution to the drag
is usually due to network forces; the pressure contribution to the total drag
is lower but follows the same trends.
Figure 3: (Color online) Drag vs obstacle cross-section $H/d_{b}$. Images are
for obstacles with area ratio $a_{r}=10$ with flow from left to right. (a)
Vertical stadium ($a_{r}=2,3,4,6,8,10$). (b) Horizontal stadium
($a_{r}=2,4,5,6,8,10,20,30$). (c) Square ($a_{r}=2,4,6,8,10,L/R=8$). (d)
Diamond ($a_{r}=2,4,6,8,10,L/R=8$).
Figure 4: Drag force on different obstacles. (a) Drag vs shape at constant
cross-section $H/d_{b}\approx 2.1$. The pressure contribution to the drag
decreases with the rounding of the leading edge and the network contribution
decreases with the rounding of the trailing edge. (b) Drag vs roundness
$R/(L+R)$, interpolating between a square ($R=L/8$) and a circle ($L=0$) with
$a_{r}=10$. The same effect is seen as in (a). (c) Drag vs obstacle length,
measured as $(L+2R)/d_{b}$, for symmetric aerofoils with $R_{1}=R_{2}$ at
constant cross-section $H/d_{b}\approx 2.1$. The first point on the left
corresponds to a circle ($L=0$), and the second to a horizontal stadium
($L=2R$). The network contribution to the drag decreases slightly with length.
(d) Drag vs radius ratio $R_{2}/R_{1}$ for a symmetric aerofoil ($a_{r}=10,L$
varies). The pressure drag decreases when the leading edge has a smaller
radius of curvature.
To tease out the effect of obstacle shape on the two components of drag
studied here, we fix the cross-section (figure 4(a),(b)) and vary the shape.
The pressure contribution to the drag is highest when the leading edge is
blunt (vertical stadium, square), since this causes the greatest deformation
to the bubbles. Similarly, the network contribution to the drag is highest
when the trailing edge is rounded (the most “circular” case in figure 4(b)),
although this effect is weaker, since a rounded trailing edge allows more
films to collect in that area. The shape of the diamond is such that the
network drag is very low, since films can gather on the sloping sides as well
as the rounded region at the very tip of the trailing edge, while the pressure
drag is intermediate.
The length $L$ of an obstacle has only a weak effect on the drag (figure
4(c)). In particular, this is the case for a symmetric aerofoil with
$R_{2}=R_{1}$, since most of the films that touch the obstacle are
perpendicular to the direction of foam flow. By varying the ratio
$R_{2}/R_{1}$ for a symmetric aerofoil with fixed cross-section $H$ and fixed
area ratio $a_{r}=10$, we can investigate the effect of fore-aft asymmetry.
Figure 4(d) shows that the total drag varies little, emphasizing that cross-
section and rounded leading and trailing edges make the major contribution to
the drag. The pressure contribution to the drag decreases with $R_{2}/R_{1}$,
that is, as the leading edge gets smaller and bubbles are less deformed there.
Figure 5: Lift versus asymmetric aerofoil arc length, scaled by $d_{b}$. All
three of $R_{1}$, $R_{2}$ and $\theta_{1}$ are chosen to increase roughly in
the same proportion. The lift is always in the negative $y$ direction and the
network contribution is smaller than that due to pressure.
The lift is, on average, zero for all obstacles with a horizontal axis of
symmetry (as in figure 2(b)); it is only significant for the asymmetric
aerofoil, being negative and of the same order of magnitude as the drag. In
particular the lift increases with aerofoil length (figure 5), and the major
component of lift arises from the bubble pressures. It appears therefore that
the curvature of the aerofoil induces changes in bubble pressures, and that it
is this, rather than an imbalance in the number of films pulling on the top
and bottom surfaces of the object, that gives rise to the lift. We return to
the bubble pressures below.
To test the effect of obstacle position in the channel, we placed the same
asymmetric aerofoil in three different positions across the channel:
$y=0.25W,0.5W$ (reference case) and $0.75W$. No significant difference in the
drag or lift was observed (data not shown), indicating that the obstacle was
still sufficiently far from the walls that they don’t interfere with the flow
(recall that this is a elasto-plastic rather than a viscous flow, distinct
from a Newtonian fluid where the wall always has an effect in 2D) and that the
lift is not just due to the foam squeezing through the gap between wall and
obstacle.
Figure 6: Pressure fields averaged over the duration of the simulation. (a)
Square obstacle with $a_{r}=10$, showing increased pressure upstream of the
obstacle and low pressure downstream. (b) Asymmetric aerofoil, with
$R_{1}/\sqrt{d_{b}}=3$, $R_{2}/\sqrt{d_{b}}=0.75$ and $\theta_{1}=\pi/6$,
showing low pressure beneath as well as downstream. The increase of pressure
upstream is less-pronounced, and there is a pressure peak beneath the trailing
edge of the aerofoil. (c) Zoom of the typical arrangement of films around the
same aerofoil, with bubbles shaded by instantaneous pressure on a scale by
which pressure increases with grey intensity. In both representations a region
of low pressure is evident beneath the aerofoil – it is this which induces a
negative lift – as well as the pressure peak beneath the trailing edge.
### III.2 Pressure field around an obstacle
To further probe the phenomenon of negative lift in foams, in figure 6 we
compare the distribution of bubble pressures around the flat-bottomed aerofoil
with an up-down symmetric obstacle typified by the square. The Surface Evolver
calculates the bubble pressures (as Lagrange multipliers of the area
constraints) in such a way that they are all relative to the pressure of one
bubble. Thus the average pressure is subtracted from all values at each
iteration, before binning the data as above.
The bubble pressures decrease in the $x$ direction, on average, because of the
flow. The presence of an obstacle induces a region of high pressure at the
leading edge and a region of low pressure at the trailing edge. In addition,
the asymmetric aerofoil shows a region of high pressure above and low pressure
below, confirming that the pressure contribution to the lift is downwards.
## IV Conclusions
The simulations described here show that the forces on an obstacle embedded in
a flow of foam depend strongly on the shape of the obstacle. We separate two
components, due to the pressure in the bubbles and the network of soap films,
and find that the pressure contribution decreases with the rounding of the
leading edge and the network contribution decreases with the rounding of the
trailing edge. Further evidence is given in figure 7(a).
Figure 7: (Color online) (a) An interpolation between a diamond and a circle
takes the shape shown, with $a_{r}=10$. When its sense is flipped relative to
the direction of flow, the relative contributions to the pressure and network
drag change, while the total drag remains the same: a sharp leading edge and
rounded trailing edge reduces the pressure drag and increases the network
drag. (b) Instantaneous arrangement of films around a flat-bottomed aerofoil
(parameters: $a_{r}=9$, cross-section $H/d_{b}=1.59$, radius of curvature of
leading edge is $R_{1}/d_{b}=0.41$, of trailing edge is $R_{2}/d_{b}=0.17$ and
of upper side is $R_{3}/d_{b}=3.46$). The instantaneous values of drag and
lift are $F_{y}^{T}=-2.90,F_{y}^{P}=-2.36,F_{x}^{T}=2.03,F_{x}^{P}=0.42$. The
lift is again negative, both network and pressure contributions are similar,
and the total lift is of the same order of magnitude as the total drag.
In classical fluid mechanics, the presence of viscosity can give rise to
trailing vortices and circulation around an obstacle in a fluid flow. Here,
not only do we neglect viscosity, but the discrete nature of the foam probably
suppresses any possibility of circulation. Yet a lift force is still observed
for obstacles without lateral symmetry, and it arises because of the way in
which the obstacle deforms the bubbles that make up the foam. It is therefore
an effect of elasticity or, more generally, viscoelasticity Wang and Joseph
(2004, 2005), due to the normal stresses generated in the fluid, and acts in
the opposite direction to the usual sense of “lift”. A concave underside, as
in the familiar Joukowski profile and the asymmetric aerofoil described above,
is not necessary to obtain a negative lift (figure 7(b)).
It remains to determine whether a given obstacle is actually stable with
respect to rotation; that is, whether the torque on any given obstacle is
sufficient to rotate it and thereby reduce the drag and/or lift. This is a
necessary pre-cursor to using this work to determining which shapes of
obstacles offer the least resistance to foam flow. It is also of interest to
incorporate some element of viscous dissipation, perhaps using the viscous
froth model Kern et al. (2004), within the simulations, which has a
particularly significant effect on rotation Davies and Cox (2010) but also the
film motion around an obstacle. We shall return to both these issues in future
work.
###### Acknowledgements.
We thank K. Brakke for developing, distributing and supporting the Surface
Evolver, I.T. Davies for technical assistance with the simulations, and F.
Graner for useful comments. SJC acknowledges financial support from EPSRC
(EP/D071127/1).
## References
* Prud’homme and Khan (1996) R.K. Prud’homme and S.A. Khan, editors. _Foams: Theory, Measurements and Applications_ , volume 57 of _Surfactant Science Series_. Marcel Dekker, New York, 1996.
* Weaire and Hutzler (1999) D. Weaire and S. Hutzler. _The Physics of Foams_. Clarendon Press, Oxford, 1999.
* Höhler and Cohen-Addad (2005) R. Höhler and S. Cohen-Addad. Rheology of Liquid Foam. _J. Phys.: Condens. Matter_ , 17:R1041–R1069, 2005\.
* Cantat et al. (2010) I. Cantat, S. Cohen-Addad, F. Elias, F. Graner, R. Höhler, O. Pitois, F. Rouyer, and A. Saint-Jalmes. _Les mousses - structure et dynamique_. Belin, Paris, 2010.
* Stokes (1850) G.G. Stokes. On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. _Trans. Camb. Phil. Soc._ , IX:8–149, 1850.
* Cox et al. (2000) S.J. Cox, M.D. Alonso, S. Hutzler, and D. Weaire. The Stokes experiment in a foam. In P. Zitha, J. Banhart and G. Verbist, editor, _Foams, emulsions and their applications_ , pages 282–289. MIT-Verlag, Bremen, 2000.
* Asipauskas et al. (2003) M. Asipauskas, M. Aubouy, J.A. Glazier, F. Graner, and Y. Jiang. A texture tensor to quantify deformations: the example of two-dimensional flowing foams. _Granular Matter_ , 5:71–74, 2003.
* de Bruyn (2004) J.R. de Bruyn. Transient and steady-state drag in foam. _Rheol. Acta_ , 44:150–159, 2004.
* Dollet et al. (2005a) B. Dollet, M. Aubouy, and F. Graner. Inverse Lift: a signature of the elasticity of complex fluids. _Phys. Rev. Lett._ , 95:168303, 2005a.
* Dollet et al. (2005b) B. Dollet, F. Elias, C. Quilliet, C. Raufaste, M. Aubouy, and F. Graner. Two-dimensional flow of foam around an obstacle: Force measurements. _Phys. Rev. E_ , 71:031403, 2005b.
* Cantat and Pitois (2005) I. Cantat and O. Pitois. Mechanical probing of liquid foam ageing. _J. Phys.: Condens. Matter_ , 17:S3455–S3461, 2005\.
* Dollet et al. (2005c) B. Dollet, F. Elias, C. Quilliet, A. Huillier, M. Aubouy, and F. Graner. Two-dimensional flows of foam: drag exerted on circular obstacles and dissipation. _Coll. Surf. A_ , 263:101–110, 2005c.
* Cox et al. (2006) S.J. Cox, B. Dollet, and F. Graner. Foam flow around an obstacle: simulations of obstacle-wall interaction. _Rheol. Acta._ , 45:403–410, 2006.
* Cantat and Pitois (2006) I. Cantat and O. Pitois. Stokes experiment in a liquid foam. _Phys. Fluids_ , 18:083302, 2006.
* Dollet et al. (2006) B. Dollet, M. Durth, and F. Graner. Flow of foam past an elliptical obstacle. _Phys. Rev. E_ , 73:061404, 2006.
* Dollet and Graner (2007) B. Dollet and F. Graner. Two-dimensional flow of foam around a circular obstacle: local measurements of elasticity, plasticity and flow. _J. Fl. Mech._ , 585:181–211, 2007.
* Raufaste et al. (2007) C. Raufaste, B. Dollet, S. Cox, Y. Jiang, and F. Graner. Yield drag in a two-dimensional foam flow around a circular obstacle: Effect of liquid fraction. _Euro. Phys. J. E_ , 23:217–228, 2007.
* Tabuteau et al. (2007) H. Tabuteau, F.K. Oppong, J.R. de Bruyn, and P. Coussot. Drag on a sphere moving through an aging system. _Europhys. Lett._ , 78:68007, 2007.
* Wyn et al. (2008) A. Wyn, I.T. Davies, and S.J. Cox. Simulations of two-dimensional foam rheology: localization in linear couette flow and the interaction of settling discs. _Euro. Phys. J. E_ , 26:81–89, 2008.
* Davies and Cox (2009) I.T. Davies and S.J. Cox. Sedimenting discs in a two-dimensional foam. _Coll. Surf. A_ , 344:8–14, 2009.
* Bragg and Nye (1947) L. Bragg and J.F. Nye. A dynamical model of a crystal structure. _Proc. R. Soc. Lond._ , A190:474–481, 1947.
* Weaire and Rivier (1984) D. Weaire and N. Rivier. Soap, cells and statistics – random patterns in two dimensions. _Contemp. Phys._ , 25:59–99, 1984.
* Buzza et al. (1995) D.M.A. Buzza, C.-Y. D. Lu, and M.E. Cates. Linear Shear Rheology of Incompressible Foams. _J. Phys. II France_ , 5:37–52, 1995.
* Davies and Cox (2010) I.T. Davies and S.J. Cox. Sedimentation of an elliptical object in a two-dimensional foam. _J. Non-Newt. Fl. Mech._ , 165:793–799, 2010.
* Brakke (1992) K. Brakke. The Surface Evolver. _Exp. Math._ , 1:141–165, 1992.
* Brakke (1986) K. Brakke. 200,000,000 Random Voronoi Polygons. www.susqu.edu/brakke/papers/voronoi.htm, 1986. Unpublished.
* Wang and Joseph (2004) J. Wang and D.D. Joseph. Potential flow of a second-order fluid over a sphere or an ellipse. _J. Fl. Mech._ , 511:201–215, 2004.
* Wang and Joseph (2005) J. Wang and D.D. Joseph. The lift, drag and torque on an airfoil in foam modeled by the potential flow of a second-order fluid, 2005. www.aem.umn.edu/people/faculty/joseph/archive/docs/931$\\_$airfoilfoam.pdf. Unpublished.
* Kern et al. (2004) N. Kern, D. Weaire, A. Martin, S. Hutzler, and S.J. Cox. Two-dimensional viscous froth model for foam dynamics. _Phys. Rev. E_ , 70:041411, 2004.
|
arxiv-papers
| 2012-02-27T07:54:45 |
2024-09-04T02:49:27.854784
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne, Simon Cox",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1202.5843"
}
|
1202.5996
|
# Long-term Running Experience with the Silicon Micro-strip Tracker at the DØ
detector
Andreas W. Jung1,††1Primary author, email contact: ajung@fnal.gov.
M. Cherry, D. Edmunds, M. Johnson, M. Matulik, M. Utes, T. Zmuda and the SMT
group Fermilab, Batavia, IL, 60510, USA
###### Abstract
The Silicon Micro-strip Tracker (SMT) at the DØ experiment in the Fermilab
Tevatron collider has been operating since 2001. In 2006, an additional layer,
referred to as ’Layer 0’, was installed to improve impact parameter resolution
and compensate for detector degradation due to radiation damage to the
original innermost SMT layer. The SMT detector provides valuable tracking and
vertexing information for the experiment. This contribution will highlight
aspects of the long term operation of the SMT, including the impact of the
silicon readout test-stand. Due to the full integration of the test-stand into
the DØ trigger framework, this test-stand provides an advantageous tool for
training of new experts and studying subtle effects in the SMT while
minimizing impact on the global data acquisition.
###### keywords:
Silicon , micro-strip , long-term operational experience
## 1 Introduction
The Run II DØ detector has been operating since 2001. It consists of two
central tracking detectors inside a $2~{}\mathrm{T}$ solenoidal magnet;
central and forward preshower systems; liquid argon calorimeters; and muon
spectrometers including a $1.8~{}\mathrm{T}$ toroidal magnet. The Silicon
Micro-strip Tracker (SMT) is part of the central tracking system of the DØ
detector and is the innermost layer of instrumentation [1]. Thus radiation
damage is a potential issue and needs to be monitored and addressed [2]. The
SMT layout is shown in Figure 1.
Fig. 1: The upgraded SMT detector consists of 6 Barrels, 12 ’F-disks’ and 2
’H-disks’. Barrels are interspersed with ’F-disk’. Additional ’F-disks’ and
’H-disks’ are placed at both ends of the detector.
It consists of six barrels each with four-layers. These barrels are
interspersed with six disks of small radius, so-called ’F-disks’. There are
another six ’F-disks’ beyond the end of the barrels. Two (originally four)
large radius detectors, so-called ’H-disks’, are located at the ends of the
detector to enhance tracking at very large pseudo-rapidities $|\eta|<3$. The
barrels provide tracking for particles with high transverse momentum in the
central regions $|\eta|<1.5$, while the disk detectors allow for the precise
reconstruction of particles traveling with pseudo-rapidity up to $|\eta|<3$. A
major SMT upgrade took place in 2006 to install an innermost layer (Layer 0)
[3]. This single-layer detector consists of eight barrels and was installed to
mitigate the degradation of the first layer of the original SMT due to
radiation damage. One ’H-disk’ was removed from each end of the detector in
2006 to accommodate the Layer 0 readout channels.
There are two different types of readout chips used for the SMT: SVX-IIe [4]
for the original SMT and SVX4 readout chips [5] for Layer 0. The SVX-IIe
readout chips are mounted on so-called HDIs (High Density Interconnects), made
from Kapton flex circuits laminated to Beryllium substrates. The silicon
sensors are glued to them and are referred to as ’module’ [1]. There are 432
such modules for the barrel, 288 for the ’F-disks’ and 96 for the ’H-disks’,
for a total of 5712 SVX-IIe chips installed. For L0 the HDIs are ceramic
hybrids made of Beryllium oxide. Including Layer 0 there are a total of 730k
readout channels providing the largest data flow of all DØ sub-detectors.
## 2 Long-term Operational Experience
The SMT has generally operated 24 hours per day 7 days a week since 2001,
which required dedicated shift personnel and experts. In general, the
operation is very stable and the response and recovery from problems is
usually quick. Shift personnel are supported by on-call experts. Furthermore
senior experts
Fig. 2: The hardware and readout chain of the original SMT from the sensor-
HDI level to the movable counting house level (MCH) via horse shoe, cathedral
and platform level.
are available for support in various aspects. Over time many tools have been
developed to monitor low and high level information such as voltages of power
supplies or on-line cluster charge and size histograms as well as on-line
track efficiencies. The hardware and readout chain of the SMT as sketched in
Figure 2 is distributed over several physical locations. These locations are
not entirely accessible on a daily basis: the ’Horse shoe’ and ’Cathedral’
area only during longer shutdowns whereas the ’Platform’ area can be accessed
between stores with agreement of Tevatron operations.
A failure in the hardware and readout chain needs to be understood and then
traced down to its physical location with the monitoring information at hand.
In order to do that it is very important to have monitoring capabilities for
low and high level information. This includes the monitoring of voltages and
current draws of the power supplies (PS) of the detector. For example a
failure of a power supply at the ’Platform’ area typically causes lower
efficiency for approximately a couple of hours until the failure is addressed.
On average all power supply failures including the ones in the cathedral area
compromised about 2.5% of the collected data. The electrical crew developed a
robotic remote switch ’R2DØ’ to switch to a spare PS within minutes for the
’Platform’ PS failures. All SMT power supplies at the ’Platform’ area have
since been equipped with ’R2DØ’ units and resulting data quality losses are
minimized.
## 3 SMT test-stand activities
The SMT test-stand provides a small ’copy’ of the full hardware and readout
chain of the SMT. In contrast to the SMT, all parts are easily accessible
allowing for detailed studies of single components.
Fig. 3: The picture shows part of the SMT test-stand setup: sensor and a
light emitting diode (lower left), PS (upper left), optional signal delay
generator with respect to the Tevatron clock (right).
Fig. 4: The graph shows the fraction of enabled HDIs for Barrel, ’F-disk’ and
’H-disk’ sensors versus time. The shaded yellow bands reflect the shutdown
periods of the DØ experiment. For a more detailed explanation of the steps in
the fraction of enabled HDIs see text.
Figure 4 shows parts of the test-stand setup for signal-to-noise studies with
HV supply (top left), sensor with a light emitting diode (bottom left) and
optional signal delay generator with respect to the Tevatron clock (right). In
general spare boards are tested at the test-stand before they are installed.
This is also true during the longer shutdowns where problematic boards are
replaced. During these shutdowns every effort is made to improve the system
stability and efficiency. For a complex system like the SMT it is difficult to
cite a single quantity characterizing global performance. A good measure is
the fraction of enabled HDIs as a function of time as shown in Figure 4. Prior
to the 2009 shutdown there was a gradual decrease of this fraction due to
hardware issues during continued operation. There are many different types of
failures and the most common ones are individual chip failures as well as bad
cable connections at various levels as given in Table 1 (larger steps in the
fraction of enabled HDIs are explained later in the text).
Type of HDI failure | 2009 / # HDI | 2010 / # HDI
---|---|---
Adapter card | 16 | 1
Clock cables | 9 | 10
Interface Board | 15 | $-$
Reseating cables | $\approx$ 6 | 2
Bad/dead (problem inside detector) | 20 | Not re-visited
Disabling bad chips | 24 | 4
Total # HDI worked on | 90 | 17
Table 1: Detailed list of HDI defects for the years $2009-2010$.
In order to trace such a failure to its underlying cause every failure is
characterized and a record of previously tried interventions is maintained.
The shutdown periods are highlighted with shaded bands in Figure 4. They
allowed for the time-consuming task to investigate and fix these individual
failures. The efforts resulted in higher fractions of enabled HDIs after a
shutdown. Another example of a sort of failure are broken wire-bonds which
interrupted the distribution of the digital power lines to the readout chips.
As the readout chips on an individual HDI are daisy-chained, a single chip
failure caused the ’loss’ of all subsequent chips of a module consisting of
up-to 9 chips. The most prominent occasion occurred in late 2006 but it is
likely that this sort of failure also contributed to the losses of enabled
HDIs prior to that incident. The test-stand facilitated the development of a
solution to this problem by using an alternative path to distribute the
digital power using a special hardware board (new adapter card). Thus the
initial failing chip was bypassed and the readout of the remaining chips on
the module could be fully restored as implemented during the shutdown in 2007,
which increased the fraction of enabled HDIs by about 10%. Furthermore the
test-stand facilitated the development of an improved sequencer firmware
version as well as a modified version of the adapter card in order to fix a
noise problem. Both have been installed during the shutdown in 2008 and
increased the fraction of enabled HDIs. An intensive and thorough
investigation for all known sorts of failures took place during the shutdown
in 2009 and resulted in the largest fraction of enabled HDIs. The tireless
efforts during the past shutdowns allowed re-enabling of HDIs and led to an
all-time high number of enabled HDIs.
The test-stand was also used for detailed firmware studies in order to improve
signal-to-noise (S/N) for the sensors controlled and readout by the SVX-IIe
type of chips. The pedestal distribution for the ’old’ firmware and the ’new’
(improved) firmware is shown in Figure 5a) and b).
Fig. 5: Pedestal distribution for 6 chips with 128 channels per chip. First
three chips are p-side and last three chips are n-side. The pedestal
distribution for the ’Old Firmware’ is shown on the left, whereas the one for
the ’New Firmware’ is shown on the right.
By moving certain activities on control lines to a different point in time a
significant reduction of the noise level was achieved. The biggest impact in
terms of reducing the noise was achieved by moving control signals for
’PreAmp’-reset and ’RampReference’-select further away from the start of
digitization. For n-side type of sensors the noise was reduced by
approximately 20% whereas for the p-side type sensors noise level was stable.
The noise source is not coupling in the same way to all channels as it can be
seen in Figure 5a). Our interpretation is that the control signal pulse
generates noise on the chip. The previously persistent second band structure
is now completely removed as it can be seen by comparing Figure 5a) and b).
This firmware is now used for the entire SMT.
The DØ data acquisition (DAQ) is a buffered system and consequently the dead-
time or front-end busy rate (FEB)
Fig. 6: Simplified sketch of the data flow from left to right. The red arrows
indicate a busy signal at different levels if no free buffer is available.
is driven by the amount of data and the ability to process it. Figure 6 shows
a simplified sketch of the data flow in the DØ experiment. Data are processed
by means of a multi-level trigger system (L1, L2, L3). The red arrows indicate
a busy signal at different levels if no free buffer is available.
Individual SMT crates showed a very peculiar FEB pattern: one would expect
that the SMT crate leading in FEB is given by highest data processing load as
it takes more time to process more data. Instead the FEB leading SMT crate
seems to appear randomly as shown in Figure 7a). It shows FEB rates [%] of all
SMT crates (different colors) as a function of time with the two crates
showing the peculiar FEB pattern highlighted by the red circles. This happened
on an apparently random basis but more frequently at higher trigger rates. The
buffer handling is organized by a VME read-out board controller (VRBC) [6]
which controls the VME read-out boards (VRB) [6], which in turn are gathering
the data from the sequencer level as sketched in Figure 2. The VRBC firmware
was extended with monitoring capabilities for buffer management.
Fig. 7: a) shows FEB rates [%] of all SMT crates (different colors) as a
function of time without the improved buffer handling firmware. The two plots
in b) show the available buffers (blue), buffers waiting for L2 (green) or L3
decisions (red) as a function of time. As an example the buffer distribution
is shown for the two SMT crates which exhibit an increased FEB rate (top plot,
crate 0x65 & 0x67) as highlighted by the red ellipses and arrows. c) shows the
FEB rates [%] of various detector subsystem crates grouped by colors (SMT
crates are colored in red). In addition the global DØ L1 busy rate (green)
consisting of all L1 subsystems is shown. More details in the text.
The two plots in Figure 7b) show the available buffers (blue), buffers waiting
for L2 (green) or L3 decisions (red) as a function of time. The buffer
distribution is shown for the two SMT crates which exhibit the increased FEB
rate (SMT crates 0x65 and 0x67) as highlighted by the red ellipses and arrows.
A good correlation between the number of available buffers and the FEB was
seen. In general there are less available buffers at higher trigger rates. The
red circles connected by arrows in Figure 7a)-b) highlight the peculiar FEB
pattern shown by two different SMT crates. This effect was due to the sudden
reduction of available buffers (blue) causing increased dead-times for the
affected SMT crates. Figure 7c) shows the FEB rates [%] of various detector
subsystem crates (SMT crates are colored in red). In addition the global DØ L1
busy rate consisting of all L1 subsystems is plotted (green). The yellow
ellipses highlight an increase of the global L1 busy rate caused by raised FEB
rates of particular SMT crates. This illustrates how the sudden reduction of
available buffers in SMT crates affected the global L1 busy rate. At higher
trigger rates (around $-50$ minutes) the effect is not large. There is an
increase of the global L1 busy rate by approximately 2% at the same time as
the jump in FEB for a SMT crate: from 10% to approximately 12%. At lower
trigger rates (around $-5$ minutes) the effect is smaller and the global L1
busy rate increases only by about 0.4%. The latter can be understood as the
reduced number of buffers has largest impact at high data taking rates.
Fig. 8: a) shows the FEB rates [%] of all SMT crates (different colors) as a
function of time with the improved buffer handling firmware. The two plots in
b) show again available buffers (blue), buffers waiting for L2 (green) or L3
decisions (red) as an example for two different SMT crates. More details in
the text.
The SMT test-stand allowed tests at high rates of new versions of the VRBC
firmware handling buffer management. A more robust VRBC firmware version was
developed and it did not show this problem anymore. Monitoring data for this
modified VRBC firmware version are shown in Figure 8a)-b). a) shows the FEB
rates [%] of all SMT crates (different colors) as a function of time with the
improved buffer handling firmware. There are no SMT crates showing a
significantly higher FEB rate. The increased FEB rates visible at the end of
the distribution was due to a change in prescale settings, which increased the
event rates. Figure 8b) shows the available buffers (blue), buffers waiting
for L2 (green) or L3 decisions (red) as a function of time for two different
SMT crates. There are no sudden drops in the number of available buffers
anymore.
## 4 Conclusions
The SMT has been operated since 2001. Its performance and efficiency have been
enhanced using new tools such as the ’R2DØ’ units. The SMT test-stand is a
unique piece of equipment to train new experts as well as to reproduce and
understand subtle effects in the SMT while minimizing impact on global data
taking. Three examples have been presented: HDI recovery effort, optimization
of signal-to-noise and the buffer management problem. In each case the results
from the test-stand led to improved performance for the entire SMT system. The
training of new experts at the test-stand allowed for new insights into the
operation of the SMT, which in turn increased the stability and performance of
the SMT.
The SMT detector is performing very well, providing good tracking and
vertexing capabilities for the DØ experiment, which is vital for high
efficiency b-tagging and electron/photon identification.
## References
* [1] S.N. Ahmed et al, The DØ Silicon Microstrip Tracker, NIM A 634 8, [arXiv:1005.0801], 2011.
* [2] Z.Ye, TIPP2011 talk, Radiation Damage to DØ Silicon Microstrip Detector, 2011.
* [3] R. Angstadt et al, The L0 Inner Silicon Detector of the DØ experiment, NIM A, 622, 298, [arXiv:0911.2522], 2010.
* [4] I. Kipnis, S. Kleinfelder, L.Luo, O. Milgrome, M. Sarraj, R. Yarema, T. Zimmerman: A Beginners Guide to the SVXIIE, FERMILAB-TM-1892. version from 10/96.
* [5] M. Garcia-Sciveres et al, The SVX4 integrated circuit, NIM A, 511, 171, 2003.
* [6] E. Barsotti, M. Bowden, H. Gonzalez, M. Johnson, D. Mendoza, T. Zmuda, VME Readout Buffer, Fermilab Document Nr ESE-SVX-950719, 10/12/2001.
|
arxiv-papers
| 2012-02-27T16:52:46 |
2024-09-04T02:49:27.868067
|
{
"license": "Public Domain",
"authors": "Andreas W. Jung, M. Cherry, D. Edmunds, M. Johnson, M. Matulik, M.\n Utes, T. Zmuda and the SMT Group",
"submitter": "Andreas Werner Jung",
"url": "https://arxiv.org/abs/1202.5996"
}
|
1202.6027
|
# Multiscale Analysis of Collective Decision–Making in Swarms: An Advection-
Diffusion with Memory Approach
M. Raghib, S.A. Levin, I.G. Kevrekidis
###### Abstract
We propose a (time) multiscale method for the coarse-grained analysis of
self–propelled particle models of swarms comprising a mixture of ‘naïve’ and
‘informed’ individuals, used to address questions related to collective motion
and collective decision–making in animal groups. The method is based on
projecting the particle configuration onto a single ‘meta-particle’ that
consists of the group elongation and the mean group velocity and position. The
collective states of the configuration can be associated with the transient
and asymptotic transport properties of the random walk followed by the
meta–particle. These properties can be accurately predicted by an advection-
diffusion equation with memory (ADEM) whose parameters are obtained from a
mean group velocity time series obtained from a single simulation run of the
individual–based model.
##### keywords
continuous time random walks, anomalous transport, collective animal behavior,
non-Markovian stochastic processes, self–propelled particle models.
## 1 Introduction
Self-propelled particle models (SPP’s) are a class of agent–based simulations
that have been used over the last three decades to explore questions related
to various kinds of collective motion in animals, including insect swarms,
bird flocks and fish schools [1, 50, 29, 26, 17, 16, 48, 52, 44]. In these
models, each individual in the (finite) population is represented by a
particle that moves with constant speed in two or three-dimensional Euclidean
space or a 2-dimensional torus. All particles update their orientations
according to a set of local averages of the current state of the
configuration. These local averages are simplified representations of
individual behaviors that depend on ‘social interactions’ –avoidance of
collisions, attraction, and orientation alignment– which result in the
remarkable property of cohesive collective motion; i.e. the particles move
about in space, yet they appear to move as a single object, resembling the
motion of real flocks [1, 50, 16]. Errors made by the individuals as they
estimate these quantities are modeled by a random rotation of the output of
this averaging procedure.
More recently, SPP models of flocking have been introduced in the context of
collective decision-making to illuminate the question of how groups of agents
achieve consensual decisions without the need of a central control [16, 13,
15, 12, 44]. Each of these decisions can be associated with a variety of
collective states, which typically involve switching between mobile/immobile
regimes [35], rotation or milling [16], motion with a directional bias [15],
or a combination of these [42]. A directional bias is relevant when critically
important information, for instance the location of a resource, a predator or
a migratory route, is available only to a fraction of the population [15, 42].
[15] explored this situation using a modified version of earlier models of
swarming [1, 50, 29, 16], where the main innovation consisted of dividing the
population into two types. The first of these, called ‘naïve’, follow only the
social rules mentioned earlier (avoidance, attraction and alignment). The
second kind, dubbed ‘informed’, also obey the social interactions of the naïve
individuals, but weigh the social output with an orientation bias along a
single ‘preferred’ direction, which in this study is identical for all
informed individuals. This orientation bias can be regarded as a simple
representation of access to privileged information. Collective decision–making
is understood in this context in terms of the ability of the informed
sub–population to transfer their orientation bias to the whole group while
simultaneously preserving group cohesion.
Despite the recent explosion of SPP models in the literature, our
understanding of these systems still remains limited. Central challenges are
related to our ability to characterize efficiently and meaningfully the
dynamics of each collective state, and critically, their dependence on the
parameters of the individual–level model. We identify three distinct
approaches to address this problem; namely Monte-Carlo simulation, continuum
models, and ‘hybrid’ multi-scale approaches.
The first (the Lagrangian approach) is mainly computational and consists of
moving with each individual particle. Macroscopic summary statistics
describing the various collective states are obtained from averages based on a
large number of independent simulation runs, or a single time series when
ergodicity is a reasonable assumption. These average quantities usually
include the mean group velocity [17], the mean angular momentum [16], mean
switching times between mobile/immobile states [35] or the ‘accuracy’ of the
decision-making process [15, 38]. Other state variables of interest link
collective states to geometrical properties of the flock, like the group
elongation [15] or its aspect ratio [8].
The second method (the Eulerian approach) focuses on continuum models for the
density and velocity fields. It has the advantage that in some cases
analytical results linking the microscopic to the macroscopic can be
rigorously derived. In addition to this, the numerical solution of the model
for large or small population densities has the same computational cost, and
the mechanisms that generate the collective patterns can often be clearly
distinguished in the various terms in the model, which provides some degree of
parsimony that approaches based solely on Monte–Carlo simulations cannot
emulate. Continuum approximations have been used to approximate discrete SPP
models mainly to study collective motion that is not cohesive [17]; i.e. the
population lives in a spatial arena with periodic or reflecting boundaries but
does not form a single distinct group. Instead, particles move about freely
forming and dissolving groups of various sizes (i.e. fission–fusion dynamics),
and collective motion is detected as a non–vanishing population average of the
velocity. These continuum models are obtained through heuristic reasoning
based on careful observation of system symmetries, or the invocation of
conservation laws [40, 54, 55, 19].
Although substantial progress has been made with Eulerian (continuum)
approaches, particularly for swarming microbial populations [2, 51], there are
still a number of issues that preclude their widespread use. First, the use of
heuristics does not clarify the dependence of the macroscopic parameters on
the individual–level model. Although some continuum models have recently been
derived formally from the individual–based model via a limiting process
(usually large population size), the theoretical progress is made at the
expense of great simplifications which restrict strongly their biological
relevance. For instance [4] and [9] each derived continuum models in the limit
of large population sizes, but restricted the individual–level interactions to
a single type of social interaction, specified via a potential function [9],
or a velocity average [4]. Second, they usually require very large population
sizes in order to be meaningful, which is problematic for models of flocking
in groups involving tens or perhaps hundreds of individuals. In this situation
the finiteness of the population size plays a fundamental role in observed
transport properties (e.g. the group tends to move more slowly as the
population size increases) [16, 7, 8, 25, 53, 57].
The third is the hybrid multiscale approach, which attempts to bridge
Monte–Carlo simulations and continuum models. It is based on assuming the
existence of a continuum model for some relevant coarse–grained state variable
or ‘reduction coordinate’; for instance Non-linear Advection–Diffusion
Equations (NADE) with density–dependent coefficients [25], or Fokker–Planck
equations with a non-linear potential [33, 20, 42, 35, 11, 60], which serves
as a model template. The unknown fluxes and coefficients in the macroscopic
template are _estimated_ from a computational experiment, which usually
consists of a single –and relatively short– simulation run of the microscopic
model. These estimated quantities are substituted into the unknown terms in
the macroscopic model, which is then analyzed by means of the appropriate
suite of classical continuum methods, numerical or analytical.
In this study, we use this latter approach to explore the ability of
Continuous Time Random Walks (CTRW) [41, 30, 14, 3], and its associated
continuum counterpart, the Advection–Diffusion Equation with Memory (ADEM)
–also known as the Generalized Master Equation (GME)– as a model template for
the coarse–grained dynamics of cohesive collective motion and collective
decision–making in self–propelled particle models of swarms comprising a
mixture of individuals that have preferential access to critical information
–the ‘informed’ type– and those who do not ( ‘naïve’). The ADEM generalizes
the classical advection–diffusion equation to a non–local–in–time transport
model via the introduction of a ‘memory’, a time weighting function
proportional to the particle’s two–time velocity autocorrelation function. The
ADEM is a useful model of anomalous transport that arises when the underlying
random walk possesses a wide distribution of transition rates [30, 31, 14, 39,
3]. The multiscale method we propose is based on coarse–graining the full SPP
configuration into a single ‘meta–particle’, that consists of the group
elongation (as a measure that the group remains cohesive) and the mean group
velocity and position. The various types of collective states displayed by the
group can then be related to the transport properties of the meta–particle’s
random walk, under the assumption that the pdf of the transition density for
the meta-particle’s position follows an ADEM.
We illustrate the method for the case of a 2–dimensional SPP model introduced
earlier by [15] for a single informed direction, but the approach is quite
general in the sense that it can be applied to any individual–based model of
movement for which the biologically meaningful coarse variables are the mean
group position and velocity, and that the effective distribution of jump
lengths at each transition event has finite moments of all orders. The
multiscale approach for collective motion based on the ADEM complements
_local_ –in–time multiscale approaches for a similar class of individual–level
models explored earlier [35, 25, 60]. For instance, the ADEM can predict
correctly the transport properties even when the individual–level model has a
strong alignment rule, which is precisely the main limitation of the otherwise
successful method based on non-linear advection–diffusion equations [25]. This
results from temporal correlations in velocity fluctuations induced by the
alignment rule that persist over macroscopically relevant time scales, a
property that can not be captured by local–in–time Markovian models, but can
be dealt with via the introduction of a memory term.
CTRW theory generalizes the classical Random Walk (RW) as a microscopic model
better suited for problems in anomalous transport, which is usually detected
when the mean squared displacement (msd) does not scale linearly with time
over a wide range of time scales. The anomalous properties can frequently be
attributed to the presence of a wide distribution of transition rates (or also
in the jump lengths), which leads to persistent temporal correlations in
velocity fluctuations. It is the presence of time correlations in velocity
that ultimately leads to anomalous transport [41, 30, 34, 39]. The variability
in transition rates can be attributed in real systems to spatial disorder in
the medium, as is the case in tracer transport in porous media [3]. The
presence of spatial disorder in the medium creates localized structures that
can trap the particle for long periods of time, or force it to move
ballistically by confining its motion along a corridor. The resulting particle
motion consists of alternating bursts of ballistic motion, apparent brownian
motion, and a stagnant phase where the particle moves very slowly, if at all.
This resembles the dynamics of the group meta–particle in SPP models of
swarms, which typically consists of bursts of alignment in the particle
orientations that lead to advective flights at the group level (the slip
phase), alternating with regimes of slow motion when the particles lose their
alignment and the mean group velocity drops sharply (the stick phase). The
power of CTRW [14, 3] and effective medium theories of random motion in
disordered media [31], lies in that the spatial inhomogeneities in the medium
responsible for the anomalies in transport properties are not modeled
explicitly. Instead, their effect is summarized _statistically_ in terms of
the effective distributions of jump lengths and waiting times that define the
random walk. The key innovation of CTRW theory is that the random walk does
not proceed by fixed spatial and temporal increments, but these become instead
random variables, defined by two probability densities, which are usually
assumed independent in applications. The first is the distribution of jumps in
space $\lambda(\xi)$, which prescribes the length of the jumps between
locations at each transition event. The second is a clock that regulates the
times elapsed between transitions, known as the distribution of waiting times
$\psi(\tau)$. A thorough discussion of modern CTRW theory and its role in
models of anomalous transport can be found in a recent review by [39].
It can be shown [61, 41, 30, 39, 3] that when the distribution of jump lengths
can be expanded in a Taylor series and the distribution of waiting times is an
arbitrary probability density function, the transition probability density
$p(\mathbf{x},t|\mathbf{0},0)$ for finding a particle around position
$\mathbf{x}\in\mathbb{R}^{2}$ at time $t$ given that it started at the origin
at time zero, obeys a modified version of the advection–diffusion equation
that is non-local in time, known as the Advection-Diffusion Equation with
Memory (ADEM)
$\displaystyle\frac{\partial p(\mathbf{x},t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\,M(t-s)\,\left[\mathbf{v_{\lambda}}\cdot\nabla
p(\mathbf{x},s)-\mathbf{D_{\lambda}}\,:\,\nabla\,\nabla
p(\mathbf{x},s)\right]\,ds$ (1) $\displaystyle p(\mathbf{x},0^{+})$
$\displaystyle=$
$\displaystyle\delta(\mathbf{x}),~{}~{}\mathbf{x}\in\mathbb{R}^{2},~{}~{}t\in\mathbb{R}^{+}$
where $\mathbf{v_{\lambda}}$ is the effective drift vector,
$\mathbf{D_{\lambda}}$ the diffusivity tensor, and the colon operator is the
inner tensor product
$\mathbf{A}:\mathbf{B}=\mathrm{Trace}\\{\mathbf{B}^{T}\cdot\mathbf{A}\\}=\sum_{i,j}A_{ij}B_{ij}.$
The transport coefficients $\mathbf{v_{\lambda}}$ and $\mathbf{D_{\lambda}}$
are determined respectively by the ratio of the first two moments of the jump
distribution to the mean of the waiting time distribution, or the median when
$\psi(t)$ does not have a finite mean [3]. The memory function $M(t)$ has two
equivalent interpretations; it is closely related to the distribution of
waiting times [14], but it can also be shown to be proportional to the
velocity time auto–correlation function of the moving particle [28, 32, 58].
$\mathrm{E}\left[v_{1}(0)\,v_{1}(\tau)\right]=2D_{1}M(\tau),$
where $v_{1}$ is the velocity along the $x_{1}$ direction, $D_{1}$ is the
diffusivity along $x_{1}$ and $M(t)$ is the memory kernel that prescribes the
decay of correlations (see Section 4 for additional details). This model
constitutes the basis for effective medium theories of anomalous transport in
disordered media, where the spatial disorder in the medium is replaced by an
ordered model with memory of the form (1) [28, 37, 32, 3, 31].
The stochastic dynamics of the meta–particle associated with the SPP model of
flocking explored here has a striking resemblance to that which motivated the
development of the theory for anomalous transport in heterogenous media based
on the CTRW and the ADEM. In SPP models, the wide range of variability in
transition rates cannot be attributed to spatial disorder in the medium, but
arises instead from stochasticity in the alternating (slip/stick) types of
collective behavior. Even though the source of variability is quite different,
this does not seem to matter provided that transport can be modeled in terms
of _effective_ distributions of jump lengths and waiting times and their
associated ADEM. Our goal is to exploit this analogy to propose an ADEM as a
continuum ‘model template’ for the dynamics of the position pdf of a swarm
centroid. The functional form of the memory and the transport parameters in
the ADEM template can be estimated from a single mean group velocity time
series obtained from a simulation run of the SPP model. The resulting fitted
model can be then used to explore the dependence of the collective behaviors
on the parameters that determine the individual–level model, particularly the
strength of the bias of the informed sub–population, the total population
size, and the proportion of informed individuals.
In the ADEM approach, the memory is the fundamental object that encodes all
the transport coefficients, the various transport regimes and their
characteristic timescales [36, 32]. When the spatial distribution of the
disorder is known [31] or the Hamiltonian of the microscopic model [32], it is
possible to derive the memory in (1) from the microscopic dynamics. In
general, one has to resort to simulations or experiments and subsequent
function fittings, in order to obtain the velocity auto–correlation function.
The non-linearities involved in the definition of the SPP seem to preclude the
derivation of the velocity time auto–correlation function rigorously from the
microscopic swarm model. We find from simulations that the memory kernel along
the informed direction for SPP models can be very well fitted by two closely
related functions. The first corresponds to a Gamma density,
$M(t)=\frac{\tau_{a}^{\gamma-1}}{\Gamma(1-\gamma)}t^{-\gamma}\exp(-t/\tau_{a}),$
(2)
which works well in swarms where there are no informed individuals present,
but also when the proportion of informed individuals is small (and relatively
low values of the coupling strength). The initial power law decay in (2) leads
to a sub–ballistic, super–diffusive transient detectable in the mean–squared
displacement. This power law behavior has an exponential truncation at a
characteristic time scale $\tau_{a}$ that establishes the onset of the
asymptotic regime, which is dominated by diffusion in swarms with no informed
individuals and a mixture of diffusion and advection (with constant drift) for
groups that include informed individuals. We also find that the diffusion
coefficient decreases with group size, and the time scale ($\tau_{a}$) that
determines the onset of the asymptotic regime increases with group size.
The second, ‘richer’ situation, arises in informed swarms for high values of
the bias along the informed direction, where the early time super-diffusive
transient is followed by a regime where correlations oscillate before reaching
the asymptotic state, which is also classical advection–diffusion. This
additional regime requires a modification of the memory kernel (2) in order to
capture these oscillations. We find that a Mittag–Leffler function
$E_{\alpha,\beta}(z)$ with an exponential truncation [49, 59],
$M(t)=\frac{\tau_{s}+\tau_{a}^{\alpha}}{\tau_{s}\tau_{a}^{\beta}}\,t^{\beta-1}\,E_{\alpha,\beta}\left[-\left(t/\tau_{s}\right)^{\alpha}\right]\,\exp(-t/\tau_{a}),$
(3)
provides an excellent fit in this regime, at the cost of introducing two
additional parameters (the exponent $\beta$ and the time scale $\tau_{s}$). We
used these estimates together with the ADEM model in order to predict the
behavior of the mean squared displacement (msd), i.e. the second moment of the
mean group position, which can be used to characterize the various types of
collective behaviors and their characteristic time scales in terms of their
effect on the meta–particle’s transport properties. The functional forms
themselves do not seem to change with group size, but only the parameters do.
For the region of parameters where the group remains cohesive, we observed
that there are two types of collective behavior that are shared by both naïve
(no informed individuals present) and informed groups. First, there is an
anomalous super–diffusive transient at early times (the scaling exponent in
the mean squared displacement lies between one and two) due to the prevalence
of slip/stick dynamics over that domain of time scales. Asymptotically, the
msd scales linearly with time for naïve groups (diffusion–dominated), but
shows a sharp transition to quadratic scaling (advection–dominated) for
informed ones along the informed direction, which indicates that on average,
informed swarms diffuse, but also move with constant velocity over the longer
time scales. This transition from linear to quadratic scaling allows the
detection of the time scale at which the informed sub-population manages to
transfer its orientation bias to the whole group; this time scale, or time to
consensus, is a natural measure of the efficiency of the decision–making
process. The magnitude of the drift, which depends on the degree of
polarization of the particle orientations along the informed direction, is a
straightforward macroscopic parameter for the degree of consensus. We also
note that as the group size gets larger, the drift gets smaller for the same
proportion of informed individuals and informed bias strength. Finally, the
diffusion coefficient along the informed direction can be interpreted as a
measure of the precision of the collective decision–making process –since it
is a measure of the spread of an ensemble of swarm meta–particles– when
compared with that of naïve configurations.
The resulting ADEM fitted from swarm simulation time-series is self-consistent
in the sense that transport parameters estimated from the memory via a
Kubo–Green relationship [36, 24] coincide with those estimated from the
moments of the jump and waiting time pdf’s of the associated CTRW for the
three group sizes explored ($N=10,50,100$), proportions of informed
individuals, and strength of the bias along the preferred direction. We also
discuss the phase diagrams for the transport coefficients estimated from this
method, where we notice velocity–precision trade–offs: as the total group size
gets larger, the decision–making becomes more precise at the expense of a
slower mean group velocity. We also note that the time scale to consensus is
invariant with respect to group size, and depends only on the proportion of
informed individuals and the strength of the coupling along the informed
direction.
The paper is organized as follows: Section 2 introduces a slightly modified
version of the SPP model with informed individuals of [15], where we removed
the constraint on the maximum turning angle that an individual can make during
a time step. We then define the set of coarse–grained variables of interest,
namely the group elongation, the mean group position, and the mean group
velocity which we called the meta–particle. Simulation results are also shown,
focusing on the mean squared displacement of the meta–particle as well as
kernel density estimates of the probabilities of mean group speeds and
orientations, finalizing with group elongation time series that detect when
the group splits appart. These results are later used to define macroscopic
measures of collective motion and collective decision–making in terms of the
transport regimes shown in the msd. Section 3 briefly reviews known results
from the theory of continuous time random walks (CTRW) [41, 30], and its
relationship to the advection-diffusion equation with memoy (ADEM) [3] that we
use later as the macroscopic transport model for the transition density of the
mean group position. Section 4 assumes that the random walk followed by the
group meta-particle evolves according to a CTRW, and discusses the procedure
used to estimate the memory and the transport coefficients of the associated
ADEM, from a single velocity time series obtained from a run of the
individual-based model. We compare mean squared displacements obtained from
ensemble averages over simulation runs with those predicted by the fitted ADEM
for which show analytical results for the time to consensus. The method is
used to carry out a systematic exploration of the dependence of the
macroscopic parameters –the diffusivity, the drift and the time to consensus–
on the microscopic ones of immediate biological relevance; namely the relative
proportion of informed individuals, the coupling strength, and the total
population size. Some final remarks are presented in Section 5.
## 2 Self-propelled particle model (SPP) with informed individuals
Consider a population of $j=1,\ldots,N$ particles with positions
$\mathbf{x}_{j}(t)$ in 2-dimensional Euclidean space. Each particle $j$ moves
with constant speed $s$ along its orientation angle $\theta_{j}(t)$ in
$[-\pi,\pi)$. We summarize this information as the (complex) particle velocity
$z_{j}(t)=s\,e^{i\,\theta_{j}(t)}.$
The state of the population at (discrete) time $t$ is represented by the
configuration $\Phi_{t}(A)$
$\Phi_{t}(A)=\left\\{\,[\,\mathbf{x}_{j}(t),z_{j}(t)\,]\right\\},$ (4)
where $A$ is the region of observation. At each tick of the clock, the
positions and orientations of each particle are updated according to,
$\displaystyle\mathbf{x}_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\mathbf{x}_{j}(t)+s\,\left(\begin{array}[]{c}\cos[\theta_{j}(t)\,]\\\
\sin[\theta_{j}(t)\,]\end{array}\right)\Delta t$ (7)
$\displaystyle\theta_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\langle\,\Phi_{t}(D_{j})\,\rangle\,\exp(i\,\Delta Q)$
where $\Delta t$ is the time increment and $\langle\Phi_{t}(D_{j})\rangle$ is
a local average of the configuration restricted to an interaction region
$D_{j}$ centered around the $j$-th particle. The details of the averaging
procedure are described in the collective motion rule below ( Figure 1).
Errors made by the individuals in their estimates of the local state of the
configuration are modeled by rotating the updated orientation obtained from
the local average by a random angle $\Delta Q$, drawn from the wrapped
Gaussian on the unit circle $\mathcal{N}_{w}(0,\sigma^{2}\,\Delta t)$ with
mean zero and variance $\sigma^{2}\Delta t$.
The local average in (7) comprises two groups of rules. The first is based on
the classical social interactions for collective motion, with parameters
restricted to the domain in which the full configuration moves cohesively as a
single object [1, 50, 17, 21, 16]. The second is a steering rule proposed by
Couzin _et al_ [15], that attempts to lead the motion of the group along a
preferred orientation $\beta$. This additional rule is followed only by a sub-
population of ‘informed individuals’. Whereas individuals that are not
informed (called ‘naïve’) update their orientations exclusively from the
output of the social rules, informed individuals update their orientations
according to a weighted average of the social interactions with the preferred
direction. The weight of the bias along the preferred direction relative to
the social rules is given by a ‘coupling constant’ $\omega$, which is
interpreted as a simple parameterization of an ‘internal state’ of the
informed individual (e.g. starvation, detection of a predator or a resource).
Collective decision–making is then understood in terms of the ability of the
informed sub-population to transfer their orientation bias to the whole group.
Figure 1: Interaction zones for a focal individual (blue) $\mathbf{x}_{j}$
(dot) with velocity $z_{j}$ (arrow). The dots and the arrows represent other
particles in the configuration (black). The region of avoidance is the
interior of the circle of radius $r_{av}$. The particles contributing to the
region of alignment and attraction lie within the annulus of external radius
$r_{at}$ and internal radius $r_{av}$.
The three social interactions are: 1) avoidance of collisions, 2) attraction
(centering), and 3) alignment (polarization). Whereas the collective motion
interactions are followed by all $N$ particles, the steering rule is followed
only by the informed sub-population of $N_{\beta}\leq N$ particles, whose
indices $J_{\beta}=\\{j_{1},\ldots,j_{N_{\beta}}\\}$ are chosen uniformly from
the set of indices of all the particles in the configuration
$J_{\Phi}=\\{1,\ldots,N\\}$. Both the number of informed particles as well as
their indices remain fixed for all times once chosen at time zero. The
particles that are not in the informed sub-group are called called ‘naïve’.
Following Couzin _et al_ [15] we have
1. 1.
Collective motion rule
1. (a)
Avoidance of collisions
We define the neighborhood of avoidance of the $j$-th particle
$Av_{j}=B(r_{av},\mathbf{x}_{j}(t))$ as the circular domain of radius $r_{av}$
centered at $\mathbf{x}_{j}(t)$ (see Figure 1). If the configuration
restricted to the window $Av_{j}$ is not empty, the avoidance rule takes
precedence over the other interactions. The avoidance rule prevents collisions
by pointing the focal particle in the opposite direction of the centroid of
the locations of the particles found within $Av_{j}$, relative to the location
of the focal particle $\mathbf{x}_{j}(t)$. The number of neighbors of the
$j$-th particle in $\Phi_{t}(Av_{j})$ is
$N_{Av_{j}}=\sum_{k\neq j}^{N}I_{Av_{j}}\left(\mathbf{x}_{k}(t)\right)$
where the focal individual $j$ is excluded from the count and
$I_{B}(\mathbf{x})$ stands for the indicator function of some 2-D domain $B$,
$\displaystyle I_{B}(\mathbf{x})=\left\\{\begin{array}[]{cc}1,&\mbox{ if
}\mathbf{x}\in B\\\ 0&\mbox{ otherwise}.\end{array}\right.$ (10)
The vector pointing in the direction opposite to the centroid of the particles
in $A_{v_{j}}$ is
$\displaystyle\mathbf{d}_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle-\frac{1}{N_{Av_{j}}}\sum_{k\neq
j}^{N}I_{Av_{j}}\left(\mathbf{x}_{k}(t)\right)\left[\,\mathbf{x}_{k}(t)-\mathbf{x}_{j}(t)\,\right],$
(11)
The updated orientation due to avoidance is
$\displaystyle\theta_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\arg(\mathbf{d}_{j})+\Delta Q,$ (12)
where $\Delta Q$ is a random angle drawn from
$\mathcal{N}_{w}(0,\sigma^{2}\,\Delta t)$.
2. (b)
Attraction and Alignment
If the configuration restricted to $Av_{j}$ is empty, we proceed to evaluate
the alignment and attraction updating rules. The neighborhood of
attraction/alignment of the $j$-th particle is
$At_{j}=B(r_{at},\mathbf{x}_{j}(t)\,)$, where $B(r_{at},\mathbf{x}_{j}(t)\,)$
is the circular domain of radius $r_{at}$ centered at $\mathbf{x}_{j}(t)$. The
social interaction in this case is given by the normalized vector sum over the
positions (which determines the local attraction vector), and the velocities,
(which dictates the local alignment vector) of the neighbors. The number of
neighbors in $At_{j}$ is
$N_{At_{j}}=\sum_{k=1}^{N}I_{At_{j}}\left(\mathbf{x}_{k}(t)\right),$
The contribution due to attraction is given by the vector
$\mathbf{d}_{j}^{\xi}$ pointing in the direction of the centroid of the
positions of the neighbors relative to the focal individual
$\displaystyle\mathbf{d}_{j}^{\xi}(t+\Delta
t)=\frac{1}{N_{At_{j}}}\sum_{k=1}^{N}I_{At_{j}}\left(\mathbf{x}_{k}(t)\right)\,[\,\mathbf{x}_{k}(t)-\mathbf{x}_{j}(t)\,]$
(13)
and the contribution due to the alignment behavior $\mathbf{d}_{j}^{\theta}$
comes from the average orientation of all the particles in $At_{j}$
$\displaystyle\mathbf{d}_{j}^{\theta}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N}I_{At_{j}}(\mathbf{x}_{k}(t))\,z_{k}(t).$ (14)
The total contribution of the social rules is given by the vector sum of the
normalized vectors associated with the attraction (13) and alignment
contributions (14),
$\displaystyle\mathbf{d}_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\frac{\mathbf{d}_{j}^{\xi}(t+\Delta
t)}{\|\mathbf{d}_{j}^{\xi}(t+\Delta
t)\|}+\frac{\mathbf{d}_{j}^{\theta}(t+\Delta
t)}{\|\mathbf{d}_{j}^{\theta}(t+\Delta t)\|}.$ (15)
These two contributions are equally weighted in (15) but could be generalized
so as to have different weights. In what follows we explore the former, mainly
to explore the potential of the ADEM to predict the macroscopic dynamics in
the presence of strong alignment, which has been shown to be problematic for
Markovian models [25]. The updated orientation is given by the argument of the
social interactions $\mathbf{d}_{j}$ after rotating it by a small random angle
$\Delta Q$ drawn as well from the wrapped Gaussian $\mathcal{N}_{w}$
$\displaystyle\theta_{j}(t+\Delta t)$ $\displaystyle=$
$\displaystyle\arg(\mathbf{d}_{j})+\Delta Q.$ (16)
2. 2.
Steering rule for the informed sub-population
If the index of the focal particle is in the list of informed indices
$J_{\beta}$, the updated direction is given by a compromise between the output
of the social rules (16) and the informed individual’s preference to move
along the informed direction $\beta$. This is given by the weighted vector
average of these two contributions ( 2)
$\displaystyle\mathbf{d}^{\ast}_{j}(t+\Delta t)=\mathbf{u}_{j}(t+\Delta
t)+\omega\,\hat{\mathbf{b}},$ (17)
where $\omega$ is a weighting constant, $\mathbf{u}_{j}$ is the unit
orientation vector arising from the social rules (16) and $\hat{\mathbf{b}}$
is the unit vector associated with the preferred orientation $\beta$ and
$\hat{\mathbf{d}}_{j}$. The updated orientation is
$\theta_{j}(t+\Delta t)=\arg[\mathbf{d}^{\ast}_{j}].$ (18)
Figure 2: Updating rule for an informed particle. The updated direction
corresponds to the normalized vector sum $\hat{\mathbf{d}}_{j}$ of the
preferential direction vector $\hat{\mathbf{b}}=(\cos(\beta),\sin(\beta))$
rescaled by a factor $\omega$, with the unit vector pointing in the direction
of the output of the social rules
$\mathbf{u}_{j}=(\cos(\theta_{j}),\sin(\theta_{j}))$.
Once all the particle’s orientations are computed according to these social
rules, the positions are updated according to (7). A summary of the parameters
in the SPP model, together with the values used for the simulations are shown
in Table 1.
Table 1: SPP model parameters Parameter | symbol | value | units
---|---|---|---
Radius of avoidance | $r_{av}$ | 1.0 | m.
Region of avoidance | $Av$ | – | m2
Radius of attraction | $r_{at}$ | 5.0 | m.
Region of attraction/alignment | $At$ | – | m2
Particle speed | $s$ | 1.0 | m/sec.
Perception error | $\sigma$ | 0.1 | radians
Time step | $\Delta t$ | 0.1 | seconds
Total population size | $N$ | 10 – 100 | individuals
Informed population size | $N_{\beta}$ | 0 –100 | individuals
Coupling constant | $\omega$ | 0.0 – 0.6 | dimensionless
Informed orientation angle | $\beta$ | 0.0 | radians
### 2.1 Simulation results
Our coarse–grained analysis of the individual-based model consists of
projecting the full configuration (4) onto a set of summary statistics that we
dubbed the ‘meta–particle’. We associate the stochastic properties of the
meta–particle random walk to the various collective states of the full
configuration. For collective decision-making, we found that a useful
projection consists of three state variables, the group elongation
$\Lambda(t)$, the mean group velocity $\bar{\mathbf{v}}(t)$, and the mean
group position $\bar{\mathbf{x}}(t)$. The introduction of the group elongation
is necessary in order to detect situations where the informed individuals
leave the main group, something that occurs at high values of the coupling
constant. In this situation collective decision–making is not consensual,
since the informed individuals fail to lead the complete group along the
informed orientation.
#### 2.1.1 Projections of the configuration
The elongation is defined as the maximum of the set of two–point distances
among all the positions of the particles in the configuration,
$\Lambda(t):=\max\left\\{\,\|\mathbf{x}_{j}(t)-\mathbf{x}_{k}(t)\|\left|\frac{}{}\right.\forall\,j,k\in
J_{\Phi}\right\\}.$ (19)
We restrict our study to values of the coupling constant $\omega$ that
preserve cohesive collective motion, in the sense that the whole configuration
moves as a single entity [50, 16]. This is tantamount to requiring
$\Lambda(t)$ to have a constant upper bound $C$,
$\Lambda(t)<C<<\infty.$ (20)
If the property (20) is preserved, measures for consensual collective motion
and decision–making can be developed in terms of the stochastic properties of
the two other projections of the configuration which together with the
elongation, define the configuration ‘meta-particle’ $\varphi_{t}$
$\varphi_{t}=\\{\,\Lambda(t),\,\bar{\mathbf{x}}(t),\,\bar{\mathbf{v}}(t)\,\\},$
(21)
where the second element in the triplet is the mean group position, or
configuration centroid $\bar{\mathbf{x}}(t)$
Figure 3: Naïve vs. Informed elongation dynamics for $N=10$ and $1\times
10^{6}$ time steps. In both panels the blue graph corresponds to an elongation
time series (19) from a configuration with no informed individuals (called a
naïve group). In both panels, the red graph shows the result of introducing a
single informed individual, where $\omega=0.6$. In the right panel, the black
graph also corresponds to a configuration with a single informed particle, but
for a higher value of the coupling constant $\omega=0.73$. In both panels the
group elongation $\Lambda(t)$ remains bounded for all observed times for the
naive configuration and the mild coupling ($\omega=0.3$), indicating a
configuration that moves cohesively. However, further increasing the coupling
strength (black graph, $\omega=0.73$) causes the group to split, as evidenced
by an elongation that grows without bound.
$\displaystyle\bar{\mathbf{x}}(t)=\frac{1}{N}\sum_{k=1}^{N}\mathbf{x}_{k}(t),$
(22)
and the third one is the mean group velocity $\bar{\mathbf{v}}(t)$ or group
polarization
$\bar{\mathbf{v}}(t)=\frac{1}{N}\sum_{k=1}^{N}z_{k}(t).$ (23)
In our simulations we observed that there is a non-trivial region of parameter
space that preserves cohesive collective motion (20) for both naïve (no
informed individuals present, $N_{\beta}=0$) and informed (at least one
informed individual present, $N_{\beta}>1$) configurations. Figure 3 shows two
scenarios for the dynamics of the elongation $\Lambda(t)$. The left panel
shows the situation where the cohesive collective motion property is preserved
for two swarms of the same total population size ($N=10$). Blue shows the
elongation associated with a naïve configuration, and red shows a
configuration that includes an informed particle ($\omega=0.6,N_{\beta}=1$),
observed during $1\times 10^{6}$ time steps. We see that the elongation
associated with the configuration involving an informed individual tends to
take higher values than in the naïve case, but remains bounded. The right
panel shows the effect of further increasing the coupling constant,
($\omega=0.73$, black line) where the elongation remains bounded for some time
(about $1\times 10^{5}$ time steps) after which it starts to increase,
signaling that the group has broken apart.
Figure 4: Group centroid sample paths. The blue rugged line in the three
panels corresponds to an individual sample path of the configuration centroid
$\bar{\mathbf{x}}(t)$ up to $t_{f}=1\times 10^{4}$ time steps starting near
the origin. The parameters shared in all three cases are the population size
$N=10$, as well as the collective motion parameters, given by
$\sigma=0.1\,\mbox{radians},r_{at}=5.0\,\mbox{m.},\,r_{av}=1.0\,\mbox{m.},\,s=1.0\,\mbox{m./sec.},\,\Delta
t=0.1\,\mbox{secs.}$ In (a) all the individuals are naïve, in (b) there is one
informed individual with coupling constant $\omega=0.6$, and in (c) there are
5 informed individuals, also with $\omega=0.6$. The preferred direction is the
positive $x_{1}$ axis.
Figure 4 shows typical sample paths (blue lines) for the configuration
centroid $\bar{\mathbf{x}}(t)$ for a group of ten individuals and $T=1\times
10^{4}$ time steps. The full configuration at the end of the simulation is
shown in the insets at the center of each panel, where red dots represent the
locations of naïve individuals, and blue the informed ones. Panel (a)
corresponds to a configuration involving only naïve individuals
($N_{\beta}=0$), Panel (b) has one informed individual ($N_{\beta}=1$) with a
coupling constant $\omega=0.6$. Finally, the lower panel (c) shows the case
where five informed individuals are present ($N_{\beta}=5,\omega=0.6$). The
insets to the right show the same sample path over smaller spatial scales. In
the inset of panel (a) we se evidence of separate clusters –where the group
moves very slowly due to a lack of polarization (the slip phase)– connected by
advective flights due to bursts of phase alignment (the stick phase). This
behavior signals the slip/stick dynamics characteristic of these systems [35],
and resembles the behavior of tracer transport in porous media with a
preferential flow direction [3], where the corridors that confine the tracer
play a roughly similar role to the polarization bursts in SPP models that lead
to ballistic flights, alternating with traps that slow the tracer motion, akin
to the slip phase in the SPP. Panel (b) shows the result of adding one
informed individual (blue dot) with a relatively high value of the coupling
constant ($\omega=0.6$) where without loss of generality we identified the
preferred direction with the positive $x_{1}$ axis. The introduction of a
single individual is enough to break the orientation symmetry of the naïve
case, resulting in a motion bias along the preferred direction, and an
disentanglement of the clusters that appear in the naïve case. Adding more
individuals for the same coupling constant leads to a higher mean velocity. In
addition to this, the motion develops an oscillatory behavior along the
coordinate perpendicular to the informed direction.
Figure 5: Empirical argument (left) and modulus (right) pdf’s for the mean
group velocity (23) for a configuration of ten individuals. Both panels show
kernel density estimates from a single velocity time series of $3\times
10^{6}$ data points. The blue graph corresponds to the naïve configuration
($N_{\beta}=0$), red to a group with one informed individual
($N_{\beta}=1,\omega=0.3$), and black shows the results for a configuration
involving five informed individuals, and the same value of the coupling
constant ($N_{\beta}=5,\omega=0.3$).
Figure 5 shows kernel density estimates of the probability density of realized
mean group velocities $\bar{\mathbf{v}}$ obtained from a single time series
($T=3\times 10^{6}$ time steps) collected after a transient of $1\times
10^{3}$ time steps, for swarms of ten individuals and three different informed
regimes. The left panel corresponds to the probability density of mean group
orientations $\arg(\bar{\mathbf{v}})$, and the right panel to the modulus (or
mean group speeds) $|\bar{\mathbf{v}}|$. We see that in the naïve swarm (blue
graph in the left panel), mean group orientations are chosen uniformly from
$[-\pi,\pi)$ at all times. However, this rotational symmetry is broken upon
the introduction of a single informed individual, which yields a symmetric
density centered around the informed direction (red, left panel). The
existence of peaks reflects a tension between the slip/stick dynamics and the
biased motion along the informed direction $\beta$. Overall, the group moves
along the informed direction, but slip/stick bursts are strong enough to
partially counter that bias by trying to recover the rotational symmetry.
Further increasing the number of individuals (black, left panel) leads to a
unimodal density concentrated around the informed direction. Kernel density
estimates for the mean group speeds shown in the right panel
$\bar{\mathbf{v}}$ show comparatively less variability between the naïve and
informed regimes. This is to be expected, since the mean group speeds arises
mainly from the social rules, and the steering rule is designed to introduce
an orientation bias, but has little effect on the modulus. There is however a
tendency to move with higher speeds as informed individuals are introduced.
Critically, the range of variability in speeds is very wide and practically
covers the full range of possible values (the individual particle speed is
$s=1$ m/s, which constitutes an upper bound for the mean group speed).
#### 2.1.2 Coarse variables and collective behaviors
Measures for the collective behaviors that are macroscopically relevant can be
defined in terms of the scaling properties of the $k$-th moments of the
transition probability density $p(\mathbf{x},t|\mathbf{0},0)$ for finding a
configuration meta–particle centroid around position $\mathbf{x}$ at time $t$
given that it started at the origin at time zero. These are defined
componentwise by
$\displaystyle
m_{1}^{(k)}(t)=\mathrm{E}\left[\bar{x}_{1}^{\,\,k}(t)\right]\sim
t^{\delta},~{}~{}~{}k\in{1,2}$ (24)
for the $x_{1}$ coordinate, where the scaling exponent $\delta$ determines the
prevailing type of transport at the time scale under consideration. The $k$-th
moments (24) can be calculated from simulations of the individual-based model
with the estimator [27]
$\hat{m}_{1}^{(k)}(t)=\frac{1}{Z}\sum_{i=1}^{Z}\left(X_{1}^{(i)}(t)\right)^{k},$
(25)
where $i=1,\ldots,Z$ is the number of simulation runs in the ensemble and each
of the $X_{1}(t)=X_{1}(0),X_{1}(1),\ldots,X_{1}(T)$ corresponds to a single
time series of length $T$ of group centroid positions along the $x_{1}$
coordinate observed at discrete time intervals of length $\Delta t$.
The simplest possible scenario for collective–decision making occurs when the
transfer of the bias of the informed sub–population leads on average to motion
with effectively constant velocity $v_{1}$ along the informed direction, which
we identify without loss of generality with the positive $x_{1}$ axis. In this
case the mean displacement ($k=1$, in (24)) scales linearly with time after a
transient determined by the characteristic time scale $\tau_{c}$, the _time to
consensus_
$m_{1}^{(1)}(t)\sim v_{1}\,t,$ (26)
Figure 6: Mean displacement $m_{1}^{(1)}(t)$ along the informed direction
$x_{1}$ versus time for a wide range of values of the coupling constant
$\omega$. Results are averaged over $3\times 10^{3}$ independent simulation
runs. Initial configurations are given by uniformly distributed locations
within a circle of radius 0.5, and uniformly distributed orientations on the
unit circle. In all cases, the mean displacement increases asymptotically
linearly with time, indicating motion with a constant effective speed. The dip
at early times for $\omega=0.007$ (red dash-dot line, first from the bottom
upwards) constitutes a signature of the transient; however a larger ensemble
is required in this case is required to capture it accurately, since the
values involved are much smaller than those present for higher values of
$\omega$. However, for our purposes, the linear long time behavior is clearly
shown. The simulation parameters are $N=10$ individuals,
$N_{\beta}=1$,$\sigma=0.1$ radians, $r_{at}=5.0$ m, $r_{av}=1.0$ m, $s=1.0$
m/sec.
where the degree of consensus $c$ can be defined by the ratio of the mean
group velocity $v_{1}$ to the individual particle speed $s$
$c=\frac{v_{1}}{s}.$ (27)
Values of $c$ close to one result from a distribution of individual particle
orientations concentrated around the informed orientation. On the other hand,
if these tend to be distributed uniformly on $(0,\pi]$, one should expect
comparatively smaller values of $c$, since the individual particle velocities
tend to cancel each other in this regime, which we associate to poor
consensus.
Figure 7 shows the behavior of the effective velocity $v_{1}$ versus the
coupling constant $\omega$ for a SPP swarm of ten particles and various
proportions of informed individuals. We see that in all cases the degree of
consensus increases as a power law of the coupling constant $\omega$ with a
slope that decreases as informed individuals are added. This means that if
there is an optimum group velocity in some appropriately defined sense, it can
be reached collectively by two different avenues. One is to have a small
number of informed individuals at a high coupling constant, and the other is
to have a large number of informed particles with low values of the coupling
constant. The power law dependence implies that the difference in values of
the coupling constant for these two strategies can span orders of magnitude.
Therefore, if the cost of recruiting informed individuals is less than that of
leading the group, a possible optimal strategy, would consist of recruiting
additional informed individuals, each of them with comparatively smaller
values of $\omega$, instead of simply increasing the coupling constant of the
informed group size, which comes at the additional complication of increasing
the probability of having the group split apart.
Naturally, these two strategies to reach the same target group velocity, are
likely to have different accuracies. This can be more readily detected in
measures of spread along the mean value, like the second moment. For this
purpose we use the msd ($k=2$, in (24)), which can also detect very
efficiently the various types of collective behaviors, either transient or
asymptotic, that contribute to macroscopic transport. For instance, the time
to consensus can be detected sharply by a transition from linear (diffusion-
dominated) or anomalous scaling to a _quadratic_ one at the point in time
where advection begins to dominate
$m_{1}^{(2)}(t)\sim v_{1}^{2}\,t^{2}.$ (28)
Figure 8 shows the msd along the informed direction $x_{1}$ for a group of 10
individuals, one of them informed, and various values of the coupling
constant. There is a transition from linear to quadratic scaling for
non–negative values of $\omega$ at a characteristic time scale $\tau_{c}$. As
one increases the coupling constant, the informed sub–population becomes more
efficient at transferring their bias to the whole group, signaled by an
earlier time to consensus. The transient regime appears to be anomalous
(supperdiffusive) in both naïve ($\omega=0$) and informed ($\omega>0$)
configurations. The anomalous transient can be better detected by looking at
the scaling of the second order fluctuations, which requires removing the mean
value in the definition of the moments (24) for $k=2$,
$M_{1}^{(2)}(t)=\mathrm{E}\left[\left(\,\bar{x}_{1}(t)-m_{1}^{(1)}(t)\,\right)^{2}\right]\sim
t^{\alpha},$ (29)
Figure 7: Dependence of the drift velocity $v_{1}$ along the informed
direction $x_{1}$ on the coupling constant $\omega$ for various values of the
number of informed individuals $N_{\beta}=1$ (black), 3 (blue) and 5 (red) for
a swarm of $N=10$ individuals. In all cases, the drift increases as a power
law of the coupling constant, but the exponent decreases as the number of
informed individuals increases, due to the upper bound of the group velocity
imposed by the individual particle speed.
If $\alpha=1$ in (29) for some set of time scales, then the fluctuations
behave as classical diffusion [22, 56]. Values of $\alpha$ different from one
are dubbed _anomalous_ and can be of two main types: sub–diffusive or
‘trapped-diffusion’ for $0<\alpha<1$, and super-diffusive (sub–ballistic) or
‘enhanced diffusion’, if $1<\alpha<2$. These anomalous behaviors signal the
presence of fluctuations that have persistent correlations in space, time, or
both at macroscopically relevant scales [59, 39]. Slip/stick dynamics dominate
the early time behavior of the second moment of the fluctuations along the
$x_{1}$ (Figure 10 ) and $x_{2}$ (Figure 9) coordinates, where the transport
is clearly anomalous. There is a (sub-ballistic) super-diffusive transient
that eventually decays to classical diffusion at a characteristic time scale
$\tau_{a}$. This is due to the alternation of bursts of advective flights (the
slip phase) due to polarization of the orientations, that are interrupted when
the polarization is lost and the group moves much more slowly (stick). Since
the msd eventually becomes diffusive, the temporal correlations in the mean
velocity induced by the polarization eventually decay at a characteristic time
scale $\tau_{a}$ (shown in Figure 10), after which the fluctuations are
diffusive, with identical diffusion coefficients along both coordinates.
The role of the informed sub-population is more nuanced along the $x_{2}$
coordinate (see Figure 9), in the sense that for higher values of the coupling
constant, there is a clearly detectable _sub-diffusive_ regime between the
early time super-diffusive transient and the diffusive regime; this is due to
reversals in velocity that are more marked along the $x_{2}$ direction. Thus,
the introduction of informed individuals at high coupling constants induces an
anisotropy not only in the values of the diffusion coefficients but also in
the manner in which the correlations decay. Whereas the the ADEM (1) allows
for anisotropies in the diffusion coefficients along the two directions, the
memory kernel $M(t)$ must be identical along each direction. This precludes
the use of the ADEM (1) for predicting the behavior of the full 2–dimensional
density, which would require a separate treatment along each coordinate. We
can still however use it for the marginals along each direction in order to
predict the mean squared displacements, and to extract the transport
coefficients.
Other macroscopic measures of interest are related to the precision of the
decision-making process. This quantity is directly linked to the strength of
the fluctuations along the informed direction, relative to those of a fully
naïve swarm of the same total population size. Higher precision in decision-
making naturally implies a distribution of centroid positions along $x_{1}$
that is highly concentrated around the mean value. This is measured by the
magnitude of the diffusivity along $x_{1}$. Figure 10 shows estimates of the
second order fluctuations along the informed direction for various values of
the number of informed individuals $N_{\beta}=0,1,3,5,7$ and a fixed coupling
strength of $\omega=0.5$. All the fluctuations scale asymptotically linearly
and thus they are dominated by classical diffusion, but with a diffusivity
that decreases as the number of informed individuals $N_{\beta}$ increases.
This agrees with the intuition that the precision of the decision–making
process should improve as informed individuals are added to a group of fixed
size. Figure 9 shows the mean squared displacement along the direction
perpendicular to the informed orientation $x_{2}$ for the same parameters in
Figure 8. We see that the diffusivity along this coordinate is numerically
indistinguishable between naïve and informed configurations for the various
values of the coupling constant used. This suggests a definition of
decision–making precision given by the ratio of the diffusivities along the
two coordinates,
$\rho=\frac{D_{1}}{D_{2}}\leq 1.$ (30)
Values of $\rho$ close to one indicate a spread of the meta-particle positions
that is comparable along both coordinates, and thus poor precision in
decision–making, and the opposite situation occurs for values of $\rho$ that
are significantly less than one.
Figure 8: Mean squared displacement along the informed direction $x_{1}$
versus time for various values of the coupling constant $\omega$, for a total
group size $N=10$ and one informed individual for non-zero values of $\omega$.
There is an anomalous transport regime at early times evidenced by the scaling
exponent $\alpha\approx 1.7$ (29). This regime decays asymptotically to
classical diffusion (linear scaling) in the naïve configuration. When $\omega$
becomes positive the transport becomes advective at a characteristic time
scale $\tau_{c}$, signaled by the quadratic scaling of the msd, this time
scale becomes shorter with increasing coupling strength. Figure 9: Mean
squared displacement along the $x_{2}$ coordinate versus time. There is a
superdiffusive transport regime at early times followed by a _sub-diffusive_
phase that appears at relatively high values of the coupling constant
$(\omega>0.3$). The anomalous transient gives way to diffusive transport
asymptotically in all cases with numerically indistinguishable diffusion
coefficients. Figure 10: Mean squared displacement with drift removal (29)
along the informed direction $x_{1}$ versus time for various values of the
informed sub-population size and a fixed value ($\omega=0.5$) of the coupling
constant and total population size ($N=10$). The super-diffusive transient
eventually gives way to classical diffusion in all cases at a characteristic
time scale $\tau_{a}$. Both the scaling exponent $\alpha$ and the asymptotic
diffusivity decrease as the number of informed individuals increases, which
signals an increase in the precision of the decision making for fixed values
of the coupling constant, following the introduction of additional informed
individuals while the total group size remains unchanged.
## 3 Continous time random walks and the advection-diffusion equation with
memory
In this section we briefly review known results about continuous time random
walk models (CTRW), that were originally introduced to describe the random
motion of a particle on a disordered lattice (or medium). The main innovation
of CTRW theory consisted in allowing the lattice spacing and updating times to
become random variables themselves. It can be shown [41, 30] that when the
distribution of lattice spaces has finite moments of all orders, the evolution
of the transition density for the location probability density of a CTRW is
given by the generalized master equation (GME) [30, 58, 32, 31] also called
the advection-diffusion equation with memory (ADEM) when there is a drift [14,
3]. This generalization of the classical random walk leads to an evolution
equation for the transition probability density of the particle position that
is non-local in time, since the flux depends on a weighted time average over
the full past. The weighting function is commonly called a ‘memory function’,
and results from the wide range of transition rates originating from the
spatial disorder. This approach has been successfully applied in models of
anomalous diffusion [39], that typically require the memory to decay
algebraically instead of exponentially. Approaches based on the CTRW and GME
methods however are more general, and allow for any functional form of the
memory, provided that it can be normalized.
As will be discussed in Section 4, the memory function plays a fundamental
role, since it encodes macroscopic transport coefficients of interest,
together with their characteristic time scales. We will exploit these
properties of memory functions in order to estimate the transport parameters
associated with the various types of collective behaviors arising in SPP
models of collective motion, with and without informed individuals.
### 3.1 Continuous time random walks
Continuous time random walks (CTRW) [41, 30, 34] are a generalization of
classical random walks [22, 56] where the jump size $\Delta x$ and updating
time $\Delta t$ are allowed to become random variables. Sample paths are
generated by drawing the jump size $\xi$, and waiting time $\tau$ from the
joint probability density $\psi(\xi,\tau)$. The elapsed time $t_{n}$ for such
a walker after $n$ steps is,
$t=\sum_{j=1}^{n}\tau_{j},~{}~{}~{}\tau_{j}\in\mathbb{R}^{+},$
and the position $\mathbf{x}_{n}(t)$, for a 2-dimensional walk,
$\mathbf{x}_{n}(t)=\sum_{j=1}^{n}\xi_{j},~{}~{}~{}\xi_{j}\in\mathbb{R}^{2}.$
The probability of observing a walker at position $\mathbf{x}$ at time $t$
given that it started at the origin at time zero is,
$p(\mathbf{x},t)=\delta(\mathbf{x})\,\Psi(t)+\int_{\mathbb{R}^{2}}\int_{0}^{t}\psi(\xi,\tau)\,p(\mathbf{x}-\xi,t-\tau)\,d\xi\,d\tau,$
(31)
where the survival function $\Psi(t)$ is the cumulative of the waiting time
marginal density of $\psi(\xi,\tau)$
$\Psi(t)=1-\int_{\mathbb{R}^{2}}\int_{0}^{t}\psi(\xi,\tau)\,d\xi\,d\tau.$ (32)
For the particular situation were the jumps and waiting times are decoupled,
the joint density $\psi(\xi,\tau)$ can be rewritten as
$\psi(\tau)\lambda(\xi)$, where $\psi(\tau)$ is the distribution of waiting
times, and $\lambda(\xi)$ is the distribution of jumps. This assumption
simplifies (31) to
$p(\mathbf{x},t)=\delta(\mathbf{x})\,\Psi(t)+\int_{\mathbb{R}^{2}}\lambda(\xi)\int_{0}^{t}\psi(\tau)\,p(\mathbf{x}-\xi,t-\tau)\,d\xi\,d\tau.$
(33)
If the jump density $\lambda(\xi)$ has finite moments, and $p(\mathbf{x},t)$
can be expanded in a Taylor series, it can be shown [3] that the differential
version of (33) corresponds to an advection–diffusion equation generalized to
non-local time,
$\displaystyle\frac{\partial p(\mathbf{x},t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\,M(t-s)\,\left[\mathbf{v_{\lambda}}\cdot\nabla
p(\mathbf{x},s)-\mathbf{D_{\lambda}}\,:\,\nabla\,\nabla
p(\mathbf{x},s)\right]\,ds$ (34) $\displaystyle p(\mathbf{x},0^{+})$
$\displaystyle=$
$\displaystyle\delta(\mathbf{x}),~{}~{}\mathbf{x}\in\mathbb{R}^{2},~{}~{}t\in\mathbb{R}^{+}$
where the memory term $M(t)$ can alternatively be defined in terms of the
Laplace transform of the waiting time density, or as the kernel of the
velocity time autocorrelation (divided by 2D so that it integrates to one). In
the former case we have,
$\widetilde{M}(\epsilon)=\frac{\bar{t}\epsilon\tilde{\psi}(\epsilon)}{1-\tilde{\psi}(\epsilon)}$
(35)
and $\tilde{f}(\epsilon)$ denotes the Laplace transform of a function $f(t)$
with $\epsilon$ being the Laplace variable, and $\bar{t}$ is the
characteristic time between transitions
$\bar{t}=\int_{0}^{\infty}\tau\,\psi(\tau)\,d\tau.$ (36)
The drift term $\mathbf{v_{\lambda}}$ in (34) is related to the first moment
of the jump pdf $\lambda(\xi)$,
$\mathbf{v_{\lambda}}=\frac{1}{\bar{t}}\int_{\mathbb{R}^{2}}\mathbf{x}\,\lambda(\mathbf{x})d\mathbf{x},$
(37)
and the diffusivity tensor $\mathbf{D_{\lambda}}$ is given by the second
moment of $\lambda(\xi)$
$\mathbf{D_{\lambda}}=\frac{1}{2\,\bar{t}}\int_{\mathbb{R}^{2}}\mathbf{x}\,\mathbf{x}^{T}\,\lambda(\mathbf{x})d\mathbf{x}.$
(38)
In the drift-free case, the Laplace domain solution of the ADEM (34) is given
by
$\tilde{p}(\mathbf{x},\epsilon)=\frac{1}{2\,\pi\,\widetilde{M}(\epsilon)\sqrt{D_{1}\,D_{2}}}\,K_{0}\left(\sqrt{\frac{\epsilon}{\widetilde{M}(\epsilon)}\left[\frac{x_{1}^{2}}{D_{1}}+\frac{x_{2}^{2}}{D_{2}}\right]}\,\right),$
(39)
which assumes that the off–diagonal components of the diffusivity tensor are
zero, $D_{1}$ and $D_{2}$ are the diffusivities along the $x_{1}$ and $x_{2}$
coordinates, $K_{0}$ is the modified Bessel function and $\tilde{M}(\epsilon)$
is the Laplace transform of the memory with $\epsilon$ being the Laplace
variable. If the drift vector has a single non-vanishing component which
coincides with the $x_{1}$ direction, the solution is [3]
$\tilde{p}(\mathbf{x},\epsilon)=\frac{1}{2\pi\,\widetilde{M}(\epsilon)\sqrt{D_{1}D_{2}}}\exp\left(\frac{x_{1}\,v_{1}}{2D_{1}}\right)\,K_{0}\left(\frac{v_{1}}{2\,D_{1}}\sqrt{x_{1}^{2}+\frac{D_{1}}{D_{2}}x_{2}^{2}\left[1+4\frac{\epsilon\,D_{1}}{\widetilde{M}(\epsilon)\,v_{1}^{2}}\right]}\right),$
(40)
where $v_{1}$ is the magnitude of the drift along the $x_{1}$ coordinate.
Finally, the Laplace domain expression for the mean squared displacement of
the ADEM (34) along the direction of the drift is
$\tilde{m}^{(2)}_{1}(\epsilon)=\frac{2\,v_{1}^{2}}{\epsilon^{3}}\,\widetilde{M}_{1}^{2}(\epsilon)+\frac{2\,D_{1}}{\epsilon^{2}}\,\widetilde{M}_{1}(\epsilon),$
(41)
which yields all the characteristic time scales after Laplace inversion.
## 4 Multiscale Method
We start with the assumption that the meta-particle random walk follows an
unknown CTRW with independent jump and waiting time distributions. In this
case one can assert that after the velocity–autocorrelation equilibrates, the
evolution of the transition density for the meta–particle location
$p(\mathbf{x},t|\mathbf{0},0)$ is given by an advection–diffusion equation
with memory (34) under relatively mild assumptions. The ADEM would be fully
specified if analytical forms of the jump and waiting time densities were
known on the basis of the SPP formulation. Unfortunately, this is not the
case. We instead _estimate_ them from a single velocity and centroid position
time series obtained from a simulation run of the spp model with a combination
of non–parametric and parametric methods. We show that this simple estimation
procedure predicts mean squared displacements that are indistinguishable
numerically from those estimated from an ensemble average over a large number
of simulation runs. We exploit these results in order to explore a wide region
of the parameter space, and obtain analytical results for the time to
consensus based on the functional forms used in the parametric estimation of
the memory. These results are exact for the case of exponential and Gamma
density (44) memories, but only approximate for the truncated Mittag–Leffler
case (46).
### 4.1 Estimation of $M(t)$
Although the memory in (34) is defined in terms of the Laplace transform of
the distribution of waiting times (35), a more convenient definition relates
it to the time velocity autocorrelation of the random walker [59, 58, 31]
$M(t)=\frac{1}{2(D_{1}+D_{2})}\,\mathrm{E}\left[\left(\mathbf{v}(\tau)-\mu\right)\cdot\left(\mathbf{v}(\tau+t)-\mu\right)\right]$
(42)
where $\mu$ is the expected value of the random velocity $\mathbf{v}(t)$. The
definition of the ADEM (34) allows different diffusivities along each
coordinate, but not anisotropies in which the correlations decay differently
along each component of the velocity. In general though, one should consider
the memory separately along each component for an accurate description of the
evolution of the transition pdf. Unfortunately, this turns out to be the case
for informed swarms at high coupling constants, as can by seen after observing
the differences for high values of the coupling constant in the mean squared
displacements along $x_{2}$ (Figure 9) and the drift-corrected msd along
$x_{1}$ (Figure 10). The former has a distinct sub-difussive regime that is
not apparent in the latter. For the purposes of collective–decision making, it
suffices to focus on the behavior of the msd (41) along the informed
direction, and the macroscopic transport coefficients $D_{1},D_{2}$ and
$v_{1}$.
We first compute a non-parametric estimate of the velocity auto-correlation
function from a velocity time series $\\{v_{1},v_{2},\ldots,v_{T}\\}$ obtained
from a single simulation run of the SPP, where each of the $v_{i},i=1,\ldots
T$ is the component of the meta-particle velocity along the informed
direction, sampled at discrete time intervals $\Delta t$, and $T$ is the
length of the time series. We used the unbiased estimator [47]
$\widehat{C}(\tau)=\frac{1}{T-\tau}\sum_{i=1}^{T-\tau}\left(v_{i}-\bar{v}\right)\,\left(v_{i+\tau}-\bar{v}\right),~{}~{}~{}\tau=0,\ldots,T-1,$
(43)
where $\tau$ is the time lag and $\bar{v}$ is the sample mean,
$\bar{v}=\frac{1}{T}\sum_{i=1}^{T}v_{i}.$
The tabulated function that results from the non-parametric estimate (43) is
fed to a non-linear least squares routine that yields a _parametric_ estimate
of the memory (see below). The Laplace transform of this function is then
substituted into the expression for the mean squared displacement (41), or the
transition pdfs (39) and (40), all of which can then be inverted numerically.
The parametric estimate requires a ‘template’ function for the velocity
auto–correlation that fits the data well and has a known analytical Laplace
transform. We identified two functions that provide remarkably good fit and
have very simple transforms. For lower values of the coupling constant this
template is the Gamma density (Figure 11),
$f(t)=\frac{\tau_{a}^{\beta-1}}{\Gamma(1-\beta)}\,t^{-\beta}\,e^{-t/\tau_{a}}$
(44)
where $\tau$ controls the exponential decay, and the exponent $\beta$ controls
the initial algebraic decay. The Laplace transform of (44) is simply
$\tilde{f}_{1}(\epsilon)=\left(\frac{1}{\tau_{a}}+\epsilon\right)^{\beta-1}.$
(45)
The second function is appropriate for higher values of the coupling constant
$\omega$ which leads to oscillations (see Figure 11). In this function the
initial power law decay in the Gamma density in (44) is substituted by an
exponentially truncated Mittag–Leffler function [49, 59],
$g(t)=\frac{\tau_{\epsilon}+\tau_{a}^{\alpha}}{\tau_{\epsilon}\tau_{a}^{\beta}}\,t^{\beta-1}\,E_{\alpha,\beta}(-t^{\alpha}/\tau_{\epsilon})\,\exp(-t/\tau_{a}),$
(46)
where the Mittag–Leffler function
$E_{\alpha,\beta}(-t^{\alpha}/\tau_{\epsilon})$ is defined as
$E_{\alpha,\beta}(-t^{\alpha}/\tau_{\epsilon})=\sum_{k=0}^{\infty}\frac{(-1)^{k}(t^{\alpha}/\tau_{\epsilon})^{k}}{\Gamma(\alpha
k+\beta)}$ (47)
where $\alpha$ and $\beta$ are shape parameters and $\tau_{\epsilon}$ controls
the transition between the early time and the asymptotic regime. The Laplace
transform of the truncated Mittag–Leffler function (46) is also very simple
[49]
$\mathcal{L}\left[\frac{}{}t^{\beta-1}\,E_{\alpha,\beta}(-a\,t^{\alpha})\,e^{-b\,t}\right](\epsilon)=\frac{(b+\epsilon)^{\alpha-\beta}}{\left(b+\epsilon\right)^{\alpha}+a}.$
(48)
Figure 11 shows the results the estimation procedure for a swarm of $N=10$
individuals, one of them informed. The black marks show the non-parametric
estimates based on (43) and the blue lines show the parametric fits using the
Gamma density (44) for lower values of the coupling constant ($\omega=0.1$ and
0.3), and the truncated Mittag–Leffler function for higher values
($\omega=0.45$ and 0.6). In all cases, the template functions provide
remarkably good fit, including the oscillation that appears for higher values
of $\omega$. The functions eventually decay to a constant value that
corresponds to the drift squared $v_{1}^{2}$, which we do not remove from the
estimator, in order to be able to resolve the changes in the qualitative
behavior of the correlation function for various values of $\omega$. Table 2
shows the parameter estimates in all four cases, together with goodness of fit
values.
$\omega$ | $D_{1}$ | $v_{1}$ | $\tau_{\epsilon}^{\ast}$ | $\tau_{a}$ | $\alpha$ | $\beta$ | $R^{2}$ | SSE
---|---|---|---|---|---|---|---|---
$0.10$ | 0.0373 | 0.012 | - | 8.33 | - | 0.20 | 0.996 | $2.3\times 10^{-6}$
$0.30$ | 0.0373 | 0.038 | - | 0.94 | - | 0.24 | 0.998 | $2.9\times 10^{-7}$
$0.45$ | 0.0326 | 0.068 | 0.55 | 0.26 | 1.09 | 0.86 | 0.999 | $4.4\times 10^{-7}$
$0.60$ | 0.0282 | 0.094 | 0.62 | 0.37 | 1.61 | 0.79 | 0.999 | $2.3\times 10^{-6}$
Table 2: Parameter estimates and goodness of fit values for the correlation
functions in Figure 11 using the Gamma density (44) and truncated Mittag-
Leffler function (46) as fitting templates. ∗ The time scale $\tau_{\epsilon}$
is displayed in units of time for comparison with the exponential relaxation
$\tau_{a}$. SSE stands for Sum of the Squared Errors. Figure 11: Estimated
velocity autocorrelation function from a single time series of $1x10^{7}$ time
steps (marks) and fitted functions (blue lines). The group size in the
simulation was 10 individuals, one of them informed. The black markers
correspond to the non-parametric estimates, for various values of $\omega$.
The continuous lines show the parametric fits with a Gamma kernel
$f(\tau)+\bar{v}_{1}^{2}$ ( 44) for $\omega=0.1$ and 0.3, and a truncated
Mittag–Leffler function $g(\tau)+\bar{v}_{1}^{2}$ (46) for $\omega=0.45$ and
0.6. Parameter estimates and goodness of fit values can be found in Table 2
Data for the velocity time series starts being collected after a transient of
1000 time steps, after which time the time series becomes second–order
stationary. Figure 12 shows that after a very short transient of a few hundred
time steps, the estimators become very narrowly bounded and no trend with time
is evident.
Figure 12: Velocity autocorrelation function at zero lag $C_{0}(\tau_{0})$
computed from a window of fixed length $T=1\times 10^{4}$ time steps, and
shifting the origin of the first data point in the window $\tau_{0}$ time
steps from the absolute origin of the simulation run. Each graph corresponds
to a different value of the coupling constant. The swarm simulation consisted
of a total group size of $N=10$ individuals, of which one is informed. The
arrow indicates the point at which data started to be collected for the
estimates of the macroscopic transport parameters ($\tau_{0}=1000$ time
steps.)
### 4.2 Estimation of $\mathbf{v_{\lambda}}$ and $\mathbf{D_{\lambda}}$
The drift coefficient $v_{1}$ can be estimated in two ways. The most
straightforward is from the sample mean of the velocity time series
$\\{v_{1},v_{2},\ldots,v_{T}\\}$,
$\hat{v}_{1}^{\ast}=\frac{1}{T}\sum_{i=1}^{T}v_{i}$ (49)
and the other is based on the first moment of the jump kernel,
$v_{\lambda}=\frac{1}{\bar{t}}\int_{\mathbb{R}}x_{1}\,\lambda_{1}(x_{1})dx_{1}$
(50)
where $\bar{t}$ is the mean time between transitions (36), and $\lambda_{1}$
is the marginal of the jump kernel $\lambda(\mathbf{x})$ along the informed
coordinate. Since the characteristic time $\bar{t}$ is not known, the
estimator requires sampling the jump kernel $\lambda$ at various lags $\tau$.
The characteristic time will be the value of $\tau$ for which the estimator
saturates,
$\hat{\mu}(\tau)=\frac{1}{T(\tau)}\sum_{i=1}^{T(\tau)}\Delta(x_{1};\tau)$ (51)
where $\Delta(x_{1};\tau)$ is the sub–series of position differences along the
direction $x_{1}$ sampled at time lag $\tau$ from the _position_ time series
$\\{x_{1},x_{2},\ldots,x_{T}\\}$, where $T$ is the total length of the series,
and $T(\tau)$ is the length of the sub-series sampled at lag $\tau$. Of
course, the quality of the estimator decreases with $\tau$, because the length
of each sub-series is twice as short as the preceding one. The lag dependent
drift is then given by
$\hat{v}_{\lambda}(\tau)=\frac{\hat{\mu}}{\tau}$
and the characteristic time can be calculated as the smallest value of the lag
$\tau^{\ast}$ for which the equality
$\hat{v}_{\lambda}(\tau^{\ast})=\hat{v}_{1}^{\ast}$
that relates both estimators holds. Since a parametric form of the memory is
already available (see Section 4.1), the diffusivity can be estimated from the
Kubo–Green relationship [37, 32, 31] that relates transport parameters to time
correlation functions,
$D_{1}=\int_{0}^{\infty}\mathrm{E}\left[\left(v_{1}(0)-\bar{v}_{1}\right)\left(v_{1}(\tau)-\bar{v}_{1}\right)\right]\,d\tau.$
(52)
It can also be estimated from the second moments of the jump kernel
$D_{1}=\frac{1}{2\,\bar{t}}\int_{\mathbb{R}}(x_{1}-\mu_{1})^{2}\,\lambda_{1}(x_{1})\,dx_{1},$
(53)
where an estimator of $D_{1}$ is developed in a similar vein as that of the
drift $\hat{v}_{\lambda}$
$\widehat{D}_{1}(\tau)=\frac{1}{2\,\tau\,(T(\tau)-1)}\sum_{i=1}^{T(\tau)}\left(\Delta(x_{1};\tau)-\mu(\tau)\right)^{2},$
(54)
the diffusion coefficient is the value for which $\hat{D_{1}}(\tau)$ reaches a
plateau. The behavior of both estimators for the swarm meta–particle is shown
in figure 14. The upper panel shows the results for naïve configurations of
various total population sizes, $N=10$ (red), $N=50$ (green) and $N=100$
(blue). The dotted black line corresponds to the estimate of the diffusivity
from the velocity time auto–correlation using the Kubo–Green relationship (52)
and the rugged lines of various colors correspond the estimates of the
diffusivity based on (54) that vary with the sampling lag $\tau$. The lower
panel shows the comparisons between both methods for informed configurations
of the same total population sizes as in the upper panel, but including
informed individuals for the same coupling constants. In all the cases the
_proportion_ of informed individuals $p=N/N_{\beta}=0.3$ was kept constant. We
observe that both methods converge to approximately the same value, in both
naïve and informed configurations. We note that the characteristic time –the
time at which the estimator saturates– increases with group size. The width of
the oscillations in the estimator (54) increases with the lag $\tau$ due to
the finite size of the location time series, since for larger values of
$\tau$, the number of data points used in the estimator decreases.
Figure 13: Behavior of the estimator of the diffusion coefficient $\hat{D}(T)$
based on the Kubo–Green relationship (52) versus the length $T$ of the
meta–particle velocity time series. Open squares denote the value of the
estimator using a truncated Mittag–Leffler kernel template for the velocity
auto–correlation, and black circles correspond to a Gamma density. In both
cases the dotted lines are the 95% confidence intervals. Figure 14: Diffusion
coefficients estimated via the Kubo–Green relationship (52) (dotted black
lines), and from the variance of the jump kernel (54) sampled at various time
lags $\tau$ . Panel (a) shows estimates for purely naïve swarms of total
population size $N=10$ (red), $N=50$ (green) and $N=100$ (blue). Panel (b)
shows the estimates for informed configurations. The three cases share the
same fraction of informed individuals $p=0.3$ and coupling constant
$\omega=0.3$, the total population sizes are color coded as in panel (a). We
note that both methods succeed in providing the asymptotic value of $D$. There
is an overall reduction in diffusivity as the total group size increases. The
diffusivity also decreases in informed groups compared with naive ones of the
same total size.
### 4.3 Estimation of the time to consensus $\tau_{c}$
Figure 15: Comparison between the mean squared displacement along the informed
direction $x_{1}$ (24) estimated from an ensemble of 3000 simulation runs
(black marks) and that obtained from the inverse Laplace transform of the msd
(41) based on the fitted ADEM (blue continuous lines), with parameters
estimated from a single simulation run. We used a Gamma density memory kernel
for the lower values of the coupling constant ($\omega=0,\omega=0.1$) and an
exponentially truncated Mittag–Leffler function for the remainder cases
($\omega=0.3,\omega=0.6$). In all cases the total population size consisted of
$N=10$ individuals, and informed configurations consisted of one informed
individual in all cases.
We defined crudely the time to consensus $\tau_{c}$ as the time scale that
determines the onset of the quadratic scaling in the mean squared displacement
(Figure 8) along the informed direction, which in the Laplace domain is given
by
$\tilde{m}^{(2)}_{1}(\epsilon)=\frac{2\,v_{1}^{2}}{\epsilon^{3}}\,\tilde{M}_{1}^{2}(\epsilon)+\frac{2\,D_{1}}{\epsilon^{2}}\,\tilde{M}_{1}(\epsilon),$
(55)
where the coefficients $v_{1}$ and $D_{1}$ can be determined from (49) and
(52) respectively. The parameters of the memory are calculated by the method
described in Section 4.1. Given that the analytical Laplace transforms are
known for both memory templates (44) and (46), substituting the Laplace
transform of the Gamma memory (45) into (55) leads to
$\tilde{m}^{(2)}_{\Gamma}(\epsilon)=\frac{2}{\epsilon^{3}}\left(\tau_{a}^{-1}+\epsilon\right)^{\beta-2}\left[D_{1}\epsilon\left(\tau_{a}^{-1}+\epsilon\right)+v_{1}^{2}\left(\tau_{a}^{-1}+\epsilon\right)^{\beta}\right].$
(56)
Likewise, for the substituting the Laplace transform of the truncated
Mittag–Leffler function (48) yields
$\tilde{m}^{(2)}_{E}(\epsilon)=\frac{2\tau_{\epsilon}\left(\tau_{a}^{-1}+\epsilon\right)^{\alpha-2\beta}}{\epsilon^{3}\left(1+\tau_{\epsilon}\left(\tau_{a}^{-1}+\epsilon\right)^{\alpha}\right)^{2}}\left(D_{1}\,\epsilon\left[\tau_{a}^{-1}+\epsilon\right]^{\beta}+\tau_{\epsilon}\left[\tau_{a}^{-1}+\epsilon\right]^{\alpha}\left[v_{1}^{2}+D_{1}\,\epsilon\left(\tau_{a}^{-1}+\epsilon\right)^{\beta}\right]\right).$
(57)
In the case of the msd for the Gamma density memory (56), it is possible to
invert analytically the Laplace transform. The msd with the truncated
Mittag–Leffler memory can be inverted numerically using the inversion
algorithm of de Hoog [18]. Before we can use the results of the analytical and
numerical inversions of (56) and (57) we show in Figure 15 comparisons between
the msd obtained from an ensemble of simulation runs of the swarm meta-
particle (black marks) and that obtained by inversion of the Laplace
transforms of the mean squared displacements (56) and (57) (blue lines) based
on the ADEM assumption, with parameters estimated from a single simulation run
of the SPP, using the method outlined in Sections 4.2 and 4.1. In all cases
the method based in the ADEM is able to capture accurately both the transient
and the asymptotic behavior. In order to use these results to calculate the
time to consensus, $\tau_{c}$ we first note that in the simpler case of a
memory of the form of a Dirac distribution $\delta(t)$, Laplace inversion of
(55) is straightforward,
$m^{(2)}_{1}(t)=v_{1}^{2}\,t^{2}+2D_{1}\,t,$
in which case $\tau_{c}$ is the smallest time scale for which the contribution
due to advection is larger than that of diffusion,
$v_{1}^{2}\,t^{2}>2D_{1}\,t,$
which leads to
$\tau_{c}=\frac{2D_{1}}{v_{1}^{2}}.$ (58)
A similar procedure can be carried out for non-trivial choices for the memory.
The first of these is an exponential memory with a relaxation time scale
$\tau_{a}=1/b$. This functional form dominates the asymptotic behavior in both
the Gamma and the truncated Mittag–Leffler memory kernels if the anomalous
time scale $\tau_{\epsilon}$ in the latter is sufficiently fast compared with
$1/b$. An analogous procedure yields the time to consensus
$\tau_{c}\approx\frac{2D_{1}\left(1-\frac{1-\exp(-b\,\tau_{c})}{b\,\tau_{c}}\right)}{v_{1}^{2}\left(1+\frac{6}{b^{2}\,\tau_{c}^{2}}-\frac{4}{b\,\tau_{c}}+\left[\frac{b\,\tau_{c}-6}{b^{2}\tau_{c}^{2}}\right]\,\exp(-b\,\tau_{c})\right),}$
(59)
which requires an iterative solution. The full Gamma kernel (44) results in
$\tau_{c}=\frac{2D_{1}\left(1+\frac{\beta-1}{b\tau_{c}}+\frac{(b\tau_{c})^{-\beta}}{\Gamma(1-\beta)}\left[\exp(-b\tau_{c})-(\beta+b\tau_{c}-1)\,\mathcal{E}_{\beta}(b\tau_{c})\right]\right)}{v_{1}^{2}\left(1+\frac{6+4b\tau_{c}(\beta-1)+2\beta(2\beta-5)}{b^{2}\tau_{c}^{2}}+\frac{(b\tau_{c})^{2(1-\beta)}}{\Gamma(4-2\beta)}\left[\exp(-b\tau_{c})(2\beta+b\tau_{c}-1)-R(b,\beta,\tau_{c})\right]\right)}$
(60)
where
$R(b,\beta,\tau_{c})=(6+b^{2}\tau_{c}^{2}+4b\tau_{c}(\beta-1))+2\beta(2\beta-5)\,\mathcal{E}_{2\beta-3}(b\tau_{c})$
(61)
and $\mathcal{E}_{\alpha}(x)$ is the exponential integral
$\mathcal{E}_{\alpha}(x)=\int_{1}^{\infty}\frac{e^{-x\,t}}{t^{\alpha}}\,dt.$
Unfortunately, we were unable to find an analytical inversion of the Laplace
transform of (57) for a Mittag–Leffler memory kernel. However, both memory
kernels are dominated asymptotically by the exponential truncation. For
simplicity, we used the exponential approximation (59) of the time to
consensus for the macroscopic analysis of the efficiency of collective
decision making for various group sizes, values of the coupling constants and
proportions of informed individuals.
### 4.4 Results
Figure 16 shows estimates of the three key macroscopic parameters of swarm
meta-particles. The magnitude of the diffusivity $D_{1}$ along the informed
direction (left column), the drift $v_{1}$ (center column), and the time to
consensus $\tau_{c}$ (right column, logarithmic scale) for three total
population sizes $N=10$ (top row), $N=50$ (center row) and $N=100$ (lower
row). In all the graphs the horizontal axis corresponds to the coupling
constant $\omega\in[0,0.6]$, and the vertical axis to the relative fraction
$p$ of the informed population size to the whole group. We see that the
precision of the collective decision, measured by the ratio of the
diffusivities along both coordinates (30) increases with the coupling constant
and the number of informed individuals. Similarly, the degree of consensus
(27), measured by the ratio of the drift $v_{1}$ to the individual particle
speed, increases as well with the coupling constant and the informed fraction.
Smaller groups move faster than larger ones, but at the cost of a loss in
precision. Finite size effects are of paramount importance in this class of
problems. Given that the diffusivity decreases with group size as was also
detected before [25], traditional approaches where macroscopic quantities are
calculated in the limit of very large population sizes are not particularly
useful in this context. The time to consensus $\tau_{c}$ decreases with
increasing number of informed individuals and coupling strength. This is not
surprising since it is tied to first order to the ratio $D_{1}/v_{1}^{2}$.
Interestingly, it appears to be invariant to group size and controlled by the
time scale of the exponential relaxation $\tau_{a}$ which increases as the
group size grows.
Figure 16: Estimates of the diffusion coefficient (left column) along the
informed direction $D$, the mean group speed (center column) $v$, and time to
consensus (right) $\tau$ for various values of the proportion of informed
individuals $p$ (vertical axis), coupling constant $\omega$ (horizontal axis),
and total population sizes. The first row ($D_{1},v_{1},\tau_{1}$) corresponds
to the case $N=10$, the second ($D_{2},v_{2},\tau_{2}$) to $N=50$ and the
third ($D_{3},v_{3},\tau_{3}$) to $N=100$.
## 5 Final comments
This study suggests that both the transient and the asymptotic regimes of
swarming populations –with strong alignment and in the presence of an
orientation bias– can be concisely approximated by an advection–diffusion
equation with memory. The presence of an orientation bias together with
macroscopic bursts of alignment, alternating with an unpolarized phase, lead
to quite non-trivial time correlations in the mean group velocity, which
persist over macroscopically relevant time scales. These must be explicitly
accounted for in order to capture accurately the macroscopic parameters that
typify the various collective states together with their characteristic time
scales. This observation is consistent with recent results by Grünbaum _et al_
[25] who found that local-in-time advection-diffusion equations even with
density dependent coefficients could not fully capture the fluxes of
individual-based models of swarming populations when alignment was an
important contributor to the dynamics at the level of the individual particle.
That study focused on looking at the fluxes of fission–fusion populations,
without informed individuals, for various values of the density in order to
try to find a functional form that fitted the dependence of the transport
coefficients on the population density. We explored a much more limited range
of population sizes, but instead looked in more detail at the _temporal_
dependence of the mean squared displacement, and the various transport
behaviors shown at each time scale. Of course, both methods are not in
opposition but complement each other. In the future, we would like to
integrate both approaches in such a way that both the density–dependence and
memory effects are included in a single transport model of swarming
populations with alignment.
We find that the mean group velocity increases as a power law of the coupling
constant, and that the exponent of the power law decreases as the number of
informed individuals increases. We also find –in agreement with earlier work
[15]– that the total group size has a dramatic impact in the collective
transport properties. Smaller groups tend to move with higher velocities, but
at the expense of a higher diffusivity and thus less precise decisions. This
may have important implications for evolutionary studies of simple models of
collective–decision making, where there is presumably costs associated with
recruiting informed individuals into the population, by having a relatively
high value of the coupling constant and by making erroneous decisions
(Vishwesha Guttal _et al_ , personal communication). If some value of the mean
group velocity along the informed direction is optimal in a way that maximizes
a measure of individual–level fitness, there are a number of possible ways to
achieve it. One possible path is to have a small number of informed
individuals, each with a relatively high coupling strength, while another is
to have a larger number of informed individuals but with a much smaller
coupling strength. A very rich trade–off space is likely to occur in this
class of systems, particularly if one allows for variability in total
population size.
Remarkably, the efficiency of collective decision–making, understood as the
time scale at which an effective drift becomes detectable over the diffusive
component of the meta–particle random walk, seems to be invariant with respect
to group size. What seems to determine the efficiency is a combination of the
fraction of informed individuals and the strength of the orientation bias.
This arises from the fact that this quantity ultimately depends on the ratio
$D/v^{2}$ and the characteristic time scale $\tau_{a}$ of the exponential
decay in the memory (59).
The time velocity auto–correlation emerges from the ADEM approach as the key
macroscopic summary statistic. It quantifies the relative contributions to
macroscopic transport from each collective behavior, and allows the
specification of their characteristic time scales. Although the ability of
time correlation functions to connect microscopic dynamics with observed
macroscopic regimes has been known in non–equilibrium statistical physics for
at least four decades since the seminal work of Kubo [36], Mori [43], Green
[24], Zwanzig [61], Montroll [41] and Kenkre [30], to our knowledge it is a
relatively unexplored concept in movement and spatial ecology, where Markovian
models have dominated the scene [46], perhaps with the notable exception of
correlated random walks [10, 23, 45]. We would like to emphasize a subtle
point though, which is that the temporal memory of the ADEM does not
necessarily imply that the individual walker has information about the past in
order to make movement decisions about the future. The memory arises naturally
as a result of the ensemble average of a continuous time random walk in the
presence of a wide range of transition rates. These can result from internal
properties –like an updating clock with a ‘fat tail’ instead of an exponential
one– or external factors such as behavioral variability due to complicated
social interactions or spatial structure in the landscape that results in
slip/stick dynamics; these can occur quite naturally if there are corridors
with preferential directions of motion alternating with regions where movement
can be described with Brownian motion. We believe that this ecological
interpretation of the time velocity auto–correlation function is likely to be
useful not only to unravel the connections between individual–based models of
movement and dispersal and their continuum approximations as we have seen in
this study, but also for other areas of ecology where interdependencies
between an individual organism’s dispersal strategy, spatial heterogeneity in
the landscape, and temporal variability in resource availability become
intertwined in observed individual trajectories, particularly in the nascent
field of movement ecology.
Future work will be devoted to a generalization of the SPP model to density-
dependent asynchronous updating, in the sense that each of the social
interactions is associated with an exponential clock that is parameterized by
the local density, in a similar way to what is done in locally regulated
models of plant population dynamics with spatial structure [5, 6]. Given that
the CTRW-ADEM can predict the the full density and not just the first moments,
future work will be devoted to this issue in order to explore first passage
times. We will also explore the situation when there are two conflicting
preferential directions, where it remains to be seen whether the ADEM has the
capability of capturing the bifurcations that have been detected in
individual–based simulations [15]. This will require generalizations of the
ADEM involving anisotropy in the memory kernel.
## 6 Acknowledgements
The authors are grateful for the support received from the National Science
Foundation (Award ID:EF-0434319) and DARPA (Award ID:HR001-05-1-0057).
Insightful discussions with I. D. Couzin, L.Giuggioli, V.M. Kenkre and
F.Bartumeus are gratefully acknowledged.
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|
arxiv-papers
| 2012-02-27T19:03:54 |
2024-09-04T02:49:27.877280
|
{
"license": "Public Domain",
"authors": "Michael Raghib, Simon A. Levin and Ioannis G. Kevrekidis",
"submitter": "Michael Raghib",
"url": "https://arxiv.org/abs/1202.6027"
}
|
1202.6074
|
knandhap@uwyo.edu (K.N.Premnath)
# Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation:
Implications for Filtered Turbulence Simulation
Kannan N. Premnath 1 and Sanjoy Banerjee 2 11affiliationmark: Department
of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, U.S.A.
22affiliationmark: Department of Chemical Engineering, City College of New
York, City University of New York, New York, NY 10031, U.S.A.
###### Abstract
We present a systematic derivation of a model based on the central moment
lattice Boltzmann equation that rigorously maintains Galilean invariance of
forces to simulate inertial frame independent flow fields. In this regard, the
central moments, i.e. moments shifted by the local fluid velocity, of the
discrete source terms of the lattice Boltzmann equation are obtained by
matching those of the continuous full Boltzmann equation of various orders.
This results in an exact hierarchical identity between the central moments of
the source terms of a given order and the components of the central moments of
the distribution functions and sources of lower orders. The corresponding
source terms in velocity space are then obtained from an exact inverse
transformation due to a suitable choice of orthogonal basis for moments.
Furthermore, such a central moment based kinetic model is further extended by
incorporating reduced compressibility effects to represent incompressible
flow. Moreover, the description and simulation of fluid turbulence for full or
any subset of scales or their averaged behavior should remain independent of
any inertial frame of reference. Thus, based on the above formulation, a new
approach in lattice Boltzmann framework to incorporate turbulence models for
simulation of Galilean invariant statistical averaged or filtered turbulent
fluid motion is discussed.
###### keywords:
Lattice Boltzmann Method, Central Moments, Galilean Invariance, Turbulence,
Filtering
05.20.Dd,47.27.-i,47.27.E-
## 1 Introduction
Minimal kinetic models for the Boltzmann equation, i.e. lattice Boltzmann
equation formulations, are evolving towards as alternative physically-inspired
computational techniques for various fluid mechanics and other problems.
Originally developed as an improved variant of the lattice gas automata [1] to
eliminate statistical noise [2], the lattice Boltzmann method (LBM) has
undergone a series of major refinements, in terms of its underlying physical
models as well as numerical solution schemes for various applications over the
last two decades [3, 4, 5, 6]. In particular, its rigorous connection to the
kinetic theory [7, 8, 9] has resulted in a number of recent developments,
including models that are more physically consistent for multiphase [10, 11]
and multicomponent flows [12], models for non-equilibrium phenomena beyond the
Navier-Stokes-Fourier representation [13] and an asymptotic analysis approach
to establish consistency of the LBM from a numerical point of view [14].
The stream-and-collide procedure of the LBM can be considered as a
dramatically simplified discrete representation of the continuous Boltzmann
equation. Here, the streaming step represents the advection of the
distribution of particle populations along discrete directions, which are
designed from symmetry considerations, between successive collisions. Much of
the physical effects being modeled are represented in terms of the collision
step, which also significantly influences the numerical stability of the LBM.
Most of the major developments until recently were associated with the single-
relaxation-time (SRT) model [15, 16] based on the BGK approximation [17], and
enjoys its popularity owing, mainly, to its simplicity. However, it is prone
to numerical instability. Moreover, it is inadequate in its representation of
certain physical aspects, such as independently adjustable transport
properties of thermal transport and viscoelastic phenomena.
These limitations have been significantly addressed in the multiple-
relaxation-time (MRT) collision model [18]. This, in a sense, represents a
simplified form of the relaxation LBM proposed earlier [19, 20], with an
important characteristic difference in that the collision process is carried
out in moment space [21] instead of in the usual velocity space. By separating
the relaxation time scales of different moments, obtained by using a linear
Fourier stability analysis, its numerical stability can be significantly
improved [22, 23]. Furthermore, it has resulted in significant advantages over
the SRT-LBM for computation of various classes of fluid flow problems,
including multiphase systems [24, 25, 26], turbulent flows [27, 28] and
magnetohydrodynamics [29]. It may be noted that recently a different form of
MRT model based on the orthogonal Hermite polynomial projections of the
distribution functions, which is independent of any underlying lattice
structure, allowing representation of higher order non-equilibrium effects has
been proposed [30].
The stabilization of the LBM using a single relaxation time has been addressed
from a different perspective by enforcing the H-theorem locally in the
collision step [31, 32, 33, 34]. By using the attractors of the distribution
function based on the minimization of a Lyapunov-type functional, non-linear
stability of the LBM is achieved in this Entropic LBM. This approach has
recently been significantly extended to incorporate multiple relaxation times
with efficient implementation strategies [35, 36]. Furthermore, systematic
procedures for different types of higher-order LBM have been developed [37,
38, 39]. An important element is the construction of higher-order lattices
based on symmetry considerations which have been analyzed using group theory
[40, 41]. Further progress, from a numerical aspect, is that based on the
consistency analysis [14] and a notion of structural stability [42, 43] (shown
related to the Onsager-like relation in non-equilibrium thermodynamics [44]),
convergence of the LBM to the Navier-Stokes equations has rigorously been
shown [45].
On the other hand, it is important to clearly understand in what sense the
lattice Boltzmann equation (LBE), which is generally considered as a
mesoscopic approach, inherits or maintains the various physical invariance
properties of the continuous full Boltzmann equation (which it represents as a
much simplified model) and the Navier-Stokes equations (which it represents
numerically). Careful considerations of these aspects play an important role
in ensuring the general applicability of the approach for various, especially
challenging, problems. In this regard, and to put the present work in
perspective, it should be noted that the continuum mechanics description as
well as the microscopic statistical (continuous Boltzmann) description of
fluid motion generally satisfy a larger invariance group, with inertial frame
invariance being just an important special case. The most general form among
these is the so-called the principal of material frame indifference, also
known as the objectivity principle [46]. According to this, the constitutive
equations should have the same forms in _all_ frames of reference, whether
inertial or not. While this is considered as an important axiom based on which
the continuum mechanics is formulated [46], its role from continuous kinetic
theory point of view was the subject of considerable analysis for sometime
[47, 48, 49, 50, 51, 52].
The following are the main outcomes of these studies: the continuous full
Boltzmann equation (i) is material frame indifferent in a _strong approximate
sense_ , when there is a large scale separation between the collision times
and the macroscopic flow times [49, 51, 52] (thus providing a strong support
to the axiomatic principle generally used in the continuum description), (ii)
satisfies both the inertial frame or Galilean invariance as well as the
extended Galilean invariance (i.e. invariance under arbitrary translational
accelerations of the reference frame) _exactly_ [52]. Furthermore, it was
shown that the standard procedures (e.g. Chapman-Enskog expansion, Maxwellian
iteration) lead to frame dependent higher order contributions for the
constitutive equations in non-inertial frames in the continuum limit, while
the continuous kinetic theory itself can be frame independent [49]. Careful
considerations of these principles could guide in the development of more
generally applicable models and numerical schemes for complex problems. For
example, material frame indifference (point (i)) is generally used as an
important constraint for the constitutive equations for complex fluids (e.g.
beyond Newtonian constitutive description such as polymeric fluids) and in the
development of turbulence models in continuum mechanics. As mentioned above,
this property is satisfied in a strong approximate sense by the continuous
Boltzmann equation, but not necessarily by the tools that relate the
microscopic and macroscopic descriptions(point iii). This aspects are
pertinent in the construction of complex models from the continuous Boltzmann
equation (e.g. [53, 54]). In this work, however, we limit our discussion to an
association of the properties mentioned in a part of the point (ii) for the
LBE, i.e. for the exact invariance group – the Galilean or inertial frame
invariance.
In this regard, as a discrete approximation to the continuous full Boltzmann
equation, the development of the LBE consists of simplifications at different
levels. Thus, its various elements should be analyzed carefully to ascertain
and quantify as to how well it satisfies Galilean invariance. First, in
contrast to continuous kinetic theory, due to the choice of finite lattice
velocity sets and associated symmetries, it introduces linear dependencies of
higher order moments with those of lower order moments that are supported by
the lattice set [40]. Such degeneracies can in turn lead to negative
dependence of viscosity on fluid velocity. It generally causes the Galilean
invariance to be broken by the presence of terms that are cubic in velocity
for the standard lattice configurations (with symmetries of square in 2D and
cube in 3D) and also leads to numerical instability, especially at higher Mach
numbers. This issue can be alleviated by the use of extended lattice velocity
sets, which then relegates the degeneracies among moments to even higher
orders. Second, the collision step including the forcing terms of the LBE
should be carefully constructed in such a way that they recover correct
physics which is inertial frame independent, i.e. the Navier-Stokes equations.
Here, the use of independent set of _central_ moments for a chosen lattice
provides a natural approach to maintain Galilean invariance that can be
constructed by invoking elements directly from kinetic theory. This is the
main goal of the present work (see below). A rational means to more
efficiently account for both the above aspects is discussed in the last
section of this paper. And, third, the streaming step of the LBE is generally
constructed as a discrete Lagrangian process. In the standard implementation,
this couples the particle velocity and configurations spaces, which in turn,
constrains the numerical accuracy of the LBE in the representation of the
Navier-Stokes equations. As a result, the Galilean invariance of the LBE is
limited by its overall numerical accuracy. However, it is known that such
coupling between physical and lattice symmetries is not necessary in the
discretization of the streaming operator. In fact, it can be discretized using
classical schemes such as finite-difference or finite-element methods that
alleviate this issue [55, 56, 57]. Specifically, exploiting higher order
discretization and time integration schemes (e.g. [58]) for the streaming
operator could further improve the order of accuracy (and hence the Galilean
invariance) of the LBE. Furthermore, the use of implicit schemes could enhance
the computational efficiency in this regard.
Focusing on the second aspect mentioned above, a different type of collision
operator and forcing can be devised that can maintain Galilean invariance for
a chosen lattice velocity set and a discretization scheme for the streaming
step. Specifically, central moments are relaxed in a moving frame of reference
during collision step [59], originally proposed to improve numerical
stability, but emphasized here for its better physical coherence. The use of
central moments, which are obtained by shifting the particle velocity with the
local fluid velocity [60], rigorously enforces Galilean invariance. In
particular, while other previous approaches are generally Galilean invariant
for up to second-order moments, the central moment based approach provides a
higher order frame invariance as permitted by the discrete lattice velocity
set. This approach was examined based on the concept of generalized local
equilibrium [61]. In addition, to further improve physical coherence, the
attractors for the higher order central moments were constructed as products
of the lower order central moments, leading to the factorized central moment
method [62]. Recently, a new approach to incorporate source terms using
central moments in the LBM that are Galilean invariant by construction, which
are important for computation of various physical problems, was developed
[63]. The consistency of this technique to the Navier-Stokes equations was
shown by means of the Chapman-Enskog analysis [64] and its numerical accuracy
was established. Furthermore, the method was also extended in three-dimensions
for various lattice velocity sets and validated for a class of canonical
problems [65]. As clarified in [63, 65], numerical stability of the central
moment approach can be enhanced, when it is executed in a multiple relaxation
time formulation, similar to the standard or raw moment based approaches.
Interestingly, it has been shown recently that when some classical schemes for
flow simulation are made to satisfy Galilean invariance more rigorously, they
led to more robust implementations (e.g. [66, 67]).
Turbulence remains as among the most challenging classes of flows for which
considerable effort has been focused on the development of theory and
applications using the LBM. Since its roots can be traced to kinetic theory,
the LBM has been analyzed for the development of turbulence models from a
fundamental point of view [68, 69, 70]. It has been employed for computation
of Reynolds-averaged description of turbulent flows [71, 72]. Furthermore, it
has found applications for large eddy simulation (LES) using LBM formulations
with SRT [73], and MRT [74] with multiblock approach for efficient
implementation [27]. Recently, dynamic subgrid scale (SGS) models for LES were
incorporated into the LBM framework that resulted in reduced empiricism for
description at such scales [28]. Moreover, an improved inertial-range
consistent SGS model was also proposed [75]. A theoretical formulation for a
SGS model based on an approximate deconvolution method [78] that does not rely
on the common eddy-viscosity concept for application with the LBM was also
devised recently [76]. Lastly, the closure modeling issues of kinetic and
continuum turbulence effects were reconciled in a unified statistical/filtered
description using a modified kinetic equation [77]. Effectively, this allows
the use of macroscopic turbulence models involving divergence of the Reynolds
stress in the forcing term of the kinetic equation.
An important physical consideration for any description of turbulent flow is
that it should be invariant for all inertial frames of reference. In other
words, for general applicability, representation of turbulence for all or any
subset of its scales should be Galilean invariant. Thus, in particular, all
SGS models, and associated numerical schemes for turbulence computation,
should be frame invariant. An insightful analysis of various turbulence models
was carried out from this viewpoint in [79]. A method to achieve Galilean
invariance by means of certain redefinition of turbulent stresses was
discussed in [80]. A recent review on this subject is reported in [81].
Furthermore, it should be noted that concepts based on central moments have
played an important role in the development of theoretical foundation of
turbulence physics – such as for statistical turbulence theory [82] and
turbulence modeling [83].
Thus, in this paper, we develop a lattice Boltzmann equation based on central
moments for Galilean invariant representation of turbulent flows.
Specifically, it allows frame-independent incorporation of general models for
turbulent Reynolds stresses in a statistical/filter averaged formulation using
LBM for turbulence simulation. Furthermore, in a general setting, it maintains
the forces and stresses to be independent of any inertial frame of reference
and could also improve numerical stability in computations. In [63], we
developed a forcing scheme based on a particular ansatz involving the local
Maxwell distribution. Here, we develop a general forcing based on central
moments by a direct examination of the continuous full Boltzmann equation
itself, which unlike [63] could also self-consistently account for non-
equilibrium effects in higher order terms. In this regard, the central moments
of the resulting source terms of the continuous and discrete counterparts are
matched successively at different orders leading to a cascaded structure. In
essence, this approach can be considered as a Galilean invariant minimal
discrete model for the full Boltzmann equation including forcing terms. The
attractors for higher order central moments in the collision step of this
computational model is based on the factorization in terms of those at lower
orders by including such general forcing terms. In addition, we further
develop this approach with reduced compressibility effects for improved
representation of turbulent flow physics in the incompressible limit. The
forcing formulation developed here for incorporating turbulence models in a
statistical/filtered formulation can be extended to other problems, such as,
for example, Galilean invariant representation of forces or stresses in
complex fluids.
The paper is organized as follows. Section 2 briefly discusses the choice of
the moment basis employed in this paper and Sec. 3 the continuous Boltzmann
equation. In Secs. 4 and 5, continuous forms of the central moments for the
distribution functions and its local equilibrium, and sources due to force
fields, respectively, are introduced. The LBE based on central moments with
the general forcing terms is presented in Sec. 6. Various discrete central
moments are presented in Sec. 7 that also specifies a matching principle to
maintain Galilean invariance and the relationships among such moments are
provided in Sec. 8. Section 9 describes various discrete raw moments and the
derivation of the source terms in terms of the discrete particle velocity
space. In Sec. 10,we present the construction of the collision operator of the
central moment based LBM. The computational procedure of this approach is
provided in Sec. 11. The derivation is extended by considering reduced
compressibility effects in Sec. 12. Furthermore, Sec. 13 discusses the use of
attractors of the higher order central moments based on the concept of their
factorization in term of those at lower orders. A natural consequence of this
overall approach is that turbulence models can be represented for Galilean
invariant filtered turbulence simulation using the LBM, which is described in
Sec. 14. Finally, the summary and conclusions of this work are discussed in
Sec. 15.
## 2 Selection of Moment Basis
An important element in the development of the central moment based LBM is the
specification of a suitable basis for moments. In this work, to elucidate our
approach, the two-dimensional, nine velocity (D2Q9) lattice model (see Fig. 1)
is considered, for which the moment basis used in [63] is adopted. It should,
however, be noted that the procedure described henceforth is of general
nature, and can be extended for other lattice models and in three dimensions.
The particle velocity for this lattice model $\overrightarrow{e}_{\alpha}$ is
given by
$\overrightarrow{e_{\alpha}}=\left\\{\begin{array}[]{ll}{(0,0)}&{\alpha=0}\\\
{(\pm 1,0),(0,\pm 1)}&{\alpha=1,\cdots,4}\\\ {(\pm 1,\pm
1)}&{\alpha=5,\cdots,8}\end{array}\right.$ (1)
Figure 1: Two-dimensional, nine-velocity (D2Q9) Lattice.
For convenience, we employ Dirac’s bra-ket notion to represent the basis
vectors, and Greek and Latin subscripts for particle velocity directions and
Cartesian coordinate directions, respectively. Noting that moments in the LBM
are discrete integral properties of the distribution function $f_{\alpha}$,
i.e. $\sum_{\alpha=0}^{8}e_{\alpha x}^{m}e_{\alpha y}^{n}f_{\alpha}$, where
$m$ and $n$ are integers, we begin with the following nine non-orthogonal
independent basis vectors obtained by combining monomials $e_{\alpha
x}^{m}e_{\alpha y}^{n}$ in an ascending order. That is,
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}}$
$\displaystyle=$ $\displaystyle\left(1,1,1,1,1,1,1,1,1\right)^{T},$
$\displaystyle\ket{e_{\alpha x}}$ $\displaystyle=$
$\displaystyle\left(0,1,0,-1,0,1,-1,-1,1\right)^{T},$
$\displaystyle\ket{e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,1,0,-1,1,1,-1,-1\right)^{T},$
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,1,1,1,1,2,2,2,2\right)^{T},$
$\displaystyle\ket{e_{\alpha x}^{2}-e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,1,-1,1,-1,0,0,0,0\right)^{T},$ (2)
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,-1,1,-1\right)^{T},$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,1,-1,-1\right)^{T},$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,-1,-1,1\right)^{T},$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,1,1,1\right)^{T},$
where the superscript ‘$T$’ represents the transpose operator.
For an efficient implementation, the above non-orthogonal basis set is
transformed into an equivalent orthogonal set through the Gram-Schmidt
procedure in the increasing order of the monomials of the products of the
Cartesian components of the particle velocities [63]:
$\displaystyle\ket{K_{0}}$ $\displaystyle=$ $\displaystyle\ket{\rho},$
$\displaystyle\ket{K_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
$\displaystyle\ket{K_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha y}},$
$\displaystyle\ket{K_{3}}$ $\displaystyle=$ $\displaystyle 3\ket{e_{\alpha
x}^{2}+e_{\alpha y}^{2}}-4\ket{\rho},$ $\displaystyle\ket{K_{4}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha x}^{2}-e_{\alpha y}^{2}},$ (3)
$\displaystyle\ket{K_{5}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}},$ $\displaystyle\ket{K_{6}}$ $\displaystyle=$
$\displaystyle-3\ket{e_{\alpha x}^{2}e_{\alpha y}}+2\ket{e_{\alpha y}},$
$\displaystyle\ket{K_{7}}$ $\displaystyle=$ $\displaystyle-3\ket{e_{\alpha
x}e_{\alpha y}^{2}}+2\ket{e_{\alpha x}},$ $\displaystyle\ket{K_{8}}$
$\displaystyle=$ $\displaystyle 9\ket{e_{\alpha x}^{2}e_{\alpha
y}^{2}}-6\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}}+4\ket{\rho}.$
This can be written explicitly in term of a matrix given by
$\mathcal{K}=\left[\begin{array}[]{rrrrrrrrr}1&0&0&-4&0&0&0&0&4\\\
1&1&0&-1&1&0&0&2&-2\\\ 1&0&1&-1&-1&0&2&0&-2\\\ 1&-1&0&-1&1&0&0&-2&-2\\\
1&0&-1&-1&-1&0&-2&0&-2\\\ 1&1&1&2&0&1&-1&-1&1\\\ 1&-1&1&2&0&-1&-1&1&1\\\
1&-1&-1&2&0&1&1&1&1\\\ 1&1&-1&2&0&-1&1&-1&1\\\ \end{array}\right],$ (4)
where we have used
$\mathcal{K}=\left[\ket{K_{0}},\ket{K_{1}},\ket{K_{2}},\ket{K_{3}},\ket{K_{4}},\ket{K_{5}},\ket{K_{6}},\ket{K_{7}},\ket{K_{8}}\right].$
(5)
## 3 Continuous Boltzmann Equation
We consider the two-dimensional (2D) continuous Boltzmann equation, for which
we aim to develop a Galilean invariant discrete model using the above basis
vectors. It represents the evolution of the continuous density distribution
function $f=f(x,y,\xi_{x},\xi_{y})$ in continuous phase space
$(x,y,\xi_{x},\xi_{y})$ subjected to a local force field
$\overrightarrow{F}=(F_{x},F_{y})$, whose origin could be internal or external
to the system. By definition, the averaged effects of $f$, weighted by various
powers of the continuous particle velocity $(\xi_{x},\xi_{y})$, i.e. its
moments, are considered to characterize the various physical processes
inherent in the motion of athermal fluids. In particular, the evolution of the
slow hydrodynamical processes are described by the local macroscopic fluid
density $\rho$ and fluid velocity $\overrightarrow{u}=(u_{x},u_{y})$. The
continuous Boltzmann equation may be written as [64, 84]
$\frac{\partial f}{\partial
t}+\overrightarrow{\xi}\cdot\overrightarrow{\nabla}_{\overrightarrow{x}}f+\frac{\overrightarrow{F}}{\rho}\cdot\overrightarrow{\nabla}_{\overrightarrow{\xi}}f=\Omega(f,f),$
(6)
where
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}fd\xi_{x}d\xi_{y},$
(7) $\displaystyle\rho\overrightarrow{u}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f\overrightarrow{\xi}d\xi_{x}d\xi_{y}.$
(8)
Here, $\Omega(f,f)$ is the collision term, which represents the cumulative
effect of binary collision of particles. The force fields modify the
distribution function exactly by the term
$-\frac{\overrightarrow{F}}{\rho}\cdot\overrightarrow{\nabla}_{\overrightarrow{\xi}}f$,
which is obtained by moving the last term on the left hand side of Eq. (6) to
its right to serve as a source term. It was shown by Grad (1949) [21] that
that solution of Eq. (6) can be approximated by the evolution equations for a
hierarchical set of moments. Here, we seek to obtain a dramatically
discretized version of this continuous Boltzmann equation by means of a
hierarchy of central moments, focusing, in particular, on the forcing term, to
obtain Galilean invariant representation of the dynamics of fluid motion.
## 4 Continuous Central Moments: Distribution Function and its Local
Attractor
We now consider the integral properties of the distribution function $f$ given
in terms of its central moments, i.e. those shifted by the macroscopic fluid
velocity. In particular, we define _continuous_ central moment of $f$ of order
$(m+n)$ as
$\widehat{\Pi}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}.$
(9)
Here, and in the rest of this paper, the use of “hat” over a symbol represents
quantities in the space of moments. The distribution function for an athermal
fluid has a local equilibrium state in the _continuous_ particle velocity
space $(\xi_{x},\xi_{y})$, which is given by the Maxwellian as [84]
$f^{\mathcal{M}}\equiv
f^{\mathcal{M}}(\rho,\overrightarrow{u},\xi_{x},\xi_{y})=\frac{\rho}{2\pi
c_{s}^{2}}\exp\left[-\frac{\left(\overrightarrow{\xi}-\overrightarrow{u}\right)^{2}}{2c_{s}^{2}}\right],$
(10)
where $c_{s}^{2}=1/3$. Analogously, we can define the corresponding central
moment of the Maxwell distribution of order $(m+n)$ as
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\mathcal{M}}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}.$
(11)
By virtue of the fact that $f^{\mathcal{M}}$ being an even function,
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}\neq 0$ when $m$ and $n$ are even and
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}=0$ when $m$ or $n$ odd. Here and
henceforth, the subscripts $x^{m}y^{n}$ mean $xxx\cdots m\mbox{-times}$ and
$yyy\cdots n\mbox{-times}$. Evaluation of the central moments of the
Maxwellian, to different orders of increasing powers, yields
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{x}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{y}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xx}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{yy}$
$\displaystyle=$ $\displaystyle c_{s}^{2}\rho,$ (12)
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xxy}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xyy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xxyy}$
$\displaystyle=$ $\displaystyle c_{s}^{4}\rho.$
## 5 Continuous Central Moments: Forcing
In the presence of a force field $\overrightarrow{F}$, in view of Eq. (6) and
as discussed in Sec. 3, the distribution function will be exactly modified by
the source term
$\delta
f^{F}=-\frac{\overrightarrow{F}}{\rho}\cdot\overrightarrow{\nabla}_{\overrightarrow{\xi}}f.$
(13)
Now we define a corresponding _continuous_ central moment of order $(m+n)$ due
to change in the distribution function as a result of a force field as
$\widehat{\Gamma}^{F}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\delta
f^{F}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}.$ (14)
Substituting Eq. (13) in Eq. (14) and integrating by parts by making use of
the asymptotic limit assumptions
$lim_{\xi_{x}\rightarrow\pm\infty}(\xi_{x}-u_{x})^{m}f(x,y,\xi_{x},\xi_{y})=0$
and
$lim_{\xi_{y}\rightarrow\pm\infty}(\xi_{y}-u_{y})^{n}f(x,y,\xi_{x},\xi_{y})=0$,
for $m,n\geq 0$, we get
$\displaystyle\widehat{\Gamma}^{F}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle
m\frac{F_{x}}{\rho}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(\xi_{x}-u_{x})^{m-1}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}+$
(15) $\displaystyle
n\frac{F_{y}}{\rho}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n-1}d\xi_{x}d\xi_{y}.$
From the definition given in Eq. (9), Eq. (15) reduces to an _exact_ identity
between continuous central moment of the source term of a given order to the
components of the continuous central moment of the distribution function of an
order lower acted upon by a force field:
$\widehat{\Gamma}^{F}_{x^{m}y^{n}}=m\frac{F_{x}}{\rho}\widehat{\Pi}_{x^{m-1}y^{n}}+n\frac{F_{y}}{\rho}\widehat{\Pi}_{x^{m}y^{n-1}},$
(16)
and for the special case of the zeroth central moment of the source as
$\widehat{\Gamma}^{F}_{0}=0$. This is a key result based on which the rest of
the derivation follows. Thus, we can enumerate the _exact_ values of the
central moments of sources in an increasing order as
$\displaystyle\widehat{\Gamma}^{F}_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Gamma}^{F}_{x}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\Pi}_{0},$
$\displaystyle\widehat{\Gamma}^{F}_{y}$ $\displaystyle=$
$\displaystyle\frac{F_{y}}{\rho}\widehat{\Pi}_{0},$
$\displaystyle\widehat{\Gamma}^{F}_{xx}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\Pi}_{x},$
$\displaystyle\widehat{\Gamma}^{F}_{yy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{y}}{\rho}\widehat{\Pi}_{y},$ (17)
$\displaystyle\widehat{\Gamma}^{F}_{xy}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\Pi}_{y}+\frac{F_{y}}{\rho}\widehat{\Pi}_{x},$
$\displaystyle\widehat{\Gamma}^{F}_{xxy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\Pi}_{xy}+\frac{F_{y}}{\rho}\widehat{\Pi}_{xx},$
$\displaystyle\widehat{\Gamma}^{F}_{xyy}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\Pi}_{yy}+2\frac{F_{y}}{\rho}\widehat{\Pi}_{xy},$
$\displaystyle\widehat{\Gamma}^{F}_{xxyy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\Pi}_{xyy}+2\frac{F_{y}}{\rho}\widehat{\Pi}_{xxy}.$
Note that if we set
$\widehat{\Pi}_{x^{m}y^{n}}=\widehat{\Pi}_{x^{m}y^{n}}^{\mathcal{M}}$ in Eq.
(17), i.e. ignore non-equilibrium effects, we arrive at the the derivation
given in [63] as a special case.
## 6 Cascaded Central Moment Lattice-Boltzmann Method with Forcing Terms
Defining a _discrete_ distribution function supported by the discrete particle
velocity set $\overrightarrow{e}_{\alpha}$ as
$\mathbf{f}=\ket{f_{\alpha}}=(f_{0},f_{1},f_{2},\ldots,f_{8})^{T}$, and a
cascaded collision operator as
$\bm{\Omega}^{c}=\ket{\Omega_{\alpha}^{c}}=(\Omega_{0}^{c},\Omega_{1}^{c},\Omega_{2}^{c},\ldots,\Omega_{8}^{c})^{T}$
as well as a source term as
$\mathbf{S}=\ket{S_{\alpha}}=(S_{0},S_{1},S_{2},\ldots,S_{8})^{T}$ based on
central moments, we obtain the lattice Boltzmann equation (LBE) as a discrete
version of Eq. (6) by temporally integrating along particle characteristics as
follows [63]:
$f_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=f_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+\int_{t}^{t+1}S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\theta,t+\theta)}d\theta.$
(18)
Here, the collision operator is written as
$\Omega_{\alpha}^{c}\equiv\Omega_{\alpha}^{c}(\mathbf{f},\mathbf{\widehat{g}})=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha},$
(19)
where
$\mathbf{\widehat{g}}=\ket{\widehat{g}_{\alpha}}=(\widehat{g}_{0},\widehat{g}_{1},\widehat{g}_{2},\ldots,\widehat{g}_{8})^{T}$.
The hydrodynamic fields are obtained from the distribution function as
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}=\braket{f_{\alpha}}{\rho},$ (20)
$\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}e_{\alpha
i}=\braket{f_{\alpha}}{e_{\alpha i}},i\in{x,y}.$ (21)
For improved accuracy in recovering Navier-Stokes solution, using a semi-
implicit representation for the source term, i.e. the last term in the above
equation (Eq. (18)) as
$\int_{t}^{t+1}S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\theta,t+\theta)}d\theta=\frac{1}{2}\left[S_{{\alpha}(\overrightarrow{x},t)}+S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)}\right]$,
so that Eq. (18) is made effectively explicit by using the transformation
$\overline{f}_{\alpha}=f_{\alpha}-\frac{1}{2}S_{\alpha}$ to reduce it to [63]
$\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)}.$
(22)
It may be noted that, as in [59], we first represent collision as a cascaded
process in which the effect of collision on lower order central moments
successively influence those at higher orders in a cascaded manner. That is,
in general,
$\widehat{g}_{\alpha}\equiv\widehat{g}_{\alpha}(\mathbf{f},\widehat{g}_{\beta}),\beta=0,1,2,\ldots,\alpha-1$.
Furthermore, the form of the source term is derived to rigorously enforce
Galilean invariance. The explicit expressions for $S_{\alpha}$ and
$\mathbf{\widehat{g}}$ will be determined later in Secs. 9 and 10,
respectively. Since the main focus of this work is on improving the collision
(including forcing) step with features independent of inertial frames, we have
only considered the standard discretization for the streaming operator.
However, as discussed in the Introduction, other types of discretization
schemes could be considered to improve the order of accuracy.
## 7 Various Discrete Central Moments and Galilean Invariance Matching
Principle
For determining the structure of the cascaded collision operator
$\mathbf{\widehat{g}}$ and the source terms $S_{\alpha}$, we first need to
define the following _discrete_ central moments of the distribution function,
Maxwellian, and source term, respectively:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}=\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{f_{\alpha}},$ (23)
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{\mathcal{M}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}^{\mathcal{M}}(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}=\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{f_{\alpha}^{\mathcal{M}}},$ (24)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}=\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{S_{\alpha}}.$ (25)
where the exact expression for the discrete $f_{\alpha}^{\mathcal{M}}$ is not
yet known, but can be determined as a result of the derivation discussed
later. To maintain physical consistency at the discrete level, we now equate
the _discrete_ central moments of the distribution function, the Maxwellian
and the source terms, defined above, with their corresponding _continuous_
central moments, whose forms are known exactly. That is, according to this
matching principle
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}_{x^{m}y^{n}},$ (26)
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{\mathcal{M}}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}},$ (27)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\widehat{\Gamma}^{F}_{x^{m}y^{n}}.$ (28)
In particular, the discrete central moments of various orders for both the
Maxwellian and the source terms, respectively, become
$\displaystyle\widehat{\kappa}^{\mathcal{M}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle\widehat{\kappa}^{\mathcal{M}}_{x}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\kappa}^{\mathcal{M}}_{y}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\kappa}^{\mathcal{M}}_{xx}$
$\displaystyle=$ $\displaystyle c_{s}^{2}\rho,$
$\displaystyle\widehat{\kappa}^{\mathcal{M}}_{yy}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ (29)
$\displaystyle\widehat{\kappa}^{\mathcal{M}}_{xy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\kappa}^{\mathcal{M}}_{xxy}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\kappa}^{\mathcal{M}}_{xyy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\kappa}^{\mathcal{M}}_{xxyy}$
$\displaystyle=$ $\displaystyle c_{s}^{4}\rho,$
and
$\displaystyle\widehat{\sigma}_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\sigma}_{x}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\kappa}_{0},$
$\displaystyle\widehat{\sigma}_{y}$ $\displaystyle=$
$\displaystyle\frac{F_{y}}{\rho}\widehat{\kappa}_{0},$
$\displaystyle\widehat{\sigma}_{xx}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\kappa}_{x},$ $\displaystyle\widehat{\sigma}_{yy}$
$\displaystyle=$ $\displaystyle 2\frac{F_{y}}{\rho}\widehat{\kappa}_{y},$ (30)
$\displaystyle\widehat{\sigma}_{xy}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\kappa}_{y}+\frac{F_{y}}{\rho}\widehat{\kappa}_{x},$
$\displaystyle\widehat{\sigma}_{xxy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\kappa}_{xy}+\frac{F_{y}}{\rho}\widehat{\kappa}_{xx},$
$\displaystyle\widehat{\sigma}_{xyy}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\kappa}_{yy}+2\frac{F_{y}}{\rho}\widehat{\kappa}_{xy},$
$\displaystyle\widehat{\sigma}_{xxyy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\kappa}_{xyy}+2\frac{F_{y}}{\rho}\widehat{\kappa}_{xxy}.$
We also define a _discrete_ central moment in terms of the transformed
distribution function to facilitate subsequent developments as
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\sum_{\alpha}\overline{f}_{\alpha}(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}=\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{\overline{f}_{\alpha}}.$ (31)
Owing to the transformation discussed in Sec. 6, it follows that
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\widehat{\kappa}_{x^{m}y^{n}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}.$
(32)
## 8 Relation Between Various Discrete Central Moments
Equation (30) is given in terms of the discrete moments of the original
distribution function $f_{\alpha}$. However, the cascaded central moment LBM
with forcing term provides evolution in terms of transformed distribution
function $\overline{f}_{\alpha}$ (Eq. (22)). Thus, it is important to write
all the expressions in terms of the central moments of
$\overline{f}_{\alpha}$, or, equivalently,
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}$. Thus, by recursive application of
Eq. (32) using Eq. (30) to successively higher orders, we get the following
exact relations up to the third-order central moments as
$\displaystyle\widehat{\kappa}_{0}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{x}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{x}+\frac{1}{2}\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{y}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{y}+\frac{1}{2}\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xx}+\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{x}+\frac{1}{2}\frac{F_{x}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{yy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{yy}+\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{y}+\frac{1}{2}\frac{F_{y}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{xy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xy}+\frac{1}{2}\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{y}+\frac{1}{2}\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{x}+\frac{1}{2}\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{xxy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}+\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xy}+\frac{1}{2}\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xx}+\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{x}+\frac{1}{2}\frac{F_{x}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{y}+\frac{3}{4}\frac{F_{x}^{2}F_{y}}{\rho^{3}}\widehat{\overline{\kappa}}_{0},$
$\displaystyle\widehat{\kappa}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}+\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xy}+\frac{1}{2}\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{yy}+\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{y}+\frac{1}{2}\frac{F_{y}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{x}+\frac{3}{4}\frac{F_{x}F_{y}^{2}}{\rho^{3}}\widehat{\overline{\kappa}}_{0}.$
That is, the central moment of the distribution function of a given order can
be written as a function of the central moment of the transformed distribution
function of the same order and successively lower orders as well. This can be
further simplified by considering the three of the lowest order central
moments, i.e., conservative moments, which by definition are
$\widehat{\kappa}_{0}=\rho$, $\widehat{\kappa}_{x}=\widehat{\kappa}_{y}=0$.
This, in turn, leads to $\widehat{\overline{\kappa}}_{0}=\rho$,
$\widehat{\overline{\kappa}}_{x}=-1/2F_{x}$,
$\widehat{\overline{\kappa}}_{y}=-1/2F_{y}$. As a result, we have the
following relations for the non-conserved central moments up to third-order:
$\displaystyle\widehat{\kappa}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xx},$
$\displaystyle\widehat{\kappa}_{yy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{yy},$
$\displaystyle\widehat{\kappa}_{xy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xy},$
$\displaystyle\widehat{\kappa}_{xxy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}+\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xy}+\frac{1}{2}\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xx},$
$\displaystyle\widehat{\kappa}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}+\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xy}+\frac{1}{2}\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{yy}.$
Thus, we can finally write the central moments of the source term in Eq. (30)
in terms of the central moments of the transformed distribution function as
$\displaystyle\widehat{\sigma}_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\sigma}_{x}$ $\displaystyle=$ $\displaystyle F_{x},$
$\displaystyle\widehat{\sigma}_{y}$ $\displaystyle=$ $\displaystyle F_{y},$
$\displaystyle\widehat{\sigma}_{xx}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\sigma}_{yy}$ $\displaystyle=$ $\displaystyle 0,$ (33)
$\displaystyle\widehat{\sigma}_{xy}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\sigma}_{xxy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xy}+\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xx},$
$\displaystyle\widehat{\sigma}_{xyy}$ $\displaystyle=$
$\displaystyle\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{yy}+2\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xy},$
$\displaystyle\widehat{\sigma}_{xxyy}$ $\displaystyle=$ $\displaystyle
2\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}+2\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}+\frac{F_{y}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{xx}+\frac{F_{x}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{yy}+4\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{xy}.$
Thus, higher order non-equilibrium effects in
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}$ and non-linear effect in
$F_{x}^{p}F_{y}^{q}$ are evident for the central moments of the source terms
that are third- and higher orders. Let us now explicitly write the central
moments of the transformed discrete Maxwellian by means of Eq. (32) using Eqs.
(29) and (33) to yield
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{x}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}F_{x},$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{y}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}F_{y},$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{xx}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{yy}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ (34)
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{xy}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{xxy}$
$\displaystyle=$
$\displaystyle-\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xy}-\frac{F_{y}}{2\rho}\widehat{\overline{\kappa}}_{xx},$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{xyy}$
$\displaystyle=$
$\displaystyle-\frac{F_{x}}{2\rho}\widehat{\overline{\kappa}}_{yy}-\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xy},$
$\displaystyle\widehat{\overline{\kappa}}^{\mathcal{M}}_{xxyy}$
$\displaystyle=$ $\displaystyle
c_{s}^{4}\rho-\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}-\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}-\frac{F_{y}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{xx}-\frac{F_{x}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{yy}-2\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{xy}.$
The main idea in the determination of the collision operator for the cascaded
version of the central moment method is to relax the central moments of the
transformed distribution function to its corresponding local attractor,
successively at various orders as given in Eq. (34) (see Sec. 10). Before
proceeding further to do this, we first need certain quantities in the rest or
lattice frame of reference, i.e. the raw moments, in which the computations
are actually performed. These are obtained in the next section.
## 9 Various Discrete Raw Moments and Source Terms in Particle Velocity Space
The raw moments, i.e. those in the rest frame of reference, can be related to
the central moments by means of the binomial theorem [85, 86]. For any state
variable $\varphi$ supported by the discrete particle velocity set, the
transformation relation between the two reference frames is thus given by [63]
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{\varphi}$ $\displaystyle=$ $\displaystyle\braket{e_{\alpha
x}^{m}e_{\alpha y}^{n}}{\varphi}+\braket{e_{\alpha
x}^{m}\left[\sum_{j=1}^{n}C^{n}_{j}e_{\alpha
y}^{n-j}(-1)^{j}u_{y}^{j}\right]}{\varphi}+$ (35)
$\displaystyle\braket{e_{\alpha y}^{n}\left[\sum_{i=1}^{m}C^{m}_{i}e_{\alpha
x}^{m-i}(-1)^{i}u_{x}^{i}\right]}{\varphi}+$
$\displaystyle\braket{\left[\sum_{i=1}^{m}C^{m}_{i}e_{\alpha
x}^{m-i}(-1)^{i}u_{x}^{i}\right]\left[\sum_{j=1}^{n}C^{n}_{j}e_{\alpha
y}^{n-j}(-1)^{j}u_{y}^{j}\right]}{\varphi}$
where $C^{p}_{q}=p!/(q!(p-q)!)$. We now define the following notations for
depicting various _discrete raw_ moments, based on which an operational LBE
will be devised later:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{f_{\alpha}},$ (36)
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}\overline{f}_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}}{\overline{f}_{\alpha}},$ (37)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{S_{\alpha}}.$ (38)
Note that the superscript “prime” (′) is used to distinguish the raw moments
from the central moments that are designated without the primes. Here,
analogous to Eq. (32), the relation
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}=\widehat{\kappa}_{x^{m}y^{n}}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$
holds. Let us first find expressions for
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}=\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}}$ to proceed further. As in [63], for convenience, we
define the following operator acting on the transformed distribution function
$\overline{f}_{\alpha}$ in this regard:
$a(\overline{f}_{\alpha_{1}}+\overline{f}_{\alpha_{3}}+\overline{f}_{\alpha_{3}}+\cdots)+b(\overline{f}_{\beta_{1}}+\overline{f}_{\beta_{2}}+\overline{f}_{\beta_{3}}+\cdots)+\cdots=\left(a\sum_{\alpha}^{A}+b\sum_{\alpha}^{B}\cdots\right)\otimes\overline{f}_{\alpha},$
(39)
where $A=\left\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots\right\\}$,
$B=\left\\{\beta_{1},\beta_{2},\beta_{3},\cdots\right\\}$,$\cdots$. For
conserved basis vectors, we write them in terms of the hydrodynamic variables
and force fields as
$\displaystyle\widehat{\overline{\kappa}}_{0}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{\rho}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}=\rho,$
(40) $\displaystyle\widehat{\overline{\kappa}}_{x}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}=\rho
u_{x}-\frac{1}{2}F_{x},$ (41)
$\displaystyle\widehat{\overline{\kappa}}_{y}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha y}=\rho
u_{y}-\frac{1}{2}F_{y},$ (42)
and, for the non-conserved basis vectors, using Eq. (39) in terms of subsets
of particle velocity directions as
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
x}^{2}=\left(\sum_{\alpha}^{A_{3}}\right)\otimes\overline{f}_{\alpha},$ (43)
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{4}}\right)\otimes\overline{f}_{\alpha},$ (44)
$\displaystyle\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha},$
(45) $\displaystyle\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)\otimes\overline{f}_{\alpha},$
(46) $\displaystyle\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)\otimes\overline{f}_{\alpha},$
(47) $\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{8}}\right)\otimes\overline{f}_{\alpha},$ (48)
where
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\left\\{1,3,5,6,7,8\right\\},$ $\displaystyle A_{4}$
$\displaystyle=$ $\displaystyle\left\\{2,4,5,6,7,8\right\\},$ $\displaystyle
A_{5}$ $\displaystyle=$
$\displaystyle\left\\{5,7\right\\},B_{5}=\left\\{6,8\right\\},$ $\displaystyle
A_{6}$ $\displaystyle=$
$\displaystyle\left\\{5,6\right\\},B_{6}=\left\\{7,8\right\\},$ $\displaystyle
A_{7}$ $\displaystyle=$
$\displaystyle\left\\{5,8\right\\},B_{7}=\left\\{6,7\right\\},$ $\displaystyle
A_{8}$ $\displaystyle=$ $\displaystyle\left\\{5,6,7,8\right\\}.$
Now, we transform the central moments of the source terms (Eq. (30)) to the
corresponding raw moments by considering Eq. (25) and using the frame
transformation relation (Eq. (35)). This yields
$\displaystyle\widehat{\sigma}_{0}^{{}^{\prime}}=\braket{S_{\alpha}}{\rho}=0,$
(49)
$\displaystyle\widehat{\sigma}_{x}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}}=F_{x},$ (50)
$\displaystyle\widehat{\sigma}_{y}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}}=F_{y},$ (51)
$\displaystyle\widehat{\sigma}_{xx}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}}=2F_{x}u_{x},$ (52)
$\displaystyle\widehat{\sigma}_{yy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}^{2}}=2F_{x}u_{y},$ (53)
$\displaystyle\widehat{\sigma}_{xy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}}=F_{x}u_{y}+F_{y}u_{x},$ (54)
$\displaystyle\widehat{\sigma}_{xxy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha
y}}=2F_{x}\left(\frac{\widehat{\overline{\kappa}}_{xy}}{\rho}+u_{x}u_{y}\right)+F_{y}\left(\frac{\widehat{\overline{\kappa}}_{xx}}{\rho}+u_{x}^{2}\right),$
(55)
$\displaystyle\widehat{\sigma}_{xyy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha
y}^{2}}=F_{x}\left(\frac{\widehat{\overline{\kappa}}_{yy}}{\rho}+u_{y}^{2}\right)+2F_{y}\left(\frac{\widehat{\overline{\kappa}}_{xy}}{\rho}+u_{x}u_{y}\right),$
(56)
$\displaystyle\widehat{\sigma}_{xxyy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha
y}^{2}}=2F_{x}u_{x}\left(\frac{\widehat{\overline{\kappa}}_{yy}}{\rho}+u_{y}^{2}\right)+2F_{y}u_{y}\left(\frac{\widehat{\overline{\kappa}}_{xx}}{\rho}+u_{x}^{2}\right)+$
$\displaystyle\frac{2F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}+\frac{2F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}+\frac{F_{x}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{yy}+\frac{F_{y}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{xx}+$
$\displaystyle
4\left[\frac{F_{x}u_{y}}{\rho}+\frac{F_{y}u_{x}}{\rho}+\frac{F_{x}F_{y}}{\rho^{2}}\right]\widehat{\overline{\kappa}}_{xy}.$
(57)
Clearly, the raw moments of source terms for third-order or higher contain
non-equilibrium and non-linear contributions. Eqs. (55)-(57) require explicit
expressions for central moments of transformed distributions such as
$\widehat{\overline{\kappa}}_{xx}$, $\widehat{\overline{\kappa}}_{yy}$,
$\widehat{\overline{\kappa}}_{xy}$, $\widehat{\overline{\kappa}}_{xxy}$ and
$\widehat{\overline{\kappa}}_{xyy}$, in terms of raw moments to facilitate
computation. They can be readily obtained in terms of raw moments from their
respective definitions and by using the binomial theorem (Eq. (35)) and
subsequent simplification as
$\displaystyle\widehat{\overline{\kappa}}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+F_{x}u_{x}-\rho
u_{x}^{2},$ (58) $\displaystyle\widehat{\overline{\kappa}}_{yy}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+F_{y}u_{y}-\rho
u_{y}^{2},$ (59) $\displaystyle\widehat{\overline{\kappa}}_{xy}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}+\frac{1}{2}(F_{x}u_{y}+F_{y}u_{x})-\rho
u_{x}u_{y},$ (60)
for second-order and
$\displaystyle\widehat{\overline{\kappa}}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\frac{1}{2}F_{x}u_{y}^{2}-F_{y}u_{x}u_{y}+2\rho
u_{x}u_{y}^{2},$ (61) $\displaystyle\widehat{\overline{\kappa}}_{xxy}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\frac{1}{2}F_{y}u_{x}^{2}-F_{x}u_{x}u_{y}+2\rho
u_{x}^{2}u_{y},$ (62)
for third-order moments. Based on the above, we now obtain the source terms
projected to the orthogonal moment basis vectors, i.e.
$\braket{K_{\beta}}{S_{\alpha}}$, $\beta=0,1,2,\ldots,8$, which would then
provide corresponding explicit expressions in terms of the particle velocity
space. Thus, from Eqs. (5) and (49)-(57), the following projected source
moments are derived:
$\displaystyle\widehat{m}^{s}_{0}=\braket{K_{0}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 0,$ (63)
$\displaystyle\widehat{m}^{s}_{1}=\braket{K_{1}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{x},$ (64)
$\displaystyle\widehat{m}^{s}_{2}=\braket{K_{2}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{y},$ (65)
$\displaystyle\widehat{m}^{s}_{3}=\braket{K_{3}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 6(F_{x}u_{x}+F_{y}u_{y}),$ (66)
$\displaystyle\widehat{m}^{s}_{4}=\braket{K_{4}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}-F_{y}u_{y}),$ (67)
$\displaystyle\widehat{m}^{s}_{5}=\braket{K_{5}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{y}+F_{y}u_{x}),$ (68)
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle-6F_{x}\left(\frac{\widehat{\overline{\kappa}}_{xy}}{\rho}+u_{x}u_{y}\right)-3F_{y}\left(\frac{\widehat{\overline{\kappa}}_{xx}}{\rho}+u_{x}^{2}-\frac{2}{3}\right),$
(69) $\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle-3F_{x}\left(\frac{\widehat{\overline{\kappa}}_{yy}}{\rho}+u_{y}^{2}-\frac{2}{3}\right)-6F_{y}\left(\frac{\widehat{\overline{\kappa}}_{xy}}{\rho}+u_{x}u_{y}\right),$
(70) $\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
18F_{x}u_{x}\left(\frac{\widehat{\overline{\kappa}}_{yy}}{\rho}+u_{y}^{2}-\frac{2}{3}\right)+18F_{y}u_{y}\left(\frac{\widehat{\overline{\kappa}}_{xx}}{\rho}+u_{x}^{2}-\frac{2}{3}\right)+$
(71) $\displaystyle
18\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}+18\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}+9\frac{F_{x}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{yy}+9\frac{F_{y}^{2}}{\rho^{2}}\widehat{\overline{\kappa}}_{xx}+$
$\displaystyle
36\left[\frac{F_{x}u_{y}}{\rho}+\frac{F_{y}u_{x}}{\rho}+\frac{F_{x}F_{y}}{\rho^{2}}\right]\widehat{\overline{\kappa}}_{xy}.$
In Eqs. (69)-(71), $\widehat{\overline{\kappa}}_{xx}$,
$\widehat{\overline{\kappa}}_{yy}$, $\widehat{\overline{\kappa}}_{xy}$,
$\widehat{\overline{\kappa}}_{xyy}$ and $\widehat{\overline{\kappa}}_{xxy}$
can be obtained from Eqs. (58)-(62), respectively. This can be written in
matrix form as
$\displaystyle\mathcal{K}^{T}\mathbf{S}=(\mathcal{K}\cdot\mathbf{S})_{\alpha}$
$\displaystyle=$
$\displaystyle(\braket{K_{0}}{S_{\alpha}},\braket{K_{1}}{S_{\alpha}},\braket{K_{2}}{S_{\alpha}},\ldots,\braket{K_{8}}{S_{\alpha}})$
(72) $\displaystyle=$
$\displaystyle(\widehat{m}^{s}_{0},\widehat{m}^{s}_{1},\widehat{m}^{s}_{2},\ldots,\widehat{m}^{s}_{8})^{T}\equiv\ket{\widehat{m}^{s}_{\alpha}}.$
Now, by exploiting the orthogonal property of $\mathcal{K}$ [63], i.e.
$\mathcal{K}^{-1}=\mathcal{K}^{T}\cdot D^{-1}$, where the diagonal matrix is
$D=\mbox{diag}(\braket{K_{0}}{K_{0}},\braket{K_{1}}{K_{1}},\braket{K_{2}}{K_{2}},\ldots,\braket{K_{8}}{K_{8}})=\mbox{diag}(9,6,6,36,4,4,12,12,36)$,
we exactly invert Eq. (72) to finally obtain source terms in velocity space
$S_{\alpha}$ as
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\left[-\widehat{m}^{s}_{3}+\widehat{m}^{s}_{8}\right],$
(73) $\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}-\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{4}+6\widehat{m}^{s}_{7}-2\widehat{m}^{s}_{8}\right],$
(74) $\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{4}+6\widehat{m}^{s}_{6}-2\widehat{m}^{s}_{8}\right],$
(75) $\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}-\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{4}-6\widehat{m}^{s}_{7}-2\widehat{m}^{s}_{8}\right],$
(76) $\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{4}-6\widehat{m}^{s}_{6}-2\widehat{m}^{s}_{8}\right],$
(77) $\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}+6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{5}-3\widehat{m}^{s}_{6}-3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(78) $\displaystyle S_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}+6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{5}-3\widehat{m}^{s}_{6}+3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(79) $\displaystyle S_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}-6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{5}+3\widehat{m}^{s}_{6}+3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(80) $\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}-6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{5}+3\widehat{m}^{s}_{6}-3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right].$
(81)
This is the explicit set of expressions for the source terms in velocity space
$S_{\alpha}$ given in terms of $\overrightarrow{F}$, $\overrightarrow{u}$ and
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}$, with $2\leq(m+n)\leq 3$ and $0\leq
m,n\leq 2$.
Again, using the orthogonal property of $\mathcal{K}$, we can obtain the raw
moments of the collision kernel
$\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}}\widehat{g}_{\beta},$ (82)
which is of central importance in the subsequent derivation. Note that for
collision invariants, $\widehat{g}_{0}=\widehat{g}_{1}=\widehat{g}_{2}=0$. We
get
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=\sum_{\beta}\braket{K_{\beta}}{\rho}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}+2\widehat{g}_{4},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}-2\widehat{g}_{4},$ (83)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 4\widehat{g}_{5},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{6},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{7},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{3}+4\widehat{g}_{8}.$
Finally, the LBE in Eq. (22) can be rewritten in terms of collision and
streaming steps, respectively, as
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t)$
$\displaystyle=$
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)},$
(84)
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)$
$\displaystyle=$
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t),$ (85)
where the symbol “tilde” ($\sim$) in the first equation refers to the post-
collision state. In terms of the transformed distribution, the hydrodynamic
fields can be computed by means of the following:
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}=\braket{\overline{f}_{\alpha}}{\rho},$
(86) $\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
i}+\frac{1}{2}F_{i}=\braket{\overline{f}_{\alpha}}{e_{\alpha
i}}+\frac{1}{2}F_{i},\qquad i\in{x,y}.$ (87)
## 10 Structure of the Collision Operator: Cascaded Central Moments
Let us now arrive at the expressions for the cascaded formulation of the
collision operator using central moments in the presence of forcing terms
based on the results obtained in the last few sections. The basic procedure
can be stated as follows. Beginning from the lowest order central moments that
are non-collisional invariants (i.e. $\widehat{\overline{\kappa}}_{xx}$ and
higher), they are successively set equal to their local attractors based on
the transformed Maxwellians (Eq. (34)). This step provides tentative
expressions for $\widehat{g}_{\alpha}$ based on the equilibrium assumption. We
then modify them to allow for relaxation during collision by multiplying them
with corresponding relaxation parameters [59]. In this step, given the
cascaded nature of the collision (i.e.
$\widehat{g}_{\alpha}\equiv\widehat{g}_{\alpha}(\mathbf{f},\widehat{g}_{\beta}),\beta=0,1,2,\ldots,\alpha-1$,
or the dependence of higher order terms on those that are lower orders), care
needs to be exercised to multiply the relaxation parameters only with those
terms that are not yet in post-collision states (i.e. terms not involving
$\widehat{g}_{\beta},\beta=0,1,2,\ldots,\alpha-1$ for $\widehat{g}_{\alpha}$).
Various details involved in this procedure are given in [63]. For brevity,
here we summarize the final results which are as follows:
$\displaystyle\widehat{g}_{3}$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{12}\left\\{\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(88) $\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{\omega_{4}}{4}\left\\{\rho(u_{x}^{2}-u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(89) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{\omega_{5}}{4}\left\\{\rho
u_{x}u_{y}-\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xy}^{{}^{\prime}}\right\\},$
(90) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{6}}{4}\left\\{2\rho
u_{x}^{2}u_{y}+\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xxy}\right\\}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4})$
(91) $\displaystyle-2u_{x}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{7}$
$\displaystyle=$ $\displaystyle\frac{\omega_{7}}{4}\left\\{2\rho
u_{x}u_{y}^{2}+\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xyy}\right\\}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4})$
(92) $\displaystyle-2u_{y}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{8}$
$\displaystyle=$
$\displaystyle\frac{\omega_{8}}{4}\left\\{\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-\left[\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}\right.\right.$
(93)
$\displaystyle\left.\left.+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}\right]-\frac{1}{2}\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right\\}-2\widehat{g}_{3}-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})$
$\displaystyle-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7}.$
In the above, the raw moments of various orders, i.e.
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}$ for different $m$ and
$n$ are required, which may be obtained from Eqs. (40)-(48). Similarly, the
raw moments of sources of various orders, i.e.
$\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$ needed in the above are given in
Eqs. (49)-(57) (see Sec. 9). Here, $\omega_{\beta}$, where
$\beta=3,4,5,\ldots,8$, are the relaxation parameters, satisfying
$0<\omega_{\beta}<2$. When a multiscale Chapman-Enskog expansion [64] is
applied to this central moment LBM based on central moments, it recovers the
Navier-Stokes equation with the relaxation parameters
$\omega_{3}=\omega^{\chi}$ and $\omega_{4}=\omega_{5}=\omega^{\nu}$
controlling bulk and shear viscosities, respectively (e.g.,
$\nu=c_{s}^{2}\left(\frac{1}{\omega^{\nu}}-\frac{1}{2}\right)$) [63]. The rest
of the parameters can be adjusted to improve numerical stability.
## 11 Cascaded Central Moment Lattice Boltzmann Equation
The collision step with the addition of forcing terms (see Eq. (19) and Eq.
(84)) in the stream-and-collide procedure of the LBM, obtained by matching
those of the continuous Boltzmann equation as discussed in the previous
sections, is expanded element-wise and can be summarized as follows:
$\displaystyle\widetilde{\overline{f}}_{0}$ $\displaystyle=$
$\displaystyle\overline{f}_{0}+\left[\widehat{g}_{0}-4(\widehat{g}_{3}-\widehat{g}_{8})\right]+S_{0},$
$\displaystyle\widetilde{\overline{f}}_{1}$ $\displaystyle=$
$\displaystyle\overline{f}_{1}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}+2(\widehat{g}_{7}-\widehat{g}_{8})\right]+S_{1},$
$\displaystyle\widetilde{\overline{f}}_{2}$ $\displaystyle=$
$\displaystyle\overline{f}_{2}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}+2(\widehat{g}_{6}-\widehat{g}_{8})\right]+S_{2},$
$\displaystyle\widetilde{\overline{f}}_{3}$ $\displaystyle=$
$\displaystyle\overline{f}_{3}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}-2(\widehat{g}_{7}+\widehat{g}_{8})\right]+S_{3},$
$\displaystyle\widetilde{\overline{f}}_{4}$ $\displaystyle=$
$\displaystyle\overline{f}_{4}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}-2(\widehat{g}_{6}+\widehat{g}_{8})\right]+S_{4},$
(94) $\displaystyle\widetilde{\overline{f}}_{5}$ $\displaystyle=$
$\displaystyle\overline{f}_{5}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}-\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{5},$
$\displaystyle\widetilde{\overline{f}}_{6}$ $\displaystyle=$
$\displaystyle\overline{f}_{6}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{6},$
$\displaystyle\widetilde{\overline{f}}_{7}$ $\displaystyle=$
$\displaystyle\overline{f}_{7}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{7},$
$\displaystyle\widetilde{\overline{f}}_{8}$ $\displaystyle=$
$\displaystyle\overline{f}_{8}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}+\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{8}.$
The collision kernel $\widehat{g}_{\beta}$ needed here can be computed from
the expressions given in the previous section (see Sec. 10). The source terms
in the velocity space can be obtained from Eqs. (73)-(81) (see Sec. 9). The
remaining streaming step is carried out as usual by using the post collision
values $\widetilde{\overline{f}}_{\alpha}$ obtained from above. Once the local
distribution function is known, macroscopic fluid density and velocity fields
satisfying the Galilean invariant Navier-Stokes equations in the presence of
force fields can be obtained from Eqs. (86) and (87), respectively.
## 12 Cascaded Collision Operator with Reduced Compressibility Effects
While a main goal of this work is the introduction of a self-consistent
approach based on the continuous Boltzmann equation to incorporate non-
equilibrium effects into the central moment approach for general
applicability, it is also useful to consider its limiting cases. For example,
the incompressible limit of fluid flow corresponds to considering very small
deviations from the local equilibrium, a special case with various
applications. In particular, this would allow simple representation of
incompressible turbulence considered later in this work.
Being a kinetic approach, the lattice Boltzmann method is inherently
compressible in nature. On the other hand, when it is desired to reproduce the
“incompressible” Navier-Stokes equations as mentioned above, it is important
to reduce such compressibility effects. An approach in this regard was
introduced earlier in [87]. Here, we will extend it further in the context of
the central moment LBM in the presence of forcing. It may be noted that the
fundamental expressions for the continuous central moments for the local
equilibrium as well as the forcing given in Secs. 4 and 5, respectively, from
which their discrete counterparts are derived, remains unchanged for this
case. However, the key element to incorporate a systematic reduction of
compressibility effects lies in the following careful definition of the raw
moments of the hydrodynamic fields:
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha},$
(95) $\displaystyle\rho_{0}\overrightarrow{u}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}\overrightarrow{e}_{\alpha}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}\overrightarrow{e}_{\alpha}+\frac{1}{2}\overrightarrow{F}.$
(96)
where $\rho=\rho_{0}+\delta\rho$. Here $\rho_{0}$ and $\delta\rho$ are the
constant reference value and fluctuations of density, respectively. That is,
in the above, contributions of density fluctuations are eliminated from first-
order moments representing the components of momentum. Thus, we get
$\displaystyle\braket{\widetilde{\overline{f}}_{\alpha}}{\rho}$
$\displaystyle=$ $\displaystyle\rho,$ (97)
$\displaystyle\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha x}}$
$\displaystyle=$ $\displaystyle\rho_{0}u_{x}+\frac{1}{2}F_{x},$ (98)
$\displaystyle\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha y}}$
$\displaystyle=$ $\displaystyle\rho_{0}u_{y}+\frac{1}{2}F_{y}.$ (99)
Using the procedure discussed in the previous sections and with the above
specialized re-definition of the conserved moments, we obtain, after some
simplification, the cascaded collision operator with reduced compressibility
effects. They are reported here in the following:
$\displaystyle\widehat{g}_{3}$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{12}\left\\{\frac{2}{3}\rho+(\rho_{0}-\delta\rho)(u_{x}^{2}+u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(100) $\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{\omega_{4}}{4}\left\\{(\rho_{0}-\delta\rho)(u_{x}^{2}-u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(101) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{\omega_{5}}{4}\left\\{(\rho_{0}-\delta\rho)u_{x}u_{y}-\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xy}^{{}^{\prime}}\right\\},$
(102) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{6}}{4}\left\\{(2\rho_{0}-\delta\rho)u_{x}^{2}u_{y}+\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xxy}\right\\}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4})$
(103) $\displaystyle-2u_{x}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{7}$
$\displaystyle=$
$\displaystyle\frac{\omega_{7}}{4}\left\\{(2\rho_{0}-\delta\rho)u_{x}u_{y}^{2}+\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xyy}\right\\}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4})$
(104) $\displaystyle-2u_{y}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{8}$
$\displaystyle=$
$\displaystyle\frac{\omega_{8}}{4}\left\\{\frac{1}{9}\rho+(3\rho_{0}-\delta\rho)u_{x}^{2}u_{y}^{2}-\left[\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}\right.\right.$
(105)
$\displaystyle\left.\left.+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}\right]-\frac{1}{2}\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right\\}-2\widehat{g}_{3}-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})$
$\displaystyle-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7}.$
The above collision operator that selectively introduces density fluctuations
where necessary can reduce compressibility effects for a inertial frame
invariant flow field while retaining its linear acoustics. It can thus allow
for a better comparison with “incompressible” macroscopic fluid dynamic
equations, particularly for turbulent flows as discussed later.
## 13 Factorized Central Moment Model for Collision
In this section, we will derive an alternative form of the central moment LBE
with forcing terms based on a different choice of the local attractor in the
collision step for improved physical coherence. Continuous kinetic theory, as
originally initiated by Maxwell [88], features two important properties for
the local equilibrium or the Maxwell distribution – Galilean invariance and
factorization in Cartesian components of the particle velocity. As discussed
recently [62, 39], it could prove useful to inherent these properties at the
discrete particle velocity level. The use of central moments maintains
Galilean invariance by construction. Factorization property of the
distribution function implies that particle velocities are random variables.
An extension of the factorization idea beyond equilibrium as a model for
describing the relaxation process during collision was proposed to construct
local attractors [62]. Specifically, the basic postulate behind this model is
that the Cartesian products of the post-collision values of the orthogonal
central moments of lower orders that are not in equilibrium forms as the basis
for the attractors of the higher order moments. Here, we further extend this
to include source terms so that the model can incorporate force fields. Thus,
the attractors for central moments of different orders are given as
$\displaystyle\widehat{\kappa}_{x}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{x}=0,$ (106)
$\displaystyle\widehat{\kappa}_{y}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{y}=0,$ (107)
$\displaystyle\widehat{\kappa}_{xy}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{x}\widetilde{\widehat{\kappa}}_{y}=0,$
(108) $\displaystyle\widehat{\kappa}_{xxy}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{y}=0,$
(109) $\displaystyle\widehat{\kappa}_{xyy}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{x}\widetilde{\widehat{\kappa}}_{yy}=0,$
(110) $\displaystyle\widehat{\kappa}_{xxyy}^{at}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy},$
(111)
while the second-order longitudinal central moments are obtained from
Maxwellian as given in an earlier section (see Sec. 7), i.e.
$\widehat{\kappa}_{xx}^{at}=\widehat{\kappa}_{yy}^{at}=\rho c_{s}^{2}$. In
essence, the distinguishing feature of the factorized central moment lies in
the use of modified attractors for third and higher order moments. Now, using
the following central moment identity of the post-collision state
$\widetilde{\widehat{\kappa}}_{x^{m}y^{n}}=\widehat{\kappa}_{x^{m}y^{n}}+\sum_{\beta}\braket{K_{\alpha}}{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}\widehat{g}_{\beta}$, for $m=2,n=0$ and
$m=0,n=2$, we get
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}_{xx}+(6\widehat{g}_{3}+2\widehat{g}_{4}),$
(112) $\displaystyle\widetilde{\widehat{\kappa}}_{yy}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}_{yy}+(6\widehat{g}_{3}-2\widehat{g}_{4}).$
(113)
Note that it also follows that
$\widehat{\kappa}_{xx}=\widehat{\overline{\kappa}}_{xx}$ and
$\widehat{\kappa}_{yy}=\widehat{\overline{\kappa}}_{yy}$. We can then rewrite
everything in terms of transformed raw moments, i.e.
$\widehat{\overline{\kappa}}_{xx}=\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}+F_{x}u_{x}$ and
$\widehat{\overline{\kappa}}_{yy}=\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}+F_{y}u_{y}$. These yield
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}+F_{x}u_{x}+(6\widehat{g}_{3}+2\widehat{g}_{4}),$ (114)
$\displaystyle\widetilde{\widehat{\kappa}}_{yy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}+F_{y}u_{y}+(6\widehat{g}_{3}-2\widehat{g}_{4}).$ (115)
In effect, the attractor for the fourth-order moment, i.e. Eq. (111) reduces
to
$\displaystyle\widehat{\kappa}_{xxyy}^{at}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}+F_{x}u_{x}+(6\widehat{g}_{3}+2\widehat{g}_{4})\right]\times$ (116)
$\displaystyle\left[\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}+F_{y}u_{y}+(6\widehat{g}_{3}-2\widehat{g}_{4})\right].$
Now, to obtain an operational step in terms of the transformed variables, we
use the relation
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{at}=\widehat{\kappa}_{x^{m}y^{n}}^{at}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}$
to finally get the following expression for the fourth-order central moment
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{at}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}+F_{x}u_{x}+(6\widehat{g}_{3}+2\widehat{g}_{4})\right]\left[\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}+F_{y}u_{y}+(6\widehat{g}_{3}-2\widehat{g}_{4})\right]$ (117)
$\displaystyle-\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}-\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}-\frac{F_{y}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{xx}-\frac{F_{x}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{yy}-2\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{xy}.$
By replacing $\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{\mathcal{M}}$ with
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{at}$ as given above, we can derive
the collision kernel with factorized central moments as attractors. It follows
that the expressions in Sec. 10 for $\widehat{g}_{\beta}$, $\beta=3,4,5,6,7$
are the same as before with the exception for $\widehat{g}_{8}$. The
expression for $\widehat{g}_{8}$ in Eq. (93) is modified such that term
$\frac{1}{9}\rho(=\widehat{\overline{\kappa}}_{xxyy}^{\mathcal{M}})$ in this
equation is now replaced by $\widehat{\overline{\kappa}}_{xxyy}^{at}$ given in
Eq. (117). In a similar vein, the above expression can be modified for reduced
compressibility effects (see Sec. 12) as
$\displaystyle\widehat{\kappa}_{xxyy}^{at}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-(\rho_{0}-\delta\rho)u_{x}^{2}+F_{x}u_{x}+(6\widehat{g}_{3}+2\widehat{g}_{4})\right]\times$
(118)
$\displaystyle\left[\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-(\rho_{0}-\delta\rho)u_{y}^{2}+F_{y}u_{y}+(6\widehat{g}_{3}-2\widehat{g}_{4})\right]$
$\displaystyle-\frac{F_{x}}{\rho}\widehat{\overline{\kappa}}_{xyy}-\frac{F_{y}}{\rho}\widehat{\overline{\kappa}}_{xxy}-\frac{F_{y}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{xx}-\frac{F_{x}^{2}}{2\rho^{2}}\widehat{\overline{\kappa}}_{yy}-2\frac{F_{x}F_{y}}{\rho^{2}}\widehat{\overline{\kappa}}_{xy}$
to modify $\widehat{g}_{8}$ in Eq. (105).
## 14 Galilean Invariant Filtered Turbulence Representation using Lattice
Kinetic Framework
Based on the various elements derived in the previous sections, we are now in
a position to construct an approach for simulation of Galilean invariant
turbulent flow field by incorporating appropriate turbulence models in the
LBM. The starting point in the statistical continuum description of turbulence
is the Reynolds decomposition of the velocity field of the fluid into
‘resolved’ and ‘unresolved’ parts. The resolved part is obtained by applying
either some averaging in space or time (in the Reynolds Averaged Navier-Stokes
(RANS) context) or by applying a filter (in the LES). Application of this
decomposition to the Navier-Stokes (NS) equation leads to additional unknown
terms involving products of the unresolved fields, which are Reynolds stresses
(in RANS) or the subgrid stresses (in LES). This closure problem then becomes
the main focus of turbulence modeling. Due to the scale invariance property of
the NS equations [83], the averaged and the filtered equations, as well as the
additional stress-like closure terms have similar forms. Thus, a unified
statistical approach may be adopted for turbulence modeling. It is interesting
to note that ideas based on kinetic theory provided the original inspiration
for the Reynolds decomposition [89] as well as early works on developing
turbulence models.
The underlying motivation here is to develop a unified statistical averaged
description (for RANS) or formal spatial filtered representation (for LES) of
_inertial frame invariant_ turbulence in a kinetic approach based on the LBM
derived in earlier sections. This would also allow reconciliation of continuum
and non-continuum effects on turbulence as discussed recently [77]. The
following notation for Reynolds decomposition is adopted here. For any scalar
$\phi$, vector $\overrightarrow{v}$ and tensor $T_{ij}$, we have
$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\underline{\phi}+\phi^{{}^{\prime}},\quad\mbox{with}\quad\underline{\phi^{{}^{\prime}}}=0,$
$\displaystyle\overrightarrow{v}$ $\displaystyle=$
$\displaystyle\underline{\overrightarrow{v}}+\overrightarrow{v}^{{}^{\prime}},\quad\mbox{with}\quad\underline{\overrightarrow{v}^{{}^{\prime}}}=0,$
$\displaystyle T_{ij}$ $\displaystyle=$
$\displaystyle\underline{T_{ij}}+T_{ij}^{{}^{\prime}},\quad\mbox{with}\quad\underline{T_{ij}^{{}^{\prime}}}=0,$
where $\underline{(\cdot)}$ is an operator representing either some form of
statistical average or filter to obtain the resolved part and the symbols with
primes denote the unresolved parts of the field. As discussed in [77],
application of the above decomposition directly to the continuous Boltzmann
equation (Eq. (6)) leads to certain difficulties. In particular, using
$f=\underline{f}+f^{{}^{\prime}}$ for the distribution function in Eq. (6),
which leads to a statistically averaged kinetic equation, does not provide a
clear interpretation of turbulence physics. The local collision term needs to
model all essential physics, including the non-linear and non-local momentum
transfer effects of turbulence. Moreover, the use of the averaged attractor
based on the Maxwellian $\underline{f^{\mathcal{M}}}$ within the collision
term $\underline{\Omega(f,f)}$ leads to modeling difficulties since
$\underline{\exp\left[-\frac{(\overrightarrow{\xi}-\overrightarrow{u})^{2}}{2c_{s}^{2}}\right]}\neq\exp\left[-\frac{(\overrightarrow{\xi}-\underline{\overrightarrow{u}})^{2}}{2c_{s}^{2}}\right]$.
Thus, an alternative approach is needed to coherently represent
continuum/kinetic effects on turbulence.
To circumvent these issues, a transformation for the velocities was recently
suggested [77], which is adopted here to provide Galilean invariant turbulence
representation. The key element is to clearly separate the advective
turbulence effects due to unresolved velocity field
$\overrightarrow{u}^{{}^{\prime}}$ from the dissipative collision that
represent microscopic effects. This is accomplished by an inspection of the
local Maxwellian given in terms of the microscopic particle velocity
$\overrightarrow{\xi}$ and the macroscopic fluid velocity
$\overrightarrow{u}$. That is, it consists of the term involving the peculiar
velocity
$\overrightarrow{\xi}-\overrightarrow{u}$
as its argument which should be made independent of the unresolved part of the
macroscopic fluid velocity $\overrightarrow{u}^{{}^{\prime}}$, when the
averaging operator is applied. That is,
$\overrightarrow{\xi}-(\underline{\overrightarrow{u}}+\overrightarrow{u}^{{}^{\prime}})=(\overrightarrow{\xi}-\overrightarrow{u}^{{}^{\prime}})-\underline{\overrightarrow{u}}$
should be transformed appropriately, which can be accomplished by defining a
new variable $\overrightarrow{\eta}$ as
$\overrightarrow{\eta}=\overrightarrow{\xi}-\overrightarrow{u}^{{}^{\prime}}.$
(119)
Now, the Maxwellian in the transformed peculiar velocity
$\overrightarrow{\eta}-\underline{\overrightarrow{u}}$ commutes with the
operator for averaging or filtering. That is,
$\underline{\exp\left[-\frac{(\overrightarrow{\eta}-\overrightarrow{u})^{2}}{2c_{s}^{2}}\right]}=\exp\left[-\frac{(\overrightarrow{\eta}-\underline{\overrightarrow{u}})^{2}}{2c_{s}^{2}}\right].$
This facilitates the separation of various aspects of turbulence physics
modeling. Based on this a new distribution function
$h(\overrightarrow{x},\overrightarrow{\eta},t)$ and its local Maxwellian are
defined by
$h(\overrightarrow{x},\overrightarrow{\eta},t)=f(\overrightarrow{x},\overrightarrow{\xi},t),$
(120)
and
$h^{\mathcal{M}}(\overrightarrow{\eta},\overrightarrow{u})=\frac{\rho}{2\pi
c_{s}^{2}}\exp\left[-\frac{(\overrightarrow{\eta}-\underline{\overrightarrow{u}})^{2}}{2c_{s}^{2}}\right],$
(121)
respectively. The continuous Boltzmann equation, i.e. Eq. (6) (without the
forcing term for simplicity) is then transformed into a modified kinetic
equation in terms of $\eta$ and $h$ as follows. From Eq. (120),
$\overrightarrow{\nabla}_{\eta}h=\overrightarrow{\nabla}_{\xi}f$. When
$\eta=\mbox{constant}$, we have
$(\overrightarrow{\nabla}_{x}\overrightarrow{\xi})_{\eta}=\overrightarrow{\nabla}_{x}\overrightarrow{u}^{{}^{\prime}}$
and
$(\partial_{t}\overrightarrow{\xi})_{\eta}=\partial_{t}\overrightarrow{u}^{{}^{\prime}}$.
Hence, the derivatives in new variables are
$\displaystyle\overrightarrow{\nabla}_{x}h$ $\displaystyle=$
$\displaystyle\overrightarrow{\nabla}_{x}f+(\overrightarrow{\nabla}_{x}\overrightarrow{\xi})_{\eta}\cdot\overrightarrow{\nabla}_{\xi}f=\overrightarrow{\nabla}_{x}f+(\overrightarrow{\nabla}_{x}\overrightarrow{u}^{{}^{\prime}})_{\eta}\cdot\overrightarrow{\nabla}_{\eta}h,$
$\displaystyle\partial_{t}h$ $\displaystyle=$
$\displaystyle\partial_{t}f+(\partial_{t}\overrightarrow{\xi})_{\eta}\cdot\overrightarrow{\nabla}_{\xi}f=\partial_{t}f+(\partial_{t}\overrightarrow{u}^{{}^{\prime}})_{\eta}\cdot\overrightarrow{\nabla}_{\eta}h.$
The continuous Boltzmann equation is thus modified to [77]
$\partial_{t}h+\overrightarrow{\eta}\cdot\overrightarrow{\nabla}_{x}h+\overrightarrow{u}^{{}^{\prime}}\cdot\overrightarrow{\nabla}_{x}h-\overrightarrow{a}^{{}^{\prime}}\cdot\overrightarrow{\nabla}_{\eta}h=\Omega(h,h),$
(122)
where
$\overrightarrow{a}^{{}^{\prime}}=\partial_{t}\overrightarrow{u}^{{}^{\prime}}+\eta\cdot\overrightarrow{\nabla}_{x}\overrightarrow{u}^{{}^{\prime}}+\overrightarrow{u}^{{}^{\prime}}\cdot\overrightarrow{\nabla}_{x}\overrightarrow{u}^{{}^{\prime}}$.
Considering incompressible flows, where the unresolved velocity field
satisfies $\overrightarrow{\nabla}\cdot\overrightarrow{u}^{{}^{\prime}}=0$, we
get
$\overrightarrow{u}^{{}^{\prime}}\cdot\overrightarrow{\nabla}_{x}h=\overrightarrow{\nabla}_{x}(h\overrightarrow{u}^{{}^{\prime}})$
and
$\overrightarrow{a}^{{}^{\prime}}\cdot\overrightarrow{\nabla}_{x}h=\overrightarrow{\nabla}_{x}(h\overrightarrow{a}^{{}^{\prime}})$.
As a result, Eq. (122) is further simplified to
$\partial_{t}h+\overrightarrow{\eta}\cdot\overrightarrow{\nabla}_{x}h+\overrightarrow{\nabla}_{x}(h\overrightarrow{u}^{{}^{\prime}})-\overrightarrow{\nabla}_{x}(h\overrightarrow{a}^{{}^{\prime}})=\Omega(h,h).$
(123)
Now, applying the statistical averaging or filtering operator on Eq. (123) and
using $\underline{\Omega(h,h)}=\Omega(\underline{h},\underline{h})$, we get
the new kinetic equation
$\partial_{t}\underline{h}+\overrightarrow{\eta}\cdot\overrightarrow{\nabla}_{x}\underline{h}+\overrightarrow{\nabla}_{x}(\underline{h\overrightarrow{u}^{{}^{\prime}}})-\overrightarrow{\nabla}_{x}(\underline{h\overrightarrow{a}^{{}^{\prime}}})=\Omega(\underline{h},\underline{h}).$
(124)
The averaged density and momentum can then be obtained by taking moments of
$\overline{h}$. That is,
$\underline{\rho}=\int\underline{h}d\overrightarrow{\eta},\underline{\rho\overrightarrow{u}}=\int\underline{h}\overrightarrow{\eta}d\overrightarrow{\eta}.$
Now, $\underline{h(\eta)}$ using Eq. (124) is better suited to represent
turbulence physics for the following reasons. The term involving
$\underline{h\overrightarrow{u}^{{}^{\prime}}}$ represents transport in
physical space, i.e. redistributes $\underline{h}$ to smoothen any gradients.
$\underline{h\overrightarrow{a}^{{}^{\prime}}}$ represents transport in
velocity phase space, and acts as a source/sink for energy cascade [77]. In
particular, both these quantities can be directly related to continuum based
closure models. On the other hand, the role of collision operator is then to
simply represent averaged effect of irreversible molecular collisions. The
averaged kinetic equation, Eq. (124), can be further simplified by considering
the following simple microscopic closure [77]
$\displaystyle\underline{h\overrightarrow{u}^{{}^{\prime}}}$
$\displaystyle\approx$
$\displaystyle\underline{h}\thinspace\thinspace\underline{\overrightarrow{u}^{{}^{\prime}}}=0,$
(125) $\displaystyle\underline{h\overrightarrow{a}^{{}^{\prime}}}$
$\displaystyle\approx$
$\displaystyle\underline{h}\thinspace\thinspace\underline{\overrightarrow{a}^{{}^{\prime}}}=\underline{h}\overrightarrow{\nabla}_{x}\cdot(\underline{\overrightarrow{u}^{{}^{\prime}}\overrightarrow{u}^{{}^{\prime}}})$
(126)
That is, $\underline{h}$ is uncorrelated with both
$\underline{u}^{{}^{\prime}}_{i}$ and $\underline{a}^{{}^{\prime}}_{i}$, which
reproduces the averaged momentum equations with additional Reynolds stress
terms $\underline{u_{i}^{{}^{\prime}}u_{j}^{{}^{\prime}}}$ that can be closed
by means of any conventional macroscopic turbulence models. Thus, Eq. (124)
can now be rewritten as
$\partial_{t}\underline{h}+\overrightarrow{\eta}\cdot\overrightarrow{\nabla}_{x}\underline{h}=\Omega(\underline{h},\underline{h})+\overrightarrow{\nabla}_{x}\cdot(\underline{\overrightarrow{u}^{{}^{\prime}}\overrightarrow{u}^{{}^{\prime}}})\cdot\overrightarrow{\nabla}_{\eta}\underline{h},$
(127)
which represents the evolution of the statistical averaged/filtered turbulence
field by means of the Reynolds stresses that appear as a forcing term in a
kinetic framework.
Let us now develop a Galilean invariant lattice kinetic equation, i.e. which
provides inertial frame invariant representation with respect to the
_resolved_ velocity field obtained by statistical averaging/filtering. For
brevity and to avoid the use of additional new notations, let us rewrite Eq.
(127) by replacing $\underline{h}$ by $\underline{f}$ (and $\overline{\eta}$
and $\overline{\xi}$) to make use of the developments of the previous
sections. That is,
$\partial_{t}\underline{f}+\overrightarrow{\xi}\cdot\overrightarrow{\nabla}_{x}\underline{f}=\Omega(\underline{f},\underline{f})+\overrightarrow{\nabla}_{x}\cdot(\underline{\overrightarrow{u}^{{}^{\prime}}\overrightarrow{u}^{{}^{\prime}}})\cdot\overrightarrow{\nabla}_{\xi}\underline{f}.$
(128)
from which the resolved hydrodynamic fields can be defined as follows:
$\underline{\rho}=\int\underline{f}d\overrightarrow{\xi},\qquad\underline{\rho\overrightarrow{u}}=\int\underline{f}\overrightarrow{\xi}d\overrightarrow{\xi}.$
(129)
Now, we define operator averaged continuous central moments as
$\underline{\widehat{\Pi}}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\underline{f}(\xi_{x}-\underline{u_{x}})^{m}(\xi_{y}-\underline{u_{y}})^{n}d\xi_{x}d\xi_{y}$
(130)
and similarly for the continuous central moments of the Maxwellian
$\underline{\widehat{\Pi}}_{x^{m}y^{n}}^{\mathcal{M}}$ based on replacing
$h^{\mathcal{M}}$ and $\overrightarrow{\eta}$ by $f^{\mathcal{M}}$ and
$\overrightarrow{\xi}$, respectively. The Cartesian components of the
unresolved turbulent Reynolds stresses may be written as
$\displaystyle\underline{a_{x}^{{}^{\prime}}}$ $\displaystyle=$
$\displaystyle-\partial_{x}(\underline{u_{x}^{{}^{\prime}}u_{x}^{{}^{\prime}}})-\partial_{y}(\underline{u_{x}^{{}^{\prime}}u_{y}^{{}^{\prime}}}),$
(131) $\displaystyle\underline{a_{y}^{{}^{\prime}}}$ $\displaystyle=$
$\displaystyle-\partial_{x}(\underline{u_{x}^{{}^{\prime}}u_{y}^{{}^{\prime}}})-\partial_{y}(\underline{u_{y}^{{}^{\prime}}u_{y}^{{}^{\prime}}}),$
(132)
where
$\underline{\overrightarrow{a}^{{}^{\prime}}}=(\underline{a_{x}^{{}^{\prime}}},\underline{a_{y}^{{}^{\prime}}})$,
from which we analogously define a source/sink continuous central moment as
$\underline{\widehat{\Gamma}^{a}_{x^{m}y^{n}}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\underline{\delta
f^{a^{{}^{\prime}}}}(\xi_{x}-\underline{u_{x}})^{m}(\xi_{y}-\underline{u_{y}})^{n}d\xi_{x}d\xi_{y}.$
(133)
Here, $\underline{\delta
f^{a^{{}^{\prime}}}}=-\underline{\overrightarrow{a}^{{}^{\prime}}}\cdot\overrightarrow{\nabla}_{\xi}\underline{f}$.
It readily follows from Eq. (16) that
$\underline{\widehat{\Gamma}}^{a}_{x^{m}y^{n}}$ also satisfies the following
exact identity
$\underline{\widehat{\Gamma}}^{a}_{x^{m}y^{n}}=ma_{x}^{{}^{\prime}}\underline{\widehat{\Pi}}_{x^{m-1}y^{n}}+na_{y}^{{}^{\prime}}\underline{\widehat{\Pi}}_{x^{m}y^{n-1}}$.
That is, the statistical averaged/filtered central moment of sources/sinks due
to unresolved fields of a given order is dependent on the product of the
Cartesian components of the gradients of turbulent stresses with the lower
order central moments of the averaged/filtered distribution function. The
corresponding discrete central moment LBM can be devised by considering the
following averaged representation of discrete vectors supported by the
particle velocity set:
$\mathbf{\underline{f}}=\ket{\underline{f_{\alpha}}}=(\underline{f}_{0},\underline{f}_{1},\underline{f}_{2},\ldots,\underline{f}_{8})^{T}$,
$\mathbf{\widehat{\underline{g}}}=\ket{\widehat{\underline{g}}_{\alpha}}=(\widehat{\underline{g}}_{0},\widehat{\underline{g}}_{1},\widehat{\underline{g}}_{2},\ldots,\widehat{\underline{g}}_{8})^{T}$,
$\mathbf{\underline{S}}=\ket{\underline{S_{\alpha}}}=(\underline{S}_{0},\underline{S}_{1},\underline{S}_{2},\ldots,\underline{S}_{8})^{T}$,
and
$\bm{\underline{\Omega}}^{c}\equiv\bm{\underline{\Omega}}^{c}(\underline{\mathbf{f}},\mathbf{\underline{\widehat{g}}})=(\mathcal{K}\cdot\mathbf{\underline{\widehat{g}}})=(\underline{\Omega}_{0}^{c},\underline{\Omega}_{1}^{c},\underline{\Omega}_{2}^{c},\ldots,\underline{\Omega}_{8}^{c})^{T}$,
and invoking Galilean invariance matching principle, i.e. matching the
continuous and discrete central moments of various quantities at successively
higher orders as discussed in earlier sections. In particular, the statistical
averaged/filtered discrete collision operator $\bm{\underline{\Omega}}^{c}$
can be obtained by considering reduced compressibility effects and factorized
attractors as in Sec. 13. Furthermore, the corresponding source terms in
velocity space $\mathbf{\underline{S}}$ can be constructed using the procedure
outlined in Sec. 9. The operator averaged LBE, in terms of the transformed
distribution function $\underline{\overline{f}}_{\alpha}$ for improved
accuracy, can be finally written as
$\underline{\overline{f}}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=\underline{\overline{f}}_{\alpha}(\overrightarrow{x},t)+\underline{\Omega}_{{\alpha}(\overrightarrow{x},t)}^{c}+\underline{S}_{{\alpha}(\overrightarrow{x},t)},$
(134)
where
$\underline{\overline{f}}_{\alpha}=\underline{f}_{\alpha}-\frac{1}{2}\underline{S}_{\alpha}$.
Here, as before, we have adopted the standard discretization for the streaming
step (see the comment following Eq. (22)). The resolved hydrodynamic fields in
the reduced compressibility formulation can then be obtained as
$\displaystyle\underline{\rho}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\underline{\overline{f}}_{\alpha}=\braket{\underline{\overline{f}}_{\alpha}}{\rho},$
(135) $\displaystyle\rho_{0}\underline{u}_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\underline{\overline{f}}_{\alpha}e_{\alpha
i}+\frac{1}{2}\underline{\rho}\thinspace\thinspace\underline{a^{{}^{\prime}}_{i}}=\braket{\underline{\overline{f}}_{\alpha}}{e_{\alpha
i}}+\frac{1}{2}\underline{\rho}\thinspace\thinspace\underline{a^{{}^{\prime}}_{i}}.\qquad
i\in{x,y}$ (136)
This provides a minimal lattice kinetic equation for incorporating turbulence
models, where the unresolved turbulent motion are inertial frame invariant
with respect to the resolved fluid motion. Here, we clarify the meaning of
this statement as follows. Unlike other areas in fluid mechanics, where models
have been developed starting from continuous kinetic theory, its role for
fluid turbulence has been more limited. This is mainly due to the fact that
kinetic theory generally considers distinct scale separation of physical
processes. On the other hand, turbulence is a flow phenomenon intrinsically
containing a continuous spectrum of scales with no scale separation. As such,
therefore, turbulence modeling developments have to rely much on phenomenology
whose mathematical forms are then constrained by invariance principles (e.g.
material frame indifference and inertial frame invariance mentioned earlier in
the introduction) and realizability considerations [81, 90]. Thus, except for
some early models such as those based on mixing length concepts and derivation
of some recent phenomenological models (e.g. [54]) based on kinetic theory,
turbulence modeling developments are generally based on macroscopic models.
The ultimate goal of our central moment approach for the filtered kinetic
equation discussed above, is, then, to simulate resolved turbulent fields
which are inertial frame independent, when an appropriate macroscopic
turbulence model for the unresolve field is used in the forcing term.
## 15 Summary and Conclusions
A discrete lattice kinetic model for the continuous Boltzmann equation,
including forcing, based on central moments is derived. The collision operator
as well as the source term of this lattice Boltzmann equation is constructed
by matching the corresponding continuous and discrete central moments
successively at various orders. The local attractor of the collision operator
is constructed to satisfy the factorization property of the Maxwellian during
relaxation process. An exact hierarchical identity of the central moment of
sources, that incorporates non-equilibrium effects, is maintained at the
discrete level. The resulting approach provides Galilean invariant
hydrodynamic fields in the presence of any external or self-consistent
internal forces in a discrete kinetic framework. It is further extended to
incorporate reduced compressibility effects for better representation of
incompressible flow, a limiting case. An important physical characteristic of
turbulent flows is that it is inertial frame independent for all or any subset
of scales. In consequence, for general applicability, all turbulence models
and their simulation approaches, should satisfy this requirement. A
statistical averaged/filtered lattice kinetic equation based on central
moments that maintains Galilean invariant representation of unresolved fluid
motion with respect to the resolved fields of turbulent flow is thus
constructed. The formalism presented here can extended to other lattice
velocity sets and in three-dimensions as well as to other physical problems
such as complex fluids.
In this regard, we make the following remark on the development of more
efficient schemes for the former aspect. Symmetry and finiteness of the
standard lattice sets lead to degeneracies of higher order moments in terms
those at lower orders that can result in frame dependent contributions to
viscous stresses. This necessitates considerations of extended lattice sets.
In this case, it is proposed that the _central moments_ relaxation (as well as
forcing) be considered _only_ up to those higher order moments that have
bearing on the physics of _hydrodynamics_ , such as stress tensors and heat
flux vectors. In turn, this would impose Galilean invariance of the
macroscopic description of the fluid motion. The relaxation of the rest of the
higher moments (including forcing) related to the fast _kinetic or ghost
modes_ can be considered in terms of the standard or _raw moments_. Here, the
form of the hierarchical identity for the sources derived in this paper for
those higher (kinetic) moments would be the same with the simple replacement
of the central moment terms by the corresponding raw moments. It is envisaged
that such mixed central/raw moment approach would exhibit interesting
mathematical structures for the resulting collision operator and the sources,
as well as being computationally more effective. This strategy is currently
under investigation.
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|
arxiv-papers
| 2012-02-27T21:19:30 |
2024-09-04T02:49:27.898774
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kannan N. Premnath and Sanjoy Banerjee",
"submitter": "Kannan Premnath",
"url": "https://arxiv.org/abs/1202.6074"
}
|
1202.6081
|
# On the Three-dimensional Central Moment Lattice Boltzmann Method
Kannan N. Premnath knandhap@uwyo.edu Sanjoy Banerjee Department of
Mechanical Engineering, University of Wyoming, Laramie, WY 82071 Department
of Chemical Engineering, City College of New York, City University of New
York, New York, NY 10031 Corresponding author.
###### Abstract
A three-dimensional (3D) lattice Boltzmann method based on central moments is
derived. Two main elements are the local attractors in the collision term and
the source terms representing the effect of external and/or self-consistent
internal forces. For suitable choices of the orthogonal moment basis for the
three-dimensional, twenty seven velocity (D3Q27), and, its subset, fifteen
velocity (D3Q15) lattice models, attractors are expressed in terms of
factorization of lower order moments as suggested in an earlier work; the
corresponding source terms are specified to correctly influence lower order
hydrodynamic fields, while avoiding aliasing effects for higher order moments.
These are achieved by successively matching the corresponding continuous and
discrete central moments at various orders, with the final expressions written
in terms of raw moments via a transformation based on the binomial theorem.
Furthermore, to alleviate the discrete effects with the source terms, they are
treated to be temporally semi-implicit and second-order, with the implicitness
subsequently removed by means of a transformation. As a result, the approach
is frame-invariant by construction and its emergent dynamics describing fully
3D fluid motion in the presence of force fields is Galilean invariant.
Numerical experiments for a set of benchmark problems demonstrate its
accuracy.
###### keywords:
Lattice Boltzmann Method , Central moments , Galilean invariance
###### PACS:
05.20.Dd , 47.11.-j
## 1 Introduction
The use of discrete velocity models based on kinetic theory is a powerful
theoretical approach and forms the basis of a modern computational method for
fluid mechanics. While the work of Broadwell [1] represents an early effort in
this direction, careful exploitation of symmetries and local conservation laws
to construct such models for discrete configuration spaces underpinned the
recent approaches, starting from the work of Frisch _et al_ [2]. The latter
led to the development of the lattice Boltzmann method (LBM) [3], albeit
without any direct connection to kinetic theory in its initial stages. Indeed,
formal demonstration of this approach as a simplified model for the continuous
Boltzmann equation [4, 5, 6], provided much impetus for recent developments,
particularly for complex fluids [7, 8, 9] and for representation beyond
continuum description [10], among others (see [11, 12, 13, 14] for general
reviews on the LBM).
The basic procedure involved in the LBM is represented by the synchronous
free-streaming of particle distribution functions along discrete directions
followed by collision, represented as a relaxation process. The latter has
major influence on the physical fidelity as well as numerical stability. A
popular approach is based on the single-relaxation time (SRT) model [15, 16].
While it is successful in many applications, it is prone to numerical
instability for situations with relatively low viscosities and is inadequate
for representing certain physical phenomena (e.g. viscoelasticity and thermal
transport) and in correctly accounting for kinetic layers near boundaries. In
contrast, the use of multiple relaxation time (MRT) models [17], which are
simplified versions of the relaxation LBM [18, 19], have addressed these
aspects significantly. Its characteristic feature is that the relaxation
process is carried out in moment space [20]. In particular, the relaxation
times for the kinetic modes can be independently adjusted by means of a linear
stability analysis to improve numerical stability [21, 22]. Furthermore, based
on the notion of duality between hydrodynamic and kinetic modes, a procedure
for construction of matrix based LBM has been proposed recently [23]. From a
different standpoint, non-linear stability can be enforced with a discrete
H-theorem locally in the collision step using the SRT model [24, 25, 26]. In
this Entropic LBM, minimization of a convex H-function with hydrodynamic
conservation constraints yields transcendental local attractors. It was also
shown that the choice of the H-function in this context can be determined by
enforcing Galilean invariance [27]. The construction based on the minimization
of a convex function has been generalized to include a larger set of
constraints that includes second-order moments yielding quasi-equilibrium
attractors and thus allowing for a two-relaxation time Entropic LBM via a
continuous H-theorem [28, 29]. A theoretical basis for such an approach based
on factorization symmetry considerations has been presented in [30]. This
work, along with others [31, 32, 33], also provides rational procedures for
constructing higher order models.
For general applicability of models and numerical schemes, it is necessary
that their description of fluid behavior be the same in all inertial frames of
reference. This important physical requirement of Galilean invariance, when
not met can also lead to numerical instability in the context of the LBM. The
latter arises from the fact that the degeneracies due to the finiteness of the
standard lattice velocity sets can lead to linear dependence of higher order
moments in terms of those at lower orders, which, in turn, can result in
negative dependence of viscosity on fluid velocity [34]. Thus, it becomes
necessary to consider large lattice velocity sets, which, however, by
themselves do not guarantee in strictly observing Galilean invariance, as they
can only lead to kinematically complete models [35]. Proper selection of the
collision model provides the sufficient or the dynamically complete condition
in this regard to recover the correct physics, such as the Navier-Stokes
equations. This can be seen by the use of unwieldy fitting of parameters [34]
or elaborate construction procedures [32] for the attractors in the collision
model with such extended lattice sets. Thus, the collision process still needs
to be carefully designed and has an important role to play in the proper
observation of Galilean invariance.
In this context, relaxation in a moving frame of reference, i.e. in terms of
moments obtained by shifting the particle velocity with the local hydrodynamic
velocity, or central moments [36], provides a natural setting and a simple
construction procedure to maintain Galilean invariance for a given velocity
set. That is, the relaxation process is constructed to observe inertial frame
invariance to a degree as permitted by the chosen lattice velocity set. We
consider this specific meaning when we use the term Galilean invariance in
this paper. Also, when different central moments are relaxed at different
rates, i.e. formulated as a MRT model, it can enhance numerical stability by
providing additional numerical dissipation similar to standard MRT models
based on raw moments. It is noted that the ideas and procedures based on
central moment relaxation are not restricted to standard lattice velocity
sets, but can be used for any extended or kinematically complete velocity sets
as well. The central moment approach exhibits a cascaded structure, which was
shown to be equivalent to considering a generalized equilibrium in the lattice
or rest frame of reference [38]. Furthermore, to further improve the physical
fidelity, the local attractors for the central moments given in terms of their
factorization into lower order moments has been proposed [39]. To incorporate
the effect of force fields, which are important for numerous physical
applications, a new approach for the source terms based on central moments was
recently developed for a two-dimensional (2D) lattice [40]. In addition, a
detailed theoretical basis for the central moment method, including a
consistency analysis of the emergent fluid motion, was also provided [40].
The objective of this work is the derivation and validation of a 3D central
moment lattice Boltzmann method, with a particular focus on deriving Galilean
invariant source terms, which are important, for example, in situations
involving multiphase/multicomponent flows or turbulence modeling. In this
regard, three-dimensional, twenty seven velocity (D3Q27) and its minimal
subset, i.e. fifteen velocity (D3Q15) velocity lattices that can recover
Navier-Stokes behavior are considered, and the overall procedure and notations
used in [40] are adopted. The D3Q27 lattice is chosen so that our results
provide the forcing scheme based on central moments to the overall formulation
considered in [36]. It is noted that the notations and the details provided in
that work [36] are cumbersome even for the collision model without forcing for
implementation. On the other hand, in practice, the computational complexity
is considerably reduced with the use of the D3Q15 lattice. Hence, the details
with both the lattices are provided, with the smaller lattice set used in most
of the computations in our validation studies. The overall procedure is as
follows. Starting from suitable choices of the orthogonal moment basis for
these lattice velocity sets, the continuous and discrete central moments of
the local attractors and source terms at different orders are successively
matched. The results are then transformed in terms of raw moments by means of
the binomial theorem. To maintain physical coherence for the discrete velocity
set, factorized local attractors for higher order central moments and
temporally second-order accurate treatment of source terms are considered.
This construction yields Galilean invariant representation of 3D fluid
dynamics in the presence of general external or self-consistent internal
forces. The computational approach thus derived is then assessed by comparison
of its results for a set of canonical problems involving forcing for which
analytical solutions are available.
The paper is organized as follows. Its main body containing the derivation
focuses only on the essential steps involved in the derivation, choosing the
D3Q27 lattice as an example, with the attendant details presented in various
appendices (see Appendices A-G; the computational scheme for the D3Q15 lattice
is presented in Appendix G). Section 2 discusses the choice made for the
orthogonal moment basis corresponding to the D3Q27 lattice. The ansatz for the
continuous central moments for the distribution functions, local attractors
and sources due to the force fields are presented in Secs. 3. Section 4
provides the corresponding 3D lattice Boltzmann equation (LBE) with source
terms based on central moments. Various discrete central moments needed for
the construction of the central moment method are defined and the matching
principle to preserve Galilean invariance is stated in Sec. 5. Section 6
obtains expressions for various discrete raw moments using the matching
principle via the binomial theorem, including the derivation of the source
terms in particle velocity space. The construction of the collision kernel is
presented in Sec. 7 and the overall procedure of the central moment LBM is
provided in Sec. 8. Validation studies involving various canonical problems
are discussed in Sec. 9. The conclusions are finally summarized in Sec. 10.
## 2 Selection of Moment Basis
We now discuss the moment basis, which is an important element on which the
central moment LBM is constructed, corresponding to the three-dimensional,
twenty seven velocity (D3Q27) lattice model (see Fig. 1). The particle
velocity for this lattice model $\overrightarrow{e}_{\alpha}$ is given by
$\overrightarrow{e_{\alpha}}=\left\\{\begin{array}[]{ll}{(0,0,0),}&{\alpha=0}\\\
{(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1),}&{\alpha=1,\cdots,6}\\\ {(\pm 1,\pm
1,0),(\pm 1,0,\pm 1),(0,\pm 1,\pm 1),}&{\alpha=7,\cdots,18}\\\ {(\pm 1,\pm
1,\pm 1),}&{\alpha=19,\cdots,26}\end{array}\right.$ (1)
Figure 1: Three-dimensional, twenty seven particle velocity (D3Q27) lattice.
For convenience, as in [40], we use Dirac’s bra-ket notion to represent the
basis vectors, and Greek and Latin subscripts for particle velocity directions
and Cartesian coordinate directions, respectively. By definition, the moments
in the LBM are discrete integral properties of the distribution function
$f_{\alpha}$, i.e. $\sum_{\alpha=0}^{26}e_{\alpha x}^{m}e_{\alpha
y}^{n}e_{\alpha z}^{p}f_{\alpha}$, where $m$, $n$ and $p$ are integers, in 3D.
As a result, we begin with the following twenty-seven non-orthogonal
independent basis vectors obtained by combining monomials $e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}$ and arranged in an ascending order.
First, the nominal basis for the conserved (zeroth and first order) moments
follows immediately:
$\displaystyle\ket{T_{0}}$ $\displaystyle=$
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}},$
$\displaystyle\ket{T_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
$\displaystyle\ket{T_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha y}},$
$\displaystyle\ket{T_{3}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha z}}.$
The basis for second-order moments are chosen such that it allows correct
representation of the momentum flux (based on the Maxwell distribution, see
below in Sec. 3) in the hydrodynamic equations. Three off-diagonal components
($\ket{T_{4}}$–$\ket{T_{6}}$) and three diagonal components
($\ket{T_{7}}$–$\ket{T_{9}}$) with $max(m,n,p)=1$ and $max(m,n,p)=2$,
respectively, while satisfying $m+n+p=2$ are considered:
$\displaystyle\ket{T_{4}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}},$ $\displaystyle\ket{T_{5}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha z}},$ $\displaystyle\ket{T_{6}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{T_{7}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}-e_{\alpha y}^{2}},$ $\displaystyle\ket{T_{8}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}-e_{\alpha z}^{2}},$
$\displaystyle\ket{T_{9}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}+e_{\alpha y}^{2}+e_{\alpha z}^{2}}.$
The following six third-order basis vectors for moments are chosen
($\ket{T_{10}}$–$\ket{T_{15}}$ with $max(m,n,p)=2$ and $\ket{T_{16}}$ with
$max(m,n,p)=1$, while satisfying $m+n+p=3$):
$\displaystyle\ket{T_{10}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}^{2}+e_{\alpha x}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{11}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}+e_{\alpha
y}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{12}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha z}+e_{\alpha y}^{2}e_{\alpha
z}},$ $\displaystyle\ket{T_{13}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}^{2}-e_{\alpha x}e_{\alpha
z}^{2}},$ $\displaystyle\ket{T_{14}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}-e_{\alpha y}e_{\alpha
z}^{2}},$ $\displaystyle\ket{T_{15}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha z}-e_{\alpha y}^{2}e_{\alpha
z}},$ $\displaystyle\ket{T_{16}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}}.$
For the fourth-order basis vectors, we consider the following six of them
($\ket{T_{17}}$–$\ket{T_{22}}$ with $max(m,n,p)=2$ for $m+n+p=4$):
$\displaystyle\ket{T_{17}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}+e_{\alpha x}^{2}e_{\alpha z}^{2}+e_{\alpha
y}^{2}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{18}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}+e_{\alpha x}^{2}e_{\alpha
z}^{2}-e_{\alpha y}^{2}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{19}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}-e_{\alpha
x}^{2}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{20}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{T_{21}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}},$ $\displaystyle\ket{T_{22}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}^{2}}.$
Finally, three fifth-order basis vectors ($\ket{T_{23}}$–$\ket{T_{25}}$) and
one sixth-order basis vector ($\ket{T_{26}}$) are considered to complete
moment basis corresponding to the D3Q27 model. In the above, in each case
$max(m,n,p)=2$, with $m+n+p=5$ and $m+n+p=6$, respectively. Thus,
$\displaystyle\ket{T_{23}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{24}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}e_{\alpha
z}^{2}},$ $\displaystyle\ket{T_{25}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}e_{\alpha z}},$
$\displaystyle\ket{T_{26}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha z}^{2}}.$ (2)
Note that due to the finiteness of the particle velocity set, higher order
longitudinal moments, i.e. $\ket{e_{\alpha i}^{m}}$ with $m\geq 3$ are
eliminated from consideration in the above. The components of the basis
vectors for the conserved moments corresponding to Eq. (1) may be written as
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}}$
$\displaystyle=$
$\displaystyle\left(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,\right.$
$\displaystyle\left.1,1,1,1,1,1\right)^{\dagger},$
$\displaystyle\ket{e_{\alpha x}}$ $\displaystyle=$
$\displaystyle\left(0,1,-1,0,0,0,0,1,-1,1,-1,1,-1,1,-1,0,0,0,0,1,-1,\right.$
$\displaystyle\left.1,-1,1,-1,1,-1\right)^{\dagger},$
$\displaystyle\ket{e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,1,-1,0,0,1,1,-1,-1,0,0,0,0,1,-1,1,-1,1,1,\right.$
$\displaystyle\left.-1,-1,1,1,-1,-1\right)^{\dagger},$
$\displaystyle\ket{e_{\alpha z}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,-1,0,0,0,0,1,1,-1,-1,1,1,-1,-1,1,1,\right.$
$\displaystyle\left.1,1,-1,-1,-1,-1\right)^{\dagger}.$
Here and henceforth, the superscript ‘$\dagger$’ represents the transpose
operator. The next key step is to transform the above non-orthogonal nominal
basis set into an equivalent orthogonal set that would allow an efficient
implementation [17]. This is accomplished by means of the standard Gram-
Schmidt procedure for the above arrangement, i.e. in the increasing order of
the monomials of the products of the Cartesian components of the particle
velocities. As a result the components of the orthogonal basis vectors are
given by
$\displaystyle\ket{K_{0}}$ $\displaystyle=$
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}},$
$\displaystyle\ket{K_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
$\displaystyle\ket{K_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha y}},$
$\displaystyle\ket{K_{3}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha z}},$
$\displaystyle\ket{K_{4}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}},$ $\displaystyle\ket{K_{5}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha z}},$ $\displaystyle\ket{K_{6}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{K_{7}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}-e_{\alpha y}^{2}},$ $\displaystyle\ket{K_{8}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}-3\ket{e_{\alpha z}^{2}},$ $\displaystyle\ket{K_{9}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}-2\ket{\rho},$ $\displaystyle\ket{K_{10}}$ $\displaystyle=$
$\displaystyle 3\ket{e_{\alpha x}e_{\alpha y}^{2}+e_{\alpha x}e_{\alpha
z}^{2}}-4\ket{e_{\alpha x}},$ $\displaystyle\ket{K_{11}}$ $\displaystyle=$
$\displaystyle 3\ket{e_{\alpha x}^{2}e_{\alpha y}+e_{\alpha y}e_{\alpha
z}^{2}}-4\ket{e_{\alpha y}},$ $\displaystyle\ket{K_{12}}$ $\displaystyle=$
$\displaystyle 3\ket{e_{\alpha x}^{2}e_{\alpha z}+e_{\alpha y}^{2}e_{\alpha
z}}-4\ket{e_{\alpha z}},$ $\displaystyle\ket{K_{13}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}^{2}-e_{\alpha x}e_{\alpha
z}^{2}},$ $\displaystyle\ket{K_{14}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}-e_{\alpha y}e_{\alpha
z}^{2}},$ $\displaystyle\ket{K_{15}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha z}-e_{\alpha y}^{2}e_{\alpha
z}},$ $\displaystyle\ket{K_{16}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{K_{17}}$ $\displaystyle=$ $\displaystyle 3\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}+e_{\alpha x}^{2}e_{\alpha z}^{2}+e_{\alpha
y}^{2}e_{\alpha z}^{2}}-4\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}+4\ket{\rho},$ $\displaystyle\ket{K_{18}}$ $\displaystyle=$
$\displaystyle 3\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}+e_{\alpha
x}^{2}e_{\alpha z}^{2}-2e_{\alpha y}^{2}e_{\alpha z}^{2}}-2\ket{2e_{\alpha
x}^{2}-e_{\alpha y}^{2}-e_{\alpha z}^{2}},$ $\displaystyle\ket{K_{19}}$
$\displaystyle=$ $\displaystyle 3\ket{e_{\alpha x}^{2}e_{\alpha
y}^{2}-e_{\alpha x}^{2}e_{\alpha z}^{2}}-2\ket{e_{\alpha y}^{2}-e_{\alpha
z}^{2}},$ $\displaystyle\ket{K_{20}}$ $\displaystyle=$ $\displaystyle
3\ket{e_{\alpha x}^{2}e_{\alpha y}e_{\alpha z}}-2\ket{e_{\alpha y}e_{\alpha
z}},$ $\displaystyle\ket{K_{21}}$ $\displaystyle=$ $\displaystyle
3\ket{e_{\alpha x}e_{\alpha y}^{2}e_{\alpha z}}-2\ket{e_{\alpha x}e_{\alpha
z}},$ $\displaystyle\ket{K_{22}}$ $\displaystyle=$ $\displaystyle
3\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}^{2}}-2\ket{e_{\alpha x}e_{\alpha
y}},$ $\displaystyle\ket{K_{23}}$ $\displaystyle=$ $\displaystyle
9\ket{e_{\alpha x}e_{\alpha y}^{2}e_{\alpha z}^{2}}-6\ket{e_{\alpha
x}e_{\alpha y}^{2}+e_{\alpha x}e_{\alpha z}^{2}}+4\ket{e_{\alpha x}},$
$\displaystyle\ket{K_{24}}$ $\displaystyle=$ $\displaystyle 9\ket{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha z}^{2}}-6\ket{e_{\alpha x}^{2}e_{\alpha
y}+e_{\alpha y}e_{\alpha z}^{2}}+4\ket{e_{\alpha y}},$
$\displaystyle\ket{K_{25}}$ $\displaystyle=$ $\displaystyle 9\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha z}}-6\ket{e_{\alpha x}^{2}e_{\alpha
z}+e_{\alpha y}^{2}e_{\alpha z}}+4\ket{e_{\alpha z}},$
$\displaystyle\ket{K_{26}}$ $\displaystyle=$ $\displaystyle 27\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha z}^{2}}-18\ket{e_{\alpha x}^{2}e_{\alpha
y}^{2}+e_{\alpha x}^{2}e_{\alpha z}^{2}+e_{\alpha y}^{2}e_{\alpha z}^{2}}$ (3)
$\displaystyle+12\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}-8\ket{\rho}.$
This can be explicitly written in terms of a orthogonal matrix of moment basis
$\mathcal{K}$ given by
$\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle\left[\ket{K_{0}},\ket{K_{1}},\ket{K_{2}},\ket{K_{3}},\ket{K_{4}},\ket{K_{5}},\ket{K_{6}},\ket{K_{7}},\ket{K_{8}}\right.$
(4)
$\displaystyle\left.\ket{K_{9}},\ket{K_{10}},\ket{K_{11}},\ket{K_{12}},\ket{K_{13}},\ket{K_{14}},\ket{K_{15}},\ket{K_{16}},\ket{K_{17}}\right].$
$\displaystyle\left.\ket{K_{18}},\ket{K_{19}},\ket{K_{20}},\ket{K_{21}},\ket{K_{22}},\ket{K_{23}},\ket{K_{24}},\ket{K_{25}},\ket{K_{26}}\right]$
whose components are presented in Appendix A. Note that unlike the standard
MRT formulation based on raw moments [22], which orders the basis vectors by
considering the character of moments, i.e. increasing powers of their
tensorial orders (i.e. scalars, vectors, tensors of different ranks,…), the
central moment basis vectors considered here are ordered according to their
ascending powers (i.e. zeroth order moment, first order moments, second order
moments,…). Furthermore, the details of the basis vectors considered in this
paper are different from those provided in [36].
## 3 Continuous Central Moments: Distribution Function, its Local Attractor
and Forcing
The central moment LBM, which is defined at the discrete level, should
preserve certain continuous integral properties of the distribution function
$f$ given in terms of its central moments, i.e. those shifted by the
macroscopic fluid velocity. In this regard, we first define _continuous_
central moment of $f$ of order $(m+n+p)$ as
$\widehat{\Pi}_{x^{m}y^{n}z^{p}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}(\xi_{z}-u_{z})^{p}d\xi_{x}d\xi_{y}d\xi_{z}.$
(5)
Here, and in the rest of this paper, the use of “hat” over a symbol represents
quantities in the space of moments. The effect of collision is to relax the
distribution function, or equivalently, its central moments, towards its local
attractor. The corresponding central moment local attractor may be written as
$\widehat{\Pi}_{x^{m}y^{n}z^{p}}^{at}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{at}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}(\xi_{z}-u_{z})^{p}d\xi_{x}d\xi_{y}d\xi_{z}.$
(6)
Here $f^{at}$ is as yet unknown, and its effect on the dynamics will be
determined in what follows. Similarly, the continuous central moments due to
sources may be written as
$\widehat{\Gamma}_{x^{m}y^{n}z^{p}}^{F}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Delta
f^{F}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}(\xi_{z}-u_{z})^{p}d\xi_{x}d\xi_{y}d\xi_{z},$
(7)
where $\Delta f^{F}$ is the change in the distribution function due to
forcing, which will be specified later. One possibility is to consider the
local Maxwellian as the attractor [36]. That is, consider
$f^{\mathcal{M}}\equiv
f^{\mathcal{M}}(\rho,\overrightarrow{u},\xi_{x},\xi_{y},\xi_{z})=\frac{\rho}{2\pi
c_{s}^{2}}\exp\left[-\frac{\left(\overrightarrow{\xi}-\overrightarrow{u}\right)^{2}}{2c_{s}^{2}}\right],$
(8)
where $c_{s}^{2}=1/3$, which yields corresponding continuous Maxwellian
central moments as
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}z^{p}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\mathcal{M}}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}(\xi_{z}-u_{z})^{p}d\xi_{x}d\xi_{y}d\xi_{z}.$
(9)
By virtue of the fact that $f^{\mathcal{M}}$ being an even function,
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}z^{p}}\neq 0$ when $m$, $n$ and $p$
are even and $\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}z^{p}}=0$ when $m$ or $n$
or $p$ is odd. Here and henceforth, the subscripts $x^{m}y^{n}z^{p}$ mean
$xxx\cdots m\mbox{-times}$, $yyy\cdots n\mbox{-times}$ and $zzz\cdots
p\mbox{-times}$. Thus,
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{i}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{ii}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{ij}$
$\displaystyle=$ $\displaystyle 0,\quad i\neq j,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{ijj}$ $\displaystyle=$
$\displaystyle 0,\quad i\neq j,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{ijk}$ $\displaystyle=$
$\displaystyle 0,\quad i\neq j\neq k,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{iijj}$ $\displaystyle=$
$\displaystyle c_{s}^{4}\rho,\quad i\neq j,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{iijk}$ $\displaystyle=$
$\displaystyle 0,\quad i\neq j\neq k,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{ijjkk}$ $\displaystyle=$
$\displaystyle 0,\quad i\neq j\neq k,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{iijjkk}$ $\displaystyle=$
$\displaystyle c_{s}^{6}\rho,\quad i\neq j\neq k.$ (10)
Now, as discussed in [39] using
$\widehat{\Pi}^{at}_{x^{m}y^{n}z^{p}}=\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}z^{p}}$
for all orders leads to some inconsistency in recovering the macroscopic fluid
equations. To circumvent this issue, we use a factorized form for (central
moment) attractors proposed in [39]. Essentially, in addition to satisfying
Galilean invariance, the Maxwellian (equilibrium) satisfies the factorization
property, i.e. independence along Cartesian coordinate directions, which
immediately applies to its central moments. In the factorized central moment
formulation, this property is extended to model non-equilibrium process, i.e.
relaxation towards equilibrium. In other words, the higher order central
moment attractors are given as its factorization in terms of lower order
central moments that are not yet in equilibrium [39]. To proceed further, let
us define the following post-collision continuous central moment of order
$(m+n+p)$:
$\widetilde{\widehat{\Pi}}_{x^{m}y^{n}z^{p}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\widetilde{f}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}(\xi_{z}-u_{z})^{p}d\xi_{x}d\xi_{y}d\xi_{z}.$
(11)
Then, we consider the factorized form for attractors as
$\displaystyle\widehat{\Pi}^{at}_{i}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{i}=0,$
$\displaystyle\widehat{\Pi}^{at}_{ij}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{i}\widetilde{\widehat{\Pi}}_{j}=0,$
$\displaystyle\widehat{\Pi}^{at}_{iij}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{ii}\widetilde{\widehat{\Pi}}_{j}=0,$
$\displaystyle\widehat{\Pi}^{at}_{ijk}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{i}\widetilde{\widehat{\Pi}}_{j}\widetilde{\widehat{\Pi}}_{k}=0,$
$\displaystyle\widehat{\Pi}^{at}_{iijj}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{ii}\widetilde{\widehat{\Pi}}_{jj},$
$\displaystyle\widehat{\Pi}^{at}_{iijk}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{ii}\widetilde{\widehat{\Pi}}_{jk},$
$\displaystyle\widehat{\Pi}^{at}_{iijjk}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{ii}\widetilde{\widehat{\Pi}}_{jj}\widetilde{\widehat{\Pi}}_{k}=0,$
$\displaystyle\widehat{\Pi}^{at}_{iijjkk}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{ii}\widetilde{\widehat{\Pi}}_{jj}\widetilde{\widehat{\Pi}}_{kk}.$
(12)
Now, however, to correctly recover the momentum flux and pressure tensor in
the macroscopic fluid dynamical equations, the diagonal components of the
second-order central moments should preserve those obtained from the
Maxwellian. That is, we set $\widehat{\Pi}^{at}_{ii}=c_{s}^{2}\rho$. Thus, the
$27$ independent components of the local factorized central moment attractors
can be written as
$\displaystyle\widehat{\Pi}^{at}_{0}$ $\displaystyle=$ $\displaystyle
0,\widehat{\Pi}^{at}_{x}=\widehat{\Pi}^{at}_{y}=\widehat{\Pi}^{at}_{z}=0,$
$\displaystyle\widehat{\Pi}^{at}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{yy}=\widehat{\Pi}^{at}_{zz}=c_{s}^{2}\rho,$
$\displaystyle\widehat{\Pi}^{at}_{xy}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{xz}=\widehat{\Pi}^{at}_{yz}=0,$
$\displaystyle\widehat{\Pi}^{at}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{xzz}=\widehat{\Pi}^{at}_{xxy}=\widehat{\Pi}^{at}_{yzz}=\widehat{\Pi}^{at}_{xxz}=\widehat{\Pi}^{at}_{yyz}=\widehat{\Pi}^{at}_{xyz}=0,$
$\displaystyle\widehat{\Pi}^{at}_{xxyy}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{xx}\widetilde{\widehat{\Pi}}_{yy},$
$\displaystyle\widehat{\Pi}^{at}_{xxzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{xx}\widetilde{\widehat{\Pi}}_{zz},$
$\displaystyle\widehat{\Pi}^{at}_{yyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{yy}\widetilde{\widehat{\Pi}}_{zz},$
$\displaystyle\widehat{\Pi}^{at}_{xxyz}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{xyyz}=\widehat{\Pi}^{at}_{xyzz}=0,$
$\displaystyle\widehat{\Pi}^{at}_{xyyzz}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{xxyzz}=\widehat{\Pi}^{at}_{xxyyz}=0,$
$\displaystyle\widehat{\Pi}^{at}_{xxyyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\Pi}}_{xx}\widetilde{\widehat{\Pi}}_{yy}\widetilde{\widehat{\Pi}}_{zz}.$
(13)
In essence, for the D3Q27 lattice, the fourth-order and sixth-order moments
are factorized in terms of longitudinal second-order moments. It may be noted
that symmetries in the factorization of the Maxwellian have been exploited to
construct other types of quasi-equilibrium attractors recently [30].
Similarly for the continuous source central moments due to force fields, one
possible choice is obtained by choosing that based on the local Maxwellian,
i.e. $\Delta
f^{F}=\frac{\overrightarrow{F}}{\rho}\cdot\frac{(\overrightarrow{\xi}-\overrightarrow{u})}{c_{s}^{2}}f^{\mathcal{M}}$,
which, however, leads to aliasing effects for higher order moments [40]. To
circumvent this issue, a simple choice involves de-aliasing higher order
moments while preserving its necessary effect on the first-order central
moments [40] which is extended to 3D in this work. Thus, we specify the
continuous source central moments as
$\widehat{\Gamma}_{x^{m}y^{n}z^{p}}^{F}=\left\\{\begin{array}[]{ll}{F_{x},}&{\quad
m=1,n=0,p=0}\\\ {F_{y},}&{\quad m=0,n=1,p=0}\\\ {F_{z},}&{\quad
m=0,n=0,p=1}\\\ {0,}&{\quad\mbox{Otherwise.}}\end{array}\right.$ (14)
## 4 Central Moment Lattice-Boltzmann Equation with Forcing Terms
Let us now write the central moment lattice Boltzmann equation (LBE) with
forcing terms by first defining a _discrete_ distribution function supported
by the discrete particle velocity set $\overrightarrow{e}_{\alpha}$ as
$\mathbf{f}=\ket{f_{\alpha}}=(f_{0},f_{1},f_{2},\ldots,f_{26})^{\dagger}$, a
collision operator as
$\mathbf{\Omega}^{c}=\ket{\Omega_{\alpha}^{c}}=(\Omega_{0}^{c},\Omega_{1}^{c},\Omega_{2}^{c},\ldots,\Omega_{26}^{c})$
as well as a source term as
$\mathbf{S}=\ket{S_{\alpha}}=(S_{0},S_{1},S_{2},\ldots,S_{26})^{\dagger}$
based on central moments. The LBE can then be obtained as a discrete version
of the continuous Boltzmann equation by temporally integrating along particle
characteristics as [40]
$f_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=f_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+\int_{t}^{t+1}S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\theta,t+\theta)}d\theta.$
(15)
In Eq. (15), the collision operator can be written in terms of the unknown
collision kernel $\mathbf{\widehat{g}}$ projected to the orthogonal matrix of
the moment basis as [36]
$\Omega_{\alpha}^{c}\equiv\Omega_{\alpha}^{c}(\mathbf{f},\mathbf{\widehat{g}})=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha},$
(16)
where
$\mathbf{\widehat{g}}=\ket{\widehat{g}_{\alpha}}=(\widehat{g}_{0},\widehat{g}_{1},\widehat{g}_{2},\ldots,\widehat{g}_{26})^{\dagger}$,
which will be derived later. The macroscopic conserved moments, i.e. the local
density and momentum, are obtained from the distribution function as
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{26}f_{\alpha}=\braket{f_{\alpha}}{\rho},$ (17)
$\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{26}f_{\alpha}e_{\alpha
i}=\braket{f_{\alpha}}{e_{\alpha i}},i\in{x,y,z}.$ (18)
We consider a semi-implicit representation for the source term, i.e. the last
term in the above equation, Eq. (15), to provide second-order accuracy [40],
i.e.
$\int_{t}^{t+1}S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\theta,t+\theta)}d\theta=\frac{1}{2}\left[S_{{\alpha}(\overrightarrow{x},t)}+S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)}\right]$.
This equation is then made effectively explicit by using the transformation
$\overline{f}_{\alpha}=f_{\alpha}-\frac{1}{2}S_{\alpha}$ to reduce it to [40]
$\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)}.$
(19)
The explicit expressions for the source term $S_{\alpha}$ as well as the
collision kernel $\mathbf{\widehat{g}}$ will be derived so as to rigorously
enforce Galilean invariance through a matching principle and a binomial
transformation. These are discussed in Secs. 6 and 7, respectively.
## 5 Various Discrete Central Moments and Galilean Invariance Matching
Principle
To facilitate the determination of the structure of the collision operator
kernel $\mathbf{\widehat{g}}$ and the source terms $S_{\alpha}$, we now define
the following _discrete_ central moments of the distribution function,
Maxwellian, and source term, respectively:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{f_{\alpha}},$
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}z^{p}}^{at}$ $\displaystyle=$
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{f_{\alpha}^{at}},$
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{S_{\alpha}}.$ (20)
Furthermore, the following definitions involving discrete central moments
based on post-collision ($\widetilde{f}_{\alpha}$) and transformed
($\overline{f}_{\alpha}$) distribution functions, and its combination
$\widetilde{\overline{f}}_{\alpha}$, are useful for further simplifications:
$\displaystyle\widetilde{\widehat{\kappa}}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{\widetilde{f}_{\alpha}},$
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{\overline{f}_{\alpha}},$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{x^{m}y^{n}z^{p}}$
$\displaystyle=$ $\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}(e_{\alpha z}-u_{z})^{p}}{\widetilde{\overline{f}}_{\alpha}}.$
(21)
Based on the definition of the transformed distribution function as given in
the last section, it immediately follows that
$\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}=\widehat{\kappa}_{x^{m}y^{n}z^{p}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}z^{p}}.$
(22)
In order to preserve the main physical characteristic, i.e. Galilean
invariance at the discrete level, we now invoke the key matching principle,
which is to set the _discrete_ central moments of the attractors of the
distribution function and the source terms, defined above, equal to their
corresponding _continuous_ central moments, whose forms are known exactly from
the ansatz derived in Sec. 3. In other words,
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}z^{p}}^{at}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{at}_{x^{m}y^{n}z^{p}},$ (23)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle\widehat{\Gamma}^{F}_{x^{m}y^{n}z^{p}}.$ (24)
This yields the following expressions for the discrete local central moment
attractors
$\displaystyle\widehat{\kappa}^{at}_{0}$ $\displaystyle=$ $\displaystyle
0,\widehat{\kappa}^{at}_{x}=\widehat{\kappa}^{at}_{y}=\widehat{\kappa}^{at}_{z}=0,$
$\displaystyle\widehat{\kappa}^{at}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}^{at}_{yy}=\widehat{\kappa}^{at}_{zz}=c_{s}^{2}\rho,$
$\displaystyle\widehat{\kappa}^{at}_{xy}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}^{at}_{xz}=\widehat{\kappa}^{at}_{yz}=0,$
$\displaystyle\widehat{\kappa}^{at}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}^{at}_{xzz}=\widehat{\kappa}^{at}_{xxy}=\widehat{\kappa}^{at}_{yzz}=\widehat{\kappa}^{at}_{xxz}=\widehat{\kappa}^{at}_{yyz}=\widehat{\kappa}^{at}_{xyz}=0,$
$\displaystyle\widehat{\kappa}^{at}_{xxyy}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy},$
$\displaystyle\widehat{\kappa}^{at}_{xxzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz},$
$\displaystyle\widehat{\kappa}^{at}_{yyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz},$
$\displaystyle\widehat{\kappa}^{at}_{xxyz}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}^{at}_{xyyz}=\widehat{\kappa}^{at}_{xyzz}=0,$
$\displaystyle\widehat{\kappa}^{at}_{xyyzz}$ $\displaystyle=$
$\displaystyle\widehat{\kappa}^{at}_{xxyzz}=\widehat{\kappa}^{at}_{xxyyz}=0,$
$\displaystyle\widehat{\kappa}^{at}_{xxyyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz}.$
(25)
In addition, the discrete source central moments satisfy the following
$\displaystyle\widehat{\sigma}_{0}=0,\widehat{\sigma}_{x}=F_{x},\widehat{\sigma}_{y}=F_{y},\widehat{\sigma}_{z}$
$\displaystyle=$ $\displaystyle F_{z},$
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}z^{p}}$ $\displaystyle=$
$\displaystyle 0,\quad m,n,p>1.$ (26)
Thus, finally, in view of Eq. (22), the attractors in terms of the transformed
central moments can be written as
$\displaystyle\widehat{\overline{\kappa}}^{at}_{0}$ $\displaystyle=$
$\displaystyle
0,\widehat{\overline{\kappa}}^{at}_{x}=-\frac{1}{2}F_{x},\widehat{\overline{\kappa}}^{at}_{y}=-\frac{1}{2}F_{y},\widehat{\overline{\kappa}}^{at}_{z}=-\frac{1}{2}F_{z},$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{yy}=\widehat{\overline{\kappa}}^{at}_{zz}=c_{s}^{2}\rho,$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xz}=\widehat{\overline{\kappa}}^{at}_{yz}=0,$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xzz}=\widehat{\overline{\kappa}}^{at}_{xxy}=\widehat{\overline{\kappa}}^{at}_{yzz}=\widehat{\overline{\kappa}}^{at}_{xxz}=\widehat{\overline{\kappa}}^{at}_{yyz}=\widehat{\overline{\kappa}}^{at}_{xyz}=0,$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xxyy}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy},$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xxzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz},$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{yyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz},$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xxyz}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xyyz}=\widehat{\overline{\kappa}}^{at}_{xyzz}=0,$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xyyzz}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xxyzz}=\widehat{\overline{\kappa}}^{at}_{xxyyz}=0,$
$\displaystyle\widehat{\overline{\kappa}}^{at}_{xxyyzz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz}.$
(27)
## 6 Various Discrete Raw Moments and Source Terms in Particle Velocity Space
In order to construct an executable central moment LBM, the above information
based on the central moments need to be related to the raw moments, i.e. those
in the usual lattice or rest frame of reference. This can be readily
accomplished by means of the binomial theorem applied to the orthogonal
products of the discrete quantities supported by the particle velocity set
[40]. In this regard, the following notations that specify various _discrete
raw_ moments will be useful:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}f_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}e_{\alpha z}^{p}}{f_{\alpha}},$ (28)
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}\overline{f}_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}e_{\alpha z}^{p}}{\overline{f}_{\alpha}},$ (29)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}e_{\alpha z}^{p}}{S_{\alpha}}.$ (30)
Here and in what follows, the superscript “prime” (′) is used to distinguish
the raw moments from the central moments that are designated without the
primes. Furthermore, similar to Eq. (22), the relation
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}=\widehat{\kappa}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$
is satisfied. Based on the above, first, we write the raw moments of the
distribution function of different orders supported by the particle velocity
set
$\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}=\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}}$ in terms of the known quantities. To
obtain a compact description of results, the following operator notation is
helpful [40]:
$\displaystyle
a(\overline{f}_{\alpha_{1}}+\overline{f}_{\alpha_{3}}+\overline{f}_{\alpha_{3}}+\cdots)$
$\displaystyle+$ $\displaystyle
b(\overline{f}_{\beta_{1}}+\overline{f}_{\beta_{2}}+\overline{f}_{\beta_{3}}+\cdots)+\cdots$
(31)
$\displaystyle=\left(a\sum_{\alpha}^{A}+b\sum_{\alpha}^{B}\cdots\right)\otimes\overline{f}_{\alpha},$
where $A=\left\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots\right\\}$,
$B=\left\\{\beta_{1},\beta_{2},\beta_{3},\cdots\right\\}$,$\cdots$. First, the
conserved transformed raw moments follows directly from the definition as
$\displaystyle\widehat{\overline{\kappa}}_{0}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{\rho}=\rho,\qquad\qquad\qquad\quad\widehat{\overline{\kappa}}_{x}^{{}^{\prime}}=\braket{\overline{f}_{\alpha}}{e_{\alpha
x}}=\rho u_{x}-\frac{1}{2}F_{x},$
$\displaystyle\widehat{\overline{\kappa}}_{y}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha y}}=\rho
u_{y}-\frac{1}{2}F_{y},\qquad\widehat{\overline{\kappa}}_{z}^{{}^{\prime}}=\braket{\overline{f}_{\alpha}}{e_{\alpha
z}}=\rho u_{z}-\frac{1}{2}F_{z}.$ (32)
The non-conserved transformed raw moments can be written, using the above
operator notation (Eq. (31)), in terms of the subsets of the particle velocity
directions, which are presented in Appendix B.
The next step is to transform the central moments of the source terms (Eq.
(26)) in terms of raw moments by using the definitions, i.e. Eq. (20) and
(30), which by the binomial theorem, readily yields the expressions that are
enumerated in Appendix C. These moments should be related to the discrete
source terms in particle velocity space so that an operational Galilean
invariant approach can be derived. To accomplish this, we first obtain a set
of intermediate quantities $\widehat{m}^{s}_{\beta}$, which are the
projections of the source terms to the orthogonal matrix of the moment basis
$\mathcal{K}$, i.e. $\widehat{m}^{s}_{\beta}=\braket{K_{\beta}}{S_{\alpha}}$,
$\beta=0,1,2,\ldots,26$, which can be obtained from the above using Eqs. (4)
and (66). The details of $\widehat{m}^{s}_{\beta}$ are provided in Appendix D.
It is noted that $\widehat{m}^{s}_{\beta}$ is equivalent to the following
matrix formulation
$\displaystyle\mathcal{K}^{\dagger}\mathbf{S}=(\mathcal{K}\cdot\mathbf{S})_{\alpha}$
$\displaystyle=$
$\displaystyle(\braket{K_{0}}{S_{\alpha}},\braket{K_{1}}{S_{\alpha}},\braket{K_{2}}{S_{\alpha}},\ldots,\braket{K_{26}}{S_{\alpha}})$
(33) $\displaystyle=$
$\displaystyle(\widehat{m}^{s}_{0},\widehat{m}^{s}_{1},\widehat{m}^{s}_{2},\ldots,\widehat{m}^{s}_{26})^{\dagger}\equiv\ket{\widehat{m}^{s}_{\alpha}},$
which can be exactly inverted by using the following orthogonal property of
$\mathcal{K}$, i.e.
$\mathcal{K}^{-1}=\mathcal{K}^{\dagger}\cdot\mathcal{D}^{-1}$, where
$\mathcal{D}$ is the diagonal matrix given by
$\mathcal{D}=\mbox{diag}(\braket{K_{0}}{K_{0}},\braket{K_{1}}{K_{1}},\braket{K_{2}}{K_{2}},\ldots,\braket{K_{26}}{K_{26}})$
[40]. Exploiting this fact, the linear system (Eq. (33)) can be solved exactly
to yield the expressions for the Galilean invariant source terms in velocity
space $S_{\alpha}$ in terms of $\widehat{m}^{s}_{\beta}$, or equivalently the
force $\overrightarrow{F}$ and velocity fields $\overrightarrow{u}$. The final
results of $S_{\alpha}$, where $\alpha=0,1,2,\ldots,26$ are summarized in
Appendix E.
Finally, to obtain the collision kernel $\widehat{g}_{\beta}$ in the next
section, we need to evaluate the expressions for its raw moments of various
orders projected to the moment basis matrix $\mathcal{K}$, i.e.
$\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha
z}^{p}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{m}e_{\alpha
y}^{n}e_{\alpha z}^{p}}\widehat{g}_{\beta}.$ (34)
For conserved moments, it follows by definition that
$\widehat{g}_{0}=\widehat{g}_{1}=\widehat{g}_{2}=\widehat{g}_{3}=0$. Again,
exploiting the orthogonal property of $\mathcal{K}$, the moments of the
collision kernel can be obtained which are presented in Appendix F.
The central moment LBE given in Eq. (19) can be rewritten in terms of the
collision and streaming steps, respectively, as
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t)$
$\displaystyle=$
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)},$
(35)
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)$
$\displaystyle=$
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t),$ (36)
where the symbol “tilde” ($\sim$) in the first equation refers to the post-
collision state. Furthermore, the conserved local fluid density and momentum
are finally written in terms of the moments of the transformed distribution
functions as
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{26}\overline{f}_{\alpha}=\braket{\overline{f}_{\alpha}}{\rho},$
(37) $\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
i}+\frac{1}{2}F_{i}=\braket{\overline{f}_{\alpha}}{e_{\alpha
i}}+\frac{1}{2}F_{i},\qquad i\in{x,y,z}.$ (38)
## 7 Structure of the Central Moment Collision Operator
We are now in a position to obtain the expressions for the collision kernel of
the 3D central moment LBM in the presence of source terms. In essence, the
procedure begins with the lowest order (i.e. second-order, off-diagonal) post-
collision central moments (i.e.
$\widetilde{\widehat{\overline{\kappa}}}_{xy},\widetilde{\widehat{\overline{\kappa}}}_{xz}$
and $\widetilde{\widehat{\overline{\kappa}}}_{yz}$), which are successively
set equal to the corresponding attractors given in Eq. (27) (i.e.
$\widehat{\overline{\kappa}}_{xy}^{at},\widehat{\overline{\kappa}}_{xz}^{at}$
and $\widehat{\overline{\kappa}}_{yz}^{at}$, respectively). This intermediate
step is based on an equilibrium assumption. Dropping this modeling assumption
to represent collision as a relaxation process by multiplying with a
corresponding relaxation parameter results in the collision kernels
$\widehat{g}_{\alpha}$ for a given order [36]. Here, the relaxation parameter
needs to be carefully applied to only those terms that are not yet in post-
collision states, i.e. those that do not contain $\widehat{g}_{\beta}$, where
$\beta=0,1,2,\ldots,\alpha-1$ for a given $\widehat{g}_{\alpha}$. Then the
results are transformed in terms of raw moments via the binomial theorem to
yield expressions useful for computations. The details of various elements in
obtaining the collision kernel are presented in [40]. To simplify exposition,
let us introduce the following notation:
$\widehat{\overline{\eta}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}=\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}+\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}},$
(39)
where the expressions for
$\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ and
$\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ are known from Sec. 6. In
the following, for brevity, we present only the final results. For the above
three off-diagonal central moments, we get
$\widehat{g}_{4}=\frac{\omega_{4}}{12}\left[-\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+\rho
u_{x}u_{y}+\frac{1}{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x})\right],$
(40)
$\widehat{g}_{5}=\frac{\omega_{5}}{12}\left[-\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+\rho
u_{x}u_{z}+\frac{1}{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x})\right],$
(41)
$\widehat{g}_{6}=\frac{\omega_{6}}{12}\left[-\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+\rho
u_{y}u_{z}+\frac{1}{2}(\widehat{\sigma}_{y}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{y})\right].$
(42)
where $\omega_{4}$, $\omega_{5}$ and $\omega_{6}$ are relaxation parameters.
Similarly, applying the procedure for the remaining three second-order
diagonal components with
$\widehat{\overline{\kappa}}_{xx}^{at}=\widehat{\overline{\kappa}}_{yy}^{at}=\widehat{\overline{\kappa}}_{zz}^{at}=c_{s}^{2}\rho$,
which preserves the Maxwellian values to provide the correct momentum flux and
pressure tensor, yields
$\widehat{g}_{7}=\frac{\omega_{7}}{12}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})+\rho(u_{x}^{2}-u_{y}^{2})+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}-\widehat{\sigma}_{y}^{{}^{\prime}}u_{y})\right],$
(43) $\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\frac{\omega_{8}}{36}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-2\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})+\rho(u_{x}^{2}+u_{y}^{2}-2u_{z}^{2})\right.$
(44)
$\displaystyle\left.+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}-2\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right],$
$\displaystyle\widehat{g}_{9}$ $\displaystyle=$
$\displaystyle\frac{\omega_{9}}{18}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})+\rho(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})\right.$
(45)
$\displaystyle\left.+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})+\rho\right].$
Next, carrying out the above matching, transformation, and relaxation steps
(with the last of this applicable only for the pre-collision terms)
successively to all the seven components of the third-order moments we get
$\displaystyle\widehat{g}_{10}$ $\displaystyle=$
$\displaystyle\frac{\omega_{10}}{24}\left[-(\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}})+2(u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}})+u_{x}(\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})\right.$
(46) $\displaystyle\left.-2\rho
u_{x}(u_{y}^{2}+u_{z}^{2})-\frac{1}{2}\widehat{\sigma}_{x}^{{}^{\prime}}(u_{y}^{2}+u_{z}^{2})-u_{x}(\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right]$
$\displaystyle+(u_{y}\widehat{g}_{4}+u_{z}\widehat{g}_{5})+\frac{1}{4}u_{x}(-\widehat{g}_{7}-\widehat{g}_{8}+2\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{11}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}}{24}\left[-(\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}})+2(u_{x}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+u_{y}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})\right.$
(47) $\displaystyle\left.-2\rho
u_{y}(u_{x}^{2}+u_{z}^{2})-\frac{1}{2}\widehat{\sigma}_{y}^{{}^{\prime}}(u_{x}^{2}+u_{z}^{2})-u_{y}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right]$
$\displaystyle+(u_{x}\widehat{g}_{4}+u_{z}\widehat{g}_{6})+\frac{1}{4}u_{y}(\widehat{g}_{7}-\widehat{g}_{8}+2\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{12}$ $\displaystyle=$
$\displaystyle\frac{\omega_{12}}{24}\left[-(\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}})+2(u_{x}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+u_{z}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})\right.$
(48) $\displaystyle\left.-2\rho
u_{z}(u_{x}^{2}+u_{y}^{2})-\frac{1}{2}\widehat{\sigma}_{z}^{{}^{\prime}}(u_{x}^{2}+u_{y}^{2})-u_{z}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y})\right]$
$\displaystyle+(u_{x}\widehat{g}_{5}+u_{y}\widehat{g}_{6})+\frac{1}{2}u_{z}(\widehat{g}_{8}+\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{13}$ $\displaystyle=$
$\displaystyle\frac{\omega_{13}}{8}\left[-(\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}-\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}})+2(u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}})+u_{x}(\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})\right.$
(49) $\displaystyle\left.-2\rho
u_{x}(u_{y}^{2}-u_{z}^{2})-\frac{1}{2}\widehat{\sigma}_{x}^{{}^{\prime}}(u_{y}^{2}-u_{z}^{2})-u_{x}(\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}-\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right]$
$\displaystyle+3(u_{y}\widehat{g}_{4}-u_{z}\widehat{g}_{5})+\frac{3}{4}u_{x}(-\widehat{g}_{7}+3\widehat{g}_{8}),$
$\displaystyle\widehat{g}_{14}$ $\displaystyle=$
$\displaystyle\frac{\omega_{14}}{8}\left[-(\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}})+2(u_{x}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+u_{y}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})\right.$
(50) $\displaystyle\left.-2\rho
u_{y}(u_{x}^{2}-u_{z}^{2})-\frac{1}{2}\widehat{\sigma}_{y}^{{}^{\prime}}(u_{x}^{2}-u_{z}^{2})-u_{y}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}-\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right]$
$\displaystyle+3(u_{x}\widehat{g}_{4}-u_{z}\widehat{g}_{6})+\frac{3}{4}u_{y}(\widehat{g}_{7}+3\widehat{g}_{8}),$
$\displaystyle\widehat{g}_{15}$ $\displaystyle=$
$\displaystyle\frac{\omega_{15}}{8}\left[-(\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}})+2(u_{x}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}-u_{y}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+u_{z}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})\right.$
(51) $\displaystyle\left.-2\rho
u_{z}(u_{x}^{2}-u_{y}^{2})-\frac{1}{2}\widehat{\sigma}_{z}^{{}^{\prime}}(u_{x}^{2}-u_{y}^{2})-u_{z}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}-\widehat{\sigma}_{y}^{{}^{\prime}}u_{y})\right]$
$\displaystyle+3(u_{x}\widehat{g}_{5}-u_{y}\widehat{g}_{6})+\frac{3}{2}u_{z}\widehat{g}_{7},$
$\displaystyle\widehat{g}_{16}$ $\displaystyle=$
$\displaystyle\frac{\omega_{16}}{8}\left[-\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-2\rho
u_{x}u_{y}u_{z}\right.$ (52)
$\displaystyle\left.-\frac{1}{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}u_{z}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x}u_{y})\right]$
$\displaystyle+\frac{3}{2}(u_{z}\widehat{g}_{4}+u_{y}\widehat{g}_{5}+u_{z}\widehat{g}_{6}),$
Notice that the cascaded structure is evident for the collision kernel
starting from the third-order moments. Now, the next three diagonal components
of the fourth-order central moments needs to carefully consider the non-zero
factorized attractors given in terms of second-order components, i.e.
$\widehat{\overline{\kappa}}^{at}_{xxyy}=\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}$,
$\widehat{\overline{\kappa}}^{at}_{xxzz}=\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz}$,
and
$\widehat{\overline{\kappa}}^{at}_{yyzz}=\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz}$
(see Eq. (27)). This yields the corresponding collision kernels as
$\displaystyle\widehat{g}_{17}$ $\displaystyle=$
$\displaystyle\frac{\omega_{17}}{12}\left[-(\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{xxzz}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yyzz}^{{}^{\prime}})+2\left(u_{x}(\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}})+u_{y}(\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}+\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}})+\right.\right.$
(53)
$\displaystyle\left.\left.u_{z}(\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}})\right)-u_{x}^{2}(\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})-u_{y}^{2}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})-u_{z}^{2}(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})\right.$
$\displaystyle\left.-4(u_{x}u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{x}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{y}u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+(\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}+\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz}+\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz})\right.$
$\displaystyle\left.+3\rho(u_{x}^{2}u_{y}^{2}+u_{x}^{2}u_{z}^{2}+u_{y}^{2}u_{z}^{2})+u_{x}^{2}(u_{y}\widehat{\sigma}_{y}^{{}^{\prime}}+u_{z}\widehat{\sigma}_{z}^{{}^{\prime}})+u_{y}^{2}(u_{x}\widehat{\sigma}_{x}^{{}^{\prime}}+u_{z}\widehat{\sigma}_{z}^{{}^{\prime}})\right.$
$\displaystyle\left.+u_{z}^{2}(u_{x}\widehat{\sigma}_{x}^{{}^{\prime}}+u_{y}\widehat{\sigma}_{y}^{{}^{\prime}})\right]-4u_{x}u_{y}\widehat{g}_{4}-4u_{x}u_{z}\widehat{g}_{5}-4u_{y}u_{z}\widehat{g}_{6}$
$\displaystyle+\frac{1}{2}(u_{x}^{2}-u_{y}^{2})\widehat{g}_{7}+\frac{1}{2}(u_{x}^{2}+u_{y}^{2}-2u_{z}^{2})\widehat{g}_{8}+\frac{1}{2}(-2u_{x}^{2}-2u_{y}^{2}-u_{z}^{2}-4)\widehat{g}_{9}$
$\displaystyle+4u_{x}\widehat{g}_{10}+4u_{y}\widehat{g}_{11}+4u_{z}\widehat{g}_{12},$
$\displaystyle\widehat{g}_{18}$ $\displaystyle=$
$\displaystyle\frac{\omega_{18}}{24}\left[-(\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{xxzz}^{{}^{\prime}}-2\widehat{\overline{\eta}}_{yyzz}^{{}^{\prime}})+2\left(u_{x}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}\right.\right.$
(54)
$\displaystyle\left.\left.-2(u_{y}\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}})\right)-u_{x}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}-u_{y}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-u_{z}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+2u_{y}^{2}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}+\right.$
$\displaystyle\left.+2u_{z}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-4(u_{x}u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{x}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}-2u_{y}u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}})+(\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}+\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz}-2\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz})\right.$
$\displaystyle\left.+3\rho(u_{x}^{2}u_{y}^{2}+u_{x}^{2}u_{z}^{2}-2u_{y}^{2}u_{z}^{2})+u_{x}^{2}u_{y}\widehat{\sigma}_{y}^{{}^{\prime}}+u_{x}^{2}u_{z}\widehat{\sigma}_{z}^{{}^{\prime}}+u_{y}^{2}u_{x}\widehat{\sigma}_{x}^{{}^{\prime}}+u_{z}^{2}u_{x}\widehat{\sigma}_{x}^{{}^{\prime}}\right.$
$\displaystyle\left.-2u_{y}^{2}u_{z}\widehat{\sigma}_{z}^{{}^{\prime}}-2u_{z}^{2}u_{y}\widehat{\sigma}_{y}^{{}^{\prime}})\right]-2u_{x}u_{y}\widehat{g}_{4}-2u_{x}u_{z}\widehat{g}_{5}+4u_{y}u_{z}\widehat{g}_{6}$
$\displaystyle+\frac{1}{4}(u_{x}^{2}-u_{y}^{2}-3u_{z}^{2}-2)\widehat{g}_{7}+\frac{1}{4}(u_{x}^{2}-5u_{y}^{2}+u_{z}^{2}-2)\widehat{g}_{8}+\frac{1}{4}(-2u_{x}^{2}+u_{y}^{2}$
$\displaystyle+2u_{z}^{2})\widehat{g}_{9}+2u_{x}\widehat{g}_{10}-u_{y}\widehat{g}_{11}-u_{z}\widehat{g}_{12}+u_{y}\widehat{g}_{14}+u_{z}\widehat{g}_{15},$
$\displaystyle\widehat{g}_{19}$ $\displaystyle=$
$\displaystyle\frac{\omega_{19}}{8}\left[-(\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}-\widehat{\overline{\eta}}_{xxzz}^{{}^{\prime}})+2\left(u_{x}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}-u_{x}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}-u_{z}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}\right)\right.$
(55)
$\displaystyle\left.-(u_{x}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-u_{z}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}})-4(u_{x}u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-u_{x}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}})\right.$
$\displaystyle\left.+(\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}-\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{zz})+3\rho(u_{x}^{2}u_{y}^{2}-u_{x}^{2}u_{z}^{2})\right.$
$\displaystyle+\left.(u_{x}^{2}u_{y}\widehat{\sigma}_{y}^{{}^{\prime}}-u_{x}^{2}u_{z}\widehat{\sigma}_{z}^{{}^{\prime}}+u_{y}^{2}u_{x}\widehat{\sigma}_{x}^{{}^{\prime}}-u_{z}^{2}u_{x}\widehat{\sigma}_{x}^{{}^{\prime}})\right]-6u_{x}u_{y}\widehat{g}_{4}+6u_{x}u_{z}\widehat{g}_{5}$
$\displaystyle+\frac{1}{4}(3u_{x}^{2}-3u_{y}^{2}+3u_{z}^{2}+2)\widehat{g}_{7}+\frac{1}{4}(-9u_{x}^{2}-3u_{y}^{2}+3u_{z}^{2}-6)\widehat{g}_{8}$
$\displaystyle+\frac{1}{4}(-3u_{y}^{2}-8)\widehat{g}_{9}+3u_{y}\widehat{g}_{11}-3u_{z}\widehat{g}_{12}+2u_{x}\widehat{g}_{13}+u_{y}\widehat{g}_{14}-u_{z}\widehat{g}_{15}.$
For calculating $\widehat{g}_{17}$ through $\widehat{g}_{19}$ in the above
equations, we need the post collision states
$\widetilde{\widehat{\kappa}}_{xx}$, $\widetilde{\widehat{\kappa}}_{yy}$ and
$\widetilde{\widehat{\kappa}}_{zz}$. These can be obtained from Eq. (22) as
follows.
$\displaystyle\widetilde{\widehat{\kappa}}_{xx}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{xx}+\frac{1}{2}\widehat{\sigma}_{xx},$
$\displaystyle\widetilde{\widehat{\kappa}}_{yy}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{yy}+\frac{1}{2}\widehat{\sigma}_{yy},$
$\displaystyle\widetilde{\widehat{\kappa}}_{zz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{zz}+\frac{1}{2}\widehat{\sigma}_{zz},$
where the second-order transformed central moments, in turn, can be related to
corresponding raw moments, which are known, as
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{xx}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}-F_{x}u_{x},$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{yy}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}-F_{y}u_{y},$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{zz}$ $\displaystyle=$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{zz}^{{}^{\prime}}-\rho
u_{z}^{2}-F_{z}u_{z}.$
Note that in terms of
$\widehat{\overline{\eta}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ these can also be
written as
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{xx}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+6\widehat{g}_{7}+6\widehat{g}_{8}+6\widehat{g}_{9}\right]-\rho
u_{x}^{2}-F_{x}u_{x},$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{yy}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-6\widehat{g}_{7}+6\widehat{g}_{8}+6\widehat{g}_{9}\right]-\rho
u_{y}^{2}-F_{y}u_{y},$
$\displaystyle\widetilde{\widehat{\overline{\kappa}}}_{zz}$ $\displaystyle=$
$\displaystyle\left[\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-12\widehat{g}_{8}+6\widehat{g}_{9}\right]-\rho
u_{z}^{2}-F_{z}u_{z}.$
Proceeding further for the remaining three fourth-order central moments using
$\widehat{\kappa}^{at}_{xxyz}=\widehat{\kappa}^{at}_{xyyz}=\widehat{\kappa}^{at}_{xyzz}=0$,
we get
$\displaystyle\widehat{g}_{20}$ $\displaystyle=$
$\displaystyle\frac{\omega_{20}}{8}\left[-\widehat{\overline{\eta}}_{xxyz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}-u_{y}u_{z}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-2u_{x}u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}\right.$
(56)
$\displaystyle\left.-2u_{x}u_{y}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+3\rho
u_{x}^{2}u_{y}u_{z}+\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}u_{y}u_{z}+\frac{1}{2}u_{x}^{2}(\widehat{\sigma}_{y}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{y})\right]$
$\displaystyle-3u_{x}u_{z}\widehat{g}_{4}-3u_{x}u_{y}\widehat{g}_{5}-\left(\frac{3}{2}u_{x}^{2}+1\right)\widehat{g}_{6}-\frac{3}{4}u_{y}u_{z}\widehat{g}_{7}-\frac{3}{4}u_{y}u_{z}\widehat{g}_{8}$
$\displaystyle-\frac{3}{4}u_{y}u_{z}\widehat{g}_{9}+\frac{3}{2}u_{z}\widehat{g}_{11}+\frac{3}{2}u_{y}\widehat{g}_{12}+\frac{1}{2}u_{z}\widehat{g}_{14}+\frac{1}{2}u_{y}\widehat{g}_{15}+2u_{x}\widehat{g}_{16},$
$\displaystyle\widehat{g}_{21}$ $\displaystyle=$
$\displaystyle\frac{\omega_{21}}{8}\left[-\widehat{\overline{\eta}}_{xyyz}^{{}^{\prime}}+2u_{y}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}}-u_{y}^{2}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}-2u_{y}u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}\right.$
(57)
$\displaystyle\left.-2u_{x}u_{y}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}-u_{x}u_{z}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+3\rho
u_{x}u_{y}^{2}u_{z}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x}u_{y}u_{z}+\frac{1}{2}u_{y}^{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x})\right]$
$\displaystyle-3u_{y}u_{z}\widehat{g}_{4}-\left(\frac{3}{2}u_{y}^{2}+1\right)\widehat{g}_{5}-3u_{x}u_{y}\widehat{g}_{6}+\frac{3}{4}u_{x}u_{z}\widehat{g}_{7}-\frac{3}{4}u_{x}u_{z}\widehat{g}_{8}$
$\displaystyle-\frac{3}{4}u_{x}u_{z}\widehat{g}_{9}+\frac{3}{2}u_{z}\widehat{g}_{10}+\frac{3}{2}u_{x}\widehat{g}_{12}+\frac{1}{2}u_{z}\widehat{g}_{13}-\frac{1}{2}u_{x}\widehat{g}_{15}+2u_{y}\widehat{g}_{16},$
$\displaystyle\widehat{g}_{22}$ $\displaystyle=$
$\displaystyle\frac{\omega_{22}}{8}\left[-\widehat{\overline{\eta}}_{xyzz}^{{}^{\prime}}+2u_{z}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}-u_{z}^{2}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-2u_{y}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}\right.$
(58)
$\displaystyle\left.-2u_{x}u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}-u_{x}u_{y}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}+3\rho
u_{x}u_{y}u_{z}^{2}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x}u_{y}u_{z}+\frac{1}{2}u_{z}^{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x})\right]$
$\displaystyle-\left(\frac{3}{2}u_{z}^{2}+1\right)\widehat{g}_{4}-3u_{y}u_{z}\widehat{g}_{5}-3u_{x}u_{z}\widehat{g}_{6}+\frac{3}{2}u_{x}u_{y}\widehat{g}_{8}$
$\displaystyle-\frac{3}{4}u_{x}u_{y}\widehat{g}_{9}+\frac{3}{2}u_{y}\widehat{g}_{10}+\frac{3}{2}u_{x}\widehat{g}_{11}-\frac{1}{2}u_{y}\widehat{g}_{13}-\frac{1}{2}u_{x}\widehat{g}_{14}+2u_{z}\widehat{g}_{16}.$
The collision kernels for the three fifth-order central moments follow
similarly from
$\widehat{\overline{\kappa}}^{at}_{xyyzz}=\widehat{\overline{\kappa}}^{at}_{xxyzz}=\widehat{\overline{\kappa}}^{at}_{xxyyz}=0$
as
$\displaystyle\widehat{g}_{23}$ $\displaystyle=$
$\displaystyle\frac{\omega_{23}}{8}\left[-\widehat{\overline{\eta}}_{xyyzz}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yyzz}^{{}^{\prime}}+2u_{y}\widehat{\overline{\eta}}_{xyzz}^{{}^{\prime}}+2u_{z}\widehat{\overline{\eta}}_{xyyz}^{{}^{\prime}}-u_{y}^{2}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}-u_{z}^{2}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}\right.$
(59)
$\displaystyle\left.-2u_{x}u_{y}\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}-2u_{x}u_{z}\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}}-4u_{y}u_{z}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{x}u_{z}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+u_{x}u_{y}^{2}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}\right.$
$\displaystyle+\left.2u_{y}u_{z}^{2}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+2u_{y}^{2}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+4u_{x}u_{y}u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}-4\rho
u_{x}u_{y}^{2}u_{z}^{2}-\frac{1}{2}u_{y}^{2}u_{z}^{2}\widehat{\sigma}_{x}^{{}^{\prime}}\right.$
$\displaystyle\left.-u_{x}(u_{z}^{2}u_{y}\widehat{\sigma}_{y}^{{}^{\prime}}+u_{y}^{2}u_{z}\widehat{\sigma}_{z}^{{}^{\prime}})\right]+(3u_{y}u_{z}^{2}+2u_{y})\widehat{g}_{4}+(3u_{y}^{2}u_{z}+2u_{z})\widehat{g}_{5}$
$\displaystyle+6u_{x}u_{y}u_{z}\widehat{g}_{6}+\left(\frac{3}{4}u_{x}u_{z}^{2}-\frac{1}{4}u_{x}\right)\widehat{g}_{7}+\left(\frac{3}{4}u_{x}u_{z}^{2}-\frac{3}{2}u_{x}u_{y}^{2}-\frac{1}{2}u_{x}\right)\widehat{g}_{8}$
$\displaystyle+\left(\frac{3}{4}u_{x}u_{z}^{2}+\frac{3}{4}u_{x}u_{y}^{2}+u_{x}\right)\widehat{g}_{9}+\left(-\frac{3}{2}u_{y}^{2}-\frac{3}{2}u_{z}^{2}-2\right)\widehat{g}_{10}-3u_{x}u_{y}\widehat{g}_{11}$
$\displaystyle-3u_{x}u_{z}\widehat{g}_{12}+\left(\frac{1}{2}u_{y}^{2}-\frac{1}{2}u_{z}^{2}\right)\widehat{g}_{13}+u_{x}u_{y}\widehat{g}_{14}+u_{x}u_{z}\widehat{g}_{15}-4u_{y}u_{z}\widehat{g}_{16}$
$\displaystyle+\frac{1}{2}u_{x}\widehat{g}_{17}-u_{x}\widehat{g}_{18}+2u_{z}\widehat{g}_{21}+2u_{y}\widehat{g}_{22},$
$\displaystyle\widehat{g}_{24}$ $\displaystyle=$
$\displaystyle\frac{\omega_{24}}{8}\left[-\widehat{\overline{\eta}}_{xxyzz}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xyzz}^{{}^{\prime}}+2u_{z}\widehat{\overline{\eta}}_{xxyz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xxzz}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}-u_{z}^{2}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}\right.$
(60)
$\displaystyle\left.-2u_{x}u_{y}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}-2u_{y}u_{z}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}-4u_{x}u_{z}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{y}u_{z}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+u_{x}^{2}u_{y}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}\right.$
$\displaystyle+\left.2u_{x}u_{z}^{2}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+2u_{x}^{2}u_{z}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+4u_{x}u_{y}u_{z}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}-4\rho
u_{x}^{2}u_{y}u_{z}^{2}-\frac{1}{2}u_{x}^{2}u_{z}^{2}\widehat{\sigma}_{y}^{{}^{\prime}}\right.$
$\displaystyle\left.-u_{y}(u_{x}u_{z}^{2}\widehat{\sigma}_{x}^{{}^{\prime}}+u_{x}^{2}u_{z}\widehat{\sigma}_{z}^{{}^{\prime}})\right]+(3u_{x}u_{z}^{2}+2u_{x})\widehat{g}_{4}+6u_{x}u_{y}u_{z}\widehat{g}_{5}$
$\displaystyle+(3u_{x}^{2}u_{z}+2u_{z})\widehat{g}_{6}+\left(\frac{3}{4}u_{y}u_{z}^{2}+\frac{1}{2}u_{y}\right)\widehat{g}_{7}+\left(\frac{3}{4}u_{y}u_{z}^{2}-\frac{3}{2}u_{x}^{2}u_{y}-\frac{1}{2}u_{y}\right)\widehat{g}_{8}$
$\displaystyle+\left(\frac{3}{4}u_{y}u_{z}^{2}+\frac{3}{4}u_{x}^{2}u_{y}+u_{y}\right)\widehat{g}_{9}-3u_{x}u_{y}\widehat{g}_{10}+\left(-\frac{3}{2}u_{x}^{2}-\frac{3}{2}u_{z}^{2}-2\right)\widehat{g}_{11}$
$\displaystyle-3u_{y}u_{z}\widehat{g}_{12}+u_{x}u_{y}\widehat{g}_{13}+\left(\frac{1}{2}u_{x}^{2}-\frac{1}{2}u_{z}^{2}\right)\widehat{g}_{14}-u_{y}u_{z}\widehat{g}_{15}-4u_{x}u_{z}\widehat{g}_{16}$
$\displaystyle+\frac{1}{2}u_{y}\widehat{g}_{17}+\frac{1}{2}u_{y}\widehat{g}_{18}-\frac{1}{2}u_{y}\widehat{g}_{19}+2u_{z}\widehat{g}_{20}+2u_{x}\widehat{g}_{22},$
$\displaystyle\widehat{g}_{25}$ $\displaystyle=$
$\displaystyle\frac{\omega_{25}}{8}\left[-\widehat{\overline{\eta}}_{xxyyz}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xyyz}^{{}^{\prime}}+2u_{y}\widehat{\overline{\eta}}_{xxyz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}}-u_{y}^{2}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}\right.$
(61)
$\displaystyle\left.-2u_{x}u_{z}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}-2u_{y}u_{z}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}-4u_{x}u_{y}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{y}^{2}u_{z}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+u_{x}^{2}u_{z}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}\right.$
$\displaystyle+\left.2u_{x}u_{y}^{2}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+2u_{x}^{2}u_{y}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+4u_{x}u_{y}u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-4\rho
u_{x}^{2}u_{y}^{2}u_{z}-\frac{1}{2}u_{x}^{2}u_{y}^{2}\widehat{\sigma}_{z}^{{}^{\prime}}\right.$
$\displaystyle\left.-u_{z}(u_{x}u_{y}^{2}\widehat{\sigma}_{x}^{{}^{\prime}}+u_{x}^{2}u_{y}\widehat{\sigma}_{y}^{{}^{\prime}})\right]+6u_{x}u_{y}u_{z}\widehat{g}_{4}+(3u_{x}u_{y}^{2}+2u_{x})\widehat{g}_{5}$
$\displaystyle+(3u_{x}^{2}u_{y}+2u_{y})\widehat{g}_{6}+\left(\frac{3}{4}u_{y}^{2}u_{z}-\frac{3}{4}u_{x}^{2}u_{z}\right)\widehat{g}_{7}+\left(\frac{3}{4}u_{y}^{2}u_{z}+\frac{3}{4}u_{x}^{2}u_{z}+u_{z}\right)\widehat{g}_{8}$
$\displaystyle+\left(\frac{3}{4}u_{y}^{2}u_{z}+\frac{3}{4}u_{x}^{2}u_{z}+u_{z}\right)\widehat{g}_{9}-3u_{x}u_{z}\widehat{g}_{10}-3u_{y}u_{z}\widehat{g}_{11}$
$\displaystyle+\left(-\frac{3}{2}u_{x}^{2}-\frac{3}{2}u_{y}^{2}-2\right)\widehat{g}_{12}-u_{x}u_{z}\widehat{g}_{13}-u_{y}u_{z}\widehat{g}_{14}+\left(\frac{1}{2}u_{x}^{2}-\frac{1}{2}u_{y}^{2}\right)\widehat{g}_{15}$
$\displaystyle-6u_{x}u_{y}\widehat{g}_{16}+\frac{1}{2}u_{z}\widehat{g}_{17}+\frac{1}{2}u_{z}\widehat{g}_{18}+\frac{1}{2}u_{z}\widehat{g}_{19}+2u_{y}\widehat{g}_{20}+2u_{x}\widehat{g}_{21}.$
Finally, for the one sixth-order component, we obtain the collision kernel
based on the non-zero factorized attractor (see Eq. (27)) as
$\displaystyle\widehat{g}_{26}$ $\displaystyle=$
$\displaystyle\frac{\omega_{26}}{8}\left[-\widehat{\overline{\eta}}_{xxyyzz}^{{}^{\prime}}+2\left(u_{x}\widehat{\overline{\eta}}_{xyyzz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xxyzz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xxyyz}^{{}^{\prime}}\right)-\left(u_{x}^{2}\widehat{\overline{\eta}}_{yyzz}^{{}^{\prime}}\right.\right.$
(62)
$\displaystyle+\left.\left.u_{y}^{2}\widehat{\overline{\eta}}_{xxzz}^{{}^{\prime}}+u_{z}^{2}\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}\right)-4\left(u_{x}u_{y}\widehat{\overline{\eta}}_{xyzz}^{{}^{\prime}}+u_{x}u_{z}\widehat{\overline{\eta}}_{xyyz}^{{}^{\prime}}+u_{y}u_{z}\widehat{\overline{\eta}}_{xxyz}^{{}^{\prime}}\right)\right.$
$\displaystyle\left.+2\left(u_{x}^{2}u_{y}\widehat{\overline{\eta}}_{yzz}^{{}^{\prime}}+u_{x}u_{y}^{2}\widehat{\overline{\eta}}_{xzz}^{{}^{\prime}}+u_{x}^{2}u_{z}\widehat{\overline{\eta}}_{yyz}^{{}^{\prime}}+u_{x}u_{z}^{2}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+u_{y}^{2}u_{z}\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}\right.\right.$
$\displaystyle\left.\left.+u_{y}u_{z}^{2}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}\right)+8u_{x}u_{y}u_{z}\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}-\left(u_{y}^{2}u_{z}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+u_{x}^{2}u_{z}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+u_{x}^{2}u_{y}^{2}\widehat{\overline{\eta}}_{zz}^{{}^{\prime}}\right)\right.$
$\displaystyle\left.-4u_{x}u_{y}u_{z}\left(u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}\right)+5\rho
u_{x}^{2}u_{y}^{2}u_{z}^{2}+\right.$
$\displaystyle\left.\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}\widetilde{\widehat{\kappa}}_{zz}+u_{x}u_{y}u_{z}\left(u_{x}u_{y}\widehat{\sigma}_{z}^{{}^{\prime}}+u_{x}u_{z}\widehat{\sigma}_{y}^{{}^{\prime}}+u_{y}u_{z}\widehat{\sigma}_{x}^{{}^{\prime}}\right)\right]$
$\displaystyle+\left(-4u_{x}u_{y}-6u_{x}u_{y}u_{z}^{2}\right)\widehat{g}_{4}+\left(-4u_{x}u_{z}-6u_{x}u_{y}^{2}u_{z}\right)\widehat{g}_{5}$
$\displaystyle+\left(-4u_{y}u_{z}-6u_{x}^{2}u_{y}u_{z}\right)\widehat{g}_{6}+\left(\frac{1}{2}u_{x}^{2}-\frac{1}{2}u_{y}^{2}+\frac{3}{4}u_{x}^{2}u_{z}^{2}-\frac{3}{4}u_{y}^{2}u_{z}^{2}\right)\widehat{g}_{7}$
$\displaystyle+\left(\frac{1}{2}u_{x}^{2}+\frac{1}{2}u_{y}^{2}-u_{z}^{2}-\frac{3}{4}u_{y}^{2}u_{z}^{2}-\frac{3}{4}u_{x}^{2}u_{z}^{2}+\frac{3}{2}u_{x}^{2}u_{y}^{2}\right)\widehat{g}_{8}$
$\displaystyle+\left(-u_{x}^{2}-u_{y}^{2}-u_{z}^{2}-\frac{3}{4}u_{x}^{2}u_{y}^{2}-\frac{3}{4}u_{x}^{2}u_{z}^{2}-\frac{3}{4}u_{y}^{2}u_{z}^{2}-1\right)\widehat{g}_{9}$
$\displaystyle+\left(3u_{x}u_{y}^{2}+3u_{x}u_{z}^{2}+4u_{x}\right)\widehat{g}_{10}+\left(3u_{x}^{2}u_{y}+3u_{y}u_{z}^{2}+4u_{y}\right)\widehat{g}_{11}$
$\displaystyle+\left(3u_{x}^{2}u_{z}+3u_{y}^{2}u_{z}+4u_{z}\right)\widehat{g}_{12}+\left(u_{x}u_{z}^{2}-u_{x}u_{y}^{2}\right)\widehat{g}_{13}$
$\displaystyle+\left(u_{y}u_{z}^{2}-u_{x}^{2}u_{y}\right)\widehat{g}_{14}+\left(u_{y}^{2}u_{z}-u_{x}^{2}u_{z}\right)\widehat{g}_{15}+8u_{x}u_{y}u_{z}\widehat{g}_{16}$
$\displaystyle+\left(-\frac{1}{2}u_{x}^{2}-\frac{1}{2}u_{y}^{2}-\frac{1}{2}u_{z}^{2}-1\right)\widehat{g}_{17}+\left(u_{x}^{2}-\frac{1}{2}u_{y}^{2}-\frac{1}{2}u_{z}^{2}\right)\widehat{g}_{18}$
$\displaystyle+\left(\frac{1}{2}u_{y}^{2}-\frac{1}{2}u_{z}^{2}\right)\widehat{g}_{19}-4u_{y}u_{z}\widehat{g}_{20}-4u_{x}u_{z}\widehat{g}_{21}-4u_{x}u_{y}\widehat{g}_{22}+2u_{x}\widehat{g}_{23}$
$\displaystyle+2u_{y}\widehat{g}_{24}+2u_{z}\widehat{g}_{25}.$
Note that the transformed raw moments of various orders, i.e.
$\widehat{\overline{\kappa}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ and raw source
moments, i.e. $\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ needed for
$\widehat{\overline{\eta}}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}$ for various $m$,
$n$ and $p$ combinations can be obtained from Eqs. (32) and (65) and Eq. (66),
respectively, which are given in Sec. 6. Similar to the 2D central moment LBM
with source terms [40], we can apply the Chapman-Enskog expansion to the above
3D formulation to show that its emergent dynamics corresponds to the Navier-
Stokes equations representing fluid motion in the presence of general force
fields. Some of the relaxation parameters in the collision model can be
related to the transport coefficients. For example, those corresponding to the
second-order moments control shear viscosity $\nu$ of the fluid. That is,
$\nu=c_{s}^{2}\left(\frac{1}{\omega^{\nu}}-\frac{1}{2}\right)$ where
$\omega^{\nu}=\omega_{j}$ where $j=4,5,6,7,8$. The rest of the parameters can
be set either to $1$ (i.e. equilibration) or adjusted independently to
carefully control and improve numerical stability by means of a linear
stability analysis, while all satisfying the usual bounds
$0<\omega_{\beta}<2$.
## 8 Operational Steps of the Central Moment LBM
To provide explicit expressions for the collision step in the central moment
LBM as a stream-and-collide procedure (i.e. Eq. (35) and (36)), we first
expand the elements of the matrix multiplication of $\mathcal{K}$ with
$\widehat{\mathbf{g}}$ in Eq. (16). This yields the post-collision values of
all the 27 components of the transformed distribution function in terms of the
Galilean invariant collision kernel $\widehat{g}_{\beta}$ (see Sec. 7) and
source terms $S_{\beta}$ (see Eq. (68) in Appendix. E) which can be summarized
as follows:
$\displaystyle\widetilde{\overline{f}}_{0}$ $\displaystyle=$
$\displaystyle\overline{f}_{0}+\left[\widehat{g}_{0}-2\widehat{g}_{9}+4\widehat{g}_{17}-8\widehat{g}_{26}\right]+S_{0},$
$\displaystyle\widetilde{\overline{f}}_{1}$ $\displaystyle=$
$\displaystyle\overline{f}_{1}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}-4\widehat{g}_{10}-4\widehat{g}_{18}+4\widehat{g}_{23}+4\widehat{g}_{26}\right]+S_{1},$
$\displaystyle\widetilde{\overline{f}}_{2}$ $\displaystyle=$
$\displaystyle\overline{f}_{2}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}+4\widehat{g}_{10}-4\widehat{g}_{18}-4\widehat{g}_{23}+4\widehat{g}_{26}\right]+S_{2},$
$\displaystyle\widetilde{\overline{f}}_{3}$ $\displaystyle=$
$\displaystyle\overline{f}_{3}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}-4\widehat{g}_{11}+2\widehat{g}_{18}-2\widehat{g}_{19}+4\widehat{g}_{24}+4\widehat{g}_{26}\right]$
$\displaystyle+S_{3},$ $\displaystyle\widetilde{\overline{f}}_{4}$
$\displaystyle=$
$\displaystyle\overline{f}_{4}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}+4\widehat{g}_{11}+2\widehat{g}_{18}-2\widehat{g}_{19}-4\widehat{g}_{24}+4\widehat{g}_{26}\right]$
$\displaystyle+S_{4},$ $\displaystyle\widetilde{\overline{f}}_{5}$
$\displaystyle=$
$\displaystyle\overline{f}_{5}+\left[\widehat{g}_{0}+\widehat{g}_{3}-2\widehat{g}_{8}-\widehat{g}_{9}-4\widehat{g}_{12}+2\widehat{g}_{18}+2\widehat{g}_{19}+4\widehat{g}_{25}+4\widehat{g}_{26}\right]$
$\displaystyle+S_{5},$ $\displaystyle\widetilde{\overline{f}}_{6}$
$\displaystyle=$
$\displaystyle\overline{f}_{6}+\left[\widehat{g}_{0}-\widehat{g}_{3}-2\widehat{g}_{8}-\widehat{g}_{9}+4\widehat{g}_{12}+2\widehat{g}_{18}+2\widehat{g}_{19}-4\widehat{g}_{25}+4\widehat{g}_{26}\right]$
$\displaystyle+S_{6},$ $\displaystyle\widetilde{\overline{f}}_{7}$
$\displaystyle=$
$\displaystyle\overline{f}_{7}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+\widehat{g}_{4}+2\widehat{g}_{8}-\widehat{g}_{10}-\widehat{g}_{11}+\widehat{g}_{13}+\widehat{g}_{14}-\widehat{g}_{17}+\widehat{g}_{18}+\right.$
$\displaystyle\left.\widehat{g}_{19}-2\widehat{g}_{22}-2\widehat{g}_{23}-2\widehat{g}_{24}-2\widehat{g}_{26}\right]+S_{7},$
$\displaystyle\widetilde{\overline{f}}_{8}$ $\displaystyle=$
$\displaystyle\overline{f}_{8}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}-\widehat{g}_{4}+2\widehat{g}_{8}+\widehat{g}_{10}-\widehat{g}_{11}-\widehat{g}_{13}+\widehat{g}_{14}-\widehat{g}_{17}+\widehat{g}_{18}+\right.$
$\displaystyle\left.\widehat{g}_{19}+2\widehat{g}_{22}+2\widehat{g}_{23}-2\widehat{g}_{24}-2\widehat{g}_{26}\right]+S_{8},$
$\displaystyle\widetilde{\overline{f}}_{9}$ $\displaystyle=$
$\displaystyle\overline{f}_{9}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}-\widehat{g}_{4}+2\widehat{g}_{8}-\widehat{g}_{10}+\widehat{g}_{11}+\widehat{g}_{13}-\widehat{g}_{14}-\widehat{g}_{17}+\widehat{g}_{18}+\right.$
$\displaystyle\left.\widehat{g}_{19}+2\widehat{g}_{22}-2\widehat{g}_{23}+2\widehat{g}_{24}-2\widehat{g}_{26}\right]+S_{9},$
$\displaystyle\widetilde{\overline{f}}_{10}$ $\displaystyle=$
$\displaystyle\overline{f}_{10}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+\widehat{g}_{4}+2\widehat{g}_{8}+\widehat{g}_{10}+\widehat{g}_{11}-\widehat{g}_{13}-\widehat{g}_{14}-\widehat{g}_{17}+\widehat{g}_{18}+\right.$
$\displaystyle\left.\widehat{g}_{19}-2\widehat{g}_{22}+2\widehat{g}_{23}+2\widehat{g}_{24}-2\widehat{g}_{26}\right]+S_{10},$
$\displaystyle\widetilde{\overline{f}}_{11}$ $\displaystyle=$
$\displaystyle\overline{f}_{11}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{3}+\widehat{g}_{5}+\widehat{g}_{7}-\widehat{g}_{8}-\widehat{g}_{10}-\widehat{g}_{12}-\widehat{g}_{13}+\widehat{g}_{15}-\widehat{g}_{17}+\right.$
$\displaystyle\left.\widehat{g}_{18}-\widehat{g}_{19}-2\widehat{g}_{21}-2\widehat{g}_{23}-2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{11},$
$\displaystyle\widetilde{\overline{f}}_{12}$ $\displaystyle=$
$\displaystyle\overline{f}_{12}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{3}-\widehat{g}_{5}+\widehat{g}_{7}-\widehat{g}_{8}+\widehat{g}_{10}-\widehat{g}_{12}+\widehat{g}_{13}+\widehat{g}_{15}-\widehat{g}_{17}+\right.$
$\displaystyle\left.\widehat{g}_{18}-\widehat{g}_{19}+2\widehat{g}_{21}+2\widehat{g}_{23}-2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{12},$
$\displaystyle\widetilde{\overline{f}}_{13}$ $\displaystyle=$
$\displaystyle\overline{f}_{13}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{3}-\widehat{g}_{5}+\widehat{g}_{7}-\widehat{g}_{8}-\widehat{g}_{10}+\widehat{g}_{12}-\widehat{g}_{13}-\widehat{g}_{15}-\widehat{g}_{17}+\right.$
$\displaystyle\left.\widehat{g}_{18}-\widehat{g}_{19}+2\widehat{g}_{21}-2\widehat{g}_{23}+2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{13},$
$\displaystyle\widetilde{\overline{f}}_{14}$ $\displaystyle=$
$\displaystyle\overline{f}_{14}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{5}+\widehat{g}_{7}-\widehat{g}_{8}+\widehat{g}_{10}+\widehat{g}_{12}+\widehat{g}_{13}-\widehat{g}_{15}-\widehat{g}_{17}+\right.$
$\displaystyle\left.\widehat{g}_{18}-\widehat{g}_{19}-2\widehat{g}_{21}+2\widehat{g}_{23}+2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{14},$
$\displaystyle\widetilde{\overline{f}}_{15}$ $\displaystyle=$
$\displaystyle\overline{f}_{15}+\left[\widehat{g}_{0}+\widehat{g}_{2}+\widehat{g}_{3}+\widehat{g}_{6}-\widehat{g}_{7}-\widehat{g}_{8}-\widehat{g}_{11}-\widehat{g}_{12}-\widehat{g}_{14}-\widehat{g}_{15}-\widehat{g}_{17}-\right.$
$\displaystyle\left.2\widehat{g}_{18}-2\widehat{g}_{20}-2\widehat{g}_{24}-2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{15},$
$\displaystyle\widetilde{\overline{f}}_{16}$ $\displaystyle=$
$\displaystyle\overline{f}_{16}+\left[\widehat{g}_{0}-\widehat{g}_{2}+\widehat{g}_{3}-\widehat{g}_{6}-\widehat{g}_{7}-\widehat{g}_{8}+\widehat{g}_{11}-\widehat{g}_{12}+\widehat{g}_{14}-\widehat{g}_{15}-\widehat{g}_{17}-\right.$
$\displaystyle\left.2\widehat{g}_{18}+2\widehat{g}_{20}+2\widehat{g}_{24}-2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{16},$
$\displaystyle\widetilde{\overline{f}}_{17}$ $\displaystyle=$
$\displaystyle\overline{f}_{17}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{6}-\widehat{g}_{7}-\widehat{g}_{8}-\widehat{g}_{11}+\widehat{g}_{12}-\widehat{g}_{14}+\widehat{g}_{15}-\widehat{g}_{17}-\right.$
$\displaystyle\left.2\widehat{g}_{18}+2\widehat{g}_{20}-2\widehat{g}_{24}+2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{17},$
$\displaystyle\widetilde{\overline{f}}_{18}$ $\displaystyle=$
$\displaystyle\overline{f}_{18}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{3}+\widehat{g}_{6}-\widehat{g}_{7}-\widehat{g}_{8}+\widehat{g}_{11}+\widehat{g}_{12}+\widehat{g}_{14}+\widehat{g}_{15}-\widehat{g}_{17}-\right.$
$\displaystyle\left.2\widehat{g}_{18}-2\widehat{g}_{20}+2\widehat{g}_{24}+2\widehat{g}_{25}-2\widehat{g}_{26}\right]+S_{18},$
$\displaystyle\widetilde{\overline{f}}_{19}$ $\displaystyle=$
$\displaystyle\overline{f}_{19}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+\widehat{g}_{3}+\widehat{g}_{4}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}+2\widehat{g}_{11}+2\widehat{g}_{12}+\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}+\widehat{g}_{20}+\widehat{g}_{21}+\widehat{g}_{22}+\widehat{g}_{23}+\widehat{g}_{24}+\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{19},$
$\displaystyle\widetilde{\overline{f}}_{20}$ $\displaystyle=$
$\displaystyle\overline{f}_{20}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}+\widehat{g}_{3}-\widehat{g}_{4}-\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}+2\widehat{g}_{11}+2\widehat{g}_{12}-\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}+\widehat{g}_{20}-\widehat{g}_{21}-\widehat{g}_{22}-\widehat{g}_{23}+\widehat{g}_{24}+\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{20},$
$\displaystyle\widetilde{\overline{f}}_{21}$ $\displaystyle=$
$\displaystyle\overline{f}_{21}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}+\widehat{g}_{3}-\widehat{g}_{4}+\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}-2\widehat{g}_{11}+2\widehat{g}_{12}-\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}-\widehat{g}_{20}+\widehat{g}_{21}-\widehat{g}_{22}+\widehat{g}_{23}-\widehat{g}_{24}+\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{21},$
$\displaystyle\widetilde{\overline{f}}_{22}$ $\displaystyle=$
$\displaystyle\overline{f}_{22}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+\widehat{g}_{3}+\widehat{g}_{4}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}-2\widehat{g}_{11}+2\widehat{g}_{12}+\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}-\widehat{g}_{20}-\widehat{g}_{21}+\widehat{g}_{22}-\widehat{g}_{23}-\widehat{g}_{24}+\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{22},$
$\displaystyle\widetilde{\overline{f}}_{23}$ $\displaystyle=$
$\displaystyle\overline{f}_{23}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}-\widehat{g}_{3}+\widehat{g}_{4}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}+2\widehat{g}_{11}-2\widehat{g}_{12}-\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}-\widehat{g}_{20}-\widehat{g}_{21}+\widehat{g}_{22}+\widehat{g}_{23}+\widehat{g}_{24}-\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{23},$
$\displaystyle\widetilde{\overline{f}}_{24}$ $\displaystyle=$
$\displaystyle\overline{f}_{24}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}+\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}+2\widehat{g}_{11}-2\widehat{g}_{12}+\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}-\widehat{g}_{20}+\widehat{g}_{21}-\widehat{g}_{22}-\widehat{g}_{23}+\widehat{g}_{24}-\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{24},$
$\displaystyle\widetilde{\overline{f}}_{25}$ $\displaystyle=$
$\displaystyle\overline{f}_{25}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}-\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}-2\widehat{g}_{11}-2\widehat{g}_{12}+\right.$
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}+\widehat{g}_{20}-\widehat{g}_{21}-\widehat{g}_{22}+\widehat{g}_{23}-\widehat{g}_{24}-\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{25},$
$\displaystyle\widetilde{\overline{f}}_{26}$ $\displaystyle=$
$\displaystyle\overline{f}_{26}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}-\widehat{g}_{3}+\widehat{g}_{4}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}-2\widehat{g}_{11}-2\widehat{g}_{12}-\right.$
(63)
$\displaystyle\left.\widehat{g}_{16}+\widehat{g}_{17}+\widehat{g}_{20}+\widehat{g}_{21}+\widehat{g}_{22}-\widehat{g}_{23}-\widehat{g}_{24}-\widehat{g}_{25}+\widehat{g}_{26}\right]+S_{26}.$
The above post-collision state allows completion of the streaming step via Eq.
(36), following which frame-independent fields of 3D fluid motion can be
obtained from Eqs. (37) and (38). In the implementation, various optimization
strategies such as those discussed in [22] should be fully exploited to
minimize the floating point operation count.
Following the general outline of the above derivation, the central moment LBM
was also formulated for the three-dimensional, fifteen velocity (D3Q15)
lattice, which has a much reduced computational complexity when compared with
the D3Q27 lattice. The results are summarized in Appendix G.
## 9 Numerical Tests
Both the central moment formulations including forcing terms derived earlier,
i.e. for the D3Q15 and D3Q27 lattices, were implemented and assessed. Let us
now discuss the validation studies carried out for these computational
approaches for a set of canonical problems for which analytical solutions are
available. First, we consider a fully developed flow between parallel plates
driven by a constant body force. The grid resolution was chosen to be $3\times
3\times 45$ with relaxation parameter $\omega^{\nu}=1.818$ for the second-
order moments ($\omega^{\nu}=\omega_{j}$ where $j=4,5,6,7,8$) that controls
the kinematic viscosity $\nu$ ($=0.0167$ here). Rest of the relaxation
parameters were set to be unity for this case as well as for all the simple
canonical problems considered in our present numerical accuracy study. It may
be noted that other values could be used for kinetic modes involving more
complex situations (e.g. turbulent flows) and could also be optimized to
improve numerical stability. For these parameters, three different values of
the body force were considered, i.e. $F_{x}=2\times 10^{-7},4\times 10^{-7}$
and $6\times 10^{-7}$ corresponding to Reynolds numbers (based on the
centerline velocity and half-width between plates) $3.6,7.2$ and $10.7$,
respectively. Half-way bounce back boundary condition was implemented to
impose the no-slip condition at both the walls. Figure 2 shows a comparison
between the computed results obtained using the central moment LBM implemented
for D3Q15 and D3Q27 lattices with the analytical solution
($u(z)=u_{0}(1-(z/L)^{2})$, where $L$ is the half-width and
$u_{0}=F_{x}L^{2}/(2\nu)$).
Figure 2: Flow between parallel plates with a constant body force: Comparison
of velocity profiles computed by D3Q15 (open symbols) and D3Q27 (filled
circles) formulations of the central moment LBM with forcing term with
analytical solution (lines) for different values of the body force $F_{x}$.
Excellent agreement is seen for both formulations with the benchmark
analytical solution. Since the results with D3Q15 and D3Q27 lattices are
essentially identical with the former involving considerably lower operation
count, henceforth we discuss the numerical performance only with the D3Q15
lattice. It may be noted that the advantage of the central moment formulation
for this lattice, over the SRT approach lies in its enhanced numerical
stability by independently and carefully adjusting the relaxation parameters
for the kinetic modes. This and other assets such as better representation of
kinetic layers are similar to the standard (raw moment) MRT approach.
Comparison of such different collision models are subjects for future
investigations. The central moment LBM using the D3Q15 lattice was further
assessed for the channel flow problem at higher Reynolds numbers. By
considering the same resolution as before and setting the body force as
$F_{x}=1\times 10^{-6}$, two different Reynolds numbers of $111.8$ and $447.2$
were considered by using $\omega^{\nu}=1.923$ and $1.961$, respectively.
Comparisons of computed and analytical solutions were made in Fig. 3, which
again show good agreement.
Figure 3: Flow between parallel plates with a constant body force: Comparison
of velocity profiles computed by D3Q15 formulation of the central moment LBM
with forcing term (open symbols) with analytical solution (lines) for
different values of Reynolds number $Re$.
In order to quantify the error between the computed and analytical solutions
and its variation at different resolutions, i.e. to establish the grid
convergence of the 3D central moment LBM, the following test was carried out.
We again considered channel flow with the computational domain discretized
using $3\times 3\times N$ nodes, where $N$ is the number of nodes in the wall
normal direction which was varied. The parameters were chosen so as to satisfy
diffusive scaling: the fluid velocity (or the Mach number) was made to scale
with the resolution, i.e. $u_{0}\sim\Delta x\sim 1/N$. This ensures that the
errors associated with compressibility effects also simultaneously reduce with
increase in resolution. Thus, with a fixed viscosity $\nu$ to maintain
constant Reynolds number ($Re=u_{0}L/\nu$) for different resolutions, using
$u_{0}=F_{x}L^{2}/(2\nu)$ the body force scales as $F_{x}\sim 1/L^{3}\sim
1/N^{3}$. Setting $\omega^{\nu}=1.818$ and considering $F_{x}=6.958\times
10^{-6}$ for the coarsest resolution ($N=13$) so that $Re=20.8$, the relative
errors in velocity field at different resolutions were computed using
$||\delta u||_{2}=\sum_{i}||(u_{c,i}-u_{a,i})||_{2}/\sum_{i}||u_{a,i}||_{2}$,
where $u_{c,i}$ and $u_{a,i}$ are computed and analytical solutions,
respectively, $||\cdot||_{2}$ is the standard second-norm and the subscript
$i$ represents the discrete location of the nodes. Figure 4 shows a log-log
plot of the relative error as a function of the number of grid nodes.
Figure 4: Grid convergence study of the D3Q15 formulation of the central
moment LBM with forcing term for channel flow under diffusive scaling. Symbols
represent relative (root-mean-square) error between the computed and
analytical solution. Best fit slope of computed results is $-1.96$.
It is evident that quadratic grid convergence is maintained by the 3D cascaded
LBM.
We will now consider a different canonical problem, where the imposed body
force is time dependent and thus represents a more stringent test of the
central moment formulation derived in this work. In particular, flow between
parallel plates driven by a force which varies sinusoidally in time was
computed using this approach. If $\Omega=2\pi/T$ is the angular frequency,
where $T$ is the time period of the application of the body force, it may be
represented as $F_{x}=F_{m}cos(\Omega t)$, where $F_{m}$ is its maximum
amplitude. This problem, generally termed as Womersley flow, is characterized
by the dimensionless parameter $\mathrm{Wo}=\sqrt{\frac{\Omega}{\nu}}L$, also
called as the Womersley number representing the relative effect of the
unsteady response of the fluid flow to the imposed unsteady body force. It has
the following analytical solution for the velocity profile
$u_{x}(z)=\mathcal{R}\left[\frac{iF_{m}}{\Omega}\left\\{1-\frac{cos\left(\beta\frac{z}{L}\right)}{cos(\beta)}\right\\}e^{i\Omega
t}\right]$, where $\beta=\sqrt{-i\mathrm{Wo}^{2}}$. Considering $3\times
3\times 45$ nodes and setting the maximum force amplitude $F_{m}=1\times
10^{-5}$ with a time period of $T=10,000$, two different values of the
relaxation parameter $\omega^{\nu}$, i.e. $1.724$ and $1.923$, were used,
which correspond to $Wo$ of $3.3$ and $6.6$, respectively. Figures 5 and 5
show comparisons of the computed velocity profiles with the above analytical
solution for different instants within the first half of the time period $T$
at these two Womersley numbers.
Figure 5: Flow between parallel plates with a temporally varying body force:
Comparison of velocity profiles computed by the D3Q15 formulation of the
central moment LBM with forcing term (open symbols) with analytical solution
(lines) at different instants within a time period $T$. (a) $Wo=3.3$ and (b)
$Wo=6.6$, where $Wo$ is the Womersley number.
It is clear that the central moment LBM reproduces the sharp variations in the
velocity profiles at different instants as prescribed by the analytical
solution, with very good agreement found between them. Furthermore, the
variations in both the amplitude as well as the lag of the response of the
fluid flow as seen by its velocity profiles at different Womersley numbers are
well reproduced by the computational approach presented in this work.
It may be noted that in all the problems considered above, the velocity field
has variation along only one coordinate direction normal to the direction of
the driving body force. Thus, as a final example, we consider fully developed
flow through a square duct in which the flow field has variations in both the
coordinate directions normal to the direction of application of the driving
force. It has the following analytical solution for the velocity field given
in terms of an infinite orthogonal (Fourier) series [41]
$u(y,z)=\frac{16a^{2}F_{x}}{\rho\nu\pi^{3}}\sum_{n=1}^{\infty}(-1)^{(n-1)}\left[1-\frac{\cosh\left(\frac{(2n-1)\pi
z}{2a}\right)}{\cosh\left(\frac{(2n-1)\pi}{2}\right)}\right]\frac{\cos\left(\frac{(2n-1)\pi
y}{2a}\right)}{(2n-1)^{3}},$ (64)
where $-a\leq y\leq a$ and $-a\leq z\leq a$. Here, $a$ is the duct half-width.
We considered the square duct to be resolved by using $3\times 45\times 45$
nodes. A constant body force of magnitude $F_{x}=1\times 10^{-6}$ was applied
by considering the relaxation parameter $\omega^{\nu}$ equal to $1.923$ such
that the Reynolds number (based on maximum or centerline velocity and duct
half-width) is equal to $65.7$. As before, the no-slip condition at the walls
was imposed using the half-way bounce back approach. Figures 6 and 6 show a
comparison between the surface contours of the computed and analytical
solution of the velocity field for the above condition.
Figure 6: Flow through a square duct with side length $2a$ subjected to a
constant body force: Comparison of surface contours of the velocity field for
Reynolds number $Re=65.7$ (a) computed by the D3Q15 formulation of the central
moment LBM with forcing term with (b) analytical solution (see Eq. (64)).
It is seen that the 3D central moment LBM with forcing term is able to
reproduce the distribution of the velocity field over the cross-section of the
square duct. In order to more clearly make a quantitative comparison, Fig. 7
shows plots of the computed velocity profiles at different locations in the
cross-section of the duct and their comparison with the corresponding
analytical solution (see Eq. (64))
Figure 7: Flow through a square duct with side length $2a$ subjected to a
constant body force: Comparison of velocity profiles computed by the D3Q15
formulation of the central moment LBM with forcing term (symbols) with
analytical solution (lines) (see Eq. (64)) at different locations in the duct
cross-section for Reynolds number $Re=65.7$.
Evidently, the results computed using the central moment LBM are found to be
in excellent agreement with the benchmark solution.
## 10 Summary and Conclusions
A derivation of the 3D central moment lattice Boltzmann method (LBM) in the
presence of forcing terms is presented. Suitable orthogonal moment basis for
the D3Q27 and D3Q15 lattices are chosen for the specification of the local
attractors and source terms in terms of central moments. In particular,
recently proposed factorized form of local attractors for higher moments and
de-aliased source terms that influence only conserved moments, which are
obtained from modifications of the properties of the Maxwellian are considered
in the construction of the approach. A Galilean invariance matching principle
is invoked that exactly preserves the continuous central moments of the
attractor and the source terms at the discrete level for all orders supported
by the particle velocity set. Based on these, expressions for the temporally
semi-implicit and second-order accurate sources are derived through an exact
inversion due to the orthogonal properties of the moment basis. The central
moment LBM, whose elements are equivalently expressed in terms of raw moments
using the binominal theorem, represents frame independent fluid motion in the
presence of general external or self-consistent internal forces. A set of
numerical tests was carried out for problems involving channel flow driven by
constant and temporally varying (periodic) body forces, and flow through a
square duct to assess the accuracy of the central moment LBM with forcing term
derived in this paper. It is demonstrated that the method maintains second-
order grid convergence under diffusive scaling. Comparisons of the computed
results are found to be in excellent agreement with analytical solutions for
all the benchmark problems considered.
## Appendix A Appendix: Orthogonal Matrix of the Moment Basis $\mathcal{K}$
for the D3Q27 Lattice
A main element of the central moment method is the moment basis. The
components of the orthogonal matrix of the the moment basis derived in Sec. 2
(see Eq. (4)) can be written as
$\mathcal{K}=$
$\left(\begin{array}[]{rrrrrrrrrrrrrrrrrrrrrrrrrrr}1&0&0&0&0&0&0&0&0&-2&0&0&0&0&0&0&0&4&0&0&0&0&0&0&0&0&-8\\\
1&1&0&0&0&0&0&1&1&-1&-4&0&0&0&0&0&0&0&-4&0&0&0&0&4&0&0&4\\\
1&-1&0&0&0&0&0&1&1&-1&4&0&0&0&0&0&0&0&-4&0&0&0&0&-4&0&0&4\\\
1&0&1&0&0&0&0&-1&1&-1&0&-4&0&0&0&0&0&0&2&-2&0&0&0&0&4&0&4\\\
1&0&-1&0&0&0&0&-1&1&-1&0&4&0&0&0&0&0&0&2&-2&0&0&0&0&-4&0&4\\\
1&0&0&1&0&0&0&0&-2&-1&0&0&-4&0&0&0&0&0&2&2&0&0&0&0&0&4&4\\\
1&0&0&-1&0&0&0&0&-2&-1&0&0&4&0&0&0&0&0&2&2&0&0&0&0&0&-4&4\\\
1&1&1&0&1&0&0&0&2&0&-1&-1&0&1&1&0&0&-1&1&1&0&0&-2&-2&-2&0&-2\\\
1&-1&1&0&-1&0&0&0&2&0&1&-1&0&-1&1&0&0&-1&1&1&0&0&2&2&-2&0&-2\\\
1&1&-1&0&-1&0&0&0&2&0&-1&1&0&1&-1&0&0&-1&1&1&0&0&2&-2&2&0&-2\\\
1&-1&-1&0&1&0&0&0&2&0&1&1&0&-1&-1&0&0&-1&1&1&0&0&-2&2&2&0&-2\\\
1&1&0&1&0&1&0&1&-1&0&-1&0&-1&-1&0&1&0&-1&1&-1&0&-2&0&-2&0&-2&-2\\\
1&-1&0&1&0&-1&0&1&-1&0&1&0&-1&1&0&1&0&-1&1&-1&0&2&0&2&0&-2&-2\\\
1&1&0&-1&0&-1&0&1&-1&0&-1&0&1&-1&0&-1&0&-1&1&-1&0&2&0&-2&0&2&-2\\\
1&-1&0&-1&0&1&0&1&-1&0&1&0&1&1&0&-1&0&-1&1&-1&0&-2&0&2&0&2&-2\\\
1&0&1&1&0&0&1&-1&-1&0&0&-1&-1&0&-1&-1&0&-1&-2&0&-2&0&0&0&-2&-2&-2\\\
1&0&-1&1&0&0&-1&-1&-1&0&0&1&-1&0&1&-1&0&-1&-2&0&2&0&0&0&2&-2&-2\\\
1&0&1&-1&0&0&-1&-1&-1&0&0&-1&1&0&-1&1&0&-1&-2&0&2&0&0&0&-2&2&-2\\\
1&0&-1&-1&0&0&1&-1&-1&0&0&1&1&0&1&1&0&-1&-2&0&-2&0&0&0&2&2&-2\\\
1&1&1&1&1&1&1&0&0&1&2&2&2&0&0&0&1&1&0&0&1&1&1&1&1&1&1\\\
1&-1&1&1&-1&-1&1&0&0&1&-2&2&2&0&0&0&-1&1&0&0&1&-1&-1&-1&1&1&1\\\
1&1&-1&1&-1&1&-1&0&0&1&2&-2&2&0&0&0&-1&1&0&0&-1&1&-1&1&-1&1&1\\\
1&-1&-1&1&1&-1&-1&0&0&1&-2&-2&2&0&0&0&1&1&0&0&-1&-1&1&-1&-1&1&1\\\
1&1&1&-1&1&-1&-1&0&0&1&2&2&-2&0&0&0&-1&1&0&0&-1&-1&1&1&1&-1&1\\\
1&-1&1&-1&-1&1&-1&0&0&1&-2&2&-2&0&0&0&1&1&0&0&-1&1&-1&-1&1&-1&1\\\
1&1&-1&-1&-1&-1&1&0&0&1&2&-2&-2&0&0&0&1&1&0&0&1&-1&-1&1&-1&-1&1\\\
1&-1&-1&-1&1&1&1&0&0&1&-2&-2&-2&0&0&0&-1&1&0&0&1&1&1&-1&-1&-1&1\\\
\end{array}\right)$
## Appendix B Appendix: Non-conserved Transformed Raw Moments for the D3Q27
Lattice
The non-conserved transformed raw moments of various orders are given in terms
of the subsets of the particle velocity directions as
$\displaystyle\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
y}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{4}}-\sum_{\alpha}^{B_{4}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha y}e_{\alpha
z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha y}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}^{2}=\left(\sum_{\alpha}^{A_{7}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{8}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{zz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{9}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{10}}-\sum_{\alpha}^{B_{10}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{11}}-\sum_{\alpha}^{B_{11}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{12}}-\sum_{\alpha}^{B_{12}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}e_{\alpha z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
y}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{13}}-\sum_{\alpha}^{B_{13}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{14}}-\sum_{\alpha}^{B_{14}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}e_{\alpha z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
y}^{2}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{15}}-\sum_{\alpha}^{B_{15}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha y}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{16}}-\sum_{\alpha}^{B_{16}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{17}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{18}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yyzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}e_{\alpha z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha
y}^{2}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{19}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha
z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{20}}-\sum_{\alpha}^{B_{20}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha
z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}^{2}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{21}}-\sum_{\alpha}^{B_{21}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{22}}-\sum_{\alpha}^{B_{22}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyyzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}^{2}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{23}}-\sum_{\alpha}^{B_{23}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{24}}-\sum_{\alpha}^{B_{24}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha
z}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}^{2}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{25}}-\sum_{\alpha}^{B_{25}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyyzz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{26}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}^{2}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{26}}-\sum_{\alpha}^{B_{26}}\right)\otimes\overline{f}_{\alpha},$
(65)
where
$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\left\\{7,10,19,22,23,26\right\\},B_{4}=\left\\{8,9,20,21,24,25\right\\},$
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle\left\\{11,14,19,21,24,26\right\\},B_{5}=\left\\{12,13,20,22,23,25\right\\},$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\left\\{15,18,19,20,25,26\right\\},B_{6}=\left\\{16,17,21,22,23,24\right\\},$
$\displaystyle A_{7}$ $\displaystyle=$
$\displaystyle\left\\{1,2,7,8,9,10,11,12,13,14,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{8}$ $\displaystyle=$
$\displaystyle\left\\{3,4,7,8,9,10,15,16,17,18,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{9}$ $\displaystyle=$
$\displaystyle\left\\{5,6,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{10}$ $\displaystyle=$
$\displaystyle\left\\{7,9,19,21,23,25\right\\},B_{10}=\left\\{8,10,20,22,24,26\right\\},$
$\displaystyle A_{11}$ $\displaystyle=$
$\displaystyle\left\\{11,13,19,21,23,25\right\\},B_{11}=\left\\{12,14,20,22,24,26\right\\},$
$\displaystyle A_{12}$ $\displaystyle=$
$\displaystyle\left\\{7,8,19,20,23,24\right\\},B_{12}=\left\\{9,10,21,22,25,26\right\\},$
$\displaystyle A_{13}$ $\displaystyle=$
$\displaystyle\left\\{15,17,19,20,23,24\right\\},B_{13}=\left\\{16,18,21,22,25,26\right\\},$
$\displaystyle A_{14}$ $\displaystyle=$
$\displaystyle\left\\{11,12,19,20,21,22\right\\},B_{14}=\left\\{13,14,23,24,25,26\right\\},$
$\displaystyle A_{15}$ $\displaystyle=$
$\displaystyle\left\\{15,16,19,20,21,22\right\\},B_{15}=\left\\{17,18,23,24,25,26\right\\},$
$\displaystyle A_{16}$ $\displaystyle=$
$\displaystyle\left\\{19,22,24,25\right\\},B_{16}=\left\\{20,21,23,26\right\\},$
$\displaystyle A_{17}$ $\displaystyle=$
$\displaystyle\left\\{7,8,9,10,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{18}$ $\displaystyle=$
$\displaystyle\left\\{11,12,13,14,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{19}$ $\displaystyle=$
$\displaystyle\left\\{15,16,17,18,19,20,21,22,23,24,25,26\right\\},$
$\displaystyle A_{20}$ $\displaystyle=$
$\displaystyle\left\\{19,20,25,26\right\\},B_{20}=\left\\{21,22,23,24\right\\},$
$\displaystyle A_{21}$ $\displaystyle=$
$\displaystyle\left\\{19,21,24,26\right\\},B_{21}=\left\\{20,22,23,25\right\\},$
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle\left\\{19,22,23,26\right\\},B_{22}=\left\\{20,21,24,25\right\\},$
$\displaystyle A_{23}$ $\displaystyle=$
$\displaystyle\left\\{19,21,23,25\right\\},B_{23}=\left\\{20,22,24,26\right\\},$
$\displaystyle A_{24}$ $\displaystyle=$
$\displaystyle\left\\{19,20,23,24\right\\},B_{24}=\left\\{21,22,25,26\right\\},$
$\displaystyle A_{25}$ $\displaystyle=$
$\displaystyle\left\\{19,20,21,22\right\\},B_{25}=\left\\{23,24,25,26\right\\},$
$\displaystyle A_{26}$ $\displaystyle=$
$\displaystyle\left\\{19,20,21,22,23,24,25,26\right\\}.$
## Appendix C Appendix: Raw Source Moments for the D3Q27 Lattice
The raw source moments of various orders are given in terms of the Cartesian
components of the force field as
$\displaystyle\widehat{\sigma}_{0}^{{}^{\prime}}=\braket{S_{\alpha}}{\rho}=0,$
$\displaystyle\widehat{\sigma}_{x}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}}=F_{x},$
$\displaystyle\widehat{\sigma}_{y}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}}=F_{y},$
$\displaystyle\widehat{\sigma}_{z}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
z}}=F_{z},$
$\displaystyle\widehat{\sigma}_{xx}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}}=2F_{x}u_{x},$
$\displaystyle\widehat{\sigma}_{yy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}^{2}}=2F_{y}u_{y},$
$\displaystyle\widehat{\sigma}_{zz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
z}^{2}}=2F_{z}u_{z},$
$\displaystyle\widehat{\sigma}_{xy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}}=F_{x}u_{y}+F_{y}u_{x},$
$\displaystyle\widehat{\sigma}_{xz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha z}}=F_{x}u_{z}+F_{z}u_{x},$
$\displaystyle\widehat{\sigma}_{yz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}e_{\alpha z}}=F_{y}u_{z}+F_{z}u_{y},$
$\displaystyle\widehat{\sigma}_{xyy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}}=F_{x}u_{y}^{2}+2F_{y}u_{y}u_{x},$
$\displaystyle\widehat{\sigma}_{xzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha z}^{2}}=F_{x}u_{z}^{2}+2F_{z}u_{z}u_{x},$
$\displaystyle\widehat{\sigma}_{xxy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}=F_{y}u_{x}^{2}+2F_{x}u_{x}u_{y},$
$\displaystyle\widehat{\sigma}_{yzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}e_{\alpha z}^{2}}=F_{y}u_{z}^{2}+2F_{z}u_{z}u_{y},$
$\displaystyle\widehat{\sigma}_{xxz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}=F_{z}u_{x}^{2}+2F_{x}u_{x}u_{z},$
$\displaystyle\widehat{\sigma}_{yyz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}^{2}e_{\alpha z}}=F_{z}u_{y}^{2}+2F_{y}u_{y}u_{z},$
$\displaystyle\widehat{\sigma}_{xyz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}}=F_{x}u_{y}u_{z}+F_{y}u_{x}u_{z}+F_{z}u_{x}u_{y},$
$\displaystyle\widehat{\sigma}_{xxyy}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}=2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}^{2},$
$\displaystyle\widehat{\sigma}_{xxzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha z}^{2}}=2F_{x}u_{x}u_{z}^{2}+2F_{z}u_{z}u_{x}^{2},$
$\displaystyle\widehat{\sigma}_{yyzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
y}^{2}e_{\alpha z}^{2}}=2F_{y}u_{y}u_{z}^{2}+2F_{z}u_{z}u_{y}^{2},$
$\displaystyle\widehat{\sigma}_{xxyz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha
z}}=u_{x}^{2}(F_{y}u_{z}+F_{z}u_{y})+2F_{x}u_{x}u_{y}u_{z},$
$\displaystyle\widehat{\sigma}_{xyyz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha
z}}=u_{y}^{2}(F_{x}u_{z}+F_{z}u_{x})+2F_{y}u_{y}u_{x}u_{z},$
$\displaystyle\widehat{\sigma}_{xyzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}e_{\alpha
z}^{2}}=u_{z}^{2}(F_{x}u_{y}+F_{y}u_{x})+2F_{z}u_{z}u_{x}u_{y},$
$\displaystyle\widehat{\sigma}_{xyyzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha
z}^{2}}=F_{x}u_{y}^{2}u_{z}^{2}+2u_{x}u_{y}u_{z}(F_{y}u_{z}+F_{z}u_{y}),$
$\displaystyle\widehat{\sigma}_{xxyzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha
z}^{2}}=F_{y}u_{x}^{2}u_{z}^{2}+2u_{x}u_{y}u_{z}(F_{x}u_{z}+F_{z}u_{x}),$
$\displaystyle\widehat{\sigma}_{xxyyz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha
z}}=F_{z}u_{x}^{2}u_{y}^{2}+2u_{x}u_{y}u_{z}(F_{x}u_{y}+F_{y}u_{x}),$
$\displaystyle\widehat{\sigma}_{xxyyzz}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha
z}^{2}}=2u_{x}u_{y}u_{z}(F_{x}u_{y}u_{z}+F_{y}u_{z}u_{x}+F_{z}u_{x}u_{y}).$
(66)
## Appendix D Appendix: Projections of the Raw Source Moments to the
Orthogonal Basis Vectors for the D3Q27 Lattice
The orthogonal projections of the raw source moments can be written as
$\displaystyle\widehat{m}^{s}_{0}=\braket{K_{0}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\widehat{m}^{s}_{1}=\braket{K_{1}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{x},$
$\displaystyle\widehat{m}^{s}_{2}=\braket{K_{2}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{y},$
$\displaystyle\widehat{m}^{s}_{3}=\braket{K_{3}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{z},$
$\displaystyle\widehat{m}^{s}_{4}=\braket{K_{4}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{y}+F_{y}u_{x}),$
$\displaystyle\widehat{m}^{s}_{5}=\braket{K_{5}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{z}+F_{z}u_{x}),$
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{y}u_{z}+F_{z}u_{y}),$
$\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}-F_{y}u_{y}),$
$\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}+F_{y}u_{y}-2F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{9}=\braket{K_{9}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}+F_{y}u_{y}+F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{10}=\braket{K_{10}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
3\left[F_{x}(u_{y}^{2}+u_{z}^{2})+2u_{x}(F_{y}u_{y}+F_{z}u_{z})\right]-4F_{x},$
$\displaystyle\widehat{m}^{s}_{11}=\braket{K_{11}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
3\left[F_{y}(u_{x}^{2}+u_{z}^{2})+2u_{y}(F_{x}u_{x}+F_{z}u_{z})\right]-4F_{y},$
$\displaystyle\widehat{m}^{s}_{12}=\braket{K_{12}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
3\left[F_{z}(u_{x}^{2}+u_{y}^{2})+2u_{z}(F_{x}u_{x}+F_{y}u_{y})\right]-4F_{z},$
$\displaystyle\widehat{m}^{s}_{13}=\braket{K_{13}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle\left[F_{x}(u_{y}^{2}-u_{z}^{2})+2u_{x}(F_{y}u_{y}-F_{z}u_{z})\right],$
$\displaystyle\widehat{m}^{s}_{14}=\braket{K_{14}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle\left[F_{y}(u_{x}^{2}-u_{z}^{2})+2u_{y}(F_{x}u_{x}-F_{z}u_{z})\right],$
$\displaystyle\widehat{m}^{s}_{15}=\braket{K_{15}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle\left[F_{z}(u_{x}^{2}-u_{y}^{2})+2u_{z}(F_{x}u_{x}-F_{y}u_{y})\right],$
$\displaystyle\widehat{m}^{s}_{16}=\braket{K_{16}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
F_{x}u_{y}u_{z}+F_{y}u_{x}u_{z}+F_{z}u_{x}u_{y},$
$\displaystyle\widehat{m}^{s}_{17}=\braket{K_{17}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
6\left[F_{x}u_{x}(u_{y}^{2}+u_{z}^{2})+F_{y}u_{y}(u_{x}^{2}+u_{z}^{2})+F_{z}u_{z}(u_{x}^{2}+u_{y}^{2})\right]$
$\displaystyle-8(F_{x}u_{x}+F_{y}u_{y}+F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{18}=\braket{K_{18}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
6\left[F_{x}u_{x}(u_{y}^{2}+u_{z}^{2})+F_{y}u_{y}(u_{x}^{2}-2u_{z}^{2})+F_{z}u_{z}(u_{x}^{2}-2u_{y}^{2})\right]$
$\displaystyle-4(2F_{x}u_{x}-F_{y}u_{y}-F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{19}=\braket{K_{19}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
6\left[F_{x}u_{x}(u_{y}^{2}-u_{z}^{2})+u_{x}^{2}(F_{y}u_{y}-F_{z}u_{z})\right]$
$\displaystyle-4(F_{y}u_{y}-F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{20}=\braket{K_{20}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle(3u_{x}^{2}-2)\left[F_{y}u_{z}+F_{z}u_{y}\right]+6F_{x}u_{x}u_{y}u_{z},$
$\displaystyle\widehat{m}^{s}_{21}=\braket{K_{21}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle(3u_{y}^{2}-2)\left[F_{x}u_{z}+F_{z}u_{x}\right]+6F_{y}u_{y}u_{x}u_{z},$
$\displaystyle\widehat{m}^{s}_{22}=\braket{K_{22}}{S_{\alpha}}$
$\displaystyle=$
$\displaystyle(3u_{z}^{2}-2)\left[F_{x}u_{y}+F_{y}u_{x}\right]+6F_{z}u_{z}u_{x}u_{y},$
$\displaystyle\widehat{m}^{s}_{23}=\braket{K_{23}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
9F_{x}\left[u_{y}^{2}u_{z}^{2}-\frac{2}{3}\left((u_{y}^{2}+u_{z}^{2})-\frac{2}{3}\right)\right]$
$\displaystyle+18u_{x}\left[F_{y}u_{y}u_{z}^{2}+F_{z}u_{z}u_{y}^{2}-\frac{2}{3}(F_{y}u_{y}+F_{z}u_{z})\right],$
$\displaystyle\widehat{m}^{s}_{24}=\braket{K_{24}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
9F_{y}\left[u_{x}^{2}u_{z}^{2}-\frac{2}{3}\left((u_{x}^{2}+u_{z}^{2})-\frac{2}{3}\right)\right]$
$\displaystyle+18u_{y}\left[F_{x}u_{x}u_{z}^{2}+F_{z}u_{z}u_{x}^{2}-\frac{2}{3}(F_{x}u_{x}+F_{z}u_{z})\right],$
$\displaystyle\widehat{m}^{s}_{25}=\braket{K_{25}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
9F_{z}\left[u_{x}^{2}u_{y}^{2}-\frac{2}{3}\left((u_{x}^{2}+u_{y}^{2})-\frac{2}{3}\right)\right]$
$\displaystyle+18u_{z}\left[F_{x}u_{x}u_{y}^{2}+F_{y}u_{y}u_{x}^{2}-\frac{2}{3}(F_{x}u_{x}+F_{y}u_{y})\right],$
$\displaystyle\widehat{m}^{s}_{26}=\braket{K_{26}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
F_{x}u_{x}\left[54u_{y}^{2}u_{z}^{2}-36(u_{y}^{2}+u_{z}^{2})+24\right]$ (67)
$\displaystyle+F_{y}u_{y}\left[54u_{x}^{2}u_{z}^{2}-36(u_{x}^{2}+u_{z}^{2})+24\right]$
$\displaystyle+F_{z}u_{z}\left[54u_{x}^{2}u_{y}^{2}-36(u_{x}^{2}+u_{y}^{2})+24\right].$
## Appendix E Appendix: Source Terms in Particle Velocity Space for the D3Q27
Lattice
The source terms in particle velocity space obtained by solving Eq. (33) are
given by
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-24\widehat{m}^{s}_{9}+24\widehat{m}^{s}_{17}-8\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{1}+18\widehat{m}^{s}_{7}+6\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}-12\widehat{m}^{s}_{10}-12\widehat{m}^{s}_{18}\right.$
$\displaystyle\left.+12\widehat{m}^{s}_{23}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{1}+18\widehat{m}^{s}_{7}+6\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}+12\widehat{m}^{s}_{10}-12\widehat{m}^{s}_{18}\right.$
$\displaystyle\left.-12\widehat{m}^{s}_{23}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{2}-18\widehat{m}^{s}_{7}+6\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}-12\widehat{m}^{s}_{11}+6\widehat{m}^{s}_{18}\right.$
$\displaystyle\left.-18\widehat{m}^{s}_{19}+12\widehat{m}^{s}_{24}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{2}-18\widehat{m}^{s}_{7}+6\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}+12\widehat{m}^{s}_{11}+6\widehat{m}^{s}_{18}\right.$
$\displaystyle\left.-18\widehat{m}^{s}_{19}-12\widehat{m}^{s}_{24}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{3}-12\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}-12\widehat{m}^{s}_{12}+6\widehat{m}^{s}_{18}+18\widehat{m}^{s}_{19}\right.$
$\displaystyle\left.+12\widehat{m}^{s}_{25}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{3}-12\widehat{m}^{s}_{8}-12\widehat{m}^{s}_{9}+12\widehat{m}^{s}_{12}+6\widehat{m}^{s}_{18}+18\widehat{m}^{s}_{19}\right.$
$\displaystyle\left.-12\widehat{m}^{s}_{25}+4\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{1}+12\widehat{m}^{s}_{2}+18\widehat{m}^{s}_{4}+12\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{10}-3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.+27\widehat{m}^{s}_{13}+27\widehat{m}^{s}_{14}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}+9\widehat{m}^{s}_{19}-18\widehat{m}^{s}_{22}-6\widehat{m}^{s}_{23}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{24}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{1}+12\widehat{m}^{s}_{2}-18\widehat{m}^{s}_{4}+12\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{10}-3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.-27\widehat{m}^{s}_{13}+27\widehat{m}^{s}_{14}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}+9\widehat{m}^{s}_{19}+18\widehat{m}^{s}_{22}+6\widehat{m}^{s}_{23}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{24}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{9}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{1}-12\widehat{m}^{s}_{2}-18\widehat{m}^{s}_{4}+12\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{10}+3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.+27\widehat{m}^{s}_{13}-27\widehat{m}^{s}_{14}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}+9\widehat{m}^{s}_{19}+18\widehat{m}^{s}_{22}-6\widehat{m}^{s}_{23}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{24}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{10}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{1}-12\widehat{m}^{s}_{2}+18\widehat{m}^{s}_{4}+12\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{10}+3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.-27\widehat{m}^{s}_{13}-27\widehat{m}^{s}_{14}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}+9\widehat{m}^{s}_{19}-18\widehat{m}^{s}_{22}+6\widehat{m}^{s}_{23}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{24}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{1}+12\widehat{m}^{s}_{3}+18\widehat{m}^{s}_{5}+18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{10}\right.$
$\displaystyle\left.-3\widehat{m}^{s}_{12}-27\widehat{m}^{s}_{13}+27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}-9\widehat{m}^{s}_{19}-18\widehat{m}^{s}_{21}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{23}-6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{1}+12\widehat{m}^{s}_{3}-18\widehat{m}^{s}_{5}+18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{10}\right.$
$\displaystyle\left.-3\widehat{m}^{s}_{12}+27\widehat{m}^{s}_{13}+27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}-9\widehat{m}^{s}_{19}+18\widehat{m}^{s}_{21}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{23}-6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{13}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{1}-12\widehat{m}^{s}_{3}-18\widehat{m}^{s}_{5}+18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{10}\right.$
$\displaystyle\left.+3\widehat{m}^{s}_{12}-27\widehat{m}^{s}_{13}-27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}-9\widehat{m}^{s}_{19}+18\widehat{m}^{s}_{21}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{23}+6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{14}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{1}-12\widehat{m}^{s}_{3}+18\widehat{m}^{s}_{5}+18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{10}\right.$
$\displaystyle\left.+3\widehat{m}^{s}_{12}+27\widehat{m}^{s}_{13}-27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}+3\widehat{m}^{s}_{18}-9\widehat{m}^{s}_{19}-18\widehat{m}^{s}_{21}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{23}+6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{15}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{2}+12\widehat{m}^{s}_{3}+18\widehat{m}^{s}_{6}-18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.-3\widehat{m}^{s}_{12}-27\widehat{m}^{s}_{14}-27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}-6\widehat{m}^{s}_{18}-18\widehat{m}^{s}_{20}-6\widehat{m}^{s}_{24}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{16}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{2}+12\widehat{m}^{s}_{3}-18\widehat{m}^{s}_{6}-18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.-3\widehat{m}^{s}_{12}+27\widehat{m}^{s}_{14}-27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}-6\widehat{m}^{s}_{18}+18\widehat{m}^{s}_{20}+6\widehat{m}^{s}_{24}\right.$
$\displaystyle\left.-6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{17}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12\widehat{m}^{s}_{2}-12\widehat{m}^{s}_{3}-18\widehat{m}^{s}_{6}-18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}-3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.+3\widehat{m}^{s}_{12}-27\widehat{m}^{s}_{14}+27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}-6\widehat{m}^{s}_{18}+18\widehat{m}^{s}_{20}-6\widehat{m}^{s}_{24}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{18}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}-12\widehat{m}^{s}_{2}-12\widehat{m}^{s}_{3}+18\widehat{m}^{s}_{6}-18\widehat{m}^{s}_{7}-6\widehat{m}^{s}_{8}+3\widehat{m}^{s}_{11}\right.$
$\displaystyle\left.+3\widehat{m}^{s}_{12}+27\widehat{m}^{s}_{14}+27\widehat{m}^{s}_{15}-6\widehat{m}^{s}_{17}-6\widehat{m}^{s}_{18}-18\widehat{m}^{s}_{20}+6\widehat{m}^{s}_{24}\right.$
$\displaystyle\left.+6\widehat{m}^{s}_{25}-2\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{19}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(\widehat{m}^{s}_{1}+\widehat{m}^{s}_{2}+\widehat{m}^{s}_{3})+18(\widehat{m}^{s}_{4}+\widehat{m}^{s}_{5}+\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(\widehat{m}^{s}_{10}+\widehat{m}^{s}_{11}+\widehat{m}^{s}_{12})+27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.+\widehat{m}^{s}_{21}+\widehat{m}^{s}_{22})+3(\widehat{m}^{s}_{23}+\widehat{m}^{s}_{24}+\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{20}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(-\widehat{m}^{s}_{1}+\widehat{m}^{s}_{2}+\widehat{m}^{s}_{3})+18(-\widehat{m}^{s}_{4}-\widehat{m}^{s}_{5}+\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(-\widehat{m}^{s}_{10}+\widehat{m}^{s}_{11}+\widehat{m}^{s}_{12})-27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.-\widehat{m}^{s}_{21}-\widehat{m}^{s}_{22})+3(-\widehat{m}^{s}_{23}+\widehat{m}^{s}_{24}+\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(\widehat{m}^{s}_{1}-\widehat{m}^{s}_{2}+\widehat{m}^{s}_{3})+18(-\widehat{m}^{s}_{4}+\widehat{m}^{s}_{5}-\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(\widehat{m}^{s}_{10}-\widehat{m}^{s}_{11}+\widehat{m}^{s}_{12})-27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(-\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.+\widehat{m}^{s}_{21}-\widehat{m}^{s}_{22})+3(\widehat{m}^{s}_{23}-\widehat{m}^{s}_{24}+\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(-\widehat{m}^{s}_{1}-\widehat{m}^{s}_{2}+\widehat{m}^{s}_{3})+18(\widehat{m}^{s}_{4}-\widehat{m}^{s}_{5}-\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(-\widehat{m}^{s}_{10}-\widehat{m}^{s}_{11}+\widehat{m}^{s}_{12})+27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(-\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.-\widehat{m}^{s}_{21}+\widehat{m}^{s}_{22})+3(-\widehat{m}^{s}_{23}-\widehat{m}^{s}_{24}+\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{23}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(\widehat{m}^{s}_{1}+\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3})+18(\widehat{m}^{s}_{4}-\widehat{m}^{s}_{5}-\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(\widehat{m}^{s}_{10}+\widehat{m}^{s}_{11}-\widehat{m}^{s}_{12})-27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(-\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.-\widehat{m}^{s}_{21}+\widehat{m}^{s}_{22})+3(\widehat{m}^{s}_{23}+\widehat{m}^{s}_{24}-\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{24}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(-\widehat{m}^{s}_{1}+\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3})+18(-\widehat{m}^{s}_{4}+\widehat{m}^{s}_{5}-\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(-\widehat{m}^{s}_{10}+\widehat{m}^{s}_{11}-\widehat{m}^{s}_{12})+27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(-\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.+\widehat{m}^{s}_{21}-\widehat{m}^{s}_{22})+3(-\widehat{m}^{s}_{23}+\widehat{m}^{s}_{24}-\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{25}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(\widehat{m}^{s}_{1}-\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3})+18(-\widehat{m}^{s}_{4}-\widehat{m}^{s}_{5}+\widehat{m}^{s}_{6})\right.$
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(\widehat{m}^{s}_{10}-\widehat{m}^{s}_{11}-\widehat{m}^{s}_{12})+27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.-\widehat{m}^{s}_{21}-\widehat{m}^{s}_{22})+3(\widehat{m}^{s}_{23}-\widehat{m}^{s}_{24}-\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right],$
$\displaystyle S_{26}$ $\displaystyle=$
$\displaystyle\frac{1}{216}\left[8\widehat{m}^{s}_{0}+12(-\widehat{m}^{s}_{1}-\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3})+18(\widehat{m}^{s}_{4}+\widehat{m}^{s}_{5}+\widehat{m}^{s}_{6})\right.$
(68)
$\displaystyle\left.\left.+12\widehat{m}^{s}_{9}+6(-\widehat{m}^{s}_{10}-\widehat{m}^{s}_{11}-\widehat{m}^{s}_{12})-27\widehat{m}^{s}_{16}+6\widehat{m}^{s}_{17}+9(\widehat{m}^{s}_{20}\right.\right.$
$\displaystyle\left.+\widehat{m}^{s}_{21}+\widehat{m}^{s}_{22})+3(-\widehat{m}^{s}_{23}-\widehat{m}^{s}_{24}-\widehat{m}^{s}_{25})+\widehat{m}^{s}_{26}\right].$
## Appendix F Appendix: Moments of the Collision Kernel for the D3Q27 Lattice
The moments of the collision kernel follow from the orthogonal property of the
moment basis matrix $\mathcal{K}$, which are given by
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=\sum_{\beta}\braket{K_{\beta}}{\rho}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha z}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 12\widehat{g}_{4},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 12\widehat{g}_{5},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 12\widehat{g}_{6},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle
6\widehat{g}_{7}+6\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$
$\displaystyle-6\widehat{g}_{7}+6\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha z}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle-12\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{10}+4\widehat{g}_{13},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{10}-4\widehat{g}_{13},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{11}+4\widehat{g}_{14},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}e_{\alpha
z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{11}-4\widehat{g}_{14},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{12}+4\widehat{g}_{15},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
12\widehat{g}_{12}-4\widehat{g}_{15},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 8\widehat{g}_{16},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{8}+8\widehat{g}_{9}+4\widehat{g}_{17}$
$\displaystyle+4\widehat{g}_{18}+4\widehat{g}_{19},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
4\widehat{g}_{7}-4\widehat{g}_{8}+8\widehat{g}_{9}$
$\displaystyle+4\widehat{g}_{17}+4\widehat{g}_{18}-4\widehat{g}_{19},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
y}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle-4\widehat{g}_{7}-4\widehat{g}_{8}+8\widehat{g}_{9}$
$\displaystyle+4\widehat{g}_{17}-8\widehat{g}_{18},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha z}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 8\widehat{g}_{6}+8\widehat{g}_{20},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 8\widehat{g}_{5}+8\widehat{g}_{21},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 8\widehat{g}_{4}+8\widehat{g}_{22},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}e_{\alpha y}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 16\widehat{g}_{10}+8\widehat{g}_{23},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 16\widehat{g}_{11}+8\widehat{g}_{24},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha z}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 16\widehat{g}_{12}+8\widehat{g}_{25},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}e_{\alpha
z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{9}+8\widehat{g}_{17}+8\widehat{g}_{26}.$ (69)
## Appendix G Appendix: Formulation of the Central Moment LBM for the Three-
dimensional, Fifteen Velocity (D3Q15) Lattice
### G.1 Moment Basis
The particle velocity for the D3Q15 lattice $\overrightarrow{e}_{\alpha}$ (see
Fig. 8) is given by
$\overrightarrow{e_{\alpha}}=\left\\{\begin{array}[]{ll}{(0,0,0),}&{\alpha=0}\\\
{(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1),}&{\alpha=1,\cdots,6}\\\ {(\pm 1,\pm
1,\pm 1),}&{\alpha=7,\cdots,14}\end{array}\right.$ (70)
Figure 8: Three-dimensional, fifteen particle velocity (D3Q27) lattice.
The components of the nominal moment basis chosen are
$\displaystyle\ket{T_{0}}$ $\displaystyle=$
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}},$
$\displaystyle\ket{T_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
$\displaystyle\ket{T_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha y}},$
$\displaystyle\ket{T_{3}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha z}},$
$\displaystyle\ket{T_{4}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}},$ $\displaystyle\ket{T_{5}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha z}},$ $\displaystyle\ket{T_{6}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{T_{7}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}-e_{\alpha y}^{2}},$ $\displaystyle\ket{T_{8}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}-e_{\alpha z}^{2}},$
$\displaystyle\ket{T_{9}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}+e_{\alpha y}^{2}+e_{\alpha z}^{2}},$ $\displaystyle\ket{T_{10}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha x}(e_{\alpha x}^{2}+e_{\alpha
y}^{2}+e_{\alpha z}^{2})},$ $\displaystyle\ket{T_{11}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha y}(e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2})},$ $\displaystyle\ket{T_{12}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha z}(e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2})},$ $\displaystyle\ket{T_{13}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{T_{14}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}+e_{\alpha x}^{2}e_{\alpha z}^{2}+e_{\alpha
y}^{2}e_{\alpha z}^{2}}.$
Based on the above set, the components of the orthogonal basis vectors are
obtained by means of the Gram-Schmidt procedure, which are given by
$\displaystyle\ket{K_{0}}$ $\displaystyle=$
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}},$
$\displaystyle\ket{K_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
$\displaystyle\ket{K_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha y}},$
$\displaystyle\ket{K_{3}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha z}},$
$\displaystyle\ket{K_{4}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}e_{\alpha y}},$ $\displaystyle\ket{K_{5}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha z}},$ $\displaystyle\ket{K_{6}}$
$\displaystyle=$ $\displaystyle\ket{e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{K_{7}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}-e_{\alpha y}^{2}},$ $\displaystyle\ket{K_{8}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}-3\ket{e_{\alpha z}^{2}},$ $\displaystyle\ket{K_{9}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}-2\ket{\rho},$ $\displaystyle\ket{K_{10}}$ $\displaystyle=$
$\displaystyle 5\ket{e_{\alpha x}(e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2})}-13\ket{e_{\alpha x}},$ $\displaystyle\ket{K_{11}}$ $\displaystyle=$
$\displaystyle 5\ket{e_{\alpha y}(e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2})}-13\ket{e_{\alpha y}},$ $\displaystyle\ket{K_{12}}$ $\displaystyle=$
$\displaystyle 5\ket{e_{\alpha z}(e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2})}-13\ket{e_{\alpha z}},$ $\displaystyle\ket{K_{13}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}e_{\alpha z}},$
$\displaystyle\ket{K_{14}}$ $\displaystyle=$ $\displaystyle 30\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}+e_{\alpha x}^{2}e_{\alpha z}^{2}+e_{\alpha
y}^{2}e_{\alpha z}^{2}}-40\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}+e_{\alpha
z}^{2}}+32\ket{\rho}.$ (71)
They can be written in a corresponding matrix form as $\mathcal{K}=$
$\left(\begin{array}[]{rrrrrrrrrrrrrrr}1&0&0&0&0&0&0&0&0&-2&0&0&0&0&32\\\
1&1&0&0&0&0&0&1&1&-1&-8&0&0&0&-8\\\ 1&-1&0&0&0&0&0&1&1&-1&8&0&0&0&-8\\\
1&0&1&0&0&0&0&-1&1&-1&0&-8&0&0&-8\\\ 1&0&-1&0&0&0&0&-1&1&-1&0&8&0&0&-8\\\
1&0&0&1&0&0&0&0&-2&-1&0&0&-8&0&-8\\\ 1&0&0&-1&0&0&0&0&-2&-1&0&0&8&0&-8\\\
1&1&1&1&1&1&1&0&0&1&2&2&2&1&2\\\ 1&-1&1&1&-1&-1&1&0&0&1&-2&2&2&-1&2\\\
1&1&-1&1&-1&1&-1&0&0&1&2&-2&2&-1&2\\\ 1&-1&-1&1&1&-1&-1&0&0&1&-2&-2&2&1&2\\\
1&1&1&-1&1&-1&-1&0&0&1&2&2&-2&-1&2\\\ 1&-1&1&-1&-1&1&-1&0&0&1&-2&2&-2&1&2\\\
1&1&-1&-1&-1&-1&1&0&0&1&2&-2&-2&1&2\\\ 1&-1&-1&-1&1&1&1&0&0&1&-2&-2&-2&-1&2\\\
\end{array}\right)$
where
$\displaystyle\mathcal{K}$ $\displaystyle=$
$\displaystyle\left[\ket{K_{0}},\ket{K_{1}},\ket{K_{2}},\ket{K_{3}},\ket{K_{4}},\ket{K_{5}},\ket{K_{6}},\ket{K_{7}},\ket{K_{8}}\right.$
(72)
$\displaystyle\left.\ket{K_{9}},\ket{K_{10}},\ket{K_{11}},\ket{K_{12}},\ket{K_{13}},\ket{K_{14}}\right].$
### G.2 Various Raw Moments and Source Terms in Velocity Space
The above orthogonal matrix results in a set of moments of the collision
kernel, which are needed in the construction of the collision operator, and
are given by
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=\sum_{\beta}\braket{K_{\beta}}{\rho}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha z}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 8\widehat{g}_{4},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 8\widehat{g}_{5},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 8\widehat{g}_{6},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle
2\widehat{g}_{7}+2\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$
$\displaystyle-2\widehat{g}_{7}+2\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha z}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle-4\widehat{g}_{8}+6\widehat{g}_{9},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
16\widehat{g}_{10},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
16\widehat{g}_{10},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 16\widehat{g}_{11},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}e_{\alpha
z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
16\widehat{g}_{11},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 16\widehat{g}_{12},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}e_{\alpha
z}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 16\widehat{g}_{12},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}e_{\alpha z}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}}\widehat{g}_{\beta}$ $\displaystyle=$
$\displaystyle 8\widehat{g}_{13},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{9}+16\widehat{g}_{14},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{9}+16\widehat{g}_{14},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}e_{\alpha z}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
y}^{2}e_{\alpha z}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{9}+16\widehat{g}_{14}.$ (73)
Note that unlike the D3Q27 lattice, additional degeneracies for various third
and higher order moment basis vectors exist for the D3Q15 lattice, as it
contains a more limited set of independent basis vectors.
It may be noted that the components of the raw moments of the source terms
$\widehat{\sigma}_{x^{m}y^{n}z^{p}}^{{}^{\prime}}=\braket{S_{\alpha}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}e_{\alpha z}^{p}}$ due to force fields can be obtained
in an analogous manner as determined for the D3Q27 lattice (see Appendix C).
The projections of the source terms to the orthogonal matrix of the moment
basis $\mathcal{K}$, i.e. $\braket{K_{\beta}}{S_{\alpha}}$,
$\beta=0,1,2,\ldots,14$ for this lattice yield
$\displaystyle\widehat{m}^{s}_{0}=\braket{K_{0}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\widehat{m}^{s}_{1}=\braket{K_{1}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{x},$
$\displaystyle\widehat{m}^{s}_{2}=\braket{K_{2}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{y},$
$\displaystyle\widehat{m}^{s}_{3}=\braket{K_{3}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{z},$
$\displaystyle\widehat{m}^{s}_{4}=\braket{K_{4}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{y}+F_{y}u_{x}),$
$\displaystyle\widehat{m}^{s}_{5}=\braket{K_{5}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{z}+F_{z}u_{x}),$
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{y}u_{z}+F_{z}u_{y}),$
$\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}-F_{y}u_{y}),$
$\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}+F_{y}u_{y}-2F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{9}=\braket{K_{9}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}+F_{y}u_{y}+F_{z}u_{z}),$
$\displaystyle\widehat{m}^{s}_{10}=\braket{K_{10}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
5\left[F_{x}(3u_{x}^{2}+u_{y}^{2}+u_{z}^{2})+2u_{x}(F_{y}u_{y}+F_{z}u_{z})\right]-13F_{x},$
$\displaystyle\widehat{m}^{s}_{11}=\braket{K_{11}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
5\left[F_{y}(u_{x}^{2}+3u_{y}^{2}+u_{z}^{2})+2u_{y}(F_{x}u_{x}+F_{z}u_{z})\right]-13F_{y},$
$\displaystyle\widehat{m}^{s}_{12}=\braket{K_{12}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
5\left[F_{z}(u_{x}^{2}+u_{y}^{2}+3u_{z}^{2})+2u_{z}(F_{x}u_{x}+F_{y}u_{y})\right]-13F_{z},$
$\displaystyle\widehat{m}^{s}_{13}=\braket{K_{13}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
F_{x}u_{y}u_{z}+F_{y}u_{x}u_{z}+F_{z}u_{x}u_{y},$
$\displaystyle\widehat{m}^{s}_{14}=\braket{K_{14}}{S_{\alpha}}$
$\displaystyle=$ $\displaystyle
20\left[F_{x}u_{x}\left(3(u_{y}^{2}+u_{z}^{2})-4\right)+F_{y}u_{y}\left(3(u_{x}^{2}+u_{z}^{2})-4\right)\right.$
(74)
$\displaystyle\left.+F_{z}u_{z}\left(3(u_{x}^{2}+u_{y}^{2})-4\right)\right].$
Using $\widehat{m}^{s}_{\beta}$, the source terms in velocity space can be
obtained by a procedure involving exact inversion that invokes orthogonal
properties of the collision matrix (see the discussion following Eq. (33) for
the D3Q27 lattice). The results are summarized as follows:
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{45}\left[3\widehat{m}^{s}_{0}-5\widehat{m}^{s}_{9}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}+18\widehat{m}^{s}_{1}+45\widehat{m}^{s}_{7}+15\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}-9\widehat{m}^{s}_{10}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}-18\widehat{m}^{s}_{1}+45\widehat{m}^{s}_{7}+15\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}+9\widehat{m}^{s}_{10}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}+18\widehat{m}^{s}_{2}-45\widehat{m}^{s}_{7}+15\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}-9\widehat{m}^{s}_{11}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}-18\widehat{m}^{s}_{2}-45\widehat{m}^{s}_{7}+15\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}+9\widehat{m}^{s}_{11}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}+18\widehat{m}^{s}_{3}-30\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}-9\widehat{m}^{s}_{12}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{180}\left[12\widehat{m}^{s}_{0}-18\widehat{m}^{s}_{3}-30\widehat{m}^{s}_{8}-10\widehat{m}^{s}_{9}+9\widehat{m}^{s}_{12}-\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}+72\widehat{m}^{s}_{1}+72\widehat{m}^{s}_{2}+72\widehat{m}^{s}_{3}+90\widehat{m}^{s}_{4}+90\widehat{m}^{s}_{5}+90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.+9\widehat{m}^{s}_{10}+9\widehat{m}^{s}_{11}+9\widehat{m}^{s}_{12}+90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}-72\widehat{m}^{s}_{1}+72\widehat{m}^{s}_{2}+72\widehat{m}^{s}_{3}-90\widehat{m}^{s}_{4}-90\widehat{m}^{s}_{5}+90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.-9\widehat{m}^{s}_{10}+9\widehat{m}^{s}_{11}+9\widehat{m}^{s}_{12}-90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{9}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}+72\widehat{m}^{s}_{1}-72\widehat{m}^{s}_{2}+72\widehat{m}^{s}_{3}-90\widehat{m}^{s}_{4}+90\widehat{m}^{s}_{5}-90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.+9\widehat{m}^{s}_{10}-9\widehat{m}^{s}_{11}+9\widehat{m}^{s}_{12}-90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{10}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}-72\widehat{m}^{s}_{1}-72\widehat{m}^{s}_{2}+72\widehat{m}^{s}_{3}+90\widehat{m}^{s}_{4}-90\widehat{m}^{s}_{5}-90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.-9\widehat{m}^{s}_{10}-9\widehat{m}^{s}_{11}+9\widehat{m}^{s}_{12}+90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}+72\widehat{m}^{s}_{1}+72\widehat{m}^{s}_{2}-72\widehat{m}^{s}_{3}+90\widehat{m}^{s}_{4}-90\widehat{m}^{s}_{5}-90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.+9\widehat{m}^{s}_{10}+9\widehat{m}^{s}_{11}-9\widehat{m}^{s}_{12}-90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}-72\widehat{m}^{s}_{1}+72\widehat{m}^{s}_{2}-72\widehat{m}^{s}_{3}-90\widehat{m}^{s}_{4}+90\widehat{m}^{s}_{5}-90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.-9\widehat{m}^{s}_{10}+9\widehat{m}^{s}_{11}-9\widehat{m}^{s}_{12}+90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{13}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}+72\widehat{m}^{s}_{1}-72\widehat{m}^{s}_{2}-72\widehat{m}^{s}_{3}-90\widehat{m}^{s}_{4}-90\widehat{m}^{s}_{5}+90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
$\displaystyle\left.+9\widehat{m}^{s}_{10}-9\widehat{m}^{s}_{11}-9\widehat{m}^{s}_{12}+90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right],$
$\displaystyle S_{14}$ $\displaystyle=$
$\displaystyle\frac{1}{720}\left[48\widehat{m}^{s}_{0}-72\widehat{m}^{s}_{1}-72\widehat{m}^{s}_{2}-72\widehat{m}^{s}_{3}+90\widehat{m}^{s}_{4}+90\widehat{m}^{s}_{5}+90\widehat{m}^{s}_{6}+40\widehat{m}^{s}_{9}\right.$
(75)
$\displaystyle\left.-9\widehat{m}^{s}_{10}-9\widehat{m}^{s}_{11}-9\widehat{m}^{s}_{12}-90\widehat{m}^{s}_{13}+\widehat{m}^{s}_{14}\right].$
For obtaining explicit expressions for the collision kernel, it is convenient
to express the non-conserved transformed raw moments using the operator
notation given in Eq. (31), which are given as subsets of the particle
velocity directions for the D3Q15 lattice. It follows that
$\displaystyle\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
y}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{4}}-\sum_{\alpha}^{B_{4}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
z}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha y}e_{\alpha
z}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha y}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}^{2}=\left(\sum_{\alpha}^{A_{7}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{8}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{zz}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
z}^{2}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
z}^{2}=\left(\sum_{\alpha}^{A_{9}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{10}}-\sum_{\alpha}^{B_{10}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}=\left(\sum_{\alpha}^{A_{11}}-\sum_{\alpha}^{B_{11}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha z}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{12}}-\sum_{\alpha}^{B_{12}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xyz}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}e_{\alpha y}e_{\alpha z}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}e_{\alpha y}e_{\alpha
z}=\left(\sum_{\alpha}^{A_{13}}-\sum_{\alpha}^{B_{13}}\right)\otimes\overline{f}_{\alpha},$
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}=\sum_{\alpha=0}^{14}\overline{f}_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}^{2}=\left(\sum_{\alpha}^{A_{14}}\right)\otimes\overline{f}_{\alpha},$ (76)
where
$\displaystyle A_{4}$ $\displaystyle=$
$\displaystyle\left\\{7,10,11,14\right\\},B_{4}=\left\\{8,9,12,13\right\\},$
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle\left\\{7,9,12,14\right\\},B_{5}=\left\\{8,10,11,13\right\\},$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\left\\{7,8,13,14\right\\},B_{6}=\left\\{9,10,11,12\right\\},$
$\displaystyle A_{7}$ $\displaystyle=$
$\displaystyle\left\\{1,2,7,8,9,10,11,12,13,14\right\\},$ $\displaystyle
A_{8}$ $\displaystyle=$
$\displaystyle\left\\{3,4,7,8,9,10,11,12,13,14\right\\},$ $\displaystyle
A_{9}$ $\displaystyle=$
$\displaystyle\left\\{5,6,7,8,9,10,11,12,13,14\right\\},$ $\displaystyle
A_{10}$ $\displaystyle=$
$\displaystyle\left\\{7,9,11,13\right\\},B_{10}=\left\\{8,10,12,14\right\\},$
$\displaystyle A_{11}$ $\displaystyle=$
$\displaystyle\left\\{7,8,11,12\right\\},B_{11}=\left\\{9,10,13,14\right\\},$
$\displaystyle A_{12}$ $\displaystyle=$
$\displaystyle\left\\{7,8,9,10\right\\},B_{12}=\left\\{11,12,13,14\right\\},$
$\displaystyle A_{13}$ $\displaystyle=$
$\displaystyle\left\\{7,10,12,13\right\\},B_{13}=\left\\{8,9,11,14\right\\},$
$\displaystyle A_{14}$ $\displaystyle=$
$\displaystyle\left\\{7,8,9,10,11,12,13,14\right\\}.$
### G.3 Collision Kernel
Following the same procedure and the notations as used for the D3Q27 lattice
and considering factorized attractors for the higher order moments, the
cascaded form of the central moment collision operator in the presence of
forcing terms can be constructed. The results are summarized as follows (for
collisional invariants,
$\widehat{g}_{0}=\widehat{g}_{1}=\widehat{g}_{2}=\widehat{g}_{3}=0$):
$\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{\omega_{4}}{8}\left[-\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+\rho
u_{x}u_{y}+\frac{1}{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x})\right],$
(77) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{\omega_{5}}{8}\left[-\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+\rho
u_{x}u_{z}+\frac{1}{2}(\widehat{\sigma}_{x}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x})\right],$
(78) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{6}}{8}\left[-\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+\rho
u_{y}u_{z}+\frac{1}{2}(\widehat{\sigma}_{y}^{{}^{\prime}}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{y})\right],$
(79) $\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\frac{\omega_{7}}{4}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})+\rho(u_{x}^{2}-u_{y}^{2})+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}-\widehat{\sigma}_{y}^{{}^{\prime}}u_{y})\right],$
(80) $\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\frac{\omega_{8}}{12}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-2\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})+\rho(u_{x}^{2}+u_{y}^{2}-2u_{z}^{2})\right.$
(81)
$\displaystyle\left.+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}-2\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})\right],$
$\displaystyle\widehat{g}_{9}$ $\displaystyle=$
$\displaystyle\frac{\omega_{9}}{18}\left[-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+\widehat{\overline{\eta}}_{zz}^{{}^{\prime}})+\rho(u_{x}^{2}+u_{y}^{2}+u_{z}^{2})\right.$
(82)
$\displaystyle\left.+(\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{z})+\rho\right],$
$\displaystyle\widehat{g}_{10}$ $\displaystyle=$
$\displaystyle\frac{\omega_{10}}{16}\left[-\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+2u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-2\rho
u_{x}u_{y}^{2}-\frac{1}{2}\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}^{2}-\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}u_{x}\right]$
(83)
$\displaystyle+u_{y}\widehat{g}_{4}+\frac{1}{8}u_{x}(-\widehat{g}_{7}+\widehat{g}_{8}+3\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{11}$ $\displaystyle=$
$\displaystyle\frac{\omega_{11}}{16}\left[-\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-2\rho
u_{x}^{2}u_{y}-\frac{1}{2}\widehat{\sigma}_{y}^{{}^{\prime}}u_{x}^{2}-\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}u_{y}\right]$
(84)
$\displaystyle+u_{x}\widehat{g}_{4}+\frac{1}{8}u_{y}(\widehat{g}_{7}+\widehat{g}_{8}+3\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{12}$ $\displaystyle=$
$\displaystyle\frac{\omega_{12}}{16}\left[-\widehat{\overline{\eta}}_{xxz}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-2\rho
u_{x}^{2}u_{z}-\frac{1}{2}\widehat{\sigma}_{z}^{{}^{\prime}}u_{x}^{2}-\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}u_{z}\right]$
(85)
$\displaystyle+u_{x}\widehat{g}_{5}+\frac{1}{8}u_{z}(\widehat{g}_{7}+\widehat{g}_{8}+3\widehat{g}_{9}),$
$\displaystyle\widehat{g}_{13}$ $\displaystyle=$
$\displaystyle\frac{\omega_{13}}{8}\left[-\widehat{\overline{\eta}}_{xyz}^{{}^{\prime}}+u_{x}\widehat{\overline{\eta}}_{yz}^{{}^{\prime}}+u_{y}\widehat{\overline{\eta}}_{xz}^{{}^{\prime}}+u_{z}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-2\rho
u_{x}u_{y}u_{z}-\frac{1}{2}\left(\widehat{\sigma}_{x}^{{}^{\prime}}u_{y}u_{z}\right.\right.$
(86)
$\displaystyle\left.\left.+\widehat{\sigma}_{y}^{{}^{\prime}}u_{x}u_{z}+\widehat{\sigma}_{z}^{{}^{\prime}}u_{x}u_{y}\right)\right]+u_{z}\widehat{g}_{4}+u_{y}\widehat{g}_{5}+u_{x}\widehat{g}_{6},$
$\displaystyle\widehat{g}_{14}$ $\displaystyle=$
$\displaystyle\frac{\omega_{14}}{16}\left[-\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}+2u_{x}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}+2u_{y}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}-u_{x}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-u_{y}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-4u_{x}u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}\right.$
(87)
$\displaystyle\left.+\widetilde{\widehat{\kappa}}_{xx}\widetilde{\widehat{\kappa}}_{yy}+3\rho
u_{x}^{2}u_{y}^{2}+\widehat{\sigma}_{x}^{{}^{\prime}}u_{x}u_{y}^{2}+\widehat{\sigma}_{y}^{{}^{\prime}}u_{y}u_{x}^{2}\right]-2u_{x}u_{y}\widehat{g}_{4}+\frac{1}{8}(u_{x}^{2}-u_{y}^{2})\widehat{g}_{7}$
$\displaystyle+\frac{1}{8}(-u_{x}^{2}-u_{y}^{2})\widehat{g}_{8}+\left(\frac{3}{8}(-u_{x}^{2}-u_{y}^{2})-\frac{1}{2}\right)\widehat{g}_{9}+2u_{x}\widehat{g}_{10}+2u_{y}\widehat{g}_{11},$
where $\omega_{4},\omega_{5},\ldots,\omega_{14}$ are relaxation parameters.
Note that similar to the D3Q27 lattice, we have the following relation for the
shear viscosity of the fluid
$\nu=c_{s}^{2}\left(\frac{1}{\omega^{\nu}}-\frac{1}{2}\right)$, where
$\omega^{\nu}=\omega_{j}$ and $j=4,5,6,7,8$. The remaining parameters can be
adjusted independently to control numerical stability.
### G.4 Operational Steps
Finally, by expanding the elements of the matrix multiplication of
$\mathcal{K}$ with $\widehat{\mathbf{g}}$ in Eq. (16), the post-collision
values of the distribution function augmented by source terms corresponding to
the D3Q15 lattice are
$\displaystyle\widetilde{\overline{f}}_{0}$ $\displaystyle=$
$\displaystyle\overline{f}_{0}+\left[\widehat{g}_{0}-2\widehat{g}_{9}+32\widehat{g}_{14}\right]+S_{0},$
$\displaystyle\widetilde{\overline{f}}_{1}$ $\displaystyle=$
$\displaystyle\overline{f}_{1}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}-8\widehat{g}_{10}-8\widehat{g}_{14}\right]+S_{1},$
$\displaystyle\widetilde{\overline{f}}_{2}$ $\displaystyle=$
$\displaystyle\overline{f}_{2}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}+8\widehat{g}_{10}-8\widehat{g}_{14}\right]+S_{2},$
$\displaystyle\widetilde{\overline{f}}_{3}$ $\displaystyle=$
$\displaystyle\overline{f}_{3}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}-8\widehat{g}_{11}-8\widehat{g}_{14}\right]+S_{3},$
$\displaystyle\widetilde{\overline{f}}_{4}$ $\displaystyle=$
$\displaystyle\overline{f}_{4}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{7}+\widehat{g}_{8}-\widehat{g}_{9}+8\widehat{g}_{11}-8\widehat{g}_{14}\right]+S_{4},$
$\displaystyle\widetilde{\overline{f}}_{5}$ $\displaystyle=$
$\displaystyle\overline{f}_{5}+\left[\widehat{g}_{0}+\widehat{g}_{3}-2\widehat{g}_{8}-\widehat{g}_{9}-8\widehat{g}_{12}-8\widehat{g}_{14}\right]+S_{5},$
$\displaystyle\widetilde{\overline{f}}_{6}$ $\displaystyle=$
$\displaystyle\overline{f}_{6}+\left[\widehat{g}_{0}-\widehat{g}_{3}-2\widehat{g}_{8}-\widehat{g}_{9}+8\widehat{g}_{12}-8\widehat{g}_{14}\right]+S_{6},$
$\displaystyle\widetilde{\overline{f}}_{7}$ $\displaystyle=$
$\displaystyle\overline{f}_{7}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+\widehat{g}_{3}+\widehat{g}_{4}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}+2\widehat{g}_{11}+2\widehat{g}_{12}\right.$
$\displaystyle\left.+\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{7},$
$\displaystyle\widetilde{\overline{f}}_{8}$ $\displaystyle=$
$\displaystyle\overline{f}_{8}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}+\widehat{g}_{3}-\widehat{g}_{4}-\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}+2\widehat{g}_{11}+2\widehat{g}_{12}\right.$
$\displaystyle\left.-\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{8},$
$\displaystyle\widetilde{\overline{f}}_{9}$ $\displaystyle=$
$\displaystyle\overline{f}_{9}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}+\widehat{g}_{3}-\widehat{g}_{4}+\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}-2\widehat{g}_{11}+2\widehat{g}_{12}\right.$
$\displaystyle\left.-\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{9},$
$\displaystyle\widetilde{\overline{f}}_{10}$ $\displaystyle=$
$\displaystyle\overline{f}_{10}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+\widehat{g}_{3}+\widehat{g}_{4}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}-2\widehat{g}_{11}+2\widehat{g}_{12}\right.$
$\displaystyle\left.+\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{10},$
$\displaystyle\widetilde{\overline{f}}_{11}$ $\displaystyle=$
$\displaystyle\overline{f}_{11}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}-\widehat{g}_{3}+\widehat{g}_{4}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}+2\widehat{g}_{11}-2\widehat{g}_{12}\right.$
$\displaystyle\left.-\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{11},$
$\displaystyle\widetilde{\overline{f}}_{12}$ $\displaystyle=$
$\displaystyle\overline{f}_{12}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}+\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}+2\widehat{g}_{11}-2\widehat{g}_{12}\right.$
$\displaystyle\left.+\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{12},$
$\displaystyle\widetilde{\overline{f}}_{13}$ $\displaystyle=$
$\displaystyle\overline{f}_{13}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}-\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}+2\widehat{g}_{10}-2\widehat{g}_{11}-2\widehat{g}_{12}\right.$
$\displaystyle\left.+\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{13},$
$\displaystyle\widetilde{\overline{f}}_{14}$ $\displaystyle=$
$\displaystyle\overline{f}_{14}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}-\widehat{g}_{3}+\widehat{g}_{4}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{9}-2\widehat{g}_{10}-2\widehat{g}_{11}-2\widehat{g}_{12}\right.$
(88) $\displaystyle\left.-\widehat{g}_{13}+2\widehat{g}_{14}\right]+S_{14}.$
## Acknowledgments
The authors wish to thank the anonymous referees for their helpful comments.
This research was in part supported by the National Science Foundation through
Teragrid resources provided by abe-queenbee-steele.teragrid sites under grant
number TG-CTS 110023.
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|
arxiv-papers
| 2012-02-27T22:01:32 |
2024-09-04T02:49:27.916059
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kannan N. Premnath and Sanjoy Banerjee",
"submitter": "Kannan Premnath",
"url": "https://arxiv.org/abs/1202.6081"
}
|
1202.6087
|
# Incorporating Forcing Terms in Cascaded Lattice-Boltzmann Approach by Method
of Central Moments
Kannan N. Premnath nandha@metah.com Department of Chemical Engineering,
University of California, Santa Barbara, Santa Barbara, CA 93106
MetaHeuristics LLC, 3944 State Street, Suite 350, Santa Barbara, CA 93105
Sanjoy Banerjee banerjee@engineering.ucsb.edu Department of Chemical
Engineering
Department of Mechanical Engineering
Bren School of Environmental Science and Management
University of California, Santa Barbara, Santa Barbara, CA 93106
###### Abstract
Cascaded lattice-Boltzmann method (Cascaded-LBM) employs a new class of
collision operators aiming to stabilize computations and remove certain
modeling artifacts for simulation of fluid flow on lattice grids with sizes
arbitrarily larger than the smallest physical dissipation length scale (Geier
_et al._ , Phys. Rev. E $\mathbf{63}$, 066705 (2006)). It achieves this and
distinguishes from other collision operators, such as in the standard single
or multiple relaxation time approaches, by performing relaxation process due
to collisions in terms of moments shifted by the local hydrodynamic fluid
velocity, i.e. central moments, in an ascending order-by-order at different
relaxation rates. In this paper, we propose and derive source terms in the
Cascaded-LBM to represent the effect of external or internal forces on the
dynamics of fluid motion. This is essentially achieved by matching the
continuous form of the central moments of the source or forcing terms with its
discrete version. Different forms of continuous central moments of sources,
including one that is obtained from a local Maxwellian, are considered in this
regard. As a result, the forcing terms obtained in this new formulation are
Galilean invariant by construction. To alleviate lattice artifacts due to
forcing terms in the emergent macroscopic fluid equations, they are proposed
as temporally semi-implicit and second-order, and the implicitness is
subsequently effectively removed by means of a transformation to facilitate
computation. It is shown that the impressed force field influences the
cascaded collision process in the evolution of the transformed distribution
function. The method of central moments along with the associated orthogonal
properties of the moment basis completely determines the analytical
expressions for the source terms as a function of the force and macroscopic
velocity fields. In contrast to the existing forcing schemes, it is found that
they involve higher order terms in velocity space. It is shown that the
proposed approach implies “generalization” of both local equilibrium and
source terms in the usual lattice frame of reference, which depend on the
ratio of the relaxation times of moments of different orders. An analysis by
means of the Chapman-Enskog multiscale expansion shows that the Cascaded-LBM
with forcing terms is consistent with the Navier-Stokes equations.
Computational experiments with canonical problems involving different types of
forces demonstrate its accuracy.
###### pacs:
47.11.Qr,05.20.Dd,47.27.-i
††preprint: PREPRINT
## I Introduction
Lattice-Boltzmann method (LBM), based on minimal discrete kinetic models, has
attracted considerable attention as an alternative computational approach for
fluid mechanics problems Benzi et al. (1992); Chen and Doolen (1998); Succi
(2001); Yu et al. (2003). While its origins can be traced to lattice gas
automata Frisch et al. (1986) as a means to remove its statistical noise
McNamara and Zanetti (1988), over the years, the LBM has undergone major
series of advances to improve its underlying models for better physical
fidelity and computational efficiency. Moreover, its connection to the
continuous Boltzmann equation as a dramatically simplified version He and Luo
(1997); Shan and He (1998) established it as an efficient approach in
computational kinetic theory and led to the development of asymptotic tools
Junk et al. (2005) providing a rigorous framework for numerical consistency
analysis. The LBM is based on performing stream-and-collide steps to compute
the evolution of the distribution of particle populations, such that its
averaged behavior recovers the dynamics of fluid motion. The streaming step is
a free-flight process along discrete characteristic particle directions
designed from symmetry considerations, while the collision step is generally
represented as a relaxation process of the distribution function to its
attractors, i.e. local equilibrium states. Considerable effort has been made
in developing models to account for various aspects of the collision process,
as it has paramount influence on the physical fidelity and numerical stability
of the LBM.
One of the simplest and among the most common is the single-relaxation-time
(SRT) model proposed by Chen _et al_. Chen et al. (1992) and Qian _et al_.
Qian et al. (1992), which is based on the BGK approximation Bhatnagar et al.
(1954). On the other hand, d’Humières (1992) d‘Humières (1992) proposed a
moment method, in which various moments that are integral properties of
distribution functions weighted by the Cartesian components of discrete
particle velocities of various orders are relaxed to their equilibrium states
at different rates during collision step, leading to the multiple-relaxation-
time (MRT) model. It is an important extension of the relaxation LBM proposed
earlier by Higuera _et al_ Higuera and Jiménez (1989); Higuera et al. (1989).
While it is a much simplified version of the latter, the major innovation lies
in representing the collision process in moment space Grad (1949) rather than
the usual particle velocity space. By carefully separating the relaxation
times of hydrodynamic and non-hydrodynamic moments, it has been shown that the
MRT-LBM significantly improves the numerical stability Lallemand and Luo
(2000); d‘Humières et al. (2002) and better physical representation in certain
problems such as kinetic layers near boundaries Ginzburg and d‘Humières
(2003), when compared with the SRT-LBM. Such MRT models have recently been
shown to reproduce challenging fluid mechanics problems such as complex
turbulent flows with good quantitative accuracy Premnath et al. (2009a, b). An
important and natural simplification of the MRT model is the two-relaxation-
time (TRT) model, in which the moments of even and odd orders are relaxed at
different rates Ginzburg (2005).
From a different perspective, Karlin and co-workers Karlin et al. (1999);
Boghosian et al. (2001); Ansumali and Karlin (2002); Succi et al. (2002);
Karlin et al. (2006) have developed the so-called entropic LBM in which the
collision process is modeled by assuming that distribution functions are drawn
towards their attractors, which are obtained by the minimization of a
Lyapanov-type functional, i.e. the so-called H-theorem is enforced locally,
while modulating the relaxation process with a single relaxation time to
maintain numerical stability. It may be noted that in contrast to the SRT or
MRT collision operators, which employ equilibria that are polynomials in
hydrodynamic fields, the entropic collision operator, in general, requires use
of non-polynomial or transcendental functions of hydrodynamic fields.
Recently, using this framework, a novel entropy-based MRT model was derived
Asinari and Karlin (2009) and a Galilean invariance restoration approach was
developed Prasianakis et al. (2009). In addition, there has been considerable
progress in the development of systematic procedures for high-order lattice-
Boltzmann models Shan et al. (2006); Chikatamarla and Karlin (2006).
Recently, Geier _et al_. Geier et al. (2006) introduced another novel class of
collision operator leading to the so-called Cascaded-LBM. Collision operators,
such as the standard SRT or MRT models, are generally constructed to recover
the Navier-Stokes equations (NSE), with errors that are quadratic in fluid
velocity. Such models, which are Galilean invariant up to a lower degree,
i.e., the square of Mach number, are prone to numerical instability, which can
be alleviated to a degree with the use of the latter model. Recognizing that
insufficient level of Galilean invariance is one of the main sources of
numerical instability, Geier proposed to perform collision process in a frame
of reference shifted by the macroscopic fluid velocity. Unlike other collision
operators which perform relaxation in a special rest or lattice frame of
reference, Cascaded-LBM chooses an intrinsic frame of reference obtained from
the properties of the system itself. The local hydrodynamic velocity, which is
the first moment of the distribution functions, is the center of mass in the
space of moments. A coordinate system moving locally with this velocity at
each node is a natural framework to describe the physics of collisions in the
space of moments. This could enable achieving a higher degree of Galilean
invariance than possible with the prior approaches. It may be noted that the
moments displaced by the local hydrodynamic velocity are termed as the
_central moments_ and are computed in a moving frame of reference. On the
other hand, the moments with no such shift are called the _raw moments_ ,
which are computed in a rest frame of reference.
Based on this insight, the collision operator is constructed in such a way
that each central moment can be relaxed independently with generally different
relaxation rates. However, it is computationally easier to perform operations
in terms of raw moments. Both forms of moments can be related to one another
in terms of the binomial theorem, and hence the latter plays an important role
in the construction of an operational collision step. As a result of this
theorem, central moment of a given order are algebraic combinations of raw
moments of different orders, with their highest order being equal to that of
the central moment. In effect, the evolution of lower order raw moments
influences higher order central moments and not vice versa. Thus, due to this
specific directionality of coupling between different central and raw moments,
starting from the lowest central moment, we can relax successively higher
order central moments towards their equilibrium, which are implicitly carried
out in terms of raw moments. Such structured sequential computation of
relaxation in an ascending order of moments leads to a novel cascaded
collision operator, in which the post-collision moments depend not only on the
conserved moments, but also on the non-conserved moments and on each other.
Moreover, it was found that relaxing different central moments differently,
certain artifacts such as aliasing that cause numerical instability for
computation on coarse grids, whose sizes can be arbitrarily larger than the
smallest physical or viscous dissipation length scale can be avoided. In
particular, this is achieved by setting the third-order central moments to its
equilibrium value, while allowing only the second-order moments to undergo
over-relaxtion Geier (2008a). The limit of stability is now dictated only by
the Courant-Friedrichs-Lewy condition Courant et al. (1967) typical of
explicit schemes and not by effects arising due to the discreteness of the
particle velocity set. Prevention of such ultra-violet catastrophe in under-
resolved computations could enable application of the LBM for high Reynolds
number flows or for fluid with low viscosities. Further insight into the
nature of the gain in numerical stability with Cascaded-LBM is achieved with
the recognition that unlike other collision operators which appear to
introduce de-stabilizing negative hyper-viscosity effects that are of second-
order in Mach number due to insufficient Galilean invariance, the former seems
to have stabilizing positive and smaller hyper-viscosity effects that are of
fourth-order in Mach number Geier (2008b). Recently, Asinari Asinari (2008)
showed that cascaded relaxation using multiple relaxation times is equivalent
to performing relaxation to a “generalized” local equilibrium in the rest
frame of reference. Such generalized local equilibrium is dependent on non-
conserved moments as well as the ratio of various relaxation times.
Clearly, several situations exist in which the dynamics of fluid motion is
driven or affected by the presence of external or self-consistent internal
forces. Examples include gravity, magnetohydrodynamic forces, self-consistent
internal forces in multi-phase or multi-fluid systems. Moreover, subgrid scale
(SGS) models for turbulence simulation can be explicitly introduced as body
forces in kinetic approaches Girimaji (2007); Premnath et al. (2009b). Thus,
it is important to develop a consistent approach to introduce the effect of
forces that act on the fluid flow in the Cascaded-LBM. The method for
introducing force terms in other LBM approaches are given, for example, in He
et al. (1998); Martys et al. (1998); Ladd and Verberg (2001); Guo et al.
(2002), in which notably Guo _et al_. Guo et al. (2002) developed a
consistent approach which avoided spurious effects in the macroscopic
equations resulting from the finiteness of the lattice set.
The approach proposed in this paper consists as follows. It consists of
deriving forcing terms which can be obtained by matching their discrete
central moments to their corresponding continuous version. In this regard, we
consider two different sets of ansatz for the continuous source central
moments – one based on a continuous local Maxwellian and another one which
makes specific assumptions regarding the effect of forces for higher order
moments. An important feature of our approach is that by construction the
source terms are Galilean invariant, which would be a very desirable aspect
from both physical and computational points of view. To facilitate
computation, the central source moments are related to corresponding raw
moments, which are, in turn, expressed in velocity space. Furthermore, to
improve temporal accuracy, the source terms are treated semi-implicitly. The
implicitness, then, is effectively removed by applying a transformation to the
distribution function. A detailed _a priori_ derivation of this central moment
method is given so that it provides a mathematical framework which could also
be useful for extension to other problems. We then establish the consistency
of our approach to macroscopic fluid dynamical equations by performing a
Chapman-Enskog multiscale moment expansion. It will be shown that when
Cascaded-LBM with forcing terms is reinterpreted in terms of the rest frame of
reference (as usual with other LBM), it implies considering a generalized
local equilibrium and sources, which also depend on the ratio of the
relaxation times of various moments, for their higher order moments. Numerical
experiments will also be performed to confirm the accuracy of our approach for
flows with different types of forces, where analytical solutions are
available.
This paper is structured as follows. Section II briefly discusses the choice
of moment basis employed in this paper. In Sec. III, continuous forms of
central moments for equilibrium and sources (for a specific ansatz) are
introduced. The Cascaded-LBE with forcing terms are presented in Sec. IV. In
Sec. V, we discuss the details of an analysis and the construction of the
Cascaded-LBM and the analytical expressions for source terms. Section VI
provides the details of how the computational procedure is modified with the
use of a different form of the central source moments. The computational
procedure for Cascaded-LBM with forcing is provided in Sec. VII. Results of
the computational procedure for some canonical problems are presented in Sec.
VIII. Summary and conclusions of this work are described in Sec. IX.
Consistency analysis of the central moment method with forcing terms by means
of a Chapman-Enskog multiscale moment expansion is presented in Appendix A.
Appendix B shows that Cascaded-LBM with forcing terms is equivalent to
considering a generalized local equilibrium and sources in the rest frame of
reference. Finally, Appendix C investigates the possibility of introducing
time-implicitness in the cascaded collision operator.
## II Choice of Basis Vectors for Moments
For concreteness, without losing generality, we consider, the two-dimensional,
nine velocity (D2Q9) model, which is shown in Fig. 1. The particle velocity
$\overrightarrow{e}_{\alpha}$ may be written as
$\overrightarrow{e_{\alpha}}=\left\\{\begin{array}[]{ll}{(0,0)}&{\alpha=0}\\\
{(\pm 1,0),(0,\pm 1)}&{\alpha=1,\cdots,4}\\\ {(\pm 1,\pm
1)}&{\alpha=5,\cdots,8}\end{array}\right.$ (1)
Figure 1: Two-dimensional, nine-velocity (D2Q9) Lattice.
Here and henceforth, we employ Greek and Latin subscripts for particle
velocity directions and Cartesian coordinate directions, respectively. Moments
in the LBM are discrete integral properties of the distribution function
$f_{\alpha}$, i.e. $\sum_{\alpha=0}^{8}e_{\alpha x}^{m}e_{\alpha
y}^{n}f_{\alpha}$, where $m$ and $n$ are integers. Since the theory of the
moment method draws heavily upon the associated orthogonality properties, for
convenience, we employ the Dirac’s bra-ket notation in this paper. That is, we
denote the “bra” operator $\bra{\phi}$ to represent a row vector of any state
variable $\phi$ along each of the particle directions, i.e.
$(\phi_{0},\phi_{1},\phi_{2},\ldots,\phi_{8})$, and the “ket” operator
$\ket{\phi}$ represents a column vector, i.e.
$(\phi_{0},\phi_{1},\phi_{2},\ldots,\phi_{8})^{\dagger}$, where the
superscript “†” is the transpose operator. In this notation,
$\braket{\phi}{\varphi}$ represents the inner-product, i.e.
$\sum_{\alpha=0}^{8}\phi_{\alpha}\varphi_{\alpha}$. To obtain a moment space
of the distribution functions, we start with a set of the following nine non-
orthogonal basis vectors obtained from the combinations of the monomials
$e_{\alpha x}^{m}e_{\alpha y}^{n}$ in an ascending order.
$\displaystyle\ket{\rho}\equiv\ket{|\overrightarrow{e}_{\alpha}|^{0}}$
$\displaystyle=$ $\displaystyle\left(1,1,1,1,1,1,1,1,1\right)^{\dagger},$ (2)
$\displaystyle\ket{e_{\alpha x}}$ $\displaystyle=$
$\displaystyle\left(0,1,0,-1,0,1,-1,-1,1\right)^{\dagger},$ (3)
$\displaystyle\ket{e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,1,0,-1,1,1,-1,-1\right)^{\dagger},$ (4)
$\displaystyle\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,1,1,1,1,2,2,2,2\right)^{\dagger},$ (5)
$\displaystyle\ket{e_{\alpha x}^{2}-e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,1,-1,1,-1,0,0,0,0\right)^{\dagger},$ (6)
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,-1,1,-1\right)^{\dagger},$ (7)
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,1,-1,-1\right)^{\dagger},$ (8)
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,-1,-1,1\right)^{\dagger},$ (9)
$\displaystyle\ket{e_{\alpha x}^{2}e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\left(0,0,0,0,0,1,1,1,1\right)^{\dagger}.$ (10)
To facilitate analysis, the above set of basis vectors is transformed into an
equivalent _orthogonal_ set of basis vectors by means of the standard Gram-
Schmidt procedure in the increasing order of the monomials of the products of
the Cartesian components of the particle velocities:
$\displaystyle\ket{K_{0}}$ $\displaystyle=$ $\displaystyle\ket{\rho},$ (11)
$\displaystyle\ket{K_{1}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha x}},$
(12) $\displaystyle\ket{K_{2}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
y}},$ (13) $\displaystyle\ket{K_{3}}$ $\displaystyle=$ $\displaystyle
3\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}}-4\ket{\rho},$ (14)
$\displaystyle\ket{K_{4}}$ $\displaystyle=$ $\displaystyle\ket{e_{\alpha
x}^{2}-e_{\alpha y}^{2}},$ (15) $\displaystyle\ket{K_{5}}$ $\displaystyle=$
$\displaystyle\ket{e_{\alpha x}e_{\alpha y}},$ (16) $\displaystyle\ket{K_{6}}$
$\displaystyle=$ $\displaystyle-3\ket{e_{\alpha x}^{2}e_{\alpha
y}}+2\ket{e_{\alpha y}},$ (17) $\displaystyle\ket{K_{7}}$ $\displaystyle=$
$\displaystyle-3\ket{e_{\alpha x}e_{\alpha y}^{2}}+2\ket{e_{\alpha x}},$ (18)
$\displaystyle\ket{K_{8}}$ $\displaystyle=$ $\displaystyle 9\ket{e_{\alpha
x}^{2}e_{\alpha y}^{2}}-6\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}}+4\ket{\rho}.$
(19)
This is very similar to that used by Geier _et al_. Geier et al. (2006),
except for the negative sign used in $\ket{K_{5}}$ by the latter. The purpose
of using a slightly different orthogonal basis than that considered in Geier
et al. (2006) is simply to illustrate how it changes the details of the
cascaded collision operator. It is obvious that we can define different sets
of orthogonal basis vectors that differ from one another by a constant factor
or a sign. Furthermore, it is noteworthy to compare the ordering of basis
vectors used for the central moment method with that considered by Lallemand
and Luo Lallemand and Luo (2000): Here, the ordering is based on the ascending
powers of moments (i.e. zeroth order moment, first order moments, second order
moments,$\ldots$) while Lallemand and Luo (2000) order their basis vectors
based on the character of moments, i.e. increasing powers of their tensorial
orders (i.e. scalars, vectors, tensors of different ranks,$\ldots$).
The orthogonal set of basis vectors can be written in terms of the following
matrix
$\mathcal{K}=\left[\ket{K_{0}},\ket{K_{1}},\ket{K_{2}},\ket{K_{3}},\ket{K_{4}},\ket{K_{5}},\ket{K_{6}},\ket{K_{7}},\ket{K_{8}}\right],$
(20)
which can be explicitly written as
$\mathcal{K}=\left[\begin{array}[]{rrrrrrrrr}1&0&0&-4&0&0&0&0&4\\\
1&1&0&-1&1&0&0&2&-2\\\ 1&0&1&-1&-1&0&2&0&-2\\\ 1&-1&0&-1&1&0&0&-2&-2\\\
1&0&-1&-1&-1&0&-2&0&-2\\\ 1&1&1&2&0&1&-1&-1&1\\\ 1&-1&1&2&0&-1&-1&1&1\\\
1&-1&-1&2&0&1&1&1&1\\\ 1&1&-1&2&0&-1&1&-1&1\\\ \end{array}\right].$ (21)
It possesses a number of interesting properties including a computationally
useful fact that $\mathcal{K}\mathcal{K}^{\dagger}$ is a diagonal matrix.
## III Continuous Central Moments: Equilibrium and Sources
Consider an athermal fluid in motion which is characterized by its local
hydrodynamic fields at the Cartesian coordinate $(x,y)$, i.e. density $\rho$,
hydrodynamic velocity $\overrightarrow{u}=(u_{x},u_{y})$, and subjected to a
force field $\overrightarrow{F}=(F_{x},F_{y})$, whose origin could be either
internal or external to the system. The local Maxwell-Boltzmann distribution,
or, simply, the Maxwellian in _continuous_ particle velocity space
$(\xi_{x},\xi_{y})$ is given by
$f^{\mathcal{M}}\equiv
f^{\mathcal{M}}(\rho,\overrightarrow{u},\xi_{x},\xi_{y})=\frac{\rho}{2\pi
c_{s}^{2}}\exp\left[-\frac{\left(\overrightarrow{\xi}-\overrightarrow{u}\right)^{2}}{2c_{s}^{2}}\right],$
(22)
where we choose
$c_{s}^{2}=1/3.$ (23)
Let us now define _continuous_ central moments, i.e. moments displaced by the
local hydrodynamic velocity, of order $(m+n)$:
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\mathcal{M}}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}.$
(24)
By virtue of the fact that $f^{\mathcal{M}}$ being an even function,
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}\neq 0$ when $m$ and $n$ are even and
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}=0$ when $m$ or $n$ odd. Here and
henceforth, the subscripts $x^{m}y^{n}$ mean $xxx\cdots m-\text{times}$ and
$yyy\cdots n-\text{times}$. Thus, evaluating this quantity in the increasing
order of moments gives
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{x}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{y}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xx}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{yy}$
$\displaystyle=$ $\displaystyle c_{s}^{2}\rho,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xxy}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xyy}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\widehat{\Pi}^{\mathcal{M}}_{xxyy}$
$\displaystyle=$ $\displaystyle c_{s}^{4}\rho.$
Here, and in the rest of this paper, the use of “hat” over a symbol represents
values in the space of moments.
Now, we propose that the continuous distribution function $f$ is modified by
the presence of a force field as given by the following ansatz:
$\Delta
f^{F}=\frac{\overrightarrow{F}}{\rho}\cdot\frac{(\overrightarrow{\xi}-\overrightarrow{u})}{c_{s}^{2}}f^{\mathcal{M}}$
(25)
It may be noted that He _et al_. (1998) He et al. (1998) proposed similar form
for the continuous Boltzmann equation to derive source terms for the SRT-LBE.
However, it’s influence on discrete distribution function due to cascaded
collision process via the method of central moments to establish Galilean
invariant solutions is expected to be, in general, be different. Let us now
define a corresponding _continuous_ central moment of order $(m+n)$ due to
change in the distribution function as a result of a force field as
$\widehat{\Gamma}^{F}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Delta
f^{F}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}.$ (26)
Evaluation of Eq. (26) in the increasing order of moments yields
$\displaystyle\widehat{\Gamma}^{F}_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\widehat{\Gamma}^{F}_{x}$ $\displaystyle=$ $\displaystyle
F_{x},$ $\displaystyle\widehat{\Gamma}^{F}_{y}$ $\displaystyle=$
$\displaystyle F_{y},$ $\displaystyle\widehat{\Gamma}^{F}_{xx}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\widehat{\Gamma}^{F}_{yy}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\widehat{\Gamma}^{F}_{xy}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle\widehat{\Gamma}^{F}_{xxy}$
$\displaystyle=$ $\displaystyle c_{s}^{2}F_{y},$
$\displaystyle\widehat{\Gamma}^{F}_{xyy}$ $\displaystyle=$ $\displaystyle
c_{s}^{2}F_{x},$ $\displaystyle\widehat{\Gamma}^{F}_{xxyy}$ $\displaystyle=$
$\displaystyle 0.$
## IV Cascaded Lattice-Boltzmann Method with Forcing Terms
First, let us define a _discrete_ distribution function supported by the
discrete particle velocity set $\overrightarrow{e}_{\alpha}$:
$\mathbf{f}=\ket{f_{\alpha}}=(f_{0},f_{1},f_{2},\ldots,f_{8})^{\dagger}.$ (27)
Following Geier _et al_. Geier et al. (2006), we represent collision as a
cascaded process in which the effect of collision on lower order moments
successively influences those of higher order in a cascaded manner. In
particular, we model the change in discrete distribution due to collision as
$\Omega_{\alpha}^{c}\equiv\Omega_{\alpha}^{c}(\mathbf{f},\mathbf{\widehat{g}})=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha},$
(28)
where
$\mathbf{\widehat{g}}=\ket{\widehat{g}_{\alpha}}=(\widehat{g}_{0},\widehat{g}_{1},\widehat{g}_{2},\ldots,\widehat{g}_{8})^{\dagger}$
(29)
determines the changes in _discrete_ moment space in a cascaded manner. That
is, in general,
$\widehat{g}_{\alpha}\equiv\widehat{g}_{\alpha}(\mathbf{f},\widehat{g}_{\beta}),\qquad\beta=0,1,2,\ldots,\alpha-1.$
(30)
The detailed structure of $\mathbf{\widehat{g}}$ will be determined later in
Sec. V.
We define that $f_{\alpha}$ changes due to external force field
$\overrightarrow{F}$ by the _discrete_ source term $S_{\alpha}$. That is,
$\mathbf{S}=\ket{S_{\alpha}}=(S_{0},S_{1},S_{2},\ldots,S_{8})^{\dagger}.$ (31)
We suppose that particle populations are continuously affected by this in time
as they traverse along their characteristics. The precise form of $S_{\alpha}$
is yet unknown and will be determined as part of the procedure presented in
Sec. V.
With the above definitions, the evolution of $f_{\alpha}$ in the Cascaded-LBM
can be written as
$f_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=f_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+\int_{t}^{t+1}S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\theta,t+\theta)}d\theta,$
(32)
where the fluid dynamical variables are determined by
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}=\braket{f_{\alpha}}{\rho},$ (33)
$\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}e_{\alpha
i}=\braket{f_{\alpha}}{e_{\alpha i}},i\in{x,y}.$ (34)
The last term on the right-hand-side (RHS) of Eq. (32) represents the
cumulative effect of forces as particle populations advect along their
characteristic directions. Various approaches are possible here to numerically
represent this integral, with the simplest being an explicit rule. However, in
general cases where $\overrightarrow{F}$ can have spatial and temporal
dependencies, for improved accuracy, it becomes imperative to represent it
with a higher order scheme. One common approach, which is employed here, is to
apply a second-order trapezoidal rule, which will sample both the temporal end
points, $(t,t+1)$, along the characteristic direction $\alpha$. That is,
$f_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=f_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+\frac{1}{2}\left[S_{{\alpha}(\overrightarrow{x},t)}+S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)}\right]$
(35)
Equation (35) is semi-implicit. To remove implicitness along discrete
characteristics, we apply the following transformation He et al. (1998);
Premnath and Abraham (2007):
$\overline{f}_{\alpha}=f_{\alpha}-\frac{1}{2}S_{\alpha}.$ (36)
Thus, Eq. (35) becomes
$\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)}.$
(37)
Clearly, we need to determine $\sum_{\alpha}S_{\alpha}$ and
$\sum_{\alpha}S_{\alpha}\overrightarrow{e}_{\alpha}$ to obtain $\rho$ and
$\rho\overrightarrow{u}$, respectively, in terms of the transformed variable
$\overline{f}_{\alpha}$, which will be carried out in the next section.
## V Construction of Cascaded Collision Operator and Forcing Terms
In order to determine the structure of the cascaded collision operator and the
source terms in the presence of force fields, we now define the following
_discrete_ central moments of the distribution functions and source terms,
respectively:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}=\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{f_{\alpha}},$ (38) $\displaystyle\widehat{\sigma}_{x^{m}y^{n}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}=\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{S_{\alpha}}.$ (39)
We also define a _discrete_ central moment in terms of transformed
distribution function to facilitate subsequent calculations:
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\sum_{\alpha}\overline{f}_{\alpha}(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}=\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{\overline{f}_{\alpha}}.$ (40)
Owing to Eq. (36), it follows that
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\widehat{\kappa}_{x^{m}y^{n}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}.$
(41)
Let us also suppose that $f_{\alpha}$ and $\overline{f}_{\alpha}$ have certain
local equilibrium states represented by $f_{\alpha}^{eq}$ and
$\overline{f}_{\alpha}^{eq}$, respectively, and the corresponding central
moments are
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{eq}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}^{eq}(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}=\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{f_{\alpha}^{eq}},$ (42)
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\overline{f}_{\alpha}^{eq}(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}=\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{\overline{f}_{\alpha}^{eq}}.$ (43)
Now, we take an important step by equating the _discrete_ central moments for
both the distribution functions (equilibrium) and source terms, defined above,
with the _continuous_ central moments derived in Sec. III. Thus, we have
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{eq}$ $\displaystyle=$
$\displaystyle\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}},$ (44)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}$ $\displaystyle=$
$\displaystyle\widehat{\Gamma}^{F}_{x^{m}y^{n}}.$ (45)
In other words, the discrete central moments of various orders for both the
distribution functions (equilibrium) and source terms, respectively, become
$\displaystyle\widehat{\kappa}^{eq}_{0}$ $\displaystyle=$ $\displaystyle\rho,$
(46) $\displaystyle\widehat{\kappa}^{eq}_{x}$ $\displaystyle=$ $\displaystyle
0,$ (47) $\displaystyle\widehat{\kappa}^{eq}_{y}$ $\displaystyle=$
$\displaystyle 0,$ (48) $\displaystyle\widehat{\kappa}^{eq}_{xx}$
$\displaystyle=$ $\displaystyle c_{s}^{2}\rho,$ (49)
$\displaystyle\widehat{\kappa}^{eq}_{yy}$ $\displaystyle=$ $\displaystyle
c_{s}^{2}\rho,$ (50) $\displaystyle\widehat{\kappa}^{eq}_{xy}$
$\displaystyle=$ $\displaystyle 0,$ (51)
$\displaystyle\widehat{\kappa}^{eq}_{xxy}$ $\displaystyle=$ $\displaystyle 0,$
(52) $\displaystyle\widehat{\kappa}^{eq}_{xyy}$ $\displaystyle=$
$\displaystyle 0,$ (53) $\displaystyle\widehat{\kappa}^{eq}_{xxyy}$
$\displaystyle=$ $\displaystyle c_{s}^{4}\rho,$ (54)
and
$\displaystyle\widehat{\sigma}_{0}$ $\displaystyle=$ $\displaystyle 0,$ (55)
$\displaystyle\widehat{\sigma}_{x}$ $\displaystyle=$ $\displaystyle F_{x},$
(56) $\displaystyle\widehat{\sigma}_{y}$ $\displaystyle=$ $\displaystyle
F_{y},$ (57) $\displaystyle\widehat{\sigma}_{xx}$ $\displaystyle=$
$\displaystyle 0,$ (58) $\displaystyle\widehat{\sigma}_{yy}$ $\displaystyle=$
$\displaystyle 0,$ (59) $\displaystyle\widehat{\sigma}_{xy}$ $\displaystyle=$
$\displaystyle 0,$ (60) $\displaystyle\widehat{\sigma}_{xxy}$ $\displaystyle=$
$\displaystyle c_{s}^{2}F_{y},$ (61) $\displaystyle\widehat{\sigma}_{xyy}$
$\displaystyle=$ $\displaystyle c_{s}^{2}F_{x},$ (62)
$\displaystyle\widehat{\sigma}_{xxyy}$ $\displaystyle=$ $\displaystyle 0.$
(63)
From Eq. (41), we get the following transformed central moments, which
comprises as one of the main elements for subsequent development and analysis:
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ (64) $\displaystyle\widehat{\overline{\kappa}}^{eq}_{x}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}F_{x},$ (65)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{y}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}F_{y},$ (66)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{xx}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ (67)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{yy}$ $\displaystyle=$
$\displaystyle c_{s}^{2}\rho,$ (68)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{xy}$ $\displaystyle=$
$\displaystyle 0,$ (69) $\displaystyle\widehat{\overline{\kappa}}^{eq}_{xxy}$
$\displaystyle=$ $\displaystyle-\frac{c_{s}^{2}}{2}F_{y},$ (70)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{xyy}$ $\displaystyle=$
$\displaystyle-\frac{c_{s}^{2}}{2}F_{x},$ (71)
$\displaystyle\widehat{\overline{\kappa}}^{eq}_{xxyy}$ $\displaystyle=$
$\displaystyle c_{s}^{4}\rho.$ (72)
To proceed further, we need to obtain the corresponding moments in rest or
lattice frame of reference, i.e. raw moments. The tool that we employ for this
purpose is the binomial theorem. The transformation between the central
moments and the raw moments for any state variable $\varphi$ supported by
discrete particle velocity set can be formally written as
$\displaystyle\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{\varphi}$ $\displaystyle=$ $\displaystyle\braket{e_{\alpha
x}^{m}e_{\alpha y}^{n}}{\varphi}+\braket{e_{\alpha
x}^{m}\left[\sum_{j=1}^{n}C^{n}_{j}e_{\alpha
y}^{n-j}(-1)^{j}u_{y}^{j}\right]}{\varphi}+$ (73)
$\displaystyle\braket{e_{\alpha y}^{n}\left[\sum_{i=1}^{m}C^{m}_{i}e_{\alpha
x}^{m-i}(-1)^{i}u_{x}^{i}\right]}{\varphi}+$
$\displaystyle\braket{\left[\sum_{i=1}^{m}C^{m}_{i}e_{\alpha
x}^{m-i}(-1)^{i}u_{x}^{i}\right]\left[\sum_{j=1}^{n}C^{n}_{j}e_{\alpha
y}^{n-j}(-1)^{j}u_{y}^{j}\right]}{\varphi}$
where $C^{p}_{q}=p!/(q!(p-q)!)$. In the above, commutation of the inner
product of vectors, represented using the “bra-ket” operators, with summations
and scalar products is assumed. Clearly, raw moments of equal or lesser order
in combination is equivalent to central moments of a given order.
Application of Eq. (73) to the forcing terms, i.e., using Eq. (39) and Eqs.
(55)-(63) yields analytical expressions in the rest frame of reference:
$\displaystyle\braket{S_{\alpha}}{\rho}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}=0,$ (74)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}=F_{x},$ (75)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha y}=F_{y},$ (76)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}^{2}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}^{2}=2F_{x}u_{x},$ (77)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha y}^{2}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha y}^{2}=2F_{x}u_{y},$ (78)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}e_{\alpha
y}=F_{x}u_{y}+F_{y}u_{x},$ (79) $\displaystyle\braket{S_{\alpha}}{e_{\alpha
x}^{2}e_{\alpha y}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}=\left(\frac{1}{3}+u_{x}^{2}\right)F_{y}+2F_{x}u_{x}u_{y},$ (80)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}e_{\alpha y}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}e_{\alpha
y}^{2}=\left(\frac{1}{3}+u_{y}^{2}\right)F_{x}+2F_{y}u_{y}u_{x},$ (81)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}^{2}e_{\alpha y}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha
x}^{2}e_{\alpha
y}^{2}=\left(\frac{2}{3}+2u_{y}^{2}\right)F_{x}u_{x}+\left(\frac{2}{3}+2u_{x}^{2}\right)F_{y}u_{y}.$
(82)
For subsequent procedure, we also need the raw moments of the collision kernel
$\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}}\widehat{g}_{\beta}.$ (83)
Since collisions do not change mass and momenta, which are thus called
collisional invariants, we can set
$\widehat{g}_{0}=\widehat{g}_{1}=\widehat{g}_{2}.$ (84)
Thus, we effectively need to determine the functional expressions for
$\widehat{g}_{\beta}$ for $\beta=3,4,\ldots,8$. Owing to the _orthogonal_
property of the eigenvectors of $\mathcal{K}$ by construction, i.e. Eq. (20),
we obtain
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=\sum_{\beta}\braket{K_{\beta}}{\rho}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$ (85)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$ (86)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$ (87)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}+2\widehat{g}_{4},$ (88)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}-2\widehat{g}_{4},$ (89)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 4\widehat{g}_{5},$
(90)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{6},$
(91)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{7},$
(92)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{3}+4\widehat{g}_{8}.$ (93)
Now, for computational convenience, the evolution equation, Eq. (37), of the
Cascaded-LBM with forcing term may be rewritten as
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t)$
$\displaystyle=$
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x},t)+\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}+S_{{\alpha}(\overrightarrow{x},t)},$
(94)
$\displaystyle\overline{f}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)$
$\displaystyle=$
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\overrightarrow{x},t).$ (95)
where Eq. (94) and Eq. (95) represent the collision step, augmented by forcing
term, and streaming step, respectively. Here and henceforth, the symbol
“tilde” ($\sim$) refers to the post-collision state. The hydrodynamic
variables can then be obtained as
$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}=\braket{\overline{f}_{\alpha}}{\rho},$
(96) $\displaystyle\rho u_{i}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
i}+\frac{1}{2}F_{i}=\braket{\overline{f}_{\alpha}}{e_{\alpha
i}}+\frac{1}{2}F_{i},i\in{x,y}$ (97)
in view of Eqs. (36), (74), (75) and (76).
Now, to obtain the source terms in particle velocity space, we first compute
$\braket{K_{\beta}}{S_{\alpha}}$, $\beta=0,1,2,\ldots,8$. From Eqs. (20) and
(74)-(82), we readily get
$\displaystyle\widehat{m}^{s}_{0}=\braket{K_{0}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 0,$ (98)
$\displaystyle\widehat{m}^{s}_{1}=\braket{K_{1}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{x},$ (99)
$\displaystyle\widehat{m}^{s}_{2}=\braket{K_{2}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{y},$ (100)
$\displaystyle\widehat{m}^{s}_{3}=\braket{K_{3}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 6(F_{x}u_{x}+F_{y}u_{y}),$ (101)
$\displaystyle\widehat{m}^{s}_{4}=\braket{K_{4}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}-F_{y}u_{y}),$ (102)
$\displaystyle\widehat{m}^{s}_{5}=\braket{K_{5}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{y}+F_{y}u_{x}),$ (103)
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(1-3u_{x}^{2})F_{y}-6F_{x}u_{x}u_{y},$ (104)
$\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(1-3u_{y}^{2})F_{x}-6F_{y}u_{y}u_{x},$ (105)
$\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle
3\left[(6u_{y}^{2}-2)F_{x}u_{x}+(6u_{x}^{2}-2)F_{y}u_{y}\right].$ (106)
Thus, we can write
$\displaystyle(\mathcal{K}\cdot\mathbf{S})_{\alpha}$ $\displaystyle=$
$\displaystyle(\braket{K_{0}}{S_{\alpha}},\braket{K_{1}}{S_{\alpha}},\braket{K_{2}}{S_{\alpha}},\ldots,\braket{K_{8}}{S_{\alpha}})$
(107) $\displaystyle=$
$\displaystyle(\widehat{m}^{s}_{0},\widehat{m}^{s}_{1},\widehat{m}^{s}_{2},\ldots,\widehat{m}^{s}_{8})^{T}\equiv\ket{\widehat{m}^{s}_{\alpha}}.$
By virtue of orthogonality of $\mathcal{K}$, we have
$\mathcal{K}\mathcal{K}^{\dagger}=~{}D~{}\equiv\text{diag}(\braket{K_{0}}{K_{0}},\braket{K_{1}}{K_{1}},\braket{K_{2}}{K_{2}},\ldots,\braket{K_{8}}{K_{8}})=\text{diag}(9,6,6,36,4,4,12,12,36)$.
Inverting Eq. (107) by making use of the property
$\mathcal{K}^{-1}=\mathcal{K}^{\dagger}\cdot D^{-1}$, we get explicit
expressions for $S_{\alpha}$ in terms of $\overrightarrow{F}$ and
$\overrightarrow{u}$ in particle velocity space as
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\left[-\widehat{m}^{s}_{3}+\widehat{m}^{s}_{8}\right],$
(108) $\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}-\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{4}+6\widehat{m}^{s}_{7}-2\widehat{m}^{s}_{8}\right],$
(109) $\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{4}+6\widehat{m}^{s}_{6}-2\widehat{m}^{s}_{8}\right],$
(110) $\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}-\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{4}-6\widehat{m}^{s}_{7}-2\widehat{m}^{s}_{8}\right],$
(111) $\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{2}-\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{4}-6\widehat{m}^{s}_{6}-2\widehat{m}^{s}_{8}\right],$
(112) $\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}+6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{5}-3\widehat{m}^{s}_{6}-3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(113) $\displaystyle S_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}+6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{5}-3\widehat{m}^{s}_{6}+3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(114) $\displaystyle S_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[-6\widehat{m}^{s}_{1}-6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}+9\widehat{m}^{s}_{5}+3\widehat{m}^{s}_{6}+3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right],$
(115) $\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left[6\widehat{m}^{s}_{1}-6\widehat{m}^{s}_{2}+2\widehat{m}^{s}_{3}-9\widehat{m}^{s}_{5}+3\widehat{m}^{s}_{6}-3\widehat{m}^{s}_{7}+\widehat{m}^{s}_{8}\right].$
(116)
We now need to find the expressions of
$\braket{\overline{f}_{\alpha}}{e_{\alpha x}^{m}e_{\alpha
y}^{n}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}^{m}e_{\alpha
y}^{n}$ to proceed further. In this regard, for convenience, we define the
following notation for a compact summation operator acting on the transformed
distribution function $\overline{f}_{\alpha}$:
$a(\overline{f}_{\alpha_{1}}+\overline{f}_{\alpha_{3}}+\overline{f}_{\alpha_{3}}+\cdots)+b(\overline{f}_{\beta_{1}}+\overline{f}_{\beta_{2}}+\overline{f}_{\beta_{3}}+\cdots)+\cdots=\left(a\sum_{\alpha}^{A}+b\sum_{\alpha}^{B}+\cdots\right)\otimes\overline{f}_{\alpha},$
(117)
where $A=\left\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots\right\\}$,
$B=\left\\{\beta_{1},\beta_{2},\beta_{3},\cdots\right\\}$,$\cdots$. For
conserved basis vectors, we have them in terms of collisional invariants
$\displaystyle\braket{\overline{f}_{\alpha}}{\rho}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}$
$\displaystyle=$ $\displaystyle\rho,$ (118)
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}$ $\displaystyle=$
$\displaystyle\rho u_{x}-\frac{1}{2}F_{x},$ (119)
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha y}$ $\displaystyle=$
$\displaystyle\rho u_{y}-\frac{1}{2}F_{y},$ (120)
and, for the non-conserved basis vectors, we have
$\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
x}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}^{2}$
$\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{3}}\right)\otimes\overline{f}_{\alpha},$
(121) $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha y}^{2}$
$\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{4}}\right)\otimes\overline{f}_{\alpha},$
(122) $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha y}$
$\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha},$
(123) $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}^{2}e_{\alpha
y}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha y}$
$\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)\otimes\overline{f}_{\alpha},$
(124) $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}e_{\alpha y}^{2}$
$\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)\otimes\overline{f}_{\alpha},$
(125) $\displaystyle\braket{\overline{f}_{\alpha}}{e_{\alpha x}^{2}e_{\alpha
y}^{2}}=\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha x}^{2}e_{\alpha
y}^{2}$ $\displaystyle=$
$\displaystyle\left(\sum_{\alpha}^{A_{8}}\right)\otimes\overline{f}_{\alpha},$
(126)
where
$\displaystyle A_{3}$ $\displaystyle=$
$\displaystyle\left\\{1,3,5,6,7,8\right\\},$ (127) $\displaystyle A_{4}$
$\displaystyle=$ $\displaystyle\left\\{2,4,5,6,7,8\right\\},$ (128)
$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle\left\\{5,7\right\\},B_{5}=\left\\{6,8\right\\},$ (129)
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\left\\{5,6\right\\},B_{6}=\left\\{7,8\right\\},$ (130)
$\displaystyle A_{7}$ $\displaystyle=$
$\displaystyle\left\\{5,8\right\\},B_{7}=\left\\{6,7\right\\},$ (131)
$\displaystyle A_{8}$ $\displaystyle=$
$\displaystyle\left\\{5,6,7,8\right\\}.$ (132)
With the above preliminaries, we are now in a position to determine the
structure of the cascaded collision operator in the presence of forcing terms.
Starting from the lowest order non-conservative post-collision central
moments, we successively set them equal to their corresponding equilibrium
states. Once the expressions for $\widehat{g}_{\beta}$ is determined, we
discard this equilibrium assumption and multiply it with a corresponding
relaxation parameter to allow for a relaxation process during collision Geier
et al. (2006). From Eq. (67), which is the lowest non-conserved central
moment, and applying the binomial theorem (Eq. (73)) to transform it to the
rest frame of reference, we get
$\widehat{\overline{\kappa}}_{xx}^{eq}=1/3\rho=\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha
x}^{2}}-2u_{x}\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha
x}}+u_{x}^{2}\braket{\widetilde{\overline{f}}_{\alpha}}{\rho}.$ (133)
From Eq. (94) and substituting for various expressions involving
$\braket{\overline{f}_{\alpha}}{e_{\alpha x}^{m}}$,
$\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{m}}\widehat{g}_{\beta}$ and
$\braket{S_{\alpha}}{e_{\alpha x}^{m}}$, where $m=0,1,2$ from the above,
yields
$6\widehat{g}_{3}+2\widehat{g}_{4}=\frac{1}{3}\rho-\left(\sum_{\alpha}^{A_{3}}\right)\otimes\overline{f}_{\alpha}+\rho
u_{x}^{2}-F_{x}u_{x}.$ (134)
Similarly, from Eq. (68)
$\widehat{\overline{\kappa}}_{yy}^{eq}=1/3\rho=\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha
y}^{2}}-2u_{y}\braket{\widetilde{\overline{f}}_{\alpha}}{e_{\alpha
y}}+u_{y}^{2}\braket{\widetilde{\overline{f}}_{\alpha}}{\rho},$ (135)
and using $\braket{\overline{f}_{\alpha}}{e_{\alpha y}^{m}}$,
$\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{m}}\widehat{g}_{\beta}$ and
$\braket{S_{\alpha}}{e_{\alpha y}^{m}}$, where $m=0,1,2$ from the above, via
the binomial theorem gives
$6\widehat{g}_{3}-2\widehat{g}_{4}=\frac{1}{3}\rho-\left(\sum_{\alpha}^{A_{4}}\right)\otimes\overline{f}_{\alpha}+\rho
u_{y}^{2}-F_{y}u_{y}.$ (136)
Solving Eq. (134) and (136) for $\widehat{g}_{3}$ and $\widehat{g}_{4}$ yields
$\widehat{g}_{3}=\frac{1}{12}\left\\{\frac{2}{3}\rho-\left(\sum_{\alpha}^{C_{3}}+2\sum_{\alpha}^{D_{3}}\right)\otimes\overline{f}_{\alpha}+\rho(u_{x}^{2}+u_{y}^{2})-(F_{x}u_{x}+F_{y}u_{y})\right\\},$
(137)
and
$\widehat{g}_{4}=\frac{1}{4}\left\\{\left(\sum_{\alpha}^{E_{4}}-\sum_{\alpha}^{F_{4}}\right)\otimes\overline{f}_{\alpha}+\rho(u_{x}^{2}-u_{y}^{2})-(F_{x}u_{x}-F_{y}u_{y})\right\\},$
(138)
where
$\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle\left\\{1,2,3,4\right\\},$ (139) $\displaystyle D_{3}$
$\displaystyle=$ $\displaystyle\left\\{5,6,7,8\right\\},$ (140) $\displaystyle
E_{3}$ $\displaystyle=$ $\displaystyle\left\\{2,4\right\\},$ (141)
$\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle\left\\{1,3\right\\}.$
(142)
Now, we drop the assumption of equilibration considered above applying
relaxation parameters, $\omega_{3}$ and $\omega_{4}$, to Eq. (137) and (138),
respectively, to get
$\widehat{g}_{3}=\omega_{3}\frac{1}{12}\left\\{-\left(\sum_{\alpha}^{C_{3}}+2\sum_{\alpha}^{D_{3}}\right)\otimes\overline{f}_{\alpha}+\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})-(F_{x}u_{x}+F_{y}u_{y})\right\\},$
(143)
and
$\widehat{g}_{4}=\omega_{4}\frac{1}{4}\left\\{\left(\sum_{\alpha}^{E_{4}}-\sum_{\alpha}^{F_{4}}\right)\otimes\overline{f}_{\alpha}+\rho(u_{x}^{2}-u_{y}^{2})-(F_{x}u_{x}-F_{y}u_{y})\right\\}.$
(144)
Let us now consider the central moment $\widehat{\overline{\kappa}}_{xy}^{eq}$
in Eq. (69), i.e.,
$\widehat{\overline{\kappa}}_{xy}^{eq}=0=\braket{\widetilde{\overline{f}}_{\alpha}}{(e_{\alpha
x}-u_{x})(e_{\alpha y}-u_{y})},$ (145)
and substituting the expressions for various raw moments, we get
$\widehat{g}_{5}=\frac{1}{4}\left\\{-\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha}+\rho
u_{x}u_{y}-\frac{1}{2}(F_{x}u_{y}+F_{y}u_{x})\right\\},$ (146)
and applying a corresponding relaxation parameter $\omega_{5}$ to represent
over-relaxation for this moment, we obtain,
$\widehat{g}_{5}=\omega_{5}\frac{1}{4}\left\\{-\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\otimes\overline{f}_{\alpha}+\rho
u_{x}u_{y}-\frac{1}{2}(F_{x}u_{y}+F_{y}u_{x})\right\\}.$ (147)
It is worth noting that due to a slightly different choice of the basis vector
$K_{5}$ for $\ket{e_{\alpha x}e_{\alpha y}}$ from that in Geier et al. (2006),
Eq. (147) differs from that in Geier et al. (2006) by a factor of $-1$ apart
from the presence of forcing terms.
We now consider the central moment of the next higher order, i.e.
$\widehat{\overline{\kappa}}_{xxy}^{eq}$ in Eq. (70),
$\widehat{\overline{\kappa}}_{xxy}^{eq}=-\frac{1}{6}F_{y}=\braket{\widetilde{\overline{f}}_{\alpha}}{(e_{\alpha
x}-u_{x})^{2}(e_{\alpha y}-u_{y})}$ and following the procedure as discussed
above, we get
$\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)-2u_{x}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{y}\sum_{\alpha}^{A_{3}}\right]\otimes\overline{f}_{\alpha}\right.$
(148) $\displaystyle\left.+2\rho
u_{x}^{2}u_{y}+\frac{1}{2}(1-u_{x}^{2})F_{y}-F_{x}u_{x}u_{y}\right\\}-2u_{x}\widehat{g}_{5}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4}).$
Notice that $\widehat{g}_{6}$ depends on $\widehat{g}_{\beta}$, $\beta<6$,
which are already post-collision states. So, we relax with relaxation
parameter $\omega_{6}$ only those terms that do no contain these terms,
leading to
$\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\omega_{6}\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)-2u_{x}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{y}\sum_{\alpha}^{A_{3}}\right]\otimes\overline{f}_{\alpha}\right.$
(149) $\displaystyle\left.+2\rho
u_{x}^{2}u_{y}+\frac{1}{2}(1-u_{x}^{2})F_{y}-F_{x}u_{x}u_{y}\right\\}-2u_{x}\widehat{g}_{5}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4}),$
That is,
$\widehat{g}_{6}=\widehat{g}_{6}(\left\\{\overline{f}_{\alpha}\right\\},\rho,\overrightarrow{u},\overrightarrow{F},\widehat{g}_{3},\widehat{g}_{4},\widehat{g}_{5},\omega_{6})$.
Considering next,
$\widehat{\overline{\kappa}}_{xyy}^{eq}=-\frac{1}{6}F_{x}=\braket{\widetilde{\overline{f}}_{\alpha}}{(e_{\alpha
x}-u_{x})(e_{\alpha y}-u_{y})^{2}}$ from Eq. (71) and following calculations
to transform all the quantities to raw moments, we get
$\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)-2u_{y}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{x}\sum_{\alpha}^{A_{4}}\right]\otimes\overline{f}_{\alpha}\right.$
(150) $\displaystyle\left.+2\rho
u_{x}u_{y}^{2}+\frac{1}{2}(1-u_{y}^{2})F_{x}-F_{y}u_{y}u_{x}\right\\}-2u_{y}\widehat{g}_{5}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4}),$
Again, notice that $\widehat{g}_{7}$ depends on $\widehat{g}_{\beta}$,
$\beta<6$, which are already post-collision states. So, applying the
respective relaxation parameter $\omega_{7}$ to terms that do no contain them,
yields
$\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\omega_{7}\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)-2u_{y}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{x}\sum_{\alpha}^{A_{4}}\right]\otimes\overline{f}_{\alpha}\right.$
(151) $\displaystyle\left.+2\rho
u_{x}u_{y}^{2}+\frac{1}{2}(1-u_{y}^{2})F_{x}-F_{y}u_{y}u_{x}\right\\}-2u_{y}\widehat{g}_{5}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4}),$
Thus,
$\widehat{g}_{7}=\widehat{g}_{7}(\left\\{\overline{f}_{\alpha}\right\\},\rho,\overrightarrow{u},\overrightarrow{F},\widehat{g}_{3},\widehat{g}_{4},\widehat{g}_{5},\omega_{7})$.
In other words, $\widehat{g}_{\beta}$ depends on only the lower order moments
and not on other components of the same order.
Finally, we consider the central moment of the highest order defined by the
discrete particle velocity set (Eq. (72)),
$\widehat{\overline{\kappa}}_{xxyy}^{eq}=\frac{1}{9}\rho=\braket{\widetilde{\overline{f}}_{\alpha}}{(e_{\alpha
x}-u_{x})^{2}(e_{\alpha y}-u_{y})^{2}}$, and apply the procedure as discussed
above to transform everything in terms of raw moments to obtain
$\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{-\left[\sum_{\alpha}^{A_{8}}-2u_{x}\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)-2u_{y}\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)+u_{x}^{2}\sum_{\alpha}^{A_{4}}+u_{y}^{2}\sum_{\alpha}^{A_{3}}+\right.\right.$
(152)
$\displaystyle\left.\left.4u_{x}u_{y}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\right]\otimes\overline{f}_{\alpha}+\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-(F_{x}u_{x}u_{y}^{2}+F_{y}u_{y}u_{x}^{2})\right\\}-2\widehat{g}_{3}$
$\displaystyle-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7},$
Clearly, $\widehat{g}_{8}$ depends on $\widehat{g}_{\beta}$, $\beta<7$, which
are already post-collision states and thus, we relax with the parameter
$\omega_{8}$ those terms that do not contain them to finally yield
$\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\omega_{8}\frac{1}{4}\left\\{-\left[\sum_{\alpha}^{A_{8}}-2u_{x}\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)-2u_{y}\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)+u_{x}^{2}\sum_{\alpha}^{A_{4}}+u_{y}^{2}\sum_{\alpha}^{A_{3}}+\right.\right.$
(153)
$\displaystyle\left.\left.4u_{x}u_{y}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)\right]\otimes\overline{f}_{\alpha}+\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-(F_{x}u_{x}u_{y}^{2}+F_{y}u_{y}u_{x}^{2})\right\\}-2\widehat{g}_{3}$
$\displaystyle-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7},$
In order words,
$\widehat{g}_{8}=\widehat{g}_{8}(\left\\{\overline{f}_{\alpha}\right\\},\rho,\overrightarrow{u},\widehat{g}_{3},\widehat{g}_{4},\widehat{g}_{5},\widehat{g}_{6},\widehat{g}_{7},\omega_{8})$.
It may be noted that because of a slightly different choice of the basis
vector $K_{5}$, the prefactors for $\widehat{g}_{5}$ in Eqs. (149)-(153)
differ from that in Geier et al. (2006) by $-1$. Unfortunately, in the seminal
work Geier et al. (2006), there are some typographical errors in Eqs.
(20)-(24) of that paper Geier et al. (2006) – in particular, some of the signs
in the last lines of its Eq. (20)-(23), and the expression in the last line of
its Eq. (24) are incorrect.
Thus, the general structure of cascaded collision operator for non-conserved
moments may be written as
$\widehat{g}_{\alpha}=\omega_{\alpha}\left[H_{1}(\rho,\overrightarrow{u})\star
M(\left\\{\overline{f}_{\beta}\right\\})+H_{2}(\rho,\overrightarrow{u})\circ
N(\overrightarrow{F})\right]+C(\widehat{g}_{\gamma}),$ (154)
where $\alpha=3,\ldots,8$, $\beta=0,1,2,\ldots,8$ and
$\gamma=0,1,2,\ldots,\alpha-1$, and $M$, $N$, $H_{1}$, and $H_{2}$ represent
certain functions, and $\star$ and $\circ$ represent certain operators. On the
other hand, in particular, the term $C(\widehat{g}_{\gamma})$ contains the
dependence of $\widehat{g}_{\alpha}$ on its corresponding lower order moments
leading to a cascaded structure. In other words, cascaded collision operator
markedly distinguishes from the SRT and MRT collision operators in that the
former is non-commutative. The above derivation involved the choice of a
particular form of the central moments of the sources. In the next section
(Sec. VI), it will be shown how a different choice could provide a better
representation of its effect on higher order moments.
## VI De-aliasing Higher Order Central Source Moments
Due to the specific formulation of the forcing term employed in Eq. (25), its
corresponding higher order central moments also have non-zero contributions,
even when the fluid is at rest and a homogeneous force is considered. Since
they only occur at third and higher order moments, they do not affect
consistency to the Navier-Stokes equations, which emerge at the second-order
level (see Appendix A). However, to be conceptually consistent, it is
desirable to avoid this effect. Thus, as a limiting case, we now maintain the
effect of the force field only on the components of the first-order central
source moments, and de-alias all the corresponding higher (odd) order central
moments, by setting them to zero. That is,
$\widehat{\Gamma}^{F}_{x^{m}y^{n}}=\left\\{\begin{array}[]{ll}{F_{x},}&{m=1,n=0}\\\
{F_{y},}&{m=0,n=1}\\\ {0,}&{m+n>1.}\end{array}\right.$ (155)
In effect, the transformed equilibrium central moments
$\widehat{\overline{\kappa}}^{eq}_{x^{m}y^{n}}$ used in the construction of
the collision operator are modified. Specifically, the third-order transformed
equilibrium central moments, Eqs. (70) and (71) now reduce to
$\widehat{\overline{\kappa}}^{eq}_{xxy}=\widehat{\overline{\kappa}}^{eq}_{xyy}=0,$
(156)
while all the other components are the same as before. Moreover, such de-
aliasing also modifies the raw moments of the forcing terms at higher orders.
In particular, Eqs. (80)-(82) now become
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}^{2}e_{\alpha y}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=F_{y}u_{x}^{2}+2F_{x}u_{x}u_{y},$ (157)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}e_{\alpha y}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}e_{\alpha
y}^{2}=F_{x}u_{y}^{2}+2F_{y}u_{y}u_{x},$ (158)
$\displaystyle\braket{S_{\alpha}}{e_{\alpha x}^{2}e_{\alpha y}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}^{2}.$ (159)
while the lower order moments remain unaltered. Notice that terms such as
$1/3F_{x}$ and $1/3F_{y}$ do not anymore appear in the third-order source
moments, while $2/3F_{x}u_{x}$ and $2/3F_{y}u_{y}$ are eliminated from the
fourth-order source moments as a result of the use of de-aliased central
source moments (Eq. (155)). Hence, when the fluid is rest, the force fields do
not influence the third and higher order raw source moments, which is
physically consistent.
The computation of the source terms in velocity space $S_{\alpha}$ using Eqs.
(108)-(116), which involve $\widehat{m}^{s}_{\beta}$, are also naturally
influenced by the above changes. In this regard, while
$\widehat{m}^{s}_{\beta}$, for $\beta=0,1,2,\ldots,5$ remain unmodified, the
higher order moments for $\beta=6,7,8$ are altered. The expressions for these
latter quantities now become
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(2-3u_{x}^{2})F_{y}-6F_{x}u_{x}u_{y},$ (160)
$\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(2-3u_{y}^{2})F_{x}-6F_{y}u_{y}u_{x},$ (161)
$\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle
6\left[(3u_{y}^{2}-2)F_{x}u_{x}+(3u_{x}^{2}-2)F_{y}u_{y}\right].$ (162)
The cascaded collision operator can now be constructed using the procedure
presented in Sec. V. The use of modified source moments do not alter the
collision kernel corresponding to $\widehat{g}_{\beta}$, where
$\beta=0,1,2,\ldots,5$ and $\beta=8$. They are the same as those presented in
Sec. V. On the other hand, the third-order collision kernel contributions are
modified, which are now summarized as follows:
$\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\omega_{6}\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{6}}-\sum_{\alpha}^{B_{6}}\right)-2u_{x}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{y}\sum_{\alpha}^{A_{3}}\right]\otimes\overline{f}_{\alpha}\right.$
(163) $\displaystyle\left.+2\rho
u_{x}^{2}u_{y}-\frac{1}{2}u_{x}^{2}F_{y}-F_{x}u_{x}u_{y}\right\\}-2u_{x}\widehat{g}_{5}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4}),$
and
$\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\omega_{7}\frac{1}{4}\left\\{\left[\left(\sum_{\alpha}^{A_{7}}-\sum_{\alpha}^{B_{7}}\right)-2u_{y}\left(\sum_{\alpha}^{A_{5}}-\sum_{\alpha}^{B_{5}}\right)-u_{x}\sum_{\alpha}^{A_{4}}\right]\otimes\overline{f}_{\alpha}\right.$
(164) $\displaystyle\left.+2\rho
u_{x}u_{y}^{2}-\frac{1}{2}u_{y}^{2}F_{x}-F_{y}u_{y}u_{x}\right\\}-2u_{y}\widehat{g}_{5}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4}).$
Again, evidently, when the fluid is at rest, the force fields do not have
direct influence on $\widehat{g}_{6}$ and $\widehat{g}_{7}$. Thus, the above
formulation eliminates spurious effects resulting from forcing due to the
finiteness of the lattice set for higher order moments, similar to that by Guo
_et al_. Guo et al. (2002) for other LBM approaches. Indeed, a Chapman-Enskog
multiscale moment expansion analysis carried out in Appendix A will establish
the consistency of this special formulation of the central moments based LBM
to the desired macroscopic fluid flow equations. The shear and bulk kinematic
viscosities is found to be dependent on the relaxation parameters
$\omega_{3}=\omega^{\chi}$ and $\omega_{4}=\omega_{5}=\omega^{\nu}$,
respectively. In particular, the shear viscosity satisfies
$\nu=c_{s}^{2}\left(\frac{1}{\omega^{\nu}}-\frac{1}{2}\right)$. The rest of
the relaxation parameters in this MRT cascaded formulation can be tuned to
maintain numerical stability. One particular choice suggested by Geier is to
equilibrate higher order, in particular, the third-order moments,
$\omega_{6}=\omega_{7}=\omega_{8}=1$ Geier (2008b). Other possible choices
could be also considered that involve over-relaxation of these moments at
certain carefully selected relaxation rates so as to control numerical
dissipation while maintaining computational stability. On the other hand, as
shown in Appendix B, when the central moments based LBM as derived in this
work is executed as a MRT cascaded process it implies generalization of both
equilibrium and sources in the lattice frame reference which also depend on
the ratio of various relaxation times. However, it does not affect the overall
consistency of the approach to the macroscopic equations as it influences only
higher order contributions. The discussions so far considered the cascaded
collision operator to be explicit in time. Appendix C presents with the
possibility of introducing time-implicitness in the cascaded collision
operator.
## VII Computational Procedure
The main element of the computational procedure consists of performing the
cascaded collision, including the forcing terms, i.e. Eq. (94) along with Eq.
(28), which can be expanded as follows:
$\displaystyle\widetilde{\overline{f}}_{0}$ $\displaystyle=$
$\displaystyle\overline{f}_{0}+\left[\widehat{g}_{0}-4(\widehat{g}_{3}-\widehat{g}_{8})\right]+S_{0},$
(165) $\displaystyle\widetilde{\overline{f}}_{1}$ $\displaystyle=$
$\displaystyle\overline{f}_{1}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}+2(\widehat{g}_{7}-\widehat{g}_{8})\right]+S_{1},$
(166) $\displaystyle\widetilde{\overline{f}}_{2}$ $\displaystyle=$
$\displaystyle\overline{f}_{2}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}+2(\widehat{g}_{6}-\widehat{g}_{8})\right]+S_{2},$
(167) $\displaystyle\widetilde{\overline{f}}_{3}$ $\displaystyle=$
$\displaystyle\overline{f}_{3}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}-2(\widehat{g}_{7}+\widehat{g}_{8})\right]+S_{3},$
(168) $\displaystyle\widetilde{\overline{f}}_{4}$ $\displaystyle=$
$\displaystyle\overline{f}_{4}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}-2(\widehat{g}_{6}+\widehat{g}_{8})\right]+S_{4},$
(169) $\displaystyle\widetilde{\overline{f}}_{5}$ $\displaystyle=$
$\displaystyle\overline{f}_{5}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}-\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{5},$
(170) $\displaystyle\widetilde{\overline{f}}_{6}$ $\displaystyle=$
$\displaystyle\overline{f}_{6}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{6},$
(171) $\displaystyle\widetilde{\overline{f}}_{7}$ $\displaystyle=$
$\displaystyle\overline{f}_{7}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{7},$
(172) $\displaystyle\widetilde{\overline{f}}_{8}$ $\displaystyle=$
$\displaystyle\overline{f}_{8}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}+\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{8}.$
(173)
Here, the terms $\widehat{g}_{\beta}$ can be obtained in a sequential manner,
i.e. evolving towards higher moment orders from Eqs. (143), (144), (147),
(149), (151), and (153). It consists of terms that involve summation of
$\overline{f}_{\alpha}$ over various subsets of the particle velocity set. The
source terms $S_{\beta}$ are computed from Eqs. (108)-(116). Once the post-
collision values, i.e. $\widetilde{\overline{f}}_{\alpha}$ are known, the
streaming step can be performed in the usual manner to obtain the updated
value of $\overline{f}_{\alpha}$ (Eq. (95)). Subsequently, the hydrodynamic
fields, viz., the local fluid density and velocity can be computed from Eqs.
(96) and (97), respectively. Depending on the specific choice of the ansatz
for the central source moments, appropriate expressions for
$\widehat{g}_{\beta}$ and $\widehat{m}_{\beta}^{s}$ need to be used (see Secs.
V and VI). In the above procedure, careful optimization needs to be carried
out to reduce the number of floating-point operations.
## VIII Computational Experiments
In order to validate the numerical accuracy of the new computational approach
presented in this work, we performed simulations for canonical fluid flow
problems subjected to different types of forces, where analytical solutions
are available. We will now present results obtained by employing the Cascaded-
LBM with de-aliased higher order source central moments (as discussed in Sec.
VI), which will be compared with corresponding analytical solutions. The first
problem considered is the flow between parallel plates subjected to a constant
body force. We considered $3\times 51$ lattice nodes to resolve the
computational domain, where periodic boundary conditions are imposed in the
flow direction and the no slip boundary condition at the walls is represented
by means of the standard link bounce back technique. The relaxation parameters
are given such that $\omega_{4}=\omega_{5}=1.754$, while the remaining ones
are set to unity and the computations are performed for different values of
the component of the body force in the flow direction, i.e. $F_{x}$ with
$F_{y}=0$. Figure 2 shows a comparison of the computed velocity profiles with
the standard analytical solution (Poiseuille’s parabolic profile, with the
maximum velocity $u_{0}=F_{x}L^{2}/(2\nu)$, where $L$ is the half-width
between the plates and $\nu$ is the fluid’s kinematic viscosity) for different
values of $F_{x}$. Excellent agreement is seen.
Figure 2: Flow between parallel plates with constant body force: Comparison
of velocity profiles computed by Cascaded-LBM with forcing term (symbols) with
analytical solution (lines) for different values of the body force $F_{x}$.
In order to quantify the difference between the computed and analytical
solution, the relative global error given in terms of the Euclidean (second)
norm is presented in Table I. Thus, for the above given set of parameters and
resolution, it is $O(10^{-4})$.
Magnitude of body force ($F_{x}$) | Relative global error ($||\delta u||_{2}$)
---|---
$1\times 10^{-6}$ | $3.999\times 10^{-4}$
$3\times 10^{-6}$ | $3.895\times 10^{-4}$
$5\times 10^{-6}$ | $3.837\times 10^{-4}$
$7\times 10^{-6}$ | $3.839\times 10^{-4}$
Table 1: Relative global error for the Poiseuille flow problem. $||\delta
u||_{2}=\sum_{i}||(u_{c,i}-u_{a,i})||_{2}/\sum_{i}||u_{a,i}||_{2}$, where
$u_{c,i}$ and $u_{a,i}$ are computed and analytical solutions, respectively,
and the summation is over the entire domain.
The second problem considered involves a spatially varying body force. One
classical problem in this regard is the Hartmann flow, i.e. flow between
parallel plates subjected to a magnetic field $B_{y}=B_{0}$ imposed in the
perpendicular direction to the fluid motion. If $F_{b}$ is the driving force
of the fluid due to imposed pressure gradient and $\mathrm{Ha}$ is the
Hartmann number that characterizes the ratio of force due to magnetic field
and the viscous force, then the induced magnetic field in the flow direction
$B_{x}$ is given by
$B_{x}=\frac{F_{b}L}{B_{0}}\left[\frac{sinh\left(\mathrm{Ha}\frac{y}{L}\right)}{sinh(\mathrm{Ha})}-\frac{y}{L}\right]$,
where the coordinate distance $y$ is measured from a position equidistant
between the plates. The interaction of the flow field with the magnetic field
results in a variable retarding force $F_{mx}=B_{y}\frac{dB_{x}}{dy}$ and
$F_{my}=-B_{x}\frac{dB_{x}}{dy}$, and, in turn, the net force acting on the
fluid is $F_{x}=F_{b}+F_{mx}$ and $F_{y}=F_{my}$. We considered the same
number of lattice nodes and the same values of the relaxation parameters as
before, with $F_{b}=5\times 10^{-6}$ and $B_{0}=8\times 10^{-3}$ and varied
the values of $\mathrm{Ha}$. The analytical solution for this problem is
$u_{x}=\frac{F_{b}L}{B_{0}}\sqrt{\frac{\eta}{\nu}}coth(\mathrm{Ha})\left[1-\frac{cosh\left(\mathrm{Ha}\frac{y}{L}\right)}{cosh(\mathrm{Ha})}\right]$,
where the magnetic resistivity $\eta$ is related to $\mathrm{Ha}$ through
$\eta=\frac{B_{0}^{2}L^{2}}{\mathrm{Ha}^{2}\nu}$. The computed velocity
profiles are compared with the analytical solution for different values of
$\mathrm{Ha}$ in Fig. 3.
Figure 3: Flow between parallel plates with a spatially varying body force:
Comparison of velocity profiles computed by Cascaded-LBM with forcing term
(symbols) with analytical solution (lines) for prescribed Lorentz force at
different Hartmann numbers.
As expected, the velocity profiles become more flattened with increasing
values of $\mathrm{Ha}$, while the case with $\mathrm{Ha}=0$ reduces to the
earlier problem. The computed velocity profiles are found to agree very well
with the analytical results. The relative global errors for this problem are
presented in Table II. It can be seen that they are dependent on the value of
$\mathrm{Ha}$ when the same grid resolution is used for different cases. In
particular, the relative error increases as the value of $\mathrm{Ha}$ is
increased for the same resolution. This can be explained as follows. This flow
problem is characterized by the presence of boundary layers – the Hartmann
layers – whose thickness is inversely proportional to $\sqrt{\mathrm{Ha}}$.
That is, the Hartmann layer becomes thinner as the value of $\mathrm{Ha}$ is
increased. Thus, resolution of this boundary layer would require increasingly
more number nodes that are clustered near walls as $\mathrm{Ha}$ is increased
to maintain the same accuracy. Otherwise, when the same number of grid nodes
that are uniformly distributed is employed, the relatively error norm is
expected to increase with $\mathrm{Ha}$. Indeed, local grid refinement
employing a suitable boundary layer transformation can maintain similar
accuracy for different $\mathrm{Ha}$ as was done with other LBM formulations
recently Pattison et al. (2008). Extension of the local grid refinement
approaches for the central moment based LBM to resolve boundary layers and
sharp gradients in solutions are subjects of future studies.
Hartmann number ($\mathrm{Ha}$) | Relative global error ($||\delta u||_{2}$)
---|---
$0.0$ | $3.837\times 10^{-4}$
$3.0$ | $2.140\times 10^{-3}$
$5.0$ | $5.967\times 10^{-3}$
$7.0$ | $1.091\times 10^{-2}$
Table 2: Relative global error for the Hartmann flow problem. $||\delta
u||_{2}=\sum_{i}||(u_{c,i}-u_{a,i})||_{2}/\sum_{i}||u_{a,i}||_{2}$, where
$u_{c,i}$ and $u_{a,i}$ are computed and analytical solutions, respectively,
and the summation is over the entire domain.
The last problem that we considered involves a temporally varying body force.
An important canonical problem in this regard is the flow between two parallel
plates driven by a force sinusoidally varying in time. That is, we considered
$F_{x}=F_{b}cos(\omega t)$, where $F_{b}$ is the peak value of the applied
force, while $\omega_{p}=2\pi/T$ is the angular frequency where $T$ is the
time period. This problem is characterized by
$\mathrm{Wo}=\sqrt{\frac{\omega_{p}}{\nu}}L$, a dimensionless number arising
from its original analysis by Womersley. The analytical velocity profile for
this flow is
$u_{x}=\mathcal{R}\left[\frac{iF_{b}}{\omega_{p}}\left\\{1-\frac{cos\left(\gamma\frac{y}{L}\right)}{cos(\gamma)}\right\\}e^{i\omega_{p}t}\right]$,
where $\gamma=\sqrt{-i\mathrm{Wo}^{2}}$. We considered $F_{b}=1\times 10^{-5}$
and $\mathrm{Wo}=12.71$, while maintaining the number of lattice nodes and the
values of the relaxation parameters to be same as in the first problem. Figure
4 shows a comparison of the computed velocity profiles with analytical
solution for different instants within the duration of the time period $T$ of
the cycle.
Figure 4: Flow between parallel plates with a temporally varying body force:
Comparison of velocity profiles computed by Cascaded-LBM with forcing term
(symbols) with analytical solution (lines) at different instants within a time
period $T$.
Evidently, the new computational approach is able to reproduce the complex
flow features for this problem involving the presence of Stokes layer very
well. Table III presents the relative global errors at different instants
within the time period $T$, corresponding to those in Fig. 4. The relatively
differences between computed and analytical solutions vary between different
time instants. On the other hand, they are identical for instants shifted by
the half time period implying that the computations are able to reproduce
temporal variations without any time lag as compared with analytical
solutions.
Time instant ($t$) | Relative global error ($||\delta u||_{2}$)
---|---
$0$ | $4.195\times 10^{-3}$
$0.05T$ | $1.701\times 10^{-3}$
$0.10T$ | $1.060\times 10^{-3}$
$0.15T$ | $7.548\times 10^{-4}$
$0.20T$ | $5.906\times 10^{-4}$
$0.40T$ | $1.842\times 10^{-3}$
$0.45T$ | $4.611\times 10^{-4}$
$0.50T$ | $4.195\times 10^{-3}$
$0.55T$ | $1.701\times 10^{-3}$
$0.60T$ | $1.060\times 10^{-3}$
$0.65T$ | $7.548\times 10^{-4}$
$0.70T$ | $5.906\times 10^{-3}$
$0.90T$ | $1.842\times 10^{-3}$
$0.95T$ | $4.611\times 10^{-3}$
Table 3: Relative global error for the Womersley flow problem. $||\delta
u||_{2}(t)=\sum_{i}||(u_{c,i}(t)-u_{a,i}(t))||_{2}/\sum_{i}||u_{a,i}(t)||_{2}$,
where $u_{c,i}(t)$ and $u_{a,i}(t)$ are computed and analytical solutions,
respectively, at instant $t$ within a time period $T$ and the summation is
over the entire domain.
It may be noted that for all the three benchmark problems presented above,
essentially same numerical results are obtained when the de-aliasing in the
forcing is turned off, i.e. expressions presented in Sec. V is used. This is
because both forms differ only in third and higher orders, while they are both
consistent at the second order level with the Navier-Stokes equations, from
which the analytical solutions are derived. It would be interesting to carry
out detailed numerical error analysis as well as stability analysis of the
central moment based LBM for different grid resolutions and characteristic
parameters, and for various canonical flow problems in future investigations.
## IX Summary and Conclusions
In this paper, we discussed a systematic procedure for the derivation of
forcing terms based on the central moments in the Cascaded-LBM. The main
elements involved in this regard are the binomial theorem that relates the
central moments and raw moments of various orders and the associated
orthogonal properties. The discrete source terms are obtained by matching with
the corresponding continuous central moment of a given order. For the latter,
we consider an ansatz based on the local Maxwell distribution. Its variant
involving a de-aliased higher order central source moments, which recovers
physically consistent higher order effects when the fluid is at rest, is also
derived. Effectively explicit and temporally second-order forms of forcing
terms are obtained through a transformation of the distribution function,
which contributes to the cascaded collision. When the values of the free
parameters in the continuous equilibrium (Maxwell) distribution, i.e. speed of
sound and those in the orthogonalization process of the moment basis from the
discrete velocity set are chosen, they completely determine the various
coefficients of both the cascaded collision operator and the source terms. The
equilibrium distribution and the source terms in velocity space are proper
polynomials and contain higher order terms. By construction, the source terms
are Galilean invariant. It is found that both the equilibrium and source terms
generalize when the cascaded formulation is represented as a relaxation
process in the lattice frame of reference. While the Cascaded-LBM with forcing
terms is based on a frame invariant kinetic theory, its consistency to the
Navier-Stokes equations is shown by means of a Chapman-Enskog moment expansion
analysis. It is found that the new approach reproduces analytical solutions
for canonical problems that involve either constant or spatially or temporally
varying forces with excellent quantitative accuracy. The approach presented in
this paper can be extended to other types of lattices such as the D3Q27 model
in three dimensions Premnath and Banerjee (2009).
## Appendix A Chapman-Enskog Multiscale Analysis
In this section, let us perform a Chapman-Enskog analysis of the central
moment formulation of the LBM using the consistent forcing terms derived in
Sec. VI. For ease of presentation and analysis, we will make a particular
assumption regarding the collision operator in this section. It will then be
pointed out in the next section that relaxing such assumption amounting to the
use of fully coherent cascaded collision kernel does not affect the
consistency analysis presented here. First, some preliminaries are provided.
In particular, we define a transformation matrix corresponding to the
following “nominal” moment basis on which the analysis is performed:
$\mathcal{T}=\left[\ket{\rho},\ket{e_{\alpha x}},\ket{e_{\alpha
y}},\ket{e_{\alpha x}^{2}+e_{\alpha y}^{2}},\ket{e_{\alpha x}^{2}-e_{\alpha
y}^{2}},\ket{e_{\alpha x}e_{\alpha y}},\ket{e_{\alpha x}^{2}e_{\alpha
y}},\ket{e_{\alpha x}e_{\alpha y}^{2}},\ket{e_{\alpha x}^{2}e_{\alpha
y}^{2}}\right],$ (174)
It is convenient to carry out the multiscale expansion in terms of various raw
moments. Thus, we also define the following _raw_ moments, where the
superscript “prime” symbol is used here and henceforth to designate that the
moment is of raw type:
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{f_{\alpha}},$ (175)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}S_{\alpha}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{S_{\alpha}},$ (176)
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{eq^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}f_{\alpha}^{eq}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{f_{\alpha}^{eq}},$ (177)
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{\alpha}\overline{f}_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}}{\overline{f}_{\alpha}},$ (178)
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq^{\prime}}$
$\displaystyle=$
$\displaystyle\sum_{\alpha}\overline{f}_{\alpha}^{eq}e_{\alpha x}^{m}e_{\alpha
y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha y}^{n}}{\overline{f}_{\alpha}^{eq}}.$
(179)
It follows that
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}=\widehat{\kappa}_{x^{m}y^{n}}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$
and
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq^{\prime}}=\widehat{\kappa}_{x^{m}y^{n}}^{eq^{\prime}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$.
We now re-write various different central moments in terms of their
corresponding raw moments by applying the binomial theorem. First, the non-
conserved part of the central moments can be written as functions of various
raw moments as follows:
$\displaystyle\widehat{\overline{\kappa}}_{xx}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\rho
u_{x}^{2}+F_{x}u_{x},$ (180) $\displaystyle\widehat{\overline{\kappa}}_{yy}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\rho
u_{y}^{2}+F_{y}u_{y},$ (181) $\displaystyle\widehat{\overline{\kappa}}_{xy}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\rho
u_{x}u_{y}+\frac{1}{2}(F_{x}u_{y}+F_{y}u_{x}),$ (182)
$\displaystyle\widehat{\overline{\kappa}}_{xxy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+2\rho
u_{x}^{2}u_{y}-\frac{1}{2}F_{y}u_{x}^{2}-F_{x}u_{x}u_{y},$ (183)
$\displaystyle\widehat{\overline{\kappa}}_{xyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+2\rho
u_{x}u_{y}^{2}-\frac{1}{2}F_{x}u_{y}^{2}-F_{y}u_{y}u_{x},$ (184)
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}$ $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}$
(185) $\displaystyle-3\rho
u_{x}^{2}u_{y}^{2}+F_{x}u_{x}u_{y}^{2}+F_{y}u_{y}u_{x}^{2}.$
The raw moments of the equilibrium distribution and source terms of various
order are:
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{0}$ $\displaystyle=$
$\displaystyle\rho,$ (186) $\displaystyle\widehat{\kappa}^{eq^{\prime}}_{x}$
$\displaystyle=$ $\displaystyle\rho u_{x},$ (187)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{y}$ $\displaystyle=$
$\displaystyle\rho u_{y},$ (188)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{xx}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\rho+\rho u_{x}^{2},$ (189)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{yy}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\rho+\rho u_{y}^{2},$ (190)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{xy}$ $\displaystyle=$
$\displaystyle\rho u_{x}u_{y},$ (191)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{xxy}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\rho u_{y}+\rho u_{x}^{2}u_{y},$ (192)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{xyy}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\rho u_{x}+\rho u_{x}u_{y}^{2},$ (193)
$\displaystyle\widehat{\kappa}^{eq^{\prime}}_{xxyy}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\rho+\frac{1}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}^{2},$ (194)
and
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{0}$ $\displaystyle=$
$\displaystyle 0,$ (195) $\displaystyle\widehat{\sigma}^{{}^{\prime}}_{x}$
$\displaystyle=$ $\displaystyle F_{x},$ (196)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{y}$ $\displaystyle=$
$\displaystyle F_{y},$ (197)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{xx}$ $\displaystyle=$
$\displaystyle 2F_{x}u_{x},$ (198)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{yy}$ $\displaystyle=$
$\displaystyle 2F_{y}u_{y},$ (199)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{xy}$ $\displaystyle=$
$\displaystyle F_{x}u_{y}+F_{y}u_{x},$ (200)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{xxy}$ $\displaystyle=$
$\displaystyle F_{y}u_{x}^{2}+2F_{x}u_{x}u_{y},$ (201)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{xyy}$ $\displaystyle=$
$\displaystyle F_{x}u_{y}^{2}+2F_{y}u_{y}u_{x},$ (202)
$\displaystyle\widehat{\sigma}^{{}^{\prime}}_{xxyy}$ $\displaystyle=$
$\displaystyle 2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}^{2},$ (203)
respectively.
In the above notation, the cascaded collision kernel may be more compactly
written as
$\displaystyle\widehat{g}_{3}$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{12}\left\\{\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(204) $\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{\omega_{4}}{4}\left\\{\rho(u_{x}^{2}-u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(205) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{\omega_{5}}{4}\left\\{\rho
u_{x}u_{y}-\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xy}^{{}^{\prime}}\right\\},$
(206) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{6}}{4}\left\\{2\rho
u_{x}^{2}u_{y}+\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xxy}\right\\}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4})-2u_{x}\widehat{g}_{5},$
(207) $\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\frac{\omega_{7}}{4}\left\\{2\rho
u_{x}u_{y}^{2}+\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xyy}\right\\}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4})-2u_{y}\widehat{g}_{5},$
(208) $\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\frac{\omega_{8}}{4}\left\\{\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-\left[\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}\right.\right.$
(209)
$\displaystyle\left.\left.+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}\right]-\frac{1}{2}\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right\\}-2\widehat{g}_{3}-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})$
$\displaystyle-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7}.$
Instead of considering the above collision operator, for now, in what follows,
let us specialize the collision term. In this regard, we first re-write the
cascaded collision step, Eq. (94), using Eq. (28) as
$(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=(\widetilde{\overline{f}}_{\alpha}-\overline{f}_{\alpha})+S_{{\alpha}},$
(210)
and reduce it by applying the central moment operator $\braket{(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}}{\cdot}$ on both of its sides. Thus, we
get
$\sum_{\beta}\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{K_{\beta}}\widehat{g}_{\beta}=(\widetilde{\widehat{\overline{\kappa}}}_{x^{m}y^{n}}-\widehat{\overline{\kappa}}_{x^{m}y^{n}})+\widehat{\sigma}_{x^{m}y^{n}}.$
(211)
Let us now consider a specific case when the post-collision state is in
“equilibrium state”. In this case, we set
$\widetilde{\widehat{\overline{\kappa}}}_{x^{m}y^{n}}=\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq},\widehat{\sigma}_{x^{m}y^{n}}=0\Rightarrow\widehat{g}_{\beta}=\widehat{g}_{\beta}^{*}$
(212)
so that $\widehat{g}_{\beta}$ takes certain specific values,
$\widehat{g}_{\beta}^{*}$.
Thus the specialized non-conserved collision kernel can be obtained by
expanding the LHS of Eq. (211) and using Eq. (212) for $m+n\geq 2$, which can
be written in matrix form as
$\mathcal{F}\left[\begin{array}[]{l}{\widehat{g}_{3}^{*}}\\\
{\widehat{g}_{4}^{*}}\\\ {\widehat{g}_{5}^{*}}\\\ {\widehat{g}_{6}^{*}}\\\
{\widehat{g}_{7}^{*}}\\\
{\widehat{g}_{8}^{*}}\end{array}\right]=\left[\begin{array}[]{l}{\widehat{\overline{\kappa}}_{xx}^{eq}-\widehat{\overline{\kappa}}_{xx}}\\\
{\widehat{\overline{\kappa}}_{yy}^{eq}-\widehat{\overline{\kappa}}_{yy}}\\\
{\widehat{\overline{\kappa}}_{xy}^{eq}-\widehat{\overline{\kappa}}_{xy}}\\\
{\widehat{\overline{\kappa}}_{xxy}^{eq}-\widehat{\overline{\kappa}}_{xxy}}\\\
{\widehat{\overline{\kappa}}_{xyy}^{eq}-\widehat{\overline{\kappa}}_{xyy}}\\\
{\widehat{\overline{\kappa}}_{xxyy}^{eq}-\widehat{\overline{\kappa}}_{xxyy}}\\\
\end{array}\right],$ (213)
where $\mathcal{F}\equiv\mathcal{F}(\overrightarrow{x},t)$ is a $6\times 6$
local frame transformation matrix that depends on the local fluid velocity and
is given by
$\mathcal{F}=\left[\begin{array}[]{cccccc}6&2&0&0&0&0\\\ 6&-2&0&0&0&0\\\
0&0&4&0&0&0\\\ -6u_{y}&-2u_{y}&-8u_{x}&-4&0&0\\\
-6u_{x}&2u_{x}&-8u_{y}&0&-4&0\\\
(8+6(u_{x}^{2}+u_{y}^{2}))&-2(u_{x}^{2}-u_{y}^{2})&16u_{x}u_{y}&8u_{y}&8u_{x}&4\\\
\end{array}\right].$ (214)
It may be noted that Eq. (214) has entries similar to that given in Ref.
Asinari (2008), except for the change in signs in the third column resulting
from the specific choice made for constructing $\ket{K_{5}}$ in the
orthogonalization (Gram-Schmidt) procedure. Now substituting for the
expressions in the RHS of Eq. (213) and inverting it, we get
$\widehat{g}_{\beta}^{*}$ in terms of the raw moments, hydrodynamic fields and
force fields. It may be written as
$\left[\begin{array}[]{l}{\widehat{g}_{3}^{*}}\\\ {\widehat{g}_{4}^{*}}\\\
{\widehat{g}_{5}^{*}}\\\ {\widehat{g}_{6}^{*}}\\\ {\widehat{g}_{7}^{*}}\\\
{\widehat{g}_{8}^{*}}\end{array}\right]=\left[\begin{array}[]{l}{\frac{1}{18}\rho+\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})-\frac{1}{12}(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{12}(F_{x}u_{x}+F_{y}u_{y})}\\\
{\frac{1}{4}\rho(u_{x}^{2}-u_{y}^{2})-\frac{1}{4}(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{4}(F_{x}u_{x}-F_{y}u_{y})}\\\
{\frac{1}{4}\rho
u_{x}u_{y}-\frac{1}{4}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\frac{1}{8}(F_{x}u_{y}+F_{y}u_{x})}\\\
{-\frac{1}{12}\rho u_{y}-\frac{1}{4}\rho
u_{x}^{2}u_{y}+\frac{1}{4}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+\frac{1}{4}F_{x}u_{x}u_{y}+\frac{1}{8}F_{y}u_{x}^{2}}\\\
{-\frac{1}{12}\rho u_{x}-\frac{1}{4}\rho
u_{x}u_{y}^{2}+\frac{1}{4}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}+\frac{1}{4}F_{y}u_{y}u_{x}+\frac{1}{8}F_{x}u_{y}^{2}}\\\
{-\frac{1}{12}\rho-\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})+\frac{1}{4}\rho
u_{x}^{2}u_{y}^{2}+\frac{1}{6}(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{4}\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}+q_{xxyy}}\end{array}\right],$
(215)
where
$q_{xxyy}=\frac{1}{6}(F_{x}u_{x}+F_{y}u_{y})-\frac{1}{4}(F_{x}u_{x}u_{y}^{2}+F_{y}u_{y}u_{x}^{2})$.
An alternative and a somewhat direct procedure to obtain
$\widehat{g}_{\beta}^{*}$ is to invoke the orthogonal properties of the basis
vectors $\ket{K_{\beta}}$. Accordingly, we can write
$\widehat{g}_{\beta}^{*}=\frac{\braket{\overline{f}_{\alpha}^{eq}-\overline{f}_{\alpha}}{K_{\beta}}}{\braket{K_{\beta}}{K_{\beta}}}=\frac{\braket{f_{\alpha}^{eq}-\overline{f}_{\alpha}-\frac{1}{2}S_{\alpha}}{K_{\beta}}}{\braket{K_{\beta}}{K_{\beta}}},\quad\quad\beta=3,4,5,\ldots,8,$
(216)
which gives expressions identical to that given in Eq. (215).
Equivalently, for the special case noted above (Eq. (212)), the collision
operator, Eq. (210), can also be written as
$\mathcal{K}\cdot\mathbf{\widehat{g}}^{*}=\mathbf{\overline{f}}^{eq}-\mathbf{\overline{f}}=\mathbf{f}^{eq}-\mathbf{\overline{f}}-\frac{1}{2}\mathbf{S}$,
which can be inverted to yield
$\mathbf{\widehat{g}}^{*}=\mathcal{K}^{-1}\left(\mathbf{f}^{eq}-\mathbf{\overline{f}}-\frac{1}{2}\mathbf{S}\right),$
(217)
where as before the boldface symbols represent the column vectors. Now, we
propose to “over-relax” the above special system by means of multiple
relaxation times (MRT) as a representation of collision process. That is, we
set
$\mathbf{\widehat{g}}=\Lambda\mathbf{\widehat{g}}^{*},$ (218)
where $\Lambda$ is a relaxation time matrix. Hence, combining Eqs. (217) and
(218), we can write the post-collision state in this MRT formulation as
$\displaystyle\mathbf{\widetilde{\overline{f}}}=\mathbf{\overline{f}}+\mathcal{K}\cdot\mathbf{\widehat{g}}+\mathbf{S}$
$\displaystyle=$
$\displaystyle\mathbf{\overline{f}}+\mathcal{K}\Lambda\mathbf{\widehat{g}}^{*}+\mathbf{S}$
(219) $\displaystyle=$
$\displaystyle\mathbf{\overline{f}}+\mathcal{K}\Lambda\mathcal{K}^{-1}\left(\mathbf{f}^{eq}-\mathbf{\overline{f}}-\frac{1}{2}\mathbf{S}\right)+\mathbf{S}$
Let,
$\Lambda^{*}=\mathcal{K}\Lambda\mathcal{K}^{-1}.$ (220)
Hence,
$\mathbf{\widetilde{\overline{f}}}=\mathbf{\overline{f}}+\Lambda^{*}\left(\mathbf{f}^{eq}-\mathbf{\overline{f}}\right)+\left(\mathcal{I}-\frac{1}{2}\Lambda^{*}\right)\mathbf{S}$
(221)
where $\mathcal{I}$ is the identity matrix.
We now define raw moments of distribution functions (including the transformed
one), equilibrium and sources for convenience as
$\mathbf{\widehat{\overline{f}}}=\mathcal{T}\mathbf{\overline{f}},\quad\mathbf{\widehat{f}}=\mathcal{T}\mathbf{f},\quad\mathbf{\widehat{f}}^{eq}=\mathcal{T}\mathbf{f}^{eq},\quad\mathbf{\widehat{S}}=\mathcal{T}\mathbf{S},$
(222)
where $\widehat{(\cdot)}$ represents column vectors in (raw) moment space and
the transformation matrix $\mathcal{T}$ is given in Eq. (174). That is,
$\displaystyle\mathbf{\widehat{\overline{f}}}=\left(\widehat{\overline{f}}_{0},\widehat{\overline{f}}_{1},\widehat{\overline{f}}_{2},\ldots,\widehat{\overline{f}}_{8}\right)^{{\dagger}}$
$\displaystyle=$
$\displaystyle\left(\widehat{\overline{\kappa}}_{0}^{{}^{\prime}},\widehat{\overline{\kappa}}_{x}^{{}^{\prime}},\widehat{\overline{\kappa}}_{y}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}},\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}\right)^{{\dagger}},$
$\displaystyle\mathbf{\widehat{f}}=\left(\widehat{f}_{0},\widehat{f}_{1},\widehat{f}_{2},\ldots,\widehat{f}_{8}\right)^{{\dagger}}$
$\displaystyle=$
$\displaystyle\left(\widehat{\kappa}_{0}^{{}^{\prime}},\widehat{\kappa}_{x}^{{}^{\prime}},\widehat{\kappa}_{y}^{{}^{\prime}},\widehat{\kappa}_{xx}^{{}^{\prime}}+\widehat{\kappa}_{yy}^{{}^{\prime}},\widehat{\kappa}_{xx}^{{}^{\prime}}-\widehat{\kappa}_{yy}^{{}^{\prime}},\widehat{\kappa}_{xy}^{{}^{\prime}},\widehat{\kappa}_{xxy}^{{}^{\prime}},\widehat{\kappa}_{xyy}^{{}^{\prime}},\widehat{\kappa}_{xxyy}^{{}^{\prime}}\right)^{{\dagger}},$
$\displaystyle\mathbf{\widehat{f}}^{eq}=\left(\widehat{f}_{0}^{eq},\widehat{f}_{1}^{eq},\widehat{f}_{2}^{eq},\ldots,\widehat{f}_{8}^{eq}\right)^{{\dagger}}$
$\displaystyle=$
$\displaystyle\left(\widehat{\kappa}_{0}^{eq^{\prime}},\widehat{\kappa}_{x}^{eq^{\prime}},\widehat{\kappa}_{y}^{eq^{\prime}},\widehat{\kappa}_{xx}^{eq^{\prime}}+\widehat{\kappa}_{yy}^{eq^{\prime}},\widehat{\kappa}_{xx}^{eq^{\prime}}-\widehat{\kappa}_{yy}^{eq^{\prime}},\widehat{\kappa}_{xy}^{eq^{\prime}},\widehat{\kappa}_{xxy}^{eq^{\prime}},\widehat{\kappa}_{xyy}^{eq^{\prime}},\widehat{\kappa}_{xxyy}^{eq^{\prime}}\right)^{{\dagger}},$
$\displaystyle\mathbf{\widehat{S}}=\left(\widehat{S}_{0},\widehat{S}_{1},\widehat{S}_{2},\ldots,\widehat{S}_{8}\right)^{{\dagger}}$
$\displaystyle=$
$\displaystyle\left(\widehat{\sigma}_{0}^{{}^{\prime}},\widehat{\sigma}_{x}^{{}^{\prime}},\widehat{\sigma}_{y}^{{}^{\prime}},\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}},\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}},\widehat{\sigma}_{xy}^{{}^{\prime}},\widehat{\sigma}_{xxy}^{{}^{\prime}},\widehat{\sigma}_{xyy}^{{}^{\prime}},\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right)^{{\dagger}}.$
Finally, using Eq. (222), we can rewrite the expressions for the collision and
source terms in Eq. (221) in terms of (raw) moment space. That is,
$\mathbf{\widetilde{\overline{f}}}=\mathbf{\overline{f}}+\mathcal{T}^{-1}\left[-\widehat{\Lambda}\left(\mathbf{\widehat{\overline{f}}}-\mathbf{\widehat{f}}^{eq}\right)+\left(\mathcal{I}-\frac{1}{2}\widehat{\Lambda}\right)\mathbf{\widehat{S}}\right],$
(223)
where $\widehat{\Lambda}$ is a diagonal collision matrix given by
$\widehat{\Lambda}=\mathcal{T}\Lambda^{*}\mathcal{T}^{-1}=diag(0,0,0,\omega_{3},\omega_{4},\omega_{5},\omega_{6},\omega_{7},\omega_{8}).$
(224)
It may be noted that from Eq. (222), we can obtain the discrete equilibrium
distribution functions and source terms in velocity space by means of the
inverse transformation. That is,
$\mathbf{f}^{eq}=\mathcal{T}^{-1}\mathbf{\widehat{f}}^{eq},\mathbf{S}=\mathcal{T}^{-1}\mathbf{\widehat{S}}$,
which yield
$\displaystyle f_{0}^{eq}$ $\displaystyle=$
$\displaystyle\frac{4}{9}\rho-\frac{2}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{1}^{eq}$ $\displaystyle=$
$\displaystyle\frac{1}{9}\rho+\frac{1}{3}\rho u_{x}+\frac{1}{2}\rho
u_{x}^{2}-\frac{1}{6}\rho(u_{x}^{2}+u_{y}^{2})-\frac{1}{2}\rho
u_{x}u_{y}^{2}-\frac{1}{2}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{2}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{9}\rho+\frac{1}{3}\rho
u_{y}+\frac{1}{2}\rho
u_{y}^{2}-\frac{1}{6}\rho(u_{x}^{2}+u_{y}^{2})-\frac{1}{2}\rho
u_{x}^{2}u_{y}-\frac{1}{2}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{3}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{9}\rho-\frac{1}{3}\rho
u_{x}+\frac{1}{2}\rho
u_{x}^{2}-\frac{1}{6}\rho(u_{x}^{2}+u_{y}^{2})+\frac{1}{2}\rho
u_{x}u_{y}^{2}-\frac{1}{2}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{4}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{9}\rho-\frac{1}{3}\rho
u_{y}+\frac{1}{2}\rho
u_{y}^{2}-\frac{1}{6}\rho(u_{x}^{2}+u_{y}^{2})+\frac{1}{2}\rho
u_{x}^{2}u_{y}-\frac{1}{2}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{5}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{36}\rho+\frac{1}{12}\rho
u_{x}+\frac{1}{12}\rho
u_{y}+\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})+\frac{1}{4}\rho
u_{x}u_{y}+\frac{1}{4}\rho u_{x}^{2}u_{y}+\frac{1}{4}\rho
u_{x}u_{y}^{2}+\frac{1}{4}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{6}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{36}\rho-\frac{1}{12}\rho
u_{x}+\frac{1}{12}\rho
u_{y}+\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})-\frac{1}{4}\rho
u_{x}u_{y}+\frac{1}{4}\rho u_{x}^{2}u_{y}-\frac{1}{4}\rho
u_{x}u_{y}^{2}+\frac{1}{4}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{7}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{36}\rho-\frac{1}{12}\rho
u_{x}-\frac{1}{12}\rho
u_{y}+\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})+\frac{1}{4}\rho
u_{x}u_{y}-\frac{1}{4}\rho u_{x}^{2}u_{y}-\frac{1}{4}\rho
u_{x}u_{y}^{2}+\frac{1}{4}\rho u_{x}^{2}u_{y}^{2},$ $\displaystyle f_{8}^{eq}$
$\displaystyle=$ $\displaystyle\frac{1}{36}\rho+\frac{1}{12}\rho
u_{x}-\frac{1}{12}\rho
u_{y}+\frac{1}{12}\rho(u_{x}^{2}+u_{y}^{2})-\frac{1}{4}\rho
u_{x}u_{y}-\frac{1}{4}\rho u_{x}^{2}u_{y}+\frac{1}{4}\rho
u_{x}u_{y}^{2}+\frac{1}{4}\rho u_{x}^{2}u_{y}^{2},$
and
$\displaystyle S_{0}$ $\displaystyle=$
$\displaystyle-2F_{x}u_{x}-2F_{y}u_{y}+2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{1}$ $\displaystyle=$
$\displaystyle+\frac{1}{2}F_{x}+F_{x}u_{x}-\frac{1}{2}F_{x}u_{y}^{2}-F_{y}u_{y}u_{x}-F_{x}u_{x}u_{y}^{2}-F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{2}$ $\displaystyle=$
$\displaystyle+\frac{1}{2}F_{y}+F_{y}u_{y}-\frac{1}{2}F_{y}u_{x}^{2}-F_{x}u_{x}u_{y}-F_{x}u_{x}u_{y}^{2}-F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}F_{x}+F_{x}u_{x}+\frac{1}{2}F_{x}u_{y}^{2}+F_{y}u_{y}u_{x}-F_{x}u_{x}u_{y}^{2}-F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}F_{y}+F_{y}u_{y}+\frac{1}{2}F_{y}u_{x}^{2}+F_{x}u_{x}u_{y}-F_{x}u_{x}u_{y}^{2}-F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{5}$ $\displaystyle=$
$\displaystyle+\frac{1}{4}F_{x}u_{y}+\frac{1}{4}F_{y}u_{x}+\frac{1}{4}F_{x}u_{y}^{2}+\frac{1}{4}F_{y}u_{x}^{2}+\frac{1}{2}F_{x}u_{x}u_{y}+\frac{1}{2}F_{y}u_{y}u_{x}+\frac{1}{2}F_{x}u_{x}u_{y}^{2}+\frac{1}{2}F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{6}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{x}u_{y}-\frac{1}{4}F_{y}u_{x}-\frac{1}{4}F_{x}u_{y}^{2}+\frac{1}{4}F_{y}u_{x}^{2}+\frac{1}{2}F_{x}u_{x}u_{y}-\frac{1}{2}F_{y}u_{y}u_{x}+\frac{1}{2}F_{x}u_{x}u_{y}^{2}+\frac{1}{2}F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{7}$ $\displaystyle=$
$\displaystyle+\frac{1}{4}F_{x}u_{y}+\frac{1}{4}F_{y}u_{x}-\frac{1}{4}F_{x}u_{y}^{2}-\frac{1}{4}F_{y}u_{x}^{2}-\frac{1}{2}F_{x}u_{x}u_{y}-\frac{1}{2}F_{y}u_{y}u_{x}+\frac{1}{2}F_{x}u_{x}u_{y}^{2}+\frac{1}{2}F_{y}u_{y}u_{x}^{2},$
$\displaystyle S_{8}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{x}u_{y}-\frac{1}{4}F_{y}u_{x}+\frac{1}{4}F_{x}u_{y}^{2}-\frac{1}{4}F_{y}u_{x}^{2}-\frac{1}{2}F_{x}u_{x}u_{y}+\frac{1}{2}F_{y}u_{y}u_{x}+\frac{1}{2}F_{x}u_{x}u_{y}^{2}+\frac{1}{2}F_{y}u_{y}u_{x}^{2}.$
Thus, the discrete equilibrium distribution and forcing terms in velocity
space resulting from corresponding imposed central moments are proper
polynomials containing higher order terms as compared to the standard LBM. The
specific functional expressions for $f_{\alpha}^{eq}$ and $S_{\alpha}$ depend
on the choice made for the “nominal moment basis” (Eq. (174)) from which they
are derived.
We are now in a position to perform a Chapman-Enskog multiscale expansion.
First, expand the raw moments $\mathbf{\widehat{f}}$ (untransformed ones, i.e.
without “overbar”, for simplicity) and the time derivative in terms of a small
bookkeeping perturbation parameter $\epsilon$ (which will be set to $1$ at the
end of the analysis) Premnath and Abraham (2007):
$\displaystyle\mathbf{\widehat{f}}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\epsilon^{n}\mathbf{\widehat{f}}^{(n)},$
(225) $\displaystyle\partial_{t}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\epsilon^{n}\partial_{t_{n}}.$ (226)
We use a Taylor expansion for the representation of the streaming operator,
which is carried out in its natural velocity space:
$\mathbf{f}(\overrightarrow{x}+\overrightarrow{e}_{\alpha}\epsilon,t+\epsilon)=\sum_{n=0}^{n}\frac{\epsilon^{n}}{n!}(\partial_{t}+\overrightarrow{e}_{\alpha}\cdot\overrightarrow{\nabla})\mathbf{f}(\overrightarrow{x},t).$
(227)
Substituting all the above three expansions in the LBE, with Eq. (223)
representing the post-collision, and equating terms of the same order of
successive powers of $\epsilon$ after making use of Eq. (222) and rearranging,
we get Premnath and Abraham (2007):
$\displaystyle O(\epsilon^{0}):\quad\mathbf{\widehat{f}}^{(0)}$
$\displaystyle=$ $\displaystyle\mathbf{\widehat{f}}^{eq},$ (228)
$\displaystyle
O(\epsilon^{1}):\quad(\partial_{t_{0}}+\widehat{E}_{i}\partial_{i})\mathbf{\widehat{f}}^{(0)}$
$\displaystyle=$
$\displaystyle-\widehat{\Lambda}\mathbf{\widehat{f}}^{(1)}+\mathbf{\widehat{S}},$
(229) $\displaystyle
O(\epsilon^{2}):\quad\partial_{t_{1}}\mathbf{\widehat{f}}^{(0)}+(\partial_{t_{0}}+\widehat{E}_{i}\partial_{i})\left[\mathcal{I}-\frac{1}{2}\widehat{\Lambda}\right]\mathbf{\widehat{f}}^{(1)}$
$\displaystyle=$ $\displaystyle-\widehat{\Lambda}\mathbf{\widehat{f}}^{(2)},$
(230)
where $\widehat{E}_{i}=\mathcal{T}(e_{\alpha
i}\mathcal{I})\mathcal{T}^{-1},\quad i\in{x,y}$. After substituting for
$\mathbf{\widehat{f}}^{(0)}$, $\widehat{E}_{i}$ and $\mathbf{\widehat{S}}$,
the first-order moment equations, i.e. Eq. (229) become
$\partial_{t_{0}}\rho+\partial_{x}(\rho u_{x})+\partial_{y}(\rho u_{y})=0,$
(231) $\partial_{t_{0}}\left(\rho
u_{x}\right)+\partial_{x}\left(\frac{1}{3}\rho+\rho
u_{x}^{2}\right)+\partial_{y}\left(\rho u_{x}u_{y}\right)=F_{x},$ (232)
$\partial_{t_{0}}\left(\rho u_{y}\right)+\partial_{x}\left(\rho
u_{x}u_{y}\right)+\partial_{y}\left(\frac{1}{3}\rho+\rho
u_{y}^{2}\right)=F_{y},$ (233)
$\displaystyle\partial_{t_{0}}\left(\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})\right)+\partial_{x}\left(\frac{4}{3}\rho
u_{x}+\rho u_{x}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{y}\left(\frac{4}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)$ (234) $\displaystyle=$
$\displaystyle-\omega_{3}\widehat{f}_{3}^{(1)}+2F_{x}u_{x}+2F_{y}u_{y},$
$\displaystyle\partial_{t_{0}}\left(\rho(u_{x}^{2}-u_{y}^{2})\right)+\partial_{x}\left(\frac{2}{3}\rho
u_{x}-\rho u_{x}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{y}\left(-\frac{2}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)$ (235) $\displaystyle=$
$\displaystyle-\omega_{4}\widehat{f}_{4}^{(1)}+2F_{x}u_{x}-2F_{y}u_{y},$
$\displaystyle\partial_{t_{0}}\left(\rho
u_{x}u_{y}\right)+\partial_{x}\left(\frac{1}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)$ $\displaystyle+$
$\displaystyle\partial_{y}\left(\frac{1}{3}\rho u_{x}+\rho
u_{x}u_{y}^{2}\right)$ (236) $\displaystyle=$
$\displaystyle-\omega_{5}\widehat{f}_{5}^{(1)}+F_{x}u_{y}+F_{y}u_{x},$
$\displaystyle\partial_{t_{0}}\left(\frac{1}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)+\partial_{x}\left(\rho u_{x}u_{y}\right)$
$\displaystyle+$
$\displaystyle\partial_{y}\left(\frac{1}{9}\rho+\frac{1}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}\right)$ (237) $\displaystyle=$
$\displaystyle-\omega_{6}\widehat{f}_{6}^{(1)}+F_{y}u_{x}^{2}+2F_{x}u_{x}u_{y},$
$\displaystyle\partial_{t_{0}}\left(\frac{1}{3}\rho u_{x}+\rho
u_{x}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{x}\left(\frac{1}{9}\rho+\frac{1}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}^{2}\right)+\partial_{y}\left(\rho u_{x}u_{y}\right)$ (238)
$\displaystyle=$
$\displaystyle-\omega_{7}\widehat{f}_{7}^{(1)}+F_{x}u_{y}^{2}+2F_{y}u_{y}u_{x},$
$\displaystyle\partial_{t_{0}}\left(\frac{1}{9}\rho+\frac{1}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{x}\left(\frac{1}{3}\rho u_{x}+\rho
u_{x}u_{y}^{2}\right)+\partial_{y}\left(\frac{1}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)$ (239) $\displaystyle=$
$\displaystyle-\omega_{8}\widehat{f}_{8}^{(1)}+2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}u_{x}^{2}.$
Similarly, the second-order moment equations can be derived from Eq. (230),
which can be written as
$\partial_{t_{0}}\rho=0,$ (240) $\partial_{t_{1}}\left(\rho
u_{x}\right)+\partial_{x}\left[\frac{1}{2}\left(1-\frac{1}{2}\omega_{3}\right)\widehat{f}_{3}^{(1)}+\frac{1}{2}\left(1-\frac{1}{2}\omega_{4}\right)\widehat{f}_{4}^{(1)}\right]+\partial_{y}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]=0,$
(241) $\partial_{t_{1}}\left(\rho
u_{y}\right)+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]+\partial_{y}\left[\frac{1}{2}\left(1-\frac{1}{2}\omega_{3}\right)\widehat{f}_{3}^{(1)}-\frac{1}{2}\left(1-\frac{1}{2}\omega_{4}\right)\widehat{f}_{4}^{(1)}\right]=0,$
(242)
$\displaystyle\partial_{t_{1}}\left(\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})\right)$
$\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{3}\right)\widehat{f}_{3}^{(1)}\right]+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{7}\right)\widehat{f}_{7}^{(1)}\right]$
(243) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{6}\right)\widehat{f}_{6}^{(1)}\right]=-\omega_{3}\widehat{f}_{3}^{(2)},$
$\displaystyle\partial_{t_{1}}\left(\rho(u_{x}^{2}-u_{y}^{2})\right)$
$\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{4}\right)\widehat{f}_{4}^{(1)}\right]+\partial_{x}\left[-\left(1-\frac{1}{2}\omega_{7}\right)\widehat{f}_{7}^{(1)}\right]$
(244) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{6}\right)\widehat{f}_{6}^{(1)}\right]=-\omega_{4}\widehat{f}_{4}^{(2)},$
$\displaystyle\partial_{t_{1}}\left(\rho u_{x}u_{y}\right)$ $\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{6}\right)\widehat{f}_{6}^{(1)}\right]$
(245) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{7}\right)\widehat{f}_{7}^{(1)}\right]=-\omega_{5}\widehat{f}_{5}^{(2)},$
$\displaystyle\partial_{t_{1}}\left(\frac{1}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)$ $\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{6}\right)\widehat{f}_{6}^{(1)}\right]+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]$
(246) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{8}\right)\widehat{f}_{8}^{(1)}\right]=-\omega_{6}\widehat{f}_{6}^{(2)},$
$\displaystyle\partial_{t_{1}}\left(\frac{1}{3}\rho u_{x}+\rho
u_{x}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{7}\right)\widehat{f}_{7}^{(1)}\right]+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{8}\right)\widehat{f}_{8}^{(1)}\right]$
(247) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]=-\omega_{7}\widehat{f}_{7}^{(2)},$
$\displaystyle\partial_{t_{1}}\left(\frac{1}{9}\rho+\frac{1}{3}\rho(u_{x}^{2}+u_{y}^{2})+\rho
u_{x}^{2}u_{y}^{2}\right)$ $\displaystyle+$
$\displaystyle\partial_{t_{0}}\left[\left(1-\frac{1}{2}\omega_{8}\right)\widehat{f}_{8}^{(1)}\right]+\partial_{x}\left[\left(1-\frac{1}{2}\omega_{7}\right)\widehat{f}_{7}^{(1)}\right]$
(248) $\displaystyle+$
$\displaystyle\partial_{y}\left[\left(1-\frac{1}{2}\omega_{6}\right)\widehat{f}_{6}^{(1)}\right]=-\omega_{8}\widehat{f}_{8}^{(2)}.$
Combining Eqs. (231), (232) and (233), with $\epsilon$ times Eqs. (240), (241)
and (242), respectively, and using
$\partial_{t}=\partial_{t_{0}}+\epsilon\partial_{t_{1}}$, we get the dynamical
equations for the conserved or hydrodynamic moments after setting the
parameter $\epsilon$ to unity. That is,
$\partial_{t}\rho+\partial_{x}(\rho u_{x})+\partial_{y}(\rho u_{y})=0,$ (249)
$\displaystyle\partial_{t}(\rho u_{x})+\partial_{x}(\rho
u_{x}^{2})+\partial_{y}(\rho u_{x}u_{y})$ $\displaystyle=$
$\displaystyle-\partial_{x}\left(\frac{1}{3}\rho\right)-\partial_{x}\left[\frac{1}{2}\left(1-\frac{1}{2}\omega_{3}\right)\widehat{f}_{3}^{(1)}+\frac{1}{2}\left(1-\frac{1}{2}\omega_{4}\right)\widehat{f}_{4}^{(1)}\right]$
(250)
$\displaystyle-\partial_{y}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]+F_{x},$
$\displaystyle\partial_{t}(\rho u_{y})+\partial_{x}(\rho u_{x}u_{y})$
$\displaystyle+$ $\displaystyle\partial_{y}(\rho
u_{y}^{2})=-\partial_{x}\left(\frac{1}{3}\rho\right)-\partial_{x}\left[\left(1-\frac{1}{2}\omega_{5}\right)\widehat{f}_{5}^{(1)}\right]$
(251) $\displaystyle-$
$\displaystyle\partial_{y}\left[\frac{1}{2}\left(1-\frac{1}{2}\omega_{3}\right)\widehat{f}_{3}^{(1)}-\frac{1}{2}\left(1-\frac{1}{2}\omega_{4}\right)\widehat{f}_{4}^{(1)}\right]+F_{y}.$
In the above three equations, Eqs. (249)-(251), we need the non-equilibrium
raw moments $\widehat{f}_{3}^{(1)}$, $\widehat{f}_{4}^{(1)}$ and
$\widehat{f}_{5}^{(1)}$ or
$\widehat{\pi}_{xx}^{{}^{\prime}(1)}+\widehat{\pi}_{yy}^{{}^{\prime}(1)}$,
$\widehat{\pi}_{xx}^{{}^{\prime}(1)}-\widehat{\pi}_{yy}^{{}^{\prime}(1)}$ and
$\widehat{\pi}_{xy}^{{}^{\prime}(1)}$, respectively. They can be obtained from
Eqs. (235), (236) and (237), respectively. Thus,
$\displaystyle\widehat{f}_{3}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{3}}\left[\left\\{-\partial_{t_{0}}\left(\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})\right)-\partial_{x}\left(\frac{4}{3}\rho
u_{x}+\rho u_{x}u_{y}^{2}\right)-\partial_{y}\left(\frac{4}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)\right\\}\right.$ (252)
$\displaystyle\left.+2F_{x}u_{x}+2F_{y}u_{y}\right],$
$\displaystyle\widehat{f}_{4}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{4}}\left[\left\\{-\partial_{t_{0}}\left(\rho(u_{x}^{2}-u_{y}^{2})\right)-\partial_{x}\left(\frac{2}{3}\rho
u_{x}-\rho u_{x}u_{y}^{2}\right)-\partial_{y}\left(-\frac{2}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)\right\\}\right.$ (253)
$\displaystyle\left.+2F_{x}u_{x}-2F_{y}u_{y}\right],$
$\displaystyle\widehat{f}_{5}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{5}}\left[\left\\{-\partial_{t_{0}}\left(\rho
u_{x}u_{y}\right)-\partial_{x}\left(\frac{1}{3}\rho u_{y}+\rho
u_{x}^{2}u_{y}\right)-\partial_{y}\left(\frac{1}{3}\rho u_{x}+\rho
u_{x}u_{y}^{2}\right)\right\\}\right.$ (254)
$\displaystyle\left.+F_{x}u_{y}+F_{y}u_{x}\right],$
The above three non-equilibrium moments can be simplified. In particular, by
using the first-order hydrodynamic moment equations, Eqs. (231)-(233) and
neglecting terms of $O(u^{3})$ or higher, we have $\partial_{t_{0}}(\rho
u_{x}^{2})\approx 2F_{x}u_{x}$, $\partial_{t_{0}}(\rho u_{y}^{2})\approx
2F_{y}u_{y}$ and $\partial_{t_{0}}(\rho u_{x}u_{y})\approx
F_{x}u_{y}+F_{y}u_{x}$. Substituting for these terms in Eqs. (252)-(254), and
representing the components of momentum for brevity as
$j_{x}=\rho u_{x},\quad j_{y}=\rho u_{y},$
we get
$\displaystyle\widehat{f}_{3}^{(1)}$ $\displaystyle\approx$
$\displaystyle-\frac{2}{3\omega_{3}}\overrightarrow{\nabla}\cdot\overrightarrow{j},$
(255) $\displaystyle\widehat{f}_{4}^{(1)}$ $\displaystyle\approx$
$\displaystyle-\frac{2}{3\omega_{4}}\left[\partial_{x}j_{x}-\partial_{y}j_{y}\right],$
(256) $\displaystyle\widehat{f}_{5}^{(1)}$ $\displaystyle\approx$
$\displaystyle-\frac{1}{3\omega_{5}}\left[\partial_{x}j_{y}+\partial_{y}j_{x}\right].$
(257)
Now, let
$\vartheta_{3}=\frac{1}{3}\left(\frac{1}{\omega_{3}}-\frac{1}{2}\right),\quad\vartheta_{4}=\frac{1}{3}\left(\frac{1}{\omega_{4}}-\frac{1}{2}\right),\quad\vartheta_{5}=\frac{1}{3}\left(\frac{1}{\omega_{5}}-\frac{1}{2}\right),$
(258)
and substituting the simplified expressions for the non-conserved moments,
Eqs. (255)-(257), and by using the relations for relaxation parameters given
in Eq. (258) in the conserved moment equations, Eqs. (249)-(251), we get
$\partial_{t}\rho+\overrightarrow{\nabla}\cdot\overrightarrow{j}=0,$ (259)
$\displaystyle\partial_{t}j_{x}+\partial_{x}\left(\frac{j_{x}^{2}}{\rho}\right)+\partial_{y}\left(\frac{j_{x}j_{y}}{\rho}\right)$
$\displaystyle=$
$\displaystyle-\partial_{x}p+\partial_{x}\left[\vartheta_{4}(2\partial_{x}j_{x}-\overrightarrow{\nabla}\cdot\overrightarrow{j})+\vartheta_{3}\overrightarrow{\nabla}\cdot\overrightarrow{j}\right]$
(260)
$\displaystyle+\partial_{y}\left[\vartheta_{5}(\partial_{x}j_{y}+\partial_{y}j_{x})\right]+F_{x},$
$\displaystyle\partial_{t}j_{y}+\partial_{x}\left(\frac{j_{x}j_{y}}{\rho}\right)+\partial_{y}\left(\frac{j_{y}^{2}}{\rho}\right)$
$\displaystyle=$
$\displaystyle-\partial_{y}p+\partial_{x}\left[\vartheta_{5}(\partial_{x}j_{y}+\partial_{y}j_{x})\right]$
(261)
$\displaystyle+\partial_{y}\left[\vartheta_{4}(2\partial_{y}j_{y}-\overrightarrow{\nabla}\cdot\overrightarrow{j})+\vartheta_{3}\overrightarrow{\nabla}\cdot\overrightarrow{j}\right]+F_{y},$
where $p=\frac{1}{3}\rho$ is the pressure field. Evidently, the relaxation
parameters $\omega_{4}$ and $\omega_{5}$ determine the shear kinematic
viscosity of the fluid, while $\omega_{3}$ controls its bulk viscous behavior.
Moreover, $\omega_{4}=\omega_{5}$ to maintain isotropy of the viscous stress
tensor ($\vartheta_{4}=\vartheta_{5}$). Thus, the proposed semi-implicit
procedure for incorporating forcing term based on a specialized central moment
lattice kinetic formulation is consistent with the weakly compressible Navier-
Stokes equations without resulting in any spurious effects.
It may be noted that in this work, we have employed a multiscale, or more
specifically a two time scale, expansion Chapman and Cowling (1964) to derive
the macroscopic equations. An alternative approach is to consider a single
time scale with an appropriate scaling relationship between space step and
time step to recover specific type of fluid flow behavior. This broadly leads
to two different types of consistency analysis techniques: (a) asymptotic
analysis approach Sone (2002) based on a diffusive or parabolic scaling Junk
et al. (2005) and (b) equivalent equation approach used in conjunction with
certain smoothness assumption and Taylor series expansion Lerat and Peyret
(1974); Warming and Hyett (1974) based on a convective or hyperbolic scaling
Dubois (2008). A recursive application of the LBE and an associated Taylor
series expansion without an explicit asymptotic relationship between the
lattice parameters can also be used to analyze the structure of the truncation
errors of the emergent macroscopic equations Holdych et al. (2004). Another
more recently developed approach is that based on a truncated Grad moment
expansion using appropriate scaling with a recursive substitution procedure
Asinari (2008), which has some features in common with an order of magnitude
analysis for kinetic methods Struchtrup (2005). It is expected that such
analysis tools can alternatively be applied to study the new computational
approach described in this work.
## Appendix B Generalization of Equilibrium and Sources with a Multiple
Relaxation Time Cascaded Lattice Kinetic Formulation
Let us first consider relaxation process of second-order non-conserved moments
in the rest frame of reference:
$\widehat{g}_{\beta}^{c}=\omega_{\beta}g_{\beta}^{*}=\omega_{\beta}\frac{\braket{\overline{f}_{\alpha}^{eq}-\overline{f}_{\alpha}}{K_{\beta}}}{\braket{K_{\beta}}{K_{\beta}}},\quad\beta=3,4,5.$
(262)
Here, summation of repeated indices with the subscript $\beta$ on the RHS is
not assumed and the superscript “c” for $\widehat{g}_{\beta}$ represents its
evaluation for cascaded collision process, with $g_{\beta}^{*}$ given in Eq.
(216) but restrict here to second-order moments. For convenience, we now
define the non-equilibrium (raw) moment of order $(m+n)$ as
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{(neq)^{\prime}}=\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq^{\prime}},$
(263)
or equivalently
$\widehat{\overline{f}}_{\beta}^{(neq)}=\widehat{\overline{f}}_{\beta}-\widehat{\overline{f}}_{\beta}^{eq}$,
where $\beta=m+n$. Thus,
$\displaystyle\widehat{g}_{3}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{3}}{12}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]=-\frac{\omega_{3}}{12}\widehat{\overline{f}}_{3}^{(neq)},$
(264) $\displaystyle\widehat{g}_{4}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{4}}{4}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]=-\frac{\omega_{4}}{4}\widehat{\overline{f}}_{4}^{(neq)},$
(265) $\displaystyle\widehat{g}_{5}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{5}}{4}\left[\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}}\right]=-\frac{\omega_{5}}{4}\widehat{\overline{f}}_{5}^{(neq)},$
(266)
The next step is to relax the third and higher order non-conserved moments in
the moving frame of reference, with each _central_ moment relaxing with
distinct relaxation time, in general. That is,
$\sum_{\beta}\braket{(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}}{K_{\beta}}\widehat{g}_{\beta}^{c}=\omega_{\beta}\left[\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq}-\widehat{\overline{\kappa}}_{x^{m}y^{n}}+\widehat{\sigma}_{x^{m}y^{n}}\right],\quad
m+n\geq 3.$ (267)
Clearly, this is equivalent to considering the last three rows of the
$\mathcal{F}$ matrix given in Eq. (214) to determine
$\widehat{g}_{\beta}^{c}$, for $\beta=6,7,8$ Asinari (2008). Expanding the
terms within the brackets of the RHS Eq. (267) in terms of raw moments, we get
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{eq}-\widehat{\overline{\kappa}}_{xxy}-\widehat{\sigma}_{xxy}$
$\displaystyle=$
$\displaystyle-\left[\widehat{\overline{\kappa}}_{xxy}^{(neq)^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime})}\right],$
(268)
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{eq}-\widehat{\overline{\kappa}}_{xyy}-\widehat{\sigma}_{xyy}$
$\displaystyle=$
$\displaystyle-\left[\widehat{\overline{\kappa}}_{xyy}^{(neq)^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime})}\right],$
(269)
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{eq}-\widehat{\overline{\kappa}}_{xxyy}-\widehat{\sigma}_{xxyy}$
$\displaystyle=$
$\displaystyle-\left[\widehat{\overline{\kappa}}_{xxyy}^{(neq)^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{(neq)^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{(neq)^{\prime})}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime})}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime})}\right.$
(270)
$\displaystyle\left.+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime})}\right].$
Now, in a manner analogous to the relaxation of second-order (raw) moments to
their equilibrium states, we assume relaxation of third and higher order (raw)
moments to their corresponding “equilibrium” states as well, which are as yet
unknown, but will be determined in the following consideration. That is, we
consider the ansatz
$\widehat{g}_{\beta}^{c}=\omega_{\beta}\frac{\braket{\overline{f}_{\alpha}^{eq,G}-\overline{f}_{\alpha}}{K_{\beta}}}{\braket{K_{\beta}}{K_{\beta}}},\quad\beta=6,7,8.$
(271)
Here, the superscript “G” represents the “generalized” expression, i.e.
$\overline{f}_{\alpha}^{eq,G}$ is the generalized equilibrium in the presence
of forcing terms (due to the presence of the ‘overbar’ symbol), which for
$\alpha=6,7,8$ will be determined in the following. Again, summation of
repeated indices with the subscript $\beta$ on the RHS is not assumed.
Evaluating Eq. (271) yields
$\displaystyle\widehat{g}_{6}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{6}}{4}\left[\widehat{\overline{\kappa}}_{xxy}^{eq,G^{\prime}}-\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}\right],$
(272) $\displaystyle\widehat{g}_{7}^{c}$ $\displaystyle=$
$\displaystyle-\frac{\omega_{7}}{4}\left[\widehat{\overline{\kappa}}_{xyy}^{eq,G^{\prime}}-\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}\right],$
(273) $\displaystyle\widehat{g}_{8}^{c}$ $\displaystyle=$
$\displaystyle\frac{\omega_{8}}{4}\left[\widehat{\overline{\kappa}}_{xxyy}^{eq,G^{\prime}}-\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}\right]-\frac{\omega_{8}}{4}\left[\widehat{\overline{\kappa}}_{xx}^{eq^{\prime}}-\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}\right]-\frac{\omega_{8}}{4}\left[\widehat{\overline{\kappa}}_{yy}^{eq^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}\right].$
(274)
Now substituting Eqs. (263),(268)-(270) and (271) in Eq. (267) and simplifying
and rearranging the resulting expressions yield the desired expressions for
the generalized equilibrium in the presence of forcing terms
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{eq,G^{\prime}}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxy}^{eq^{\prime}}+\varphi_{6}^{3}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{6}^{4}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{6}^{5}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}},$
(275) $\displaystyle\widehat{\overline{\kappa}}_{xyy}^{eq,G^{\prime}}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xyy}^{eq^{\prime}}+\varphi_{7}^{3}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{7}^{4}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{7}^{5}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}},$
(276) $\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{eq,G^{\prime}}$
$\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{xxyy}^{eq^{\prime}}+\varphi_{8}^{3}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{8}^{4}\left[\widehat{\overline{\kappa}}_{xx}^{(neq)^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{(neq)^{\prime}}\right]+\varphi_{8}^{5}\widehat{\overline{\kappa}}_{xy}^{(neq)^{\prime}}$
(277)
$\displaystyle+\varphi_{8}^{6}\widehat{\overline{\kappa}}_{xxy}^{(neq)^{\prime}}+\varphi_{8}^{7}\widehat{\overline{\kappa}}_{xyy}^{(neq)^{\prime}},$
where the coefficients $\varphi_{\alpha}^{\beta}$ in Eqs. (275)-(277) are
functions of the various ratios of the relaxation times of the above MRT
cascaded formalism and velocity field arising relaxing the moments in the
moving frame of reference. The coefficients for
$\widehat{\overline{\kappa}}_{xxy}^{eq,G^{\prime}}$ are
$\varphi_{6}^{3}=\frac{1}{2}\left(1-\theta_{6}^{3}\right)u_{y},\quad\varphi_{6}^{4}=\frac{1}{2}\left(1-\theta_{6}^{4}\right)u_{y},\quad\varphi_{6}^{5}=2\left(1-\theta_{6}^{5}\right)u_{x},$
(278)
and for $\widehat{\overline{\kappa}}_{xyy}^{eq,G^{\prime}}$ are
$\varphi_{7}^{3}=\frac{1}{2}\left(1-\theta_{7}^{3}\right)u_{x},\quad\varphi_{7}^{4}=-\frac{1}{2}\left(1-\theta_{7}^{4}\right)u_{x},\quad\varphi_{7}^{5}=2\left(1-\theta_{7}^{5}\right)u_{y},$
(279)
and, finally, for $\widehat{\overline{\kappa}}_{xxyy}^{eq,G^{\prime}}$ are
$\displaystyle\varphi_{8}^{3}$ $\displaystyle=$
$\displaystyle-\left\\{\left(1-\theta_{8}^{3}\right)\left[\frac{2}{3}+\frac{1}{2}(u_{x}^{2}+u_{y}^{2})\right]-\theta_{8}^{6}\left(1-\theta_{6}^{3}\right)u_{y}^{2}-\theta_{8}^{7}\left(1-\theta_{7}^{3}\right)u_{x}^{2}\right\\},$
$\displaystyle\varphi_{8}^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1-\theta_{8}^{4}\right)(u_{x}^{2}-u_{y}^{2})+\theta_{8}^{6}\left(1-\theta_{6}^{4}\right)u_{y}^{2}-\theta_{8}^{7}\left(1-\theta_{7}^{4}\right)u_{x}^{2},$
$\displaystyle\varphi_{8}^{5}$ $\displaystyle=$
$\displaystyle-4\left[\left(1-\theta_{8}^{5}\right)-\theta_{8}^{6}\left(1-\theta_{6}^{5}\right)-\theta_{8}^{7}\left(1-\theta_{7}^{5}\right)\right]u_{x}u_{y},$
(280) $\displaystyle\varphi_{8}^{6}$ $\displaystyle=$ $\displaystyle
2\left(1-\theta_{8}^{6}\right)u_{y},$ $\displaystyle\varphi_{8}^{7}$
$\displaystyle=$ $\displaystyle 2\left(1-\theta_{8}^{7}\right)u_{x}.$
Here, in Eqs. (278)-(280), the parameter $\theta_{\beta}^{\alpha}$ refers to
the ratio of relaxation times $\omega_{\alpha}$ and $\omega_{\beta}$. That is
$\theta_{\beta}^{\alpha}=\frac{\omega_{\alpha}}{\omega_{\beta}}.$ (281)
Now, in the notations of the previous section, we can rewrite
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{eq,G^{\prime}}$ in terms of
$\widehat{\overline{f}}_{\beta}^{G}$, or more explicitly, in terms of the
regular generalized equilibrium and source moments, i.e.
$\widehat{f}_{\beta}^{G}$ and $\widehat{S}_{\beta}^{G}$, respectively, using
$\widehat{\overline{f}}_{\beta}^{G}=\widehat{f}_{\beta}^{G}-\frac{1}{2}\widehat{S}_{\beta}^{G}$.
Thus, compactly, the generalized equilibrium and source moments are
$\displaystyle\widehat{f}_{\beta}^{eq,G}$ $\displaystyle=$
$\displaystyle\widehat{f}_{\beta}^{eq}+\sum_{\alpha=3}^{N_{v}}\varphi_{\beta}^{\alpha}\widehat{f}_{\alpha}^{(neq)}=\widehat{f}_{\beta}^{eq}+\sum_{\alpha=3}^{N_{v}}\varphi_{\beta}^{\alpha}\left(\widehat{f}_{\alpha}-\widehat{f}_{\alpha}^{(eq)}\right),\quad\quad\beta=6,7,8$
(282) $\displaystyle\widehat{S}_{\beta}^{G}$ $\displaystyle=$
$\displaystyle\widehat{S}_{\beta}-\sum_{\alpha=3}^{N_{v}}\varphi_{\beta}^{\alpha}\widehat{S}_{\alpha},\quad\quad\beta=6,7,8$
(283)
where $N_{v}=\left\\{\begin{array}[]{ll}{5,}&{\beta=6,7}\\\
{7,}&{\beta=8}\end{array}\right.$. It should, however, be noted that
$\widehat{f}_{\beta}^{eq,G}=\widehat{f}_{\beta}^{eq}$ and
$\widehat{S}_{\beta}^{G}=\widehat{S}_{\beta}$ for $\beta\leq 5$. This analysis
further extends that of Asinari Asinari (2008), who showed generalized
equilibrium for a particular form of Cascaded-LBM without forcing terms. Thus,
the generalized equilibrium arising from the cascaded nature of the collision
step for the third and higher order (raw) moments is a function of conserved
moments, non-equilibrium part of the lower order moments and the various
ratios of the relaxation times in the MRT formulation. Similarly, the
generalized sources for the third and higher order moments is a function of
the products of force fields and fluid velocity, as well as the ratio of
relaxation times. In view of the above, the cascaded formulation can also be
reinterpreted by defining the generalization of the equilibrium and source in
terms of the following local coefficient matrix
$\mathcal{C}\equiv\mathcal{C}(\overrightarrow{x},t)$:
$\mathcal{C}=\left[\begin{array}[]{ccccccccc}0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\
0&0&0&\varphi_{6}^{3}&\varphi_{6}^{4}&\varphi_{6}^{5}&0&0&0\\\
0&0&0&\varphi_{7}^{3}&\varphi_{7}^{4}&\varphi_{7}^{5}&0&0&0\\\
0&0&0&\varphi_{8}^{3}&\varphi_{8}^{4}&\varphi_{8}^{5}&\varphi_{8}^{6}&\varphi_{8}^{7}&0\\\
\end{array}\right].$ (284)
That is, if the information cascades from lower to higher moments during a
time interval $(t,t+1)$, the raw equilibrium and source moments in the lattice
frame of reference generalize to
$\displaystyle\widehat{\mathbf{f}}_{(\overrightarrow{x},t^{*})}^{eq,G}$
$\displaystyle=$
$\displaystyle\left(\mathcal{I}-\mathcal{C}\right)\widehat{\mathbf{f}}_{(\overrightarrow{x},t)}^{eq}+\mathcal{C}\widehat{\mathbf{f}}_{(\overrightarrow{x},t+1)},$
(285) $\displaystyle\widehat{\mathbf{S}}_{(\overrightarrow{x},t^{*})}^{G}$
$\displaystyle=$
$\displaystyle\left(\mathcal{I}-\mathcal{C}\right)\widehat{\mathbf{S}}_{(\overrightarrow{x},t)}$
(286)
where $t^{*}$ represents some intermediate time in $(t,t+1)$. Clearly, the
generalization of both equilibrium and sources degenerate to corresponding
regular forms only when the relaxation times of all the moments are the same.
That is, when the approach is reduced to the SRT formulation,
$\widehat{f}_{\beta}^{eq,G}=\widehat{f}_{\beta}^{eq}$ and
$\widehat{S}_{\beta}^{G}=\widehat{S}_{\beta}$ for all possible values of
$\beta$, since $\mathcal{C}=\bf{0}$, i.e. a null matrix in that case. In the
previous section, a consistency analysis for a special case of the central
moment method was presented. The same notation and procedure can be adopted
for the general case involving cascaded relaxation (represented as a
relaxation of non-conserved raw moments to their generalized equilibrium) with
generalized sources presented here, when $\widehat{f}_{\beta}^{eq}$ becomes
$\widehat{f}_{\beta}^{eq,G}$ and
$\left(1-\frac{1}{2}\omega_{\beta}\right)\widehat{S}_{\beta}$ becomes
$\widehat{S}_{\beta}^{G}$ for $\beta=6,7,8$. Inspection of the details of the
Chapman-Enskog moment expansion analysis presented in the earlier section
shows that the consistency of the Cascaded-LBM to the NSE remains unaffected
by the presence of generalized equilibrium and sources. In particular, the
generalized forms contain coefficients which are functions of local fluid
velocity and the ratio of various relaxation times, and terms that are non-
equilibrium part of the lower order moments, which are negligibly small in
nature for slow or weakly compressible flows, as they involve products of
various powers of hydrodynamic fields. Since for consistency purpose, we need
to retain only $O(Ma^{2})$, the presence of the generalized terms do not
affect the end result of the derivation presented in the previous section.
An interesting viewpoint to note is that the use of relaxation to generalized
equilibrium (including the effect of sources), i.e. Eq. (271) may be
considered as an alternative computational framework to actually execute the
cascaded MRT collision step. It reduces to a corresponding TRT collision step,
when $\omega^{\mathrm{even}}=\omega_{4}=\omega_{6}=\omega_{8}$ and
$\omega^{\mathrm{odd}}=\omega_{3}=\omega_{5}=\omega_{7}$. Also, a different
perspective of the generalized equilibrium, Eq. (282) can be arrived at in
light of the consistency analysis performed in the previous section. For
example, for the third-order moments, $\beta=6$ and $7$, Eq. (282) needs the
non-equilibrium moments $\widehat{f}_{3}^{(neq)}$, $\widehat{f}_{4}^{(neq)}$
and $\widehat{f}_{5}^{(neq)}$, which can be approximated by Eqs. (255), (256)
and (257), respectively, which actually provide expressions for the components
of the strain rate tensor in the cascaded formulation. Thus, we get
$\displaystyle\widehat{f}_{6}^{eq,G}$ $\displaystyle\approx$
$\displaystyle\widehat{f}_{6}^{eq}-\frac{1}{3}\left(\frac{1}{\omega_{3}}-\frac{1}{\omega_{6}}\right)u_{y}\overrightarrow{\nabla}\cdot\overrightarrow{j}-\frac{1}{3}\left(\frac{1}{\omega_{4}}-\frac{1}{\omega_{6}}\right)u_{y}\left(\partial_{x}j_{y}-\partial_{y}j_{x}\right)$
(287)
$\displaystyle-\frac{2}{3}\left(\frac{1}{\omega_{5}}-\frac{1}{\omega_{6}}\right)u_{x}\left(\partial_{x}j_{y}+\partial_{y}j_{x}\right),$
$\displaystyle\widehat{f}_{7}^{eq,G}$ $\displaystyle\approx$
$\displaystyle\widehat{f}_{7}^{eq}-\frac{1}{3}\left(\frac{1}{\omega_{3}}-\frac{1}{\omega_{7}}\right)u_{x}\overrightarrow{\nabla}\cdot\overrightarrow{j}-\frac{1}{3}\left(\frac{1}{\omega_{4}}-\frac{1}{\omega_{7}}\right)u_{x}\left(\partial_{x}j_{y}-\partial_{y}j_{x}\right)$
(288)
$\displaystyle-\frac{2}{3}\left(\frac{1}{\omega_{5}}-\frac{1}{\omega_{7}}\right)u_{y}\left(\partial_{x}j_{y}+\partial_{y}j_{x}\right).$
In other words, the generalized equilibrium is a function of density and
velocity fields and their gradients, the coefficients of the latter terms are
given as difference of relaxation times of moments of different order.
## Appendix C Introducing Time-implicitness in the Cascaded Collision
Operator
Here, let us investigate the possibility of developing an executable LBE
formulation where implicitness in time is introduced in the cascaded collision
kernel, which could be useful in certain applications. In particular, we
extend Eq. (35) such that the cascaded collision operator
$\Omega_{{\alpha}(\overrightarrow{x},t)}^{c}$ is now treated to be semi-
implicit in time:
$\displaystyle
f_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)=f_{\alpha}(\overrightarrow{x},t)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[(\mathcal{K}\cdot\mathbf{\widehat{g}})_{{\alpha}(\overrightarrow{x},t)}+(\mathcal{K}\cdot\mathbf{\widehat{g}})_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)}\right]$
(289) $\displaystyle+$
$\displaystyle\frac{1}{2}\left[S_{{\alpha}(\overrightarrow{x},t)}+S_{{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)}\right]$
In order to avoid an iterative procedure for the use of Eq. (289), we now
define the following transformation with the introduction of a new variable
$\overline{h}_{\alpha}$:
$\overline{h}_{\alpha}=f_{\alpha}-\frac{1}{2}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}-\frac{1}{2}S_{\alpha}.$
(290)
Now, substituting Eq. (290) in Eq. (289), we get
$\overline{h}_{\alpha}(\overrightarrow{x}+\overrightarrow{e}_{\alpha},t+1)-\overline{h}_{\alpha}(\overrightarrow{x},t)=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{{\alpha}(\overrightarrow{x},t)}+S_{{\alpha}(\overrightarrow{x},t)}$
(291)
As a result, Eq. (291) now becomes effectively explicit. In the new variable,
the hydrodynamic fields can be obtained as
$\rho=\sum_{\alpha=0}^{8}\overline{h}_{\alpha}$ and $\rho
u_{i}=\sum_{\alpha=0}^{8}\overline{h}_{\alpha}e_{\alpha i}+\frac{1}{2}F_{i}$.
The post-collision values, i.e. $\widetilde{\overline{h}}_{\alpha}$ can be
obtained by replacing $\overline{f}_{\alpha}$ with $\overline{h}_{\alpha}$ in
Eqs. (165)-(173). Now, to obtain the collision kernel
$(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}$ in Eq. (291) in terms of
$\overline{h}_{\alpha}$, we define the following raw moment of order $(m+n)$:
$\widehat{\overline{\eta}}_{x^{m}y^{n}}^{{}^{\prime}}=\sum_{\alpha}\overline{h}_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\braket{e_{\alpha x}^{m}e_{\alpha
y}^{n}}{\overline{h}_{\alpha}},$ (292)
where $\widehat{\overline{\eta}}_{x^{m}y^{n}}^{{}^{\prime}}$ can be
represented and computed in a manner similar to that given in Eqs.
(121)-(126). From Eqs. (290) and (292), we obtain
$\displaystyle\widehat{\overline{\eta}}_{x^{m}y^{n}}^{{}^{\prime}}$
$\displaystyle=$
$\displaystyle\widehat{\kappa}_{x^{m}y^{n}}^{{}^{\prime}}-\frac{1}{2}\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{m}e_{\alpha
y}^{n}}\widehat{g}_{\beta}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$
(293) $\displaystyle=$
$\displaystyle\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}-\frac{1}{2}\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{m}e_{\alpha y}^{n}}\widehat{g}_{\beta}$
where $\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{m}e_{\alpha
y}^{n}}\widehat{g}_{\beta}$ can be obtained by exploiting the orthogonal
properties of $\mathcal{K}$, i.e. from Eqs. (85)-(93).
Now substituting Eq. (293) in the collision kernel written in compact notation
as given in Appendix A, i.e. in Eqs. (204)-(209), and simplifying we get
$\displaystyle\widehat{g}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\frac{\omega_{3}}{\left(1+\frac{1}{2}\omega_{3}\right)}\left\\{\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}+\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(294) $\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\omega_{4}}{\left(1+\frac{1}{2}\omega_{4}\right)}\left\\{\rho(u_{x}^{2}-u_{y}^{2})-(\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\widehat{\overline{\eta}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(295) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\omega_{5}}{\left(1+\frac{1}{2}\omega_{5}\right)}\left\\{\rho
u_{x}u_{y}-\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xy}^{{}^{\prime}}\right\\},$
(296) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\omega_{6}}{\left(1+\frac{1}{2}\omega_{6}\right)}\left\\{2\rho
u_{x}^{2}u_{y}+\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xxy}\right\\}$
(297)
$\displaystyle-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4})-2u_{x}\widehat{g}_{5},$
$\displaystyle\widehat{g}_{7}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\omega_{7}}{\left(1+\frac{1}{2}\omega_{7}\right)}\left\\{2\rho
u_{x}u_{y}^{2}+\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xyy}\right\\}$
(298)
$\displaystyle-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4})-2u_{y}\widehat{g}_{5},$
$\displaystyle\widehat{g}_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\omega_{8}}{\left(1+\frac{1}{2}\omega_{8}\right)}\left\\{\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-\left[\widehat{\overline{\eta}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\eta}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\eta}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\eta}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\eta}}_{xx}^{{}^{\prime}}\right.\right.$
(299)
$\displaystyle\left.\left.+4u_{x}u_{y}\widehat{\overline{\eta}}_{xy}^{{}^{\prime}}\right]-\frac{1}{2}\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right\\}-2\widehat{g}_{3}-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})$
$\displaystyle-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7}.$
It may be noted that a Chapman-Enskog analysis, as given in Appendix A, when
performed with the above collision operator, yields the following relations
between relaxation parameters and transport coefficients (see Eq. (258)):
$\vartheta_{3}=\frac{1}{3\omega_{3}},\quad\vartheta_{4}=\frac{1}{3\omega_{4}},\quad\vartheta_{5}=\frac{1}{3\omega_{5}}$,
for the hydrodynamical equations given in Eq. (260) and (261). Thus, the above
considerations show that it is possible to introduce time-implicitness in the
cascaded collision kernel, and when a transformation is introduced to make the
computational procedure effectively explicit, it leaves the form of
$\widehat{g}_{\beta}$ unchanged with a simple re-scaling of the relaxation
parameters.
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|
arxiv-papers
| 2012-02-27T22:53:58 |
2024-09-04T02:49:27.937925
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kannan N. Premnath and Sanjoy Banerjee",
"submitter": "Kannan Premnath",
"url": "https://arxiv.org/abs/1202.6087"
}
|
1202.6092
|
# A multiscale maximum entropy moment closure for locally regulated space–time
point process models of population dynamics ††thanks: M.R. is grateful for a
postgraduate scholarship from the Principal’s development fund of the
University of Glasgow, an overseas student award granted from the Department
of Mathematics, University of Glasgow, and DARPA (Award ID:HR001-05-1-0057)
Michael Raghib Nicholas A. Hill Ulf Dieckmann
###### Abstract
The prevalence of structure in biological populations challenges fundamental
assumptions at the heart of continuum models of population dynamics based on
mean densities (local or global) only. Individual-based models (IBM’s) were
introduced over the last decade in an attempt to overcome this limitation by
following explicitly each individual in the population. Although the IBM
approach has been quite insightful, the capability to follow each individual
usually comes at the expense of analytical tractability, which limits the
generality of the statements that can be made. For the specific case of
spatial structure in populations of sessile (and identical) organisms,
space–time point processes with _local regulation_ seem to cover the middle
ground between analytical tractability and a higher degree of biological
realism. This approach has shown that simplified representations of fecundity,
local dispersal and density–dependent mortality weighted by the local
competitive environment are sufficient to generate spatial patterns that mimic
field observations. Continuum approximations of these stochastic processes try
to distill their fundamental properties, but because they keep track of not
only mean densities, but also higher order spatial correlations, they result
in infinite hierarchies of moment equations. This leads to the problem of
finding a ‘moment closure’; that is, an appropriate order of (lower order)
truncation, together with a method of expressing the highest order density not
explicitly modelled in the truncated hierarchy in terms of the lower order
densities. We use the principle of constrained maximum entropy to derive a
closure relationship at second order using normalisation and the product
densities of first and second orders as constraints, and apply it to one such
hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood
superposition approximation, or ‘power-3’ closure, but it is complemented with
previously unknown correction terms that depend on integrals over the region
for which third order correlations are irreducible. The region of irreducible
triplet correlations is found as the domain that solves an integral equation
associated with the normalisation constraint. This also serves the purpose of
a validation check, since a single, non–trivial domain can only be found if
the assumptions of the closure are consistent with the predictions of the
hierarchy. Comparisons between simulations of the point process, alternative
heuristic closures, and the maxent closure show significant improvements in
the ability of the truncated hierarchy to predict equilibrium values for
mildly aggregated spatial patterns. However, the maxent closure performs
comparatively poorly in segregated ones. Although the closure is applied in
the context of point processes, the method does not require fixed locations to
be valid, and can in principle be applied to problems where the particles
move, provided that their correlation functions are stationary in space and
time.
## 1 Introduction
One of the most widely used models in theoretical ecology is the logistic
equation [50, 56, 75]
$\displaystyle\frac{d}{dt}{m_{1}}(t)$ $\displaystyle=$ $\displaystyle
r\,{m_{1}}(t)\left(1-\frac{{m_{1}}(t)}{K}\right)$ (1)
$\displaystyle{m_{1}}(0)$ $\displaystyle=$ $\displaystyle n_{0},$
which describes the dynamics of a population in terms of a single state
variable ${m_{1}}(t)$, which can be interpreted as the total population size
or as the global density. The rate of change of the density in the logistic
model is determined by three drivers. The first two are present in the net
growth term $r=b-d$, where $b$ and $d$ are respectively the _per capita_
fecundity and intrinsic mortality rates. The third one is the density-
dependent mortality rate, which is assumed to be proportional to the density,
where the constant of proportionality $K$ is the ‘carrying capacity’, i.e. the
maximum number of individuals per unit area or volume that can be supported by
some unspecified limiting resource. This model is built on the following set
of assumptions [4, 19, 44]:
1. 1.
There are no facilitative interactions among conspecifics.
2. 2.
Contributions to mortality due to competition are pairwise additive.
3. 3.
The limiting resource is uniformly distributed in space, and shared
proportionally by all individuals.
4. 4.
There are no differences among individuals in age, size or phenotype.
5. 5.
The spatial locations of the individuals are uncorrelated.
6. 6.
Allocation to reproductive tissues is independent of the local resource
availability.
7. 7.
Density–dependent mortality occurs at the same temporal scales than fecundity
and intrinsic mortality.
These assumptions are valid only for a rather restricted set of biological
situations. For instance, facilitative interactions are known to play a
determinant role alongside competition in shaping community structure and
dynamics [9]. In plant communities, non-succesional positive interactions can
result from additional resources being made available through synergies (e.g.
hydraulic lift, microbial enhancement, mycorrhizal networks), a reduction in
the impact of climate extremes and predation [31] or a combination of these.
The assumption of pairwise additivity in density-dependent mortality enjoys
some degree of empirical support for plant populations [76], but it is still
an unresolved issue [17, 22]. Forms of population structure driven by size (or
age), phenotype or spatial pre-patterning in the abiotic substrate having an
impact on fecundity, recruitment and survivorship are ubiquitously observed
both in the field and experimental literature [57, 73] [67]. Seed dispersal
and competitive interactions are known to occur over a characteristic range of
spatial scales rather than being uniformly distributed as is commonly assumed
in the logistic model [12, 29, 63, 67, 68, 70, 10].
These limitations have motivated the search for alternatives to the logistic
equation that can address questions of broader biological interest, while
simultaneously maintaining a reasonable degree of mathematical and
computational tractability. Achieving this goal depends heavily on the
development of multiscale modeling approaches capable of linking patterns
manifested at the larger, population–level scales, to their drivers, which lie
in biological processes occurring at the level of individuals; typically
taking place over spatial and temporal scales that differ substantially from
those at which the population–level regularities are detected [4, 6, 19, 23,
43, 44, 48, 45, 61].
Among all the possible paths suggested as one relaxes these assumptions (1–7),
understanding the role of spatial structure, particularly that driven by
biological processes alone, has received a considerable amount of interest
[20, 44, 4, 5, 7, 62, 12, 8, 36]. The approaches that have been developed for
the spatial problem have a number of commonalities. They usually consist of an
individual-based model (IBM) [18, 30] which follows simplified representations
of the life histories of each individual in the population. These
representations include the biological processes believed to play a role in
driving the population–level phenomena, and typically include a combination of
fecundity, dispersal, mortality and in some cases, growth. These are modeled
in such a way that some form of density–dependent regulation is present in at
least in at least one of them. Second, the density–dependent regulation is
determined by the neighborhood configuration surrounding each focal
individual, which leads to a _local_ regulation of the process [3, 24, 26].
Third, the dynamics of the macroscopic patterns is obtained from an average of
a sufficiently large number of independent realisations of the
individual–level model. Insights about the emergence of various forms of
population structure, in particular space, are gained as these broad scale
patterns are allowed to vary with the characteristic scales that regulate the
biological processes at the level of the individual organism [4, 43, 54, 55,
77].
This approach, albeit insightful, restricts severely the statements that can
be made about how the processes present across various scales interact to
produce pattern, since typically there is an absence of a model condensing the
dynamics of pattern at the larger scale. To circumvent this deficiency,
several attempts to derive population–level models from the IBM have been
introduced in the literature. In the context of spatial pattern in plant
population dynamics [4, 43, 36, 62], these models typically take the form of
hierarchies of equations for relevant families of summary statistics where
quantities in addition to the mean density capture spatial correlations among
pairs, triplets etc, that quantify spatial pattern across a range of scales
[12, 72]. These summary statistics are closely related to the central,
factorial or raw spatial moments of the underlying spatial stochastic process.
For pair configurations in plant population models, common choices are the
spatial auto-covariance or the second order product density [12, 14, 21, 72].
A discussion of these various approaches in the development of continuum
approximations to spatio-temporal stochastic processes in ecology can be found
in a compilation edited by Dieckmann _et al_ [20].
The non-linearities due to the presence of density–dependence in spatially
explicit IBM’s inevitably result in infinite hierarchies of evolution
equations for the summary statistics, where the dynamics of the correlations
of order $k$ is tied to that of order $k+1$. If one truncates the hierarchy at
some order, the evolution equation at the order of the truncation will depend
on the _unknown_ density of the next higher order. Analysis of these
hierarchies can only proceed after truncation for some small order. This
requires the solution of two problems. The first, is identifying an
appropriate order of truncation $k$. The second is compensating for the
resulting loss of information. The order of truncation in existing models is
chosen on the basis of computational complexity, and rarely goes beyond two
[69, 4, 43]. For the second problem, the density of order $k+1$ is replaced by
a functional relationship of all the densities of order up to $k$, usually
called a ‘moment closure’. This functional dependence of higher order
quantities on lower order ones is constructed mainly on heuristic reasoning
[4, 19, 51]. For instance, when the order of truncation is two, assuming
vanishing central moments of order three leads to the so-called ‘power–1’
closure [4]. The ‘power–2’ closure arises from an analogy with the pair
approximation used in discrete spatial models [36, 19]. Assuming independence
of the three pair correlations associated with each edge of a triplet for all
spatial scales leads to the ‘power–3’ or Kirkwood superposition approximation
[41, 19]. Although higher order closures do exist , they have restricted
applicability due to the daunting computational problem that results at orders
higher than three [69].
Despite some encouraging success that resulted in analytical solutions of the
hierarchy at equilibrium for truncation at second order [4, 5, 7], and
remarkably good fit of the numerical solution of the hierarchy with
individual–based simulations with so–called _asymmetric_ versions of
previously used closures [44, 51], most predict poorly the equilibrium
densities even for situations of mild spatial correlations. In the cases where
they succeed over a broader range of regimes of spatial correlations (i.e. the
asymmetric power–2), the closure depends on tuning a set of weighting
constants whose values can presently be found only by comparison with
simulations of the stochastic process. A significant obstacle in the
widespread adoption of these continuum approximations and their closures is
that none of them is equipped with a criterion for their domain of validity
that does not depend on comparisons with simulations of the individual–based
model. Nevertheless, many of these heuristic closures do provide a better
approximation to the dynamics of a spatially structured population than the
logistic equation, and illuminate a variety of mechanisms by which
endogenously generated spatial pattern appears in plant populations.
Inspired by earlier results of Hillen [35] and Singer [69], who used the
principle of constrained maximum entropy [66, 40] [38] to respectively derive
closures for velocity jump processes [52] and the BBGKY hierarchy arising in
the statistical mechanics of fluids [41], we develop a closure scheme based on
constrained entropy maximisation for the moment hierarchy developed by Law &
Dieckmann [43], constrained to satisfy normalisation and the product densities
up to order two. In order to be able to relate the output of the entropy
maximisation to the approximating dynamical system, we also reframe the
hierarchy of Law & Dieckmann [43] in terms of product densities rather than
the spatial moments. These two kinds of sets of summary statistics are very
closely related, since the latter can be seen as estimators of the former. The
approach of Hillen [35] consists of proving that the $L^{2}$–norm over the
space of velocities of the transport equation of Othmer _et al_ [52] behaves
like an entropy, with the velocity moments acting as constraints. Singer [69]
treats the triplet product density as a probability density in order to
construct an entropy from the point of view of information theory [66, 40,
38], using consistency of the marginals as constraints. Our approach differs
from these two other maximum entropy maximisation methods in a number of ways.
First, we use the information theoretical entropy functional for point
processes [46, 16], based on the negative of the expected log–likelihood, and
includes all the orders that contribute spatial information, not just order
three. Second, the product densities which provide the constraints are
incorporated into the entropy functional by means of an expansion that allows
to express the likelihoods (or Janossy densities) in terms of product
densities and vice versa [13, 14], this allows us to establish a formal
connection between the entropy functional and the moment hierarchy. Third, our
closure is implicit, in the sense that the density of order three appears at
both sides of the closing relationship, thus allowing irreducible correlations
of third order to be explicitly included. Fourth, the method presented here
complements the Kirkwood (or power–3) closure with previously unknown
correction terms that depend on the area for which the three points in the
triplet become independent. These correction terms are important where the
three particles in the triplet configuration are close to each other, but
progressively vanish as these become separated, at which point the maximum
entropy closure reduces to the classical Kirkwood superposition approximation.
These correction terms lead to substantial improvements in the prediction of
the equilibrium density for mildly aggregated patterns. In addition, the
closure comes equipped with a criterion of validity stemming from the
normalisation constraint. This validity check comes from an ancillary integral
equation that returns the area of the domain at which the points become
independent. This equation produces a single, non-trivial root when the
correlations predicted by the moment hierarchy are consistent with the
truncation assumptions, but fails to do so otherwise. The maximum entropy
closure relationship we found is given by
$\displaystyle{m_{3}}(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle\left[{m_{2}}(\xi_{1})-\,{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{1},\xi_{2}^{\prime})\,\,d\xi_{2}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2})-{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{2},\xi_{2}-\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2}-\xi_{1})-\,{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{2}-\xi_{1}^{\prime},\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\frac{J_{0}(A_{0})}{\left[{m_{1}}-\,{|A_{0}|}\int_{A_{0}}{m_{2}}(\xi_{1}^{\prime})\,d\xi_{1}^{\prime}+\frac{{|A_{0}|}^{2}}{2}\int_{A_{0}\times
A_{0}}{m_{3}}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}\right]^{3}},$
where ${m_{1}},{m_{2}},{m_{3}}$ are the first, second and third order product
densities (the densities of the factorial moments of the underlying spatial
point process), $\xi_{1}$, and $\xi_{2}$ are vector distances respectively
linking the pairs of particles $(x_{1},x_{2})$ and $(x_{1},x_{3})$ conforming
a triplet configuration. The set $A_{0}$ is a circular domain of area
$|A_{0}|$ that establishes the spatial scale for which triplet correlations
are irreducible, and $J_{0}(A)$ is the avoidance function (i.e. the
probability of observing no points in $A$) of the spatial point process for
the window $A$ [13, 14]. This set is found as the domain of integration that
solves the normalisation condition
$\int_{A_{\epsilon}}{m_{2}}(\xi_{1}^{\prime})\,d\xi_{1}^{\prime}-\frac{1}{3}\int_{A_{\epsilon}\times
A_{\epsilon}}{m_{3}}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}=|A_{\epsilon}|\,{m_{1}}^{2}-\frac{|A_{\epsilon}|^{2}}{3}{m_{1}}^{3}$
(3)
where $A_{\epsilon}$ is a circular domain of radius $\epsilon$ centered at the
origin. The set $A_{0}$ is found by allowing the radius $\epsilon$ to take
positive real values until the equality in (3) holds. This closure is applied
if the three points in the triplet lie inside $A_{0}$, and outside this region
the classical Kirkwood closure applies. If the area of normalisation $A_{0}$
is small, the largest correction is due to the $J_{0}$ term since the integral
correction terms in the numerator and denominator tend to cancel each other,
in which case the maxent closure is simply given by
${m_{3}}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{2}-\xi_{1})}{{m_{1}}^{3}}\exp(-{m_{1}}{|A_{0}|})$
(4)
where the exponential term corresponds to the avoidance function of a Poisson
point process of mean density ${m_{1}}$ normalised with respect to the window
$A_{0}$.
The paper is organized as follows. Section 2 discusses the locally regulated
space-time point process model originally developed by Law & Dieckmann [43],
and includes a Gillespie-type simulation algorithm [28, 59], together with
known definitions and estimators for the product densities and some simulation
results included for illustration purposes only. Broader simulation results
for this process can be found elsewhere [44, 51, 58]. Section 3 reframes the
spatial moment equations of Law _et al_ [44] in terms of product densities.
Section 4 discusses the moment closure for truncation at second order based on
constrained entropy maximisation. Section 5 discusses the numerical
implementation of the closure and compares its predictions against simulations
of the point process for mildly aggregated patterns. Finally, section 6
presents a critique of the maximum entropy method method, and suggests further
areas of development.
## 2 Spatio-temporal point process model
We consider a single population of identical individuals, each of which can
occupy arbitrary locations on a 2-dimensional continuous and bounded spatial
arena $A$. The state of the population for each fixed time $t$ is modeled as a
realisation of a spatial point process, called the configuration or point
pattern [14, 72, 21],
$\varphi_{t}({A})=\left\\{x_{1},\ldots,x_{N_{t}}\right\\},$ (5)
where the $x_{i}$ are the spatial locations of all individuals found within
$A$. Alternatively we have
$N_{t}(A)=\\#\left\\{x_{1},\ldots,x_{N_{t}}\right\\}$
where $N_{t}(A)$ stands for the total population counts within $A$, and the
cardinality operator $\\#$ counts the number of elements in a set. Note that
in (5) both the locations $x_{i}$ and the total counts $N_{t}$ are random
variables. The dynamics of the population is modeled by introducing a time
component, where the updating times are also random variables, subject to
_local_ regulation [11, 14]. Two versions of this model have been introduced
independently by Bolker & Pacala [4] and Dieckmann & Law [19]. Both share the
key ingredients of non-uniform dispersal, and a density-dependent mortality
term that depends on the configuration surrounding the focal individual which
is the mechanism that introduces the local regulation. The configuration (5)
evolves in time by sampling from two exponential distributions of waiting
times that regulate the inter-event times between fecundity/dispersal and
mortality events at the individual level, where the latter is determined from
both intrinsic and density-dependent contributions.
Table 1: Point process model parameters Parameter | symbol | units
---|---|---
fecundity | $b$ | time-1
intrinsic mortality | $d$ | time-1
density-dependent mortality | $d_{N}$ | time-1 indiv-1
non-spatial carrying capacity | $K$ | individuals
dispersal scale | $\sigma_{B}$ | length
competition scale | $\sigma_{W}$ | length
initial population size | $N_{0}$ | individuals
spatial arena | $A$ | length2
### 2.1 Dispersal and fecundity
Per capita waiting times between births are assumed to be exponentially
distributed with constant parameter $b$, the _birth (or fecundity) rate_. If a
birth occurs, the newborn is displaced instantaneously from the location of
its mother $x_{i}$ to a random new location $x_{j}$, sampled from the
probability density, $B(x_{i}-x_{j};\sigma_{B})$ the _dispersal kernel_ ,
where $\sigma_{B}$ is a parameter that measures the characteristic dispersal
length. The index $i$ of the mother is chosen uniformly from the list of
indices $J_{A}=\\{1,2,\ldots,N_{t}(A)\\}$ in the configuration.
### 2.2 Mortality
The probability that a given individual $i$ at location $x_{i}$ dies in the
time interval $(t,t+dt)$ is also assumed to be exponentially distributed with
parameter $m(x_{i})$, the _total per capita mortality rate_ , given by
$m(x_{i})=d+d_{N}\sum_{j\neq i\,\in\,J_{A}}W(|x_{i}-x_{j}|;\sigma_{W}),$ (6)
where $d$, is the _intrinsic_ mortality rate, and $d_{N}$ is the
_density–dependent_ mortality rate. In order to allow comparisons with the
predictions of the logistic model (1) we defined it as $d_{N}=(b-d)/K$, where
$K$ is the non-spatial carrying capacity (the expected value at equilibrium
under complete spatial randomness). This second ‘mortality clock’ is rescaled
by a weighted average of the local configuration around the focal individual,
so that mortality due to competition is more likely to occur in locally dense
regions than in comparatively sparse ones. The contributions of neighbors to
the mortality of $x_{i}$ are assumed to decay monotonically with distance.
This is modeled by a normalized, radially symmetric weighting function
$W(|\xi|\,;\sigma_{W})$, the _mortality kernel_ , that vanishes outside a
finite interaction domain $D_{W}$ , where $\sigma_{W}$ is a parameter
associated with the characteristic length scale of competitive interactions.
This function is interpreted as an average effect that simplifies the details
of the physiology of mortality due to crowding. The parameters of the model
are summarized in Table 1.
### 2.3 Simulation algorithm
A sample path for the space-time point process with rates described in
Sections 2.1 and 2.2 can be simulated by a variant of the Gillespie algorithm
[28, 59]. The spatial arena can be identified with the unit square
$W=[0,1]\times[0,1]$ (after rescaling the parameters in the interaction
kernels), with periodic boundary conditions. The initial population consists
of $N_{0}$ individuals, and $[0,T_{\mbox{max}}]$ is the time interval of
interest.
1. 1.
Generate the configuration at time
$t=0,~{}\varphi_{0}=\\{(x_{1},y_{1});\ldots;(x_{N_{0}},y_{N_{0}})\\}$, from
two independent sets of $N_{0}$ deviates from $U(0,1)$,
$X_{0}=\\{x_{1},\ldots,x_{N_{0}}\\}$ and $Y_{0}=\\{y_{1},\ldots,y_{N_{0}}\\}$.
2. 2.
While the elapsed time $t$ is less than $T_{\mbox{max}}$ do:
1. (a)
Generate a birth waiting time $T_{b}$ from the exponential density with
parameter $b\,N_{t}$, where $N_{t}$ is the number of individuals that are
alive at time $t$.
2. (b)
Generate the set of mortality waiting times
$T_{m}=\\{\tau_{1},\dots,\tau_{N_{t}}\\}$ from a set of exponential densities,
each with parameter ${m(x_{i})}=d+d_{N}\sum_{j\neq i}W(|x_{i}-x_{j}|)$, for
each of the $i=1,\ldots N_{t}$ individuals in the configuration at time $t$
3. (c)
The time until the next event is given by $\tau_{n}=\min\\{T_{b}\cup
T_{m}\\}$.
1. i.
A birth occurs if $\tau_{n}=T_{b}$, in which case the location of the newborn
individual $x_{b}$ is given by
$x_{b}=x_{p}+\xi$
where the index of the parent $p$ is drawn uniformly from the set of indices
$J_{A}$ and the displacement $\xi$ is drawn from the dispersal kernel
$B(\xi)$. The configuration is then updated to include the newborn
$\varphi_{t+T_{b}}\rightarrow\varphi_{t}\cup\\{x_{b}\\}.$
2. ii.
If $\tau_{n}\neq T_{b}$ then the next event is a death in which case the
$i$-th individual in $T_{m}$ for which $\tau_{i}=\tau_{n}$ is removed from the
configuration
$\varphi_{t+\tau_{n}}\rightarrow\varphi_{t}\setminus\\{x_{i}\\}$
4. (d)
Update the elapsed time $t\rightarrow t+\tau_{n}$.
### 2.4 Summary statistics
The specific configurations resulting from simulations of the algorithm in
Section 2.3 are of limited interest. The fundamental question is understanding
how spatial correlations develop from an unstructured initial condition, and
how the equilibrium density departs from the logistic behavior when
considering an ensemble of simulations for various combinations of the spatial
scales of competition and dispersal [4, 43, 19]. This requires a set of
summary statistics capable of distinguishing various forms of spatial
structure for the same population size (see Figure 1). A useful set for this
task are the product densities (or densities of the factorial moments), i.e
the densities of the expected configurations involving one, two or more
_distinct_ points after removing self-configurations [72, 14, 21]. For
spatially stationary point processes, these are functions of the inter–point
distances between the points comprising an expected configuration of a certain
order $k$. The product densities are defined in terms of the population count
$N_{t}(B)$ observed through some window $B$ at time $t$ defined as [72, 11,
14]
$N_{t}(B)=\sum_{x_{i}\in\varphi_{t}}I_{B}\,(x_{i}),$ (7)
where $I_{B}(x)$ is the indicator function of the set $B$ defined by
$\displaystyle I_{B}(x)=\left\\{\begin{array}[]{l}1~{}~{}\mbox{if}~{}x\in
B,\\\ 0~{}~{}\mbox{otherwise.}\end{array}\right.$ (10)
The coarsest is the mean density (or intensity) which measures the expected
number of individuals per unit area at each time, defined as
${m_{1}}(x\,,t)=\lim_{\epsilon\downarrow
0}\frac{{\mathrm{E}}\\{N_{t}(\,S_{\epsilon}(x)\,)\\}}{|S_{\epsilon}(x)|}$ (11)
where $S_{\epsilon}(x)$ is the open ball of radius $\epsilon$ centered around
$x$, and $|A|$ is the area of the window $A$. Since the mortality and
fecundity rates do not depend specific locations but on relative distances,
and both the dispersal or competition kernels are symmetric by definition, the
spatial point process is spatially stationary and isotropic, in which case the
mean density is constant for each fixed time
${m_{1}}(x\,,t)={m_{1}}(t).$
A naïve estimator for the mean density from a single realisation is [72, 21]
$\hat{m}_{1}(t)=\frac{N_{t}(A)}{|A|}$ (12)
Figure 1: The three upper panels show different types of point patterns
sharing the same number of points $N(A)=136$, where the window $A$ is the unit
square. The left panel shows aggregation, the center panel corresponds to
complete spatial randomness and the the right panel displays a segregated
pattern. In the aggregated pattern we see the tendency of points to occur near
each other. By contrast in the regular pattern points tend to avoid each other
at short spatial scales. The lower three panels show estimates of the pair
correlation function $\hat{g}_{2}(r)$ for each of the three point patterns at
the top. The lower left panel indicates aggregation at short scales but
segregation at intermediate ones. In the lower center panel the pair
correlation function oscillates rapidly around one, which signals randomness,
and the lower right panel indicates a tendency to segregation at short scales.
where $N_{t}(A)$ is as in (7). If an ensemble of $\Omega$ independent
replicates of the process is available, this estimate can be improved by
averaging over the ensemble
$\bar{m}_{1}(t)=\frac{\langle N_{t}(A)\rangle_{\Omega}}{|A|}.$ (13)
For a Poisson process, the mean density (11) is a sufficient statistic for the
process. More general cases require keeping track of spatial correlations.
Higher order quantities are required to distinguish between aggregated (or
clustered), random and segregated (or over–dispersed) point patterns with the
same mean density (see Figure 1). For this purpose we need, at the very least,
information about two-point correlations. These are measured by the pair
correlation function, defined as the ratio
$g_{2}(\xi\,;t)=\frac{{m_{2}}(\xi,t)}{{m_{1}}^{2}(t)}$ (14)
which requires knowledge of the density of the expected number of pairs at
spatial lag $\xi$, measured by the second order product density
${m_{2}}(\xi,t)$.
${m_{2}}(\xi\,;t)=\lim_{\epsilon\downarrow
0}\frac{{\mathrm{E}}\left\\{N_{t}(S_{\epsilon}(\mathbf{0})\,)\left[\,N_{t}(S_{\epsilon}(\mathbf{0}+\xi)\,)-\delta_{\mathbf{0}}(S_{\epsilon}(\mathbf{0}+\xi))\,\right]\right\\}}{|S_{\epsilon}(\mathbf{0})|\,|S_{\epsilon}(\mathbf{0}+\xi)|}$
(15)
where $S_{\epsilon}(\mathbf{0})$ and $S_{\epsilon}(\mathbf{0}+\xi)$ are small
windows of observation respectively centered at the origin, and at distance
$\xi$ from the origin. The Dirac measure in the second factor in the numerator
removes the count at zero lag from the second window in order to avoid self-
configurations. In general, the definition (15) centers the count for each
specific location $x$, but given that in our case the process is stationary
and isotropic by construction, it can be translated to the origin without loss
of generality, in which case ${m_{2}}$ depends only on the spatial lag $\xi$.
In the case of a spatially random configuration (a Poisson point process), the
counts on non-overlapping windows are independent of each other and thus the
second order density is simply the square of the mean density. Correlations of
configurations involving $k$ points are simply the $k$-th powers of the mean
density [21, 72]. The pair correlation function (14) is the lowest order
product density that allows detection of departures from complete spatial
randomness. Thus, values of the pair correlation function greater than one for
some lag $\xi$ indicate aggregation at that scale, whereas values below one
signal segregation. Estimation of the pair correlation function requires an
estimator of the squared density [72]
$\bar{m}_{1}^{2}(t)=\frac{\langle\,N_{t}(A)\,[N_{t}(A)-1]\,\rangle_{\Omega}}{|A|^{2}},$
together with a kernel density estimator for the second order product density
[64, 71],
$\widehat{m}^{\,(h)}_{2}(r,t)=\frac{1}{2\pi r}\sum_{i}\sum_{j\neq
i}\frac{k_{h}(r-\|x_{i}-x_{j}\|)}{\,\left|A_{x_{i}}\cap A_{x_{j}}\right|}$
(16)
where $r$ is the spatial lag, $h$ is the bandwidth of the kernel density
estimate $k_{h}$, the points $x_{i}$ belong to a configuration
$\varphi_{t}(A)$ sampled at time $t$, and $\|x_{i}-x_{j}\|$ is the Euclidean
distance between the points $x_{i}$ and $x_{j}$. The denominator is an edge
corrector that rescales the count in the numerator by the area of the
intersection of the window of observation $A_{x_{i}}$ shifted so that its
centered around the point $x_{i}$, with the window $A_{x_{j}}$ shifted around
$x_{j}$ [11, 12, 72]
$A_{x_{i}}=\\{x+x_{i}:x\in A\\}.$
If an ensemble of independent realisations is available, the single
realisation estimator (16) can be improved by means of an ensemble average
$\bar{m}^{\,(h)}_{2}(r,t)=\left<\widehat{m}^{\,(h)}_{2}(r,t)\right>_{\Omega}.$
As before, the angle brackets $\left<\right>_{\Omega}$ represent an average of
the estimates across a number of independent sample paths $\Omega$. For the
smoothing kernel $k_{h}$ a common choice is the Epanechnikov kernel
$k_{h}(s)=\frac{3}{4h}\left(1-\frac{s^{2}}{h^{2}}\right)I_{(-h,h)}(s),$
where $I$ is the indicator function (10). Although empirical methods for
selection of the bandwidth $h$ are widely used, for instance the rule [71]
$h=c/\sqrt{\hat{m}_{1}(t)},\,c\in(0.1,0.2),$
data-driven methods for optimal choices of $h$ based on cross-validation have
been recently introduced [33, 34]. In general, the product density of order
$k$ is defined as [2]
$\displaystyle m_{k}(x_{1},\ldots,x_{k},t)=\lim_{\epsilon\downarrow
0}\,{\mathrm{E}}\left\\{\prod_{j=1}^{k}\frac{\left[N_{t}(S_{\epsilon}(x_{j}))-\sum_{i=1}^{j-1}\delta_{x_{i}}(S_{\epsilon}(x_{j}))\right]}{|S_{\epsilon}(x_{j})|}\right\\},$
(17)
where $\sum_{i=1}^{j-1}\delta_{x_{i}}(S_{\epsilon}(x_{j}))$ removes self
$j$-tuples for $j>i$. In the case of spatial stationarity and isotropy, the
specific locations $x_{1},\ldots,x_{k}$ can be replaced by the relative
distances $\xi_{1},\ldots,\xi_{k-1}$,
$m_{k}(\xi_{1},\ldots,\xi_{k-1},t),$
and the $k$-th correlation function becomes,
$g_{k}(\xi_{1},\ldots,\xi_{k-1};t)=\frac{m_{k}(\xi_{1},\ldots,\xi_{k-1},t)}{m_{1}^{\,k}(t)}$
which is interpreted in a similar way to the pair correlation function, but
considering $k$-plets instead of pairs.
### 2.5 Point process simulation results
For the convenience of the reader, simulation results for the point process
are shown in Figure 2, with the same parameter values as in Law _et al_ [44],
but obtained from code developed independently. The spatial arena is the unit
square, and the kernels are both radially symmetric Gaussians, but the
mortality kernel is truncated (and renormalized) at $3\,\sigma_{W}$. The left
panel shows estimates of the mean density versus time for various values of
the characteristic spatial scales of dispersal and mortality. The right panel
shows the pair correlation function at the end of the simulation for each of
the four spatial regimes for which the population persists. Both quantities
were estimated from an ensemble of 300 independent sample paths.
Case (b) in both panels corresponds to dispersal and mortality kernels with
large characteristic spatial scales ($\sigma_{B}=0.12,\,\sigma_{W}=0.12$). In
this situation there is enough mixing to destroy spatial correlations
—confirmed by the almost constant pair correlation function— and the mean
density equilibrates at a value that is very close to the non-spatial carrying
capacity ($K=200$). Case (a) shows results for a segregated (or regular)
spatial pattern that arises from very local competitive interactions, but long
range scales of dispersal ($\sigma_{B}=0.12,\,\sigma_{W}=0.02$). In this
situation local densities experienced by the focal individual are lower than
the random case (the pair correlation function is below one), which results in
equlibrium densities that equilibrate at higher values than the non– spatial
carrying capacity. This results from the ability of newborns to escape locally
crowded regions via the long range dispersal kernel. Case (c) is associated to
a segregated pattern of clusters, which is the converse situation of the
segregated pattern with very localized dispersal, and mild competition
distributed over a longer range ($\sigma_{B}=0.02,\,\sigma_{W}=0.12$). The
oscillations of the pair correlation function indicate two scales of pattern.
There is short scale aggregation, but the clusters themselves form a
segregated pattern with respect to each other, so the local crowding due to
clustering that should lead to high density-dependent mortality is compensated
by the overdispersion. Overall, the local competitive neighborhood experienced
by an individual in this situation is more crowded than in a random
distribution of points, which results in a mean density that equilibrates at
lower values than the non-spatial carrying capacity. Case (d) corresponds to a
mildly aggregated pattern ($\sigma_{B}=0.04,\,\sigma_{W}=0.04$), where there
is a single scale of aggregation. Even for small departures from complete
spatial randomness such as this one, the effect of the spatial pattern in the
dynamics of the mean density is substantial, since we see a reduction of about
$30\%$ in the equilibrium density in this case with respect to that of
complete spatial randomness. Finally, case (e) indicates an extreme case of
aggregation, with very intense, local mortality and dispersal
($\sigma_{B}=0.02,\,\sigma_{W}=0.02$), where the population goes to extinction
(exponentially) after a short growth transient.
Figure 2: The left panel shows estimates for the mean density $\bar{m}_{1}(t)$
from an ensemble of $\Omega=300$ realisations, for various characteristic
spatial scales of dispersal and density-dependent mortality. The dotted lines
are the envelopes for one standard deviation. The right panel shows the
corresponding estimates for the pair correlation function
$\bar{g}_{2}^{\ast}(r)$ at the end of the simulation. The other parameters,
$b=0.4,\,d=0.2,\,K=200$, are fixed for all cases. The spatial arena is the
unit square with periodic boundaries.
## 3 Moment equations and the closure problem
The central problem associated with the space-time point process described
earlier in Section 2.3 is to obtain a closed form expression for the finite
dimensional distributions,
${\mathbb{P}}_{k}\left\\{A_{1},\ldots,A_{k},n_{1},\ldots,n_{k};t\right\\},$
(18)
that determine the probability of observing $n_{1}$ points in the window
$A_{1}$, $n_{2}$ points in the window $A_{2}$, and so forth up to the $n_{k}$
points in $A_{k}$ at time $t$, from the definition of the space-time point
process discussed in the previous section. Unfortunately, this seems to be
remarkably difficult, due to the presence of the non-linearity in the
mortality rate in (6), and the localized nature of dispersal [24]. However,
the question of ecological interest is understanding the modifications that
should be introduced to the logistic equation (1) in order to account for the
effects of spatial correlations in the dynamics of the mean density. This can
be accomplished by deriving evolution equations for the product densities
(which are the densities of the factorial moments of (18)) from the transition
rates of the point process discussed in the previous Section. Following a
Master equation approach similar to that used by Bolker & Pacala [4] and
Dieckmann & Law [19], we derive the following hierarchy of product density
equations (see Appendix A). The first member in this hierarchy corresponds to
the modified or ‘spatial’ logistic equation [49],
$\displaystyle\frac{d}{dt}{m_{1}}(t)=r\,{m_{1}}(t)-d_{N}\int_{{\mathbb{R}}^{2}}W(\xi_{1})\,{m_{2}}(\xi_{1}^{\prime},t)\,d\xi_{1}^{\prime}.$
(19)
where $r=b-d$, $d_{N}=(b-d)/K_{s}$ and $W(\xi_{1})$ is the mortality kernel in
(6). $K_{s}$ is the spatial carrying capacity, or the number of individuals
per unit area that can be supported under random mixing
$K_{s}=\frac{K}{|A|}.$
Equation (19) shows that the required modification of the logistic equation
consists of substituting the quadratic term with an average of the second
order product density ${m_{2}}(\xi_{1},t)$ weighted by the mortality kernel
$W(\xi_{1})$. This term computes the effective number of neighbors
$n_{\mbox{eff}}$ that contribute to density–dependent mortality,
$n_{\mbox{eff}}\,(t)=\int_{{\mathbb{R}}^{2}}W(\xi_{1})\,{m_{2}}(\xi_{1}^{\prime},t)\,d\xi_{1}^{\prime}.$
Thus, the effect of mortality on the evolution of the mean density is tied to
a weighted average of the mortality kernel with the two-point spatial
correlations in the process. Equation (19) reduces to the logistic equation
for the Poisson point process, in which case ${m_{2}}(\xi_{1})={m_{1}}^{2}$.
In aggregated spatial patterns, ${m_{2}}$ exceeds ${m_{1}}^{2}$ for some
domain. If mortality is modeled by a kernel that penalizes close proximity
over the same range of scales where aggregation is detected, then the effect
of mortality due to competition is stronger in this case than that of the
logistic equation, in which case the density equilibrates below $K_{s}$
(Figure 2, cases (c),(d) and (e) ). The opposite situation occurs in
segregated patterns, where ${m_{2}}$ is less than ${m_{1}}^{2}$ at the scales
where the mortality kernel penalizes aggregation. As a result, the effect of
competition on mortality is milder than in a random spatial pattern, in which
case the mean density equilibrates at values greater than $K_{s}$ (Figure 2,
case (a)). Equation (19) depends on the unknown second order density
${m_{2}}$. A similar procedure to that used in the derivation of (19) one
obtains the evolution equation for this quantity
$\displaystyle\frac{1}{2}\,\frac{d}{dt}{m_{2}}(\xi_{1},t)$ $\displaystyle=$
$\displaystyle
b\int_{{\mathbb{R}}^{2}}B(\xi_{2})\,{m_{2}}(\xi_{1}-\xi_{2},t)\,d\xi_{2}+b\,B(\xi_{1})\,{m_{1}}(t)-d\,{m_{2}}(\xi_{1},t)$
(20) $\displaystyle-$ $\displaystyle
d_{N}W(\xi_{1})\,{m_{2}}(\xi_{1},t)-d_{N}\int_{{\mathbb{R}}^{2}}W(\xi_{2})\,{m_{3}}(\xi_{1},\xi_{2},t)\,d\xi_{2}.$
Figure 3: Schematic representation of a spatially stationary triplet
configuration. The pair densities are evaluated at each inter-event
(vectorial) distances $\xi_{1}$, $\xi_{2}$ and $\xi_{1}-\xi_{2}$
Here the role of dispersal and competition kernels as the main pattern drivers
can be clearly discerned [4, 7, 19, 44]. The first two terms in (20), related
to fecundity and dispersal, are
$b\int_{{\mathbb{R}}^{2}}B(\xi_{2})\,{m_{2}}(\xi_{1}-\xi_{2};t)\,d\xi_{2}+b\,B(\xi_{1})\,{m_{1}}(t).$
Both are nonnegative by definition for all values of $\xi_{1}$ and $t$. The
rate of change of ${m_{2}}$ increases due to their effect, and thus they drive
aggregation at the scales controlled by the characteristic spatial scale of
the dispersal kernel. The convolution measures the creation of pairs along
$\xi_{1}$ due to dispersal of the third member of the triplet along the
$\xi_{1}-\xi_{2}$ edge (Figure 3). The second term measures the creation of
pairs along the $\xi_{1}$ edge due to the dispersal events generated the
individual at the origin of $\xi_{1}$. The remaining terms due to mortality
are,
$-d\,{m_{2}}(\xi_{1},t)-d_{N}W(\xi_{1})\,{m_{2}}(\xi_{1},t)-d_{N}\int_{{\mathbb{R}}^{2}}W(\xi_{2})\,{m_{3}}(\xi_{1},\xi_{2};t)\,d\xi_{2}.$
All the three terms are negative, and thus contribute to the destruction of
pairs along the $\xi_{1}$ edge, leading to segregated patterns. The first term
measures intrinsic mortality of both members of the pair and the remaining
ones are related to density–dependent mortality. The second, measures
mortality of the pairs due to competition at the scales controlled by the
mortality kernel. The last term measures the destruction of the pair along the
$\xi_{1}$ edge due to the effect of competition with the additional member of
the triplet located along the $\xi_{2}$ edge.
These terms for both dispersal and mortality are initially calculated by
fixing the count at the origin of $\xi_{1}$ and let the count at the end of
$\xi_{1}$ vary according to the fecundity, dispersal and mortality terms.
Symmetry considerations require consideration of the reverse situation, where
the count at the end of $\xi_{1}$ is fixed, and the origin is allowed to vary.
Since these are symmetric, these additional terms lead to the factor of $1/2$
on the left hand side of the equation for the second order product density.
## 4 Moment closure by Shannon entropy maximisation
The product density equations (19) and (20) cannot be solved in that form
because the evolution equation for the second order density has a mortality
term that depends on a weighted average of the third order one. Although it is
possible to derive an additional evolution equation for this quantity, it will
involve an unknown fourth order term, leading to a system that is not closed.
In general, the evolution equation for the density of order $k$ will depend on
the density of order $k+1$. This gives rise to two problems, known together as
‘a moment closure’ [4, 43]. The first is choosing an appropriate order of
truncation $k$, and the second is finding an expression for the product
density of order $k+1$ in terms of the densities of orders up to $k$ (or $k+1$
in the case of an implicit closure).
Ideally, the order of the truncation should be based on an understanding of
the convergence properties of the hierarchy in order to establish error
bounds. In practice, the order of the truncation is determined by the
computational cost of the numerical solution, which is determined by the size
of the arrays that can be stored and operated on efficiently. Explicit
representation of third order terms already requires least $3.2$ Gb of memory
using double precision and a relatively coarse discretisation of 100 grid
points per dimension. This situation pretty much constrains to three the
highest order density that can be represented explicitly.
From an applied perspective, the first and second order terms are of greatest
interest, since these respectively encode the dynamics of the average density
and the spatial covariance. The latter can be interpreted biologically as the
average environment experienced by an individual as a function of spatial
scale [43, 44]. The shape of the second order correlation function can be used
to distinguish between aggregated, random and segregated spatial patterns
sharing the same average density (see Section 2.4).
Closure problems are pervasive in the statistical mechanics of fluids where
thermodynamic quantities are derived from the statistical properties of the
particle distributions [69][60, 32, 65, 47, 41]. Here our intent is somewhat
similar in the sense that a detailed individual-based model is used to inform
a mean-field model that does not neglect the role of spatial fluctuations in
density due to endogenously generated spatial structure structure [4, 5, 44].
Within spatial ecology, moment closures have been proposed with varying
degrees of success, using a suite of methods, among which we have:
* •
_Heuristic reasoning_ , where consistency arguments are used to construct
closing relationships [44, 19, 51, 4].
* •
_Distributional properties_ , where closures are based on assuming a
functional form for the distribution of the process [42].
* •
_Variational_ methods, where it is assumed that the unknown distribution
optimizes some meaningful functional, usually an entropy–like object [35, 69]
In order to make the paper reasonably self-contained, we shall briefly review
closures based on heuristic reasoning, which have dominated work in this
problem. Additional information can be found in a recent review by Murrell _et
al_ [51].
### 4.1 Heuristic methods of moment closure
Heuristic closures are usually based on self–consistency arguments. For
instance, they should be strictly positive and invariant under permutations of
the arguments [21, 11, 14]. Also, if correlations are assumed to decay
monotonically with distance, then there is a distance $d$ beyond which the
particles become uncorrelated and thus higher order densities become simple
powers of the mean density. Although a large number of functional forms can be
chosen in order to satisfy these minimum requirements, the simplest ones
usually involve additive combinations of various powers of the second and
first moments. For instance, if one further assumes that central third moments
vanish, the resulting expansion in terms of product densities, leads to the
_power–1_ closure, dubbed that way because the highest occurring power of the
second order density is one [4, 5, 7, 19],
Figure 4: Closure comparison. Panel (a) shows the mean density
${\hat{m}}_{1}(t)$ of the point process versus time averaged over 300 sample
paths (blue) up to a simulation of 300 time units. The continuous black line
shows the predicted mean density from the moment equations with the power–3 or
Kirkwood closure, the dashed black line corresponds to the power 2 closure.
The dash-dot line corresponds two the power 1 closure. Panel (b) shows the
pair correlation function at time $t=300$ (blue), indicating aggregation at
short scales, but segregation at intermediate ones. The black line corresponds
to the pair correlation function predicted by the solution of the moment
hierarchy with the power–3 closure, and the dashed line corresponds to the
power 2.
${m_{3}}(\xi_{1},\xi_{2})={m_{1}}\,{m_{2}}(\xi_{1})+{m_{1}}\,{m_{2}}(\xi_{2})+{m_{1}}\,{m_{2}}(\xi_{1}-\xi_{2})-2\,{m_{1}}^{3}.$
(21)
This closure has the attractive property of preserving the linearity of the
moment hierarchy, which allows the derivation of analytical results at
equlibrium [4, 5]. It is quite successful at low densities
(${m_{1}}^{\ast}\sim 20$) and 1–dimensional systems. However, at intermediate
to high densities (${m_{1}}\sim>100$) aggregated patterns, this closure
predicts extinction in situations where the point process persists (see dash-
dot line in panel (a) in Figure 4), even for mild correlation regimes. It is
nonetheless a useful benchmark result.
The _power–2_ closure is obtained as a continuous space analogue to the pair
approximation used in discrete spatial systems [61],
${m_{3}}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})}{{m_{1}}}+\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{1}-\xi_{2})}{{m_{1}}}+\frac{{m_{2}}(\xi_{1}-\xi_{2})\,{m_{2}}(\xi_{2})}{{m_{1}}}-2\,{m_{1}}^{3};$
(22)
this closure does predict a persisting population. However, it underestimates
quite strongly the second order density, which leads to overshooting the mean
density (see panel (b) in Figure 4, dashed black line). It is non-linear and
thus solutions have to be obtained numerically. There are _asymmetric_
versions of this closure that consist of rescaling each additive term in (22)
with a set of weighting constants [44, 51]. Law _et al_ [43] showed that a
particular combination of weighting constants provides a very good fit to
simulations. However, this result is difficult to generalize as there is no
theory informing how these constants are chosen, since they depend on the
details of the model [51], and can only be found by comparisons with
simulations of the IBM.
Finally, the _power–3_ or Kirkwood closure (24) has a distinguished tradition
in the statistical mechanics of fluids [41, 41]. Recently, Singer [69] showed
that this closure can be obtained in the hydrodynamic limit after invoking a
maximum entropy principle to truncate the BBGKY hierarchy. Earlier motivations
for this closure were based on the assumption that each of the pair
correlation functions associated with the three edges of the triplet
configuration (see Fig. 3) occurs independently of each other _at all spatial
scales_ ,
$g_{3}(x_{1},x_{2},x_{3})=g_{2}(x_{1},x_{2})\,g_{2}(x_{1},x_{3})\,g_{2}(x_{2},x_{3}).$
(23)
Substituting the definition of the $k$-th correlation function in terms of the
product densities (2.4) into (23) for $k=3$ yields a version of the Kirkwood
closure (23) that can be used to close the equation at second order (20)
${m_{3}}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{1}-\xi_{2})}{{m_{1}}^{3}}.$
(24)
This closure also underestimates the second order density, but less
dramatically so than the power–2 closure, which results in a slightly better
prediction of the mean density (see panel (b) in Figure 4). Despite its
appealing simplicity, the power–3 closure shares the same limitations of the
other heuristic closures, e.g. there is no criterion of validity, and it
provides poor fit to the equilibrium density even for mildly aggregated
patterns [58] [19]. Heuristic closures have reasonably good performance in
random and segregated spatial configurations, but are significantly more
limited in aggregated regimes, with the sole exception of the asymmetric
power-2. Their limitation arises from the implicit assumption that there are
no irreducible triplet correlations at any scale, in the sense that after
fixing a pair that forms an edge, for instance the points $x_{1}$ and $x_{2}$
(see Fig 3), the two other edges of the triplet formed with the third point
$x_{3}$ occur independently of how the first edge is chosen. This can only be
true when the three points are sufficiently far apart, but irreducible third
order correlations are likely to occur when the three points are close
together in aggregated patterns (Figure 6).
### 4.2 The Maxent closure
The concept of entropy from an information theoretic point of view, as opposed
to the thermodynamical definition of entropy, is tightly related to the
uncertainty (or information content) associated with an outcome of a random
variable. It can be shown that the information content of a particular outcome
$(x^{\prime}+dx^{\prime})$ of random variable $x$ with probability density
$p(x)$, is given by $\log[p(x^{\prime})dx^{\prime}]$[66, 40]. The entropy
functional is constructed by taking the expected value of the information
content over all the possible outcomes of $x$ [66, 38, 40]. To illustrate what
this means, consider the uniform distribution on an interval
$[a,b]\in{\mathbb{R}}^{+}$. It is not surprising that this distribution
maximizes the entropy functional if no constraints are introduced, since all
the values in its domain of definition have the same probability weight, thus
the uncertainty about a specific outcome of a random variable with this
distribution is maximal. The opposite situation occurs for the Dirac delta
distribution which is centered on one single value, say $x^{\prime}$. In this
situation, a single value occurs with probability one, and all the others have
probability zero, therefore the uncertainty about an outcome of this
(pathological) random variable is null.
The principle of maximum entropy is a powerful method that allows the
derivation of probability distributions when only but a few average properties
are all that is known. Maximizing the entropy functional subject to the
constraints provided by these averages, leads to probability distributions
that have the least bias with respect to the known information [38, 39, 66,
40]. For instance, maximisation of the entropy constrained to satisfy
normalisation and a given mean value leads to the exponential density.
Likewise, maximizing the entropy constrained to satisfy normalisation for a
given mean and variance leads to the Gaussian density. For point processes
[46, 16] the entropy is defined with respect to some spatial window of
observation $A$, and has two sources of uncertainty, the first is related to
the _counts_ within $A$, and the second is related to the locations of the $n$
points inside this window. Truncating the hierarchy at order two assumes that
only configurations involving up to three points possess irreducible _spatial_
information. We carry that assumption forward onto the locational component of
the full point process entropy functional, which we then maximise subject to
the constraints of normalisation and product densities up to order two, which
are given by the truncated hierarchy. We exploit formal relationships between
the product densities and the probabilistic objects used to construct the
entropy functional of a point process —the _Janossy_ densities— that allow the
incorporation of the product density constraints onto the entropy functional,
and then translate the results of the maximisation procedure in terms of
product densities in order to obtain a closure expression.
Our result differs from other maxent closures, like those of Singer [69] and
Hillen [35], in a number of ways. First, it is _implicit_ , in the sense that
the _third_ order density appears in both sides of the closing expression for
truncation at second order. We do so because the Kirkwood closure arises
naturally from independence considerations [69] for spatial scales larger than
the minimum distance for which the pair correlation function is not constant,
but it is not valid within the domain of irreducible triplet correlations,
i.e. the probability of observing a third point in the triplet depends on how
the first two are chosen. If improvements to the Kirkwood closure are to be
made, irreducible triplet correlations must appear in the closure. In the
maxent method we propose irreducible third order correlations are generated by
iteration of the closure relationship, while the first and second order
densities, generated by the hierarchy, are held fixed. Second, we assume that
these irreducible third order correlations are confined to a finite window, or
spatial scale $A_{0}$, which is found by comparison of the normalisation
condition for the correlated process with that of a Poisson process of the
same mean density. Third, in contrast to other existing approaches, we used
all the moments up to the order of the truncation (including the zeroth) to
constrain the entropy functional. This is critically important because the
zero-th moment is associated with the normalisation constraint, which allows
the determination of the domain of triplet correlations.
The variational problem is formulated in terms of the locational entropy
functional of the marginal spatial point process. In order to introduce the
product densities as constraints, we exploit known expansions of these in
terms of the Janossy densities [14, 37] that constitute the probabilistic
objects (the likelihoods) required to construct the entropy functional.
Whereas Singer [69] used the $k$-th order product density to constrain an
entropy functional, and Hillen [35], used an $L^{2}$-norm of the moment
hierarchy for this purpose, we used instead the classical definition of the
entropy functional for a point process, based on the full battery of Janossy
densities [46, 16].
The implicit, order two maxent closure (1) resembles the structure of the
power–3 or Kirkwood closure (24), but is complemented by a number of
correction terms that depend on averages of the product densities for each
scale at which triplets are irreducible. Outside this domain, these correction
terms vanish and the closure becomes identical to the power–3. There are two
scales of relevance in the closure, one where irreducible triplet correlations
are important, and another one where these can be expressed in terms of second
and first orders only.
For the sake of completeness, we first discuss known results related to the
entropy of spatial point processes in subsection 4.3, and the key expansions
of Janossy densities in terms of product densities. This is followed by the
derivation of the implicit maxent closure for truncation at order two (4.4).
### 4.3 The entropy of a point process
The Shannon (or information) entropy $H[{\mathcal{P}}]$ of a stochastic
process ${\mathcal{P}}$, interpreted as the average uncertainty (or
information content) associated with a given outcome of ${\mathcal{P}}$, is
defined as minus the expected value of the log-likelihood $L$ [14, 16, 38, 39,
40, 66],
$H[{\mathcal{P}}]=-{\mathrm{E}}\left\\{\log(L)\right\\}.$ (25)
The specialisation of the entropy (25) to point processes requires a special
form of the likelihood, given that in a realisation of a point process of the
form $\\{x_{1},\ldots,x_{n}\\}$ in a window $A$ there are two sources of
uncertainty. The first comes from uncertainty about the number of points $n$
within $A$ (the counts), which is controlled by an integer-valued probability
distribution $p_{n}=\Pr\\{N(A)=n\\}$. Conditionally on the value of $n$, the
other contribution comes from the uncertainty associated with the _locations_
of the $n$ points, which is given by a symmetric (in the sense of invariance
under permutations of the indices) probability density
$s_{n}(x_{1},\ldots,x_{n}|A)$ on $A^{(n)}$. Thus, the likelihood of a spatial
point process is the probability of finding $n$ points within $A$, each in one
of the infinitesimal locations $dx_{1},\ldots,dx_{n}$ and nowhere else within
$A$. This coincides with the definition of the local Janossy density [14, 16,
37]
$L_{A}(x_{1},\ldots,x_{n})=p_{n}s_{n}(x_{1},\ldots,x_{n}|A)=j_{n}(x_{1},\ldots,x_{n}|A).$
(26)
Separating the contributions due to the counts and those due to spatial
information, we can represent the entropy of a point process
${\mathcal{N}}_{A}$ on a window $A$ by [14, 16]
$H[{\mathcal{N}}_{A}]=-\sum_{r=0}^{\infty}p_{r}\log(r!p_{r})-\sum_{r=1}^{\infty}p_{r}\int_{A^{(r)}}s_{r}(x_{1},\ldots
x_{r})\,\log[s_{r}(x_{1},\ldots x_{r})]\,dx_{1}\cdots dx_{r},$ (27)
where the integrals calculate the contribution due to the locations, an the
sums that of the counts. If we fix the expected number of points in
$A,~{}\mu={m_{1}}\,|A|={\mathrm{E}}[N(A)]$, it can be shown that the first sum
in (27) is maximized by the Poisson distribution [16, 40, 46],
$p_{r}=\frac{\mu^{r}}{r!}\exp(-\mu).$
Conditional on the counts $r$, the second sum is maximized by the uniform
density on $A^{(r)}$
$s_{r}\equiv\frac{1}{|A|^{r}}.$
Thus, the point process of maximum entropy is the homogeneous Poisson point
process with first order density ${m_{1}}$ [15, 16]. For closure purposes we
use the definition (25) written in terms of the local Janossy densities
$H[{\mathcal{N}}_{A}]=-\sum_{n=0}^{\infty}\frac{1}{n!}\int_{A^{(n)}}j_{n}(x_{1},\ldots,x_{n}|A)\,\log[j_{n}(x_{1},\ldots,x_{n}|A)]\,dx_{1}\cdots
dx_{n},$ (28)
where division by $n!$ ensures normalisation with respect to the $n!$
permutations of the $n$ indices. Our method of closure consists of maximizing
(28) constrained to satisfy the product densities up to the order of
truncation. These can only be meaningfully incorporated as constraints if they
can be expressed in terms of integrals over $A$ of the Janossy densities. We
do this by using the expansion [14],
$m_{n}({x_{1},\ldots,x_{n}})=\sum_{q=0}^{\infty}\frac{1}{q!}\int_{A^{(q)}}j_{q+n}({x_{1},\ldots,x_{q}},y_{1},\ldots,y_{n})\,dy_{1}\dots
dy_{n},$ (29)
where the inverse relationship,
$j_{n}({x_{1},\ldots,x_{n}}\,|A)=\sum_{q=\,0}^{\infty}\frac{(-1)^{q}}{q!}\int_{A^{(q)}}m_{n+q}({x_{1},\ldots,x_{n}},y_{1},\ldots,y_{q})\,dy_{1}\dots
dy_{q},$ (30)
can be used to translate the results of the constrained optimisation procedure
in terms of product densities in order to yield a closure for the product
density hierarchy.
### 4.4 Maximum entropy closure at order $k=2$
In the case of the non-homogeneous Poisson point process, which maximizes the
entropy functional (28), all the points can in principle depend on the
specific locations, but these are uncorrelated. For this special case the
expansion of the likelihoods in terms of the product densities (30) takes the
simplified form,
$j_{n}({x_{1},\ldots,x_{n}}\,|A)=\prod_{p=1}^{n}{m_{1}}(x_{p})\sum_{q=\,0}^{\infty}\frac{(-1)^{q}}{q!}\prod_{l=0}^{q}m_{1}(y_{l})|A|^{l}.$
(31)
If the process is a spatially stationary and homogeneous Poisson point
process, then all the product densities become simple powers of the mean
density [21, 14], which further simplifies (30) to,
$j_{n}({x_{1},\ldots,x_{n}}\,|A)={m_{1}}^{n}\,\exp(-{m_{1}}|A|).$ (32)
Thus the probability of observing $n$ points within a window $A$ is
$\Pr\left[N(A)=n\right]=\frac{1}{n!}\int_{A^{(n)}}j_{n}({x_{1},\ldots,x_{n}}\,|A)\,dx_{1}\cdots
dx_{n},$ (33)
which after substituting (32) into (33) leads to the Poisson distribution
$\Pr\left[N(A)=n\right]=\frac{(m_{1}|A|)^{n}\,\exp(-m_{1}\,|A|)}{n!}.$
Figure 5: Estimated radial pair correlation functions at equilibrium
$\hat{g}_{2}^{\ast}(r)$ from simulations of the point process in Section 2.3
with dispersal and mortality kernels given by symmetric bivariate Gaussians.
Parameters lead to a mildly aggregated pattern (case b, dashed line) and a
segregated pattern of clusters (case a, continuous line). In (b) we note that
correlations decay quickly and become constant at a spatial lag $r>0.2$,
whereas in (a) there are distinct patterns in at least two spatial scales.
Aggregation in the smaller ones, and segregation at intermediate ones.
We assume somewhat crudely that the Janossy expansions of the point process
associated with the moment hierarchy have an intermediate structure between
the two extreme cases (30) where the spatial configurations of all orders are
irreducible, and the Poisson point process (32) where all the locations occur
independently. This assumption can be justified from the truncation
assumption, since truncating the hierarchy at order two implicitly assumes
that terms of order equal or higher than four do not contribute to the
formation of second and third order spatial correlations. Also we see in
estimates of the pair correlation functions for the point process discussed in
Section 2, shown in Figure (5) that there is a region in the parameters for
which the spatial correlations of second order decay quickly. Case (a)
corresponds to segregated clusters and thus the pair correlation oscillates
around one. There are two different scales with pattern there. One associated
with the clusters (the region where $g_{2}>1$) and another with the separation
between the clusters themselves ($g_{2}<1$). Case (b) on the other hand
corresponds to a simply aggregated pattern. In this latter case we see clearly
that there is a spatial scale for which the pair correlation function becomes
constant and identical to one, therefore
$m_{2}(r)=m_{1}^{2},~{}~{}~{}r\gg r_{0}$
for some spatial scale $r_{0}$. This assumption is tantamount to requiring
that the Janossy expansions of the process to have the form,
$\displaystyle{j_{n}(x_{1},\ldots,x_{n}|A)}$ $\displaystyle=$
$\displaystyle\sum_{q=\,0}^{k+1-n}\frac{(-1)^{q}}{q!}\int_{A^{(n)}}m_{n+q}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{q})\,dy_{1}\dots
dy_{q}$ (34) $\displaystyle+$
$\displaystyle\prod_{p=1}^{n}{m_{1}}(x_{p})\sum_{q>\,k+1-n}^{\infty}\frac{(-1)^{q}}{q!}\int_{A^{(q)}}\prod_{r=1}^{q}{m_{1}}(y_{r})\,dy_{r},$
where the first term corresponds to the terms that make contributions due to
spatial correlations, and the second term is the (non-homogeneous) Poisson
remainder. For $k=2$, equation (35) becomes
$\displaystyle{j_{n}(x_{1},\ldots,x_{n}|A)}$ $\displaystyle=$
$\displaystyle\sum_{q=\,0}^{3-n}\frac{(-1)^{q}}{q!}\int_{A^{(n)}}m_{n+q}(x_{1},\ldots,x_{n},y_{1},\ldots,y_{q})\,dy_{1}\dots
dy_{q}$ (35) $\displaystyle+$
$\displaystyle\prod_{p=1}^{n}{m_{1}}(x_{p})\sum_{q>\,3-n}^{\infty}\frac{(-1)^{q}}{q!}\int_{A^{(q)}}\prod_{r=1}^{q}{m_{1}}(y_{r})\,dy_{r}.$
The closure assumption implies that only the Janossy densities of order up to
$k+1$ make contributions to the _locational_ entropy, in which case the
entropy functional (28) becomes
$\displaystyle H^{(3)}_{loc}[{\mathcal{N}}_{A}]$ $\displaystyle=$
$\displaystyle-J_{0}(A)\log[J_{0}(A)]-\sum_{n=1}^{3}\sum_{1\leq
i_{1}<\dots\leq i_{n}\leq 3}\frac{(3-n)!}{3!}$ $\displaystyle\times$
$\displaystyle\int_{A^{(n)}}j_{n}(x_{i_{1}},\ldots,x_{i_{n}}|A)\,\log[j_{n}(x_{i_{1}},\ldots,x_{i_{n}}|A)]\,dx_{i_{1}}\cdots
dx_{i_{n}}$
where $J_{0}(A)$ is the avoidance probability in $A$. The first constraint
added to (4.4) is that of normalisation,
$1=\sum_{n=0}^{\infty}\frac{1}{n!}\int_{A^{(n)}}j_{n}({x_{1},\ldots,x_{n}})\,dx_{1}\cdots
dx_{n},$
which after simplification with the assumption (35) can be added to the
entropy functional
$\displaystyle+\Lambda_{0}\cdot\left(J_{0}(A)+\sum_{q=1}^{3}\sum_{1\leq
i_{1}<\dots\leq i_{q}\leq
3}\frac{(3-q)!}{3!}\int_{A^{(q)}}j_{q}(x_{i_{1}},\ldots,x_{i_{q}}\,|A)\,dx_{i_{1}}\dots
dx_{i_{q}}\right.$
$\displaystyle\left.+\sum_{n>3}^{\infty}\prod_{i=1}^{n}{m_{1}}(x_{i})\sum_{l>\,3-n}^{\infty}\frac{(-1)^{l}}{l!}\prod_{r=1}^{l}\int_{A^{(r)}}{m_{1}}(y_{r})\,dy_{r}-1\right)$
where $\Lambda_{0}$ is a (constant) Lagrange multiplier. The second constraint
is that of the first order product density ${m_{1}}(x_{i})$
$\displaystyle+$ $\displaystyle\sum_{1\leq i_{1}\leq
3}\frac{1}{3}\int_{A}\Lambda_{1}(x_{i_{1}})\left(\sum_{q=0}^{2}\sum_{1\leq
i_{1}<\dots\leq i_{q}\leq 3}\frac{(3-q)!}{3!}\right.$ (37)
$\displaystyle\times$
$\displaystyle\int_{A^{(q)}}j_{1+q}(x_{i_{1}},\ldots,x_{i_{n}},y_{i_{1}},\ldots,y_{i_{q}}\,|A)dy_{i_{1}}\dots
dy_{i_{q}}$ $\displaystyle-$
$\displaystyle\left.{m_{1}}(x_{i_{1}})\frac{}{}\right)dx_{i_{1}}.$
where $\Lambda_{1}(x_{i_{1}})$ is a vector of functional Lagrange multipliers,
each associated with the permutations in the locations $x_{1},x_{2}$ and
$x_{3}$ comprising the triplet. Finally, the constraint for the second order
product density ${m_{2}}(x_{i_{1}},x_{i_{2}})$ is
$\displaystyle+$ $\displaystyle\sum_{1\leq i_{1}<i_{2}\leq
3}\frac{1}{6!}\int_{A^{(2)}}\Lambda_{2}(x_{i_{1}},x_{i_{2}})\left(\sum_{q=0}^{1}\sum_{1\leq
i_{1}<\dots\leq i_{q}\leq 3}\right.$ (38)
$\displaystyle\frac{(3-q)!}{3!}\int_{A^{(q)}}j_{2+q}(x_{i_{1}},\ldots,x_{i_{n}},y_{i_{1}},\ldots,y_{i_{q}}\,|A)dy_{i_{1}}\dots
dy_{i_{q}}$ $\displaystyle-$
$\displaystyle\left.{m_{2}}(x_{i_{1}},x_{i_{2}})\frac{}{}\right)\,dx_{i_{1}}\,dx_{i_{2}}.$
Likewise, the $\Lambda_{2}(x_{i_{1}},x_{i_{2}})$ are the Lagrange multipliers
associated with each of the permutations of the pairs in the triplet. The
Euler–Lagrange equations of the functional (4.4)–(38) are
$\displaystyle\frac{\delta H^{(3)}}{\delta J_{0}(A)}=$ $\displaystyle-$
$\displaystyle 1-\log[J_{0}(A)]+\Lambda_{0}=0,$ $\displaystyle\frac{\delta
H^{(3)}}{\delta j_{1}(x_{i_{1}})}=$ $\displaystyle-$
$\displaystyle\frac{1}{3}(1+\log
j_{1}\left[(x_{i_{1}})\right])+\frac{1}{3}\Lambda_{0}+\frac{1}{3}\Lambda_{1}(x_{i_{1}})=0,~{}~{}~{}~{}~{}~{}~{}~{}1\leq
i_{1}\leq 3$ $\displaystyle\frac{\delta H^{(3)}}{\delta
j_{2}(x_{i_{1}},x_{i_{2}})}=$ $\displaystyle-$
$\displaystyle\frac{1}{6}(1+\log\left[j_{2}(x_{i_{1}},x_{i_{2}})\right])+\frac{1}{6}\Lambda_{0}+\frac{1}{3}\Lambda_{1}(x_{i_{1}})+\frac{1}{6}\Lambda_{2}(x_{i_{1}},x_{i_{2}})=0,~{}~{}1\leq
i_{1}\leq i_{2}\leq 3$ $\displaystyle\frac{\delta H^{(3)}}{\delta
j_{3}(x_{1},x_{2},x_{3})}=$ $\displaystyle-$
$\displaystyle\frac{1}{6}(1+\log\left[j_{3}(x_{1},x_{2},x_{3})\right])+\frac{1}{6}\Lambda_{0}+\frac{1}{2}\left[\Lambda_{1}(x_{1})+\Lambda_{1}(x_{2})\right.$
(39) $\displaystyle+$
$\displaystyle\left.\Lambda_{1}(x_{3})\right]+\frac{1}{2}\left[\Lambda_{2}(x_{1},x_{2})+\Lambda_{2}(x_{2},x_{3})+\Lambda_{2}(x_{1},x_{3})\right]=0.$
It can be seen by inspection that each of the second variations is inversely
proportional to minus the Janossy density of order $k$. Since these are all
probability densities, each of the second variations is negative and thus the
extrema given in the first variation (39) are maxima. Solving the Euler-
Lagrange equations (39) for the Lagrange multipliers yields
$\displaystyle\Lambda_{0}$ $\displaystyle=$ $\displaystyle 1+\log[J_{0}(A)]$
$\displaystyle\Lambda_{1}(x_{1})$ $\displaystyle=$
$\displaystyle\log\left[\frac{j_{1}(x_{1})}{J_{0}(A)}\right]$
$\displaystyle\Lambda_{1}(x_{2})$ $\displaystyle=$
$\displaystyle\log\left[\frac{j_{1}(x_{2})}{J_{0}(A)}\right]$
$\displaystyle\Lambda_{1}(x_{3})$ $\displaystyle=$
$\displaystyle\log\left[\frac{j_{1}(x_{3})}{J_{0}(A)}\right]$
$\displaystyle\Lambda_{2}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\log\left[\frac{J_{0}(A)\,j_{2}(x_{1},x_{2})}{j_{1}^{2}(x_{1})}\right]$
$\displaystyle\Lambda_{2}(x_{2},x_{3})$ $\displaystyle=$
$\displaystyle\log\left[\frac{J_{0}(A)\,j_{2}(x_{1},x_{3})}{j_{1}^{2}(x_{2})}\right]$
$\displaystyle\Lambda_{2}(x_{1},x_{3})$ $\displaystyle=$
$\displaystyle\log\left[\frac{J_{0}(A)\,j_{2}(x_{2},x_{3})}{j_{1}^{2}(x_{3})}\right].$
(40)
After substituting the Lagrange multipliers in (40) into the equation for the
first variation with respect to $j_{3}$ in (39) yields an expression that
relates the Janossy density of third order to the lower order ones under the
assumption of maximum entropy constrained by the moments, namely
$j_{3}(x_{1},x_{2},x_{3}|A)=\frac{j_{2}(x_{1},x_{2}|A)\,j_{2}(x_{2},x_{3}|A)\,j_{2}(x_{1},x_{3}|A)}{j_{1}(x_{1}|A)\,j_{1}(x_{2}|A\,)j_{1}(x_{3}|A)}\,J_{0}(A),$
(41)
Equation (41) is formally similar to the Kirkwood closure. However, there are
a number of important differences. First, it varies with the choice of the
window $A$, since it depends on the _local_ likelihoods (see Figure 6) rather
than the product densities used in the Kirkwood closure, which are global
properties that do not depend on the window of observation. This domain $A$
depends on the spatial scale for which the third particle in the triplet
becomes independent of the other two. Second, the closure is weighted by the
avoidance probability $J_{0}(A)$. This term is conceptually similar to the
exponential weight suggested by Meeron [47] and Salpeter [60], but now arises
from a maximum entropy consideration. The relationship (41) can be used as a
closure of the moment hierarchy after using the expansions (30) and (35) that
allow the Janossy densities to be expressed in terms of product densities.
Figure 6: The domain $A$ represents the region beyond which a third particle
becomes independent of the other two. Shifting $x^{\prime}_{3}$ to $x_{3}$,
makes that third point independent of the other two, in which case the triplet
requires only information about second and first orders density, since the two
points along the $\xi_{1}$ edge are still correlated. This corresponds to the
spatial scale for which the assumptions leading to the Kirkwood closure are
valid.
Since the underlying point process is spatially stationary by construction,
then the mean density is constant, and the densities of higher orders depend
on the relative rather than absolute distances between points. After rescaling
the product densities in the expansion by the area of the window $A$ (the
product densities that come from the hierarchy are defined in terms of the
much larger spatial window used to observe the full process) we have that the
maxent closure is given by
if $|\xi_{1}|\leq r_{0}$ and $|\xi_{2}|\leq r_{0}$ and $|\xi_{2}-\xi_{1}|\leq
r_{0}$
$\displaystyle{m_{3}}(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle\left[{m_{2}}(\xi_{1})-\,{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{1},\xi_{2}^{\prime})\,\,d\xi_{2}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2})-{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{2},\xi_{2}-\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2}-\xi_{1})-\,{|A_{0}|}\int_{A_{0}}{m_{3}}(\xi_{2}-\xi_{1}^{\prime},\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\frac{J_{0}(A_{0})}{\left[{m_{1}}-\,{|A_{0}|}\int_{A_{0}}{m_{2}}(\xi_{1}^{\prime})\,d\xi_{1}^{\prime}+\frac{{|A_{0}|}^{2}}{2}\int_{A_{0}^{(2)}}{m_{3}}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}\right]^{3}},$
else
${m_{3}}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{2}-\xi_{1})}{{m_{1}}^{3}}$
(43)
where the circular domain $A_{0}$ of radius $r_{0}$ is determined from the
normalisation constraint (described below). The avoidance function
$J_{0}(A_{0})$ is given by
$\displaystyle J_{0}(A_{0})$ $\displaystyle=$ $\displaystyle
1-m_{1}{|A_{0}|}+\frac{{|A_{0}|}}{2}\int_{A_{0}}m_{2}(\xi_{1})d\xi_{1}-\frac{{|A_{0}|}}{6}\int_{A_{0}^{(2)}}m_{3}(\xi_{1},\xi_{2})d\xi_{1}d\xi_{2}$
(44) $\displaystyle+$
$\displaystyle\sum_{n=4}^{\infty}\frac{(-1)^{n}}{n!}(m_{1}{|A_{0}|})^{n}$
and the summation term is equal to
$\sum_{n=4}^{\infty}\frac{(-1)^{n}}{n!}(m_{1}{|A_{0}|})^{n}=\exp\left(-m_{1}{|A_{0}|}\right)-1+m_{1}|A_{0}|-\frac{\left(m_{1}|A_{0}|\right)^{2}}{2}+\frac{\left(m_{1}|A_{0}|\right)^{3}}{6}.$
After simplifying we have
$\displaystyle J_{0}(A_{0})$ $\displaystyle=$
$\displaystyle\exp\left(-m_{1}{|A_{0}|}\right)+\frac{{|A_{0}|}}{2}\int_{A_{0}}m_{2}(\xi_{1})d\xi_{1}-\frac{\left(m_{1}|A_{0}|\right)^{2}}{2}-\frac{{|A_{0}|}}{6}\int_{A_{0}^{(2)}}m_{3}(\xi_{1},\xi_{2})d\xi_{1}d\xi_{2}$
$\displaystyle+$ $\displaystyle\frac{\left(m_{1}|A_{0}|\right)^{3}}{6}.$
In order to obtain the family of sets $A_{0}$ in the correction terms of the
closure, we first need to identify the spatial scale $r_{0}$ beyond which two
points become independent. This is equivalent to finding the smallest region
$A_{0}$ for which the correlated point process has the same statistics of a
Poisson process of the same mean density. This domain is obtained by comparing
the avoidance functions for each case, which must coincide for this specific
set. Since the avoidance probability for a homogeneous Poisson point process
of intensity $m_{1}$ for some reference window $B$ is equal [14] to
$J_{0}^{\ast}(B)=\exp\left(-m_{1}|B|\right),$ (46)
Thus the set $A_{0}$ must satisfy
$J_{0}(A_{0})=J_{0}^{\ast}(A_{0}).$ (47)
Substituting the rhs of (46) and (4.4) into (47) leads to the integral
equation
$\int_{A_{r}}m_{2}(\xi_{1})d\xi_{1}-m_{1}^{2}|A_{r}|-\frac{1}{3}\int_{A_{r}^{(2)}}m_{3}(\xi_{1},\xi_{2})d\xi_{1}d\xi_{2}+\frac{m_{1}^{3}|A_{r}|^{2}}{3}=0,$
(48)
where $A_{r}=B(0,r)$ is the ball of radius $r$ centered at the origin. Since
all the product densities are given by the hierarchy and the closure
relationship (4.4), the only unknown in (48) is the domain ${A_{0}}$ that
satisfies the equality (48). This can be found by evaluating the rhs of (48)
for an increasing family of domains $A_{r}$. The values of for $r$ that
satisfy the equality are the roots of interest. There are four possible
scenarios for these roots:
1. 1.
The trivial root, $r=0$ is the only solution. This is always a solution by
simple inspection.
2. 2.
A single non-trivial root $r^{\ast}$.
3. 3.
A finite number of $n$ non trivial roots
$r^{\ast}_{1},r^{\ast}_{2},\ldots,r^{\ast}_{n}$.
4. 4.
An infinite number of roots.
A criterion of validity for the closure scheme can be built on the basis of
the number of roots. Case 1 indicates that there is not a scale within the
observed range of $r$ for which correlations decay as powers of the mean
density, and thus truncation should be tried at a higher order. Case 2
indicates that there is a single Poisson domain $A_{0}$ and thus the closure
assumptions are consistent with the predicted values of the hierarchy. Case 3
indicates that there are several scales of spatial pattern, due to
correlations that oscillate as they decay, i.e. segregated clusters (see
Figure 5). In this situation each scale of pattern should be treated
separately. An infinite number of roots (case 4) indicates that the process is
indistinguishable from a Poisson process at all scales.
Although the closure expression seems complicated, we note that if the area
$a_{0}=|A_{0}|$ is small, then the integral correction terms are of similar
magnitude, and relatively small in comparison with the correction introduced
by avoidance probability, which by far dominates the closure. In this
situation we have a much simpler approximation to the exact closure, given by
${m_{3}}(\xi_{1},\xi_{2})\approx\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{2}-\xi_{1})}{{m_{1}}^{3}}\exp(-{m_{1}}{|A_{0}|}).$
(49)
## 5 Numerical implementation
The numerical solution of the hierarchy with the maxent closure requires two
separate modules of code: one for the integration of the hierarchy itself, and
the other for the iterative procedure that computes the third order density.
The first, which we call the ‘outer’ code, consists of a standard numerical
integration scheme that predicts the first and second order product densities
at a time $(t+h)$ using the first, second and third order ones at time $t$ as
input, where $h$ is a small time step. The second module, or ‘inner code’,
computes the third order density at time $(t+h)$ from the maxent closure. The
inner code starts by computing an initial value for the area of normalisation
$A_{0}^{(old)}$ using the values of the first and second order densities at
time $(t+h)$, and the third order density at time $t$ as an initial trial.
This first value $A_{0}^{(old)}$ is then substituted in the maxent closure
expression (4.4) to produce an updated value for the third order density. The
area of normalisation is recalculated with the updated third order density to
produce a new value $A_{0}^{(new)}$; if the relative difference between the
old and the new radii associated with each normalisation area falls below some
pre–specified tolerance, then the iteration stops and the final value of the
third order density at time $(t+h)$ is the one being used to calculate the
last iteration of area of normalisation. If not, the iterations continue until
the tolerance is achieved. We now propose an algorithm for the implementation
the maxent closure, and subsequently show its performance for a broad range of
parameters of the spatial scales. Our numerical results are well behaved and
convergence of the iteration scheme occurs rapidly for a sufficiently small
time step ($h=0.1$), where typically two or three iterations of the closure
are sufficient for a relative error tolerance within one percent. The problem
consists of solving the coupled system
$\displaystyle\left\\{\begin{array}[]{ccl}\frac{d}{dt}{m_{1}}(t)&=&r\,{m_{1}}(t)-d_{N}\int_{\Gamma}W(\xi_{1})\,{m_{2}}(\xi_{1},t)\,d\xi_{1}\\\
\\\
\frac{1}{2}\,\frac{d}{dt}{m_{2}}(\xi_{1},t)&=&b\int_{\Gamma}B(\xi_{2})\,{m_{2}}(\xi_{1}-\xi_{2},t)\,d\xi_{2}+b\,B(\xi_{1})\,{m_{1}}(t)-d\,{m_{2}}(\xi_{1},t)\\\
\\\
&-&d_{N}\,W(\xi_{1})\,{m_{2}}(\xi_{1},t)-d_{N}\int_{\Gamma}W(\xi_{2})\,{m_{3}}(\xi_{1},\xi_{2},t)\,d\xi_{2}.\end{array}\right.$
(55)
where $\Gamma\subset{\mathbb{R}}^{2}$ is the computational window. The initial
condition
$~{}~{}{m_{1}}(0)=n_{0},~{}{m_{2}}(\xi_{1},0)=n_{0}^{2},~{}{m_{3}}(\xi_{1},\xi_{2},0)=n_{0}^{3}.$
The window $\Gamma$ should be large enough to approximate correctly the
integral terms so that the scale for which the second and third product
densities respectively decay to $m_{1}^{2}$ and $m_{1}^{3}$ lie well within
the computational window$\Gamma$. This hierarchy can be closed at order 2 with
the maxent closure (4.4)
$\displaystyle{m_{3}}(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle\left[{m_{2}}(\xi_{1})-\,{|A_{0}|}\int_{{A_{0}}}{m_{3}}(\xi_{1},\xi_{2}^{\prime})\,\,d\xi_{2}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2})-{|A_{0}|}\int_{{A_{0}}}{m_{3}}(\xi_{2},\xi_{2}-\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2}-\xi_{1})-\,{|A_{0}|}\int_{{A_{0}}}{m_{3}}(\xi_{2}-\xi_{1}^{\prime},\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\frac{J_{0}({A_{0}})}{\left[{m_{1}}-\,{|A_{0}|}\int_{{A_{0}}}{m_{2}}(\xi_{1}^{\prime})\,d\xi_{1}^{\prime}+\frac{{|A_{0}|}^{2}}{2}\int_{{A_{0}}\times{A_{0}}}{m_{3}}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}\right]^{3}},$
which is applied if each the three distance vectors ($\xi_{1},\xi_{2}$ and
$\xi_{2}-\xi_{1}$, see Figure 6) connecting the three points in the triple
configuration fall within the normalisation domain $A_{0}$. Outside of this
region we apply the Kirkwood closure on the basis of probabilistic
independence of the third point in the triplet, as discussed in the previous
section
${m_{3}}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{2}-\xi_{1})}{{m_{1}}^{3}}.$
(57)
In the maxent closure (5) the avoidance function $J_{0}(A_{0})$ is given by
$\displaystyle J_{0}({A_{0}})=\exp\left(-m_{1}{|A_{0}|}\right).$
The circular domain $A_{0}$ is computed from the comparison between the
normalisation constraint for the truncated hierarchy and that of a Poisson
process of the same mean intensity. It is calculated by finding the value of
$r$ that satisfies
$\int_{A_{r}}m_{2}(\xi_{1}^{\prime})d\xi_{1}^{\prime}-m_{1}^{2}|A_{r}|-\frac{1}{3}\int_{A_{r}^{(2)}}m_{3}(\xi_{1}^{\prime},\xi_{2}^{\prime})d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}+\frac{m_{1}^{3}|A_{r}|^{2}}{3}=0.$
(58)
where $A_{r}$ is the 2-dimensional ball of radius $r$ centred at the origin.
### 5.1 Algorithm for the numerical implementation
The coupled system of product density equations with the maxent closure can be
solved from the following algorithm:
1. 1.
From a sequence of radii $r_{i}=0,\ldots,r_{max}$, construct an increasing
family of domains $A_{r_{i}}$.
2. 2.
At time $t=0$ the initial configuration is given by a homogeneous Poisson
point process, thus all the product densities are powers of the mean density
$N_{0}/|X|$, where $X$ is the computational spatial arena, and $N_{0}$ is the
population size at time $t=0$.
3. 3.
While the elapsed time $t<T_{max}$ do
1. (a)
Integrate forward the densities ${m_{1}}(t+h)$ and ${m_{2}}(\xi_{1},t+h)$ from
the hierarchy using a standard numerical procedure.
2. (b)
Use the value of the triplet density at the earlier time step
${m_{3}}^{(old)}(\xi_{1},\xi_{2},t)$ as the initial guess in the normalisation
condition for the Poisson area $A_{0}$. Generate a sequence of values
$f(r_{i})$ by calculating the the normalisation condition (58) for each the
domains previously constructed in Step 1 according to
$\displaystyle f(r_{i})$ $\displaystyle=$
$\displaystyle\int_{A_{r_{i}}}{m_{2}}(\xi_{1}^{\prime},t+h)\,d\xi_{1}^{\prime}-\frac{1}{3}\int_{A_{r_{i}}^{(2)}}{m_{3}}^{(old)}(\xi_{1}^{\prime},\xi_{2}^{\prime},t)\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}-{m_{1}}^{2}(t+h)\,a_{r_{i}}$
(59) $\displaystyle+$
$\displaystyle\frac{1}{3}{m_{1}}^{3}(t+h)\,a_{r_{i}}^{2}$
where the $a_{r_{i}}$ are the areas for each of the $A_{r_{i}}$.
3. (c)
Find the largest value $r_{o}$ that satisfies $f(r_{o})=0$ by linear
interpolation between the consecutive $r_{i}$ where $f(r_{i})$ changes sign.
4. (d)
Use $r_{o}$ from Step 3c to generate the estimate of the Poisson domain
$A_{0}=A_{r_{o}}$.
5. (e)
Loop the spatial arguments $\xi_{1}$ and $\xi_{2}$ over the computational
spatial arena.
6. (f)
Compute the magnitudes $d_{1}$, $d_{2}$ and $d_{3}$ of the the distance
vectors $\xi_{1}$, $\xi_{2}$ and $\xi_{2}-\xi_{1}$
7. (g)
if $d_{1}\leq r_{0}$ and $d_{2}\leq r_{0}$ and $d_{3}\leq r_{0}$ apply the
maxent closure
$\displaystyle{m_{3}}^{(new)}(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle\frac{\exp(-{m_{1}}\,{|A_{0}|})}{\left[{m_{1}}-A_{0}\int_{A_{0}}{m_{2}}(\xi_{1}^{\prime})\,d\xi_{1}^{\prime}+\frac{A_{0}^{2}}{2}\int_{A_{0}}^{(2)}{m_{3}}^{(old)}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{1}^{\prime}\,d\xi_{2}^{\prime}\right]^{3}}$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{1})-A_{0}\int_{A_{0}}{m_{3}}^{(old)}(\xi_{1}^{\prime},\xi_{2}^{\prime})\,d\xi_{2}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2})-A_{0}\int_{A_{0}}{m_{3}}^{(old)}(\xi_{2},\xi_{2}-\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right]$
$\displaystyle\times$
$\displaystyle\left[{m_{2}}(\xi_{2}-\xi_{1})-A_{0}\int_{A_{0}}{m_{3}}^{(old)}(\xi_{2}-\xi_{1}^{\prime},\xi_{1}^{\prime})\,d\xi_{1}^{\prime}\right],$
8. (h)
else use the Kirkwood closure
${m_{3}}^{(new)}(\xi_{1},\xi_{2})=\frac{{m_{2}}(\xi_{1})\,{m_{2}}(\xi_{2})\,{m_{2}}(\xi_{2}-\xi_{2})}{{m_{1}}^{3}}.$
(60)
9. (i)
Recompute the Poisson domain $A_{0}^{(new)}$ and its radius $r_{0}^{(new)}$ by
inserting the corrected triplet density ${m_{3}}^{(new)}$ from Step 3e into
the normalisation equation into Step 3c and estimate a new root $r_{n}$.
10. (j)
If the difference between the old radius and the new one falls within the
error tolerance
$\frac{\left|\,r_{o}-r_{o}^{(new)}\right|}{r_{o}}\leq\mbox{tolerance}$
then the third order density at time $(t+h)$ is the one calculated at Step 3e
${m_{3}}(\xi_{1},\xi_{1},t+h)={m_{3}}^{(new)}(\xi_{1}),\xi_{2}$
else the old third order density becomes the new third order density
${m_{3}}^{(new)}\rightarrow{m_{3}}^{{(old)}}$
and repeat Steps 3c through 3i until the error falls within the tolerance.
4. 4.
update the elapsed time
$t\rightarrow t+h.$
### 5.2 Closure performance
We applied the simulation algorithm introduced in the previous Section 5.1
using a spatial discretisation of $47$ points per linear dimension, and the
domain $B$ was the unit square $[-1/2,1/2]\times[-1/2,1/2]$. The spatial
integrals were computed using the trapezoidal rule, and the convolution in
(55) was calculated using the fast Fourier transform. For the solution of the
moment hierarchy we use a fourth–order Runge-Kutta scheme (with a time step
$h=0.1$). Convergence was checked by halving the time step and the spatial
discretisation and no significant differences were found (
$m_{1}^{\star}=168.6,\Delta x=1/47,h=0.1$ and $m_{1}^{\star}=168.9,\Delta
x=1/95,h=0.05$, for $\sigma_{W}=\sigma_{B}=0.05$).
The maxent closure is expected work well in situations where the spatial
scales of dispersal and mortality are similar, since this combination of
parameters tends to produce a single scale of spatial pattern of mild
aggregation (see Figure 5), where higher order terms are small. Figure 7
compares the dynamics of the mean density predicted by the maxent closure in a
mildly aggregated regime ($\sigma_{B}=\sigma_{W}=0.05$ ) against averages of
the point process model and the other closure methods used in the literature,
power–1, power–2 and power–3 (but the _asymmetric_ power–2 is not used in the
comparison). We see that the maxent closure outperforms the other closures. As
before, in all cases the transient is predicted poorly. This is to be expected
of the maxent method, because the locational entropy can be assumed to be
maximised only once the stochastic process has reached its stationary
distribution. For this reason, even with the correction terms, the truncated
hierarchy with the maxent closure fails at capturing the transient behavior,
which typically consists of long range spatial correlations that decay only
once the density–dependent mortality term is large enough to cause mixing at
longer scales, thus producing a shorter correlation scale.
Figure 7: Comparison between the mean density (jagged blue line) for a sample
of 300 simulations of the point process for the mildly aggregated case
$\sigma_{B}=0.05,\sigma_{W}=0.05$ (open circles) and the truncated product
density hierarchy using various closures. The the maximum entropy closure
(maxent) (continuous black line), the power–3 (dash-dot), the symmetric
power–2 (dot) and power–1 (dashed). The maximum entropy closure provides the
best fit to the equilibrium values of the IBM. However the performance of all
the closures is poor during the transient regime. Figure 8: Behavior of the
area of corrections in the maxent closure for two types of agregated spatial
patterns. The upper three panels correspond to a segregated pattern of
clusters with $\sigma_{B}=0.02,\,\sigma_{W}=0.12$, and the lower panels to a
mildly aggregated pattern with $\sigma_{B}=\sigma_{W}=0.04$. The left column
shows a single point pattern at the end of the simulation, the middle column
shows a kernel density estimate of the pair correlation function for the
pattern displayed in the left and the right column shows the temporal behavior
of the area of the set in the correction terms.
The ability of the maxent closure to predict accurately the mean density
changes dramatically when the two interaction kernels have very different
characteristic scales. This combination of parameters leads to several scales
of pattern, that can consist of short range aggregation compensated by long
range segregation, or short scale segregation compensated by long range
clustering. This occurs because the total number of pairs over sufficiently
long ranges must be equal to the density squared. Thus, extreme aggregation
over short scales must be compensated by segregation over the longer scales in
order to preserve the total number of pairs. When dispersal has a much shorter
characteristic scale than that of density–dependent mortality, the resulting
pattern consists of segregated clusters. This situation violates the closure
assumptions (that require a single scale of pattern), and we expect the
validity checks in the maxent closure to be activated in this situation. This
is illustrated for two types of aggregated patterns in Figure 8. The upper
three panels correspond to segregated clusters
($\sigma_{B}=0.02,\sigma_{W}=0.12$), and the lower three to the mild
aggregation case discussed earlier ($\sigma_{B}=\sigma_{W}=0.04$). The left
column conformed by panels (a) and (b) show typical point patterns obtained at
the same time at which the numerical solution of the hierarchy stopped,
$t=1.56$ in (a), because of the validity check, and $t=80$ in (b) which was
long enough to reach equilibirium. The center column, consisting of panels (c)
and (d), displays kernel density estimates of the pair correlation function
for the point patterns shown to the left. We see in panel (c) a very high
degree of aggregation at short scales followed by long range segregation.
Finally, panels (e) and (f) show the dynamics of the area of correlations
$A_{0}(t)$ for both regimes. We see failure of the maxent closure to find a
non-trivial root for $A_{0}$ in panel (e) after a short transient, as should
be expected due to the presence of various scales of pattern detected in the
pair correlation function in panel (c). In this situation, the extreme form of
‘checkerboard’ aggregation requires truncation at a higher order. Since the
pair correlation function is clearly not constant, but yet the normalisation
constraint only finds the trivial root zero, the validity check is activated
and the numerical solution of the hierarchy stops. By contrast, in the lower
panels when the degree of clustering is comparatively smaller, the method
succeeds in finding a single root $A_{0}$ that eventually reaches a single
equilibrium (see panel (f)).
We carried out a systematic exploration of the behavior of the maxent closure
for a wide range of combinations (441 in total) of the spatial parameters
falling within the range $[0.02,0.12]$ that correspond to those explored
earlier by Law _et al_ [44], and compare the results with the predictions of
the point process, and the product density hierarchy with the power–3 closure.
This allows the assessment of the relative importance of the correction terms
in the maxent closure. The upper limit in the parameter domain was chosen
because for that scale ($\sigma_{B}=\sigma_{W}=0.12$) there is only a very
small departure from complete spatial randomness. Figure 9 shows various
equilibrium values predicted by the product density hierarchy with the maxent
closure. Panel (a) corresponds to the mean density, panel (b) shows the
equilibrium value of the second moment at the origin, normalized by the mean
density squared, and finally, panel (c) shows the area of normalisation at
equilibrium. The removed regions (white) in panel (a) result from the
application of the validity check of normalisation, since for this parameter
the area of correlations is zero (see panel (c)), but the second order product
density indicates the existence of spatial pattern.
Figure 9: Simulation results of the product density hierarchy with the maxent
for various values of the characteristic spatial scales of dispersal
$\sigma_{B}$ (horizontal axis) and mortality $\sigma_{W}$(vertical axis). The
left panel (a) shows the equilibrium mean density $m_{1^{\ast}}$. The center
panel shows the value of the second order product density at equilibrium
evaluated at the origin, normalized by the squared mean density. In this panel
values higher than one indicate clustering at short scales, and values below
one indicate segregation. The right panel (c) shows the value at equilibrium
of the area of the domain used in the correction terms $A_{0}$.
In Figure 10 we compare the mean equilibrium density predicted from an average
of the the space–time point process (a), the maxent closure (b), and the
power–3 closure (c). The maxent closure is not a good predictor of the mean
density for intermediate to low ranges of mortality combined with long range
dispersal; in this regime both the qualitative and quantitative behavior of
the closure is poor. We see a sharp drop in the values of the mean density,
whereas in the point process model it grows monotonically before reaching the
plateau that occurs when both dispersal and mortality act over long scales.
This combination of parameters leads to segregation at short scales and long
range (albeit mild) aggregation. The maxent method detects only the scale of
aggregation, which produces comparatively larger values of the area of
correlations (see panel (c) in Figure 9). This leads to over–correction in the
maxent closure, which results in an equilibrium density that falls well below
that predicted by the point process model. In this regime, the power–3 closure
provides a much more precise prediction of the equilibrium density, both
qualitatively and quantitatively. For sufficiently short ranges of dispersal
together with short to intermediate ranges of mortality the point process
model predicts extinction, as already noted earlier by [43, 44]. In this
regime, neither the maxent closure nor the power—3 closure is capable of
predicting the persistance/extinction threshold, and the maxent validity check
does not seem to operate either. However, for intermediate ranges of
aggregation close or above the main diagonal ($\sigma_{W}=\sigma_{B}$), the
maxent closure does provide an improved prediction of the equilibrium density,
with the added benefit of the criterion of validity being activated when
dispersal is short range with long range mortality, which leads to different
scales of pattern.
We computed the relative error between the equilibrium density of the point
process, and that predicted by the moment equations with the two closures,
shown in Figure 10. Panel (a) corresponds to the maxent closure and panel (b)
to the power–3. We see that the maxent closure has larger relative error than
the power–3 for values located below the diagonal ($\sigma_{W}=\sigma_{B}$),
which are associated with segregated spatial patterns (see Figure 9, panel
(b)). In contrast, the power–3 closure performs quite well in this region. The
advantage of the maxent closure becomes more noticeable on, and above the
diagonal, which is associated with aggregated patterns. The ability to predict
correctly the equilibrium density in this regime is nearly optimal;
particularly when the two scales have similar magnitudes, even when both
dispersal and mortality act over short ranges. The regions of the parameter
space for which each of the two closures is relatively more useful are shown
in Figure 12, which displays the difference in relative error between the two
closures $\Delta E=err_{p3}-err_{maxent}$. Positive values of $\Delta E$
indicate that the error in the power-3 closure is larger than the maxent
closure, and vice versa for negative values of $\Delta E$. As discussed above
the largest improvement of the maxent closure around to the region where the
two scales are of similar magnitude.
Figure 10: Comparision of the mean density $m_{1}^{\ast}$ at equilibrium
predicted by an ensemble average of the point process model (a), the maxent
closure (b), and the power–3 or Kirkwood closure (c). In panel (b) the white
region no the upper left corner corresponds to the domain where the
normalisation constraint returns a trivial root for values of the second order
product density that indicate the presence of spatial pattern, activating the
validity check (48). Figure 11: Relative error of the maxent closure (a) and
the power–3 closure (b). We see that the maxent closure performs better than
the power–three closure for mildly aggregated patterns (lower left), but the
Kirkwood closure outperforms the maxent in segregated patterns (lower right)
Figure 12: Difference in relative error between the maxent and power–3
closures for various combinations of dispersal and mortality spatial scales.
Values higher than zero indicate that the maxent closure outperforms the
power–3 closure, whereas negative values are evidence of better precision of
the power–3 closure.
## 6 Discussion
The results of this research resonate with previous work [4, 44, 53] that
demonstrates that the analysis of stochastic, locally-regulated, individual-
based models of population dynamics in continuous space is feasible [53, 4,
44]. The numerical implementation of the maxent closure is computationally
more expensive (about twice as much) than existing closure methods, but is
nonetheless faster than resorting to direct simulation of the point process;
if one is willing to approximate, the simplified closure based solely on the
exponential correction (49) is substantially simpler to implement, and
produces very small errors in comparison with the full maxent closure.
Although a number of moment closures have been proposed in the literature,
some using heuristic arguments, and others based on constrained entropy
maximisation, very few, if any have a criterion of validity, with the
exception of Ovaskainen & Cornell [53] who were able to derive a series
expansion for the mean density of a spatially explicit metapopulation problem,
and show rigorously that their approximation to the mean density is exact in
the limit of long range interactions. The principal benefit of the maxent
method lies in the fact that the normalisation constraint used to find the
domain for the correction terms fails to find a non-trivial root when the
closure assumptions are not met. This situation occurs when higher order terms
are required in order to fully capture the dynamics, or when correlations
extend over a range that goes beyond the window of observation. This property
constitutes a validation check, not present in other proposed closure schemes.
Although the power–3 or Kirkwood closure had previously been derived from
maximum entropy arguments [69] (but using a different set of constraints and a
different definition of the entropy functional), the correction terms
presented here are new, and extend the probabilistic interpretation of the
Kirkwood closure to situations where there is a region of irreducible triplet
correlations. These correction terms introduce significant improvements in the
agreement between the simulations of the stochastic process (for mildly
aggregated patterns) and its deterministic approximation by the product
density hierarchy. It remains to be seen how the maxent closure behaves for
other functional forms of the interaction kernels, particularly for those that
have tails that decay algebraically ( i.e. power laws) instead of exponential.
Another area of further work would be related to changes in the value of the
non–spatial carrying capacity $K$. For higher densities, spatial effects
become less important.
Since the derivation of the method does not depend on the details of the
model, but only on that its equilibrium distribution is of maximum
_locational_ entropy with moment constraints, the maxent closure may be useful
beyond spatial ecology where unclosed hierarchies for particle distribution
functions are also commonly found, for instance in the statistical mechanics
of fluids where the Kirkwood closure was first introduced [69], or in problems
where the organisms move in space [1, 25, 78], provided that the correlation
functions in those models are stationary in both space and time. A limitation
of the method is its poor ability to predict the transient. This is to be
expected, since maximum entropy is a meaningful property of the _equilibrium_
distribution only when detailed balance is satisfied [74, 27, 38] and the
transitions due to fecundity and dispersal events coincide with mortality.
Other areas of current and future work include the generalisation of the
moment hierarchy and the maxent closure to an arbitrary order of truncation,
extensions to _marked_ spatial point processes for populations with both
spatial and size structure.
Appendix. Derivation of moment equations In order to derive the equation for
${m_{1}}(t)$, we start by fixing a small region of observation $dx_{1}$ (so
that the count inside $dx_{1},N(dx_{1})$ is either 0 or 1) and write a Master
equation for the probabilities of change in the count $\Delta N_{\delta
t}(dx_{1})$ during a small time interval $\delta t$, defined as
$\Delta N_{\delta t}(dx_{1})=N_{t+\delta t}(dx_{1})-N_{t}(dx_{1}).$
These come from the birth and death transitions. Births are given by the
probability that the count $N(dx_{1})$ increases by one in $\delta t$ due to a
birth in $dx_{1}$
$N\mapsto N+1,$
This probability is controlled by the fecundity rate and the dispersal kernel,
$\displaystyle f(x_{1}|\varphi_{t})$ $\displaystyle=$
$\displaystyle{\mathbb{P}}\left\\{\mbox{ one birth in }(dx_{1})\mbox{ during
}(t,t+\delta t)\,|\,\varphi_{t}(X)\right\\}.$ (61) $\displaystyle=$
$\displaystyle\left[b\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,N_{t}(dx_{n})\ell(dx_{1})\right]\delta
t+o(\delta t),$
where $b$ is the birth rate, $B(\xi)$ is the dispersal kernel, $\varphi_{t}$
is the configuration of points at time $t$ and $\ell(A)$ is the area of the
2-dimensional domain $A$. For the death of the individual in $dx_{1}$, we have
the transition
$N\mapsto N-1,$
controlled by
$\displaystyle\mu(x_{1}|\varphi_{t})$ $\displaystyle=$
$\displaystyle{\mathbb{P}}\left\\{\mbox{ death of individual }x_{1}\mbox{
during }(t,t+\delta t)\,|\,\varphi_{t}(X)\right\\}.$ $\displaystyle=$
$\displaystyle
N_{t}(dx_{1})\left[d+d_{N}\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n})\left(N_{t}(dx_{n})-\delta_{x_{1}}(dx_{n})\right)\right]\delta
t+o(\delta t),$
where $d$ and $d_{N}$ are positive constants defined in Section 2, the
density–independent, and density–dependent contributions to the mortality and
$W(\xi)$ is the mortality kernel). This probability is conditional on there
being an individual in $dx_{1}$. The change in the count $\Delta N_{\delta
t}(x_{1})$ is then given by both contributions
$\Delta N_{\delta t}(dx_{1})=f(x_{1}|\varphi_{t})-\mu(x_{1}|\varphi_{t})$
so
$\displaystyle\Delta N_{\delta t}(dx_{1})$ $\displaystyle=$
$\displaystyle\left[b\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,N_{t}(dx_{n})\,\ell(dx_{1})\right.$
$\displaystyle-$
$\displaystyle\left.N_{t}(dx_{1})\left(d+d_{N}\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n})(N_{t}(dx_{n})-\delta_{x_{1}}(dx_{n})\,\right)\right]\delta
t.$
Taking expectations (ensemble averaging) on both sides and dividing by the
duration of a small time interval $\delta t$ yields
$\displaystyle\frac{{\mathrm{E}}\\{\Delta N_{\delta t}(dx_{1})\\}}{\delta t}$
$\displaystyle=$ $\displaystyle
b\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,{\mathrm{E}}\\{N_{t}(dx_{n})\\}\,\ell(dx_{1})$
$\displaystyle-$
$\displaystyle{\mathrm{E}}\left\\{N_{t}(dx_{1})\left(d+d_{N}\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n})(N_{t}(dx_{n})-\delta_{x_{1}}(dx_{n}))\right)\right\\}.$
after rearranging the second term, dividing both sides by $\ell(dx_{1})$ and
multiplying the second sum by $\ell(dx_{n})/\ell(dx_{n})$ we get
$\displaystyle\frac{{\mathrm{E}}\\{\Delta N_{\delta
t}(dx_{1})\\}}{\ell(dx_{1})\,\delta t}$ $\displaystyle=$ $\displaystyle
b\frac{{\mathrm{E}}\\{N_{t}(dx_{n})\\}}{\ell(dx_{1})}\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,\ell(dx_{1})-d\,\frac{{\mathrm{E}}\\{N_{t}(dx_{1})\\}}{\ell(dx_{1})}$
$\displaystyle-$ $\displaystyle
d_{N}\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n})\frac{{\mathrm{E}}\left\\{N_{t}(dx_{1})\,\left(N_{t}(dx_{n})-\delta_{x_{1}}(dx_{n})\right)\right\\}}{\ell(dx_{1})\,\ell(dx_{n})}\,\ell(dx_{n}).$
taking the limits as $\ell(dx_{1})$ and $\ell(dx_{n})$ go to zero, and using
definition of the product density (17)
$\displaystyle\frac{\Delta{m_{1}}(x_{1},t)}{\delta t}$ $\displaystyle=$
$\displaystyle
b\,{m_{1}}(x_{1},t)\int_{\Re^{2}}B(x_{1}-x_{n})\,dx_{1}-d\,{m_{1}}(x_{1},t)$
$\displaystyle-$ $\displaystyle
d_{N}\int_{\Re^{2}}W(x_{1}-x_{n})\,{m_{2}}(x_{1},x_{n},t)\,dx_{n}.$
since the process is spatially stationary by construction and exploiting the
fact that the dispersal kernel integrates to unity, yields
$\displaystyle\frac{\Delta{m_{1}}(t)}{\delta
t}=b\,{m_{1}}(t)-d\,{m_{1}}(t)-d_{N}\int_{\Re^{2}}W(\xi_{1})\,{m_{2}}(\xi_{1},t)\,d\xi_{1},$
finally, after taking the limit as $\delta t\rightarrow 0$ we get,
$\displaystyle\frac{d}{dt}{m_{1}}(t)=b\,{m_{1}}(t)-d\,{m_{1}}(t)-d_{N}\int_{\Re^{2}}W(\xi_{1})\,{m_{2}}(\xi_{1},t)\,d\xi_{1}.$
(64)
On setting $r=b-d$, we get the the generalisation of the logistic equation to
the spatial case obtained by Law & Dieckmann [43] and Law _et al,law03_ , but
derived explicitly in terms of product densities,
$\displaystyle\frac{d}{dt}{m_{1}}(t)=r\,{m_{1}}(t)-d_{N}\int_{\Re^{2}}W(\xi_{1})\,{m_{2}}(\xi_{1},t)\,d\xi_{1}.$
(65)
Since ${m_{2}}$ is unknown, we need an additional evolution equation for this
object. We follow a similar procedure to that used for the mean density, but
considering the expected change of the _product_ of the counts in two
observation regions $dx_{1}$ and $dx_{2}$. This requires the consideration of
how pairs of points are created and destroyed as individuals disperse and die.
There are three possible ways in which changes to occur. The first if to fix
the count $N_{t}(dx_{1})$ and allow only $N_{t}(dx_{2})$ to change. The second
is the reverse situation, fixing $N_{t}(dx_{2})$ and allowing only
$N_{t}(dx_{1})$ to change. The third is when _both_ $N_{t}(dx_{1})$ and
$N_{t}(dx_{2})$ change in a small time interval. We have that
$\displaystyle\Delta[N_{t}(dx_{1})\,(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))]$
$\displaystyle=$ $\displaystyle
N_{t}(dx_{1})\Delta(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))$ $\displaystyle+$
$\displaystyle(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))\Delta N_{t}(dx_{1})$
$\displaystyle+$ $\displaystyle\Delta
N_{t}(dx_{1})\Delta(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))$
where the Dirac delta distribution is used to remove self-pairs. The following
derivation for the second order product densities is based on the symmetry in
the probabilities of a birth or a death event occurring at both extremes of
the distance vector linking $x_{1}$ and $x_{2}$. We also assume that a
_simultaneous_ change in both $N_{t}(dx_{2})$ and $N_{t}(dx_{1})$ is
negligible
${\mathbb{P}}\left[\Delta
N_{t}(dx_{1})\Delta(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))\right]=o(\delta t)$
and thus the transitions of second order can be written as
$\Delta[N_{t}(dx_{1})\,(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))]=2\Delta
N_{t}(dx_{1})(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2})).$ (67)
Since we already have an expression for $\Delta N_{t}(dx_{1})$, given by (6),
(67) becomes
$\displaystyle\Delta[N_{t}(dx_{1})\,(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))]$
$\displaystyle=$ $\displaystyle
2\cdot(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))\left[b\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,N_{t}(dx_{n})\,\ell(dx_{1})\right.$
$\displaystyle-$
$\displaystyle\left.N_{t}(dx_{1})\left(d+d_{N}\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n})(N_{t}(dx_{n})-\delta_{x_{1}}(dx_{n})\,\right)\right]\delta
t.$
Taking expectations, and dividing by both sides by $\delta t$ gives
$\displaystyle\frac{\Delta[N_{t}(dx_{1})\,(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))]}{2\,\delta
t}$ $\displaystyle=$ $\displaystyle
b\sum_{x_{n}\in\varphi_{t}}B(x_{1}-x_{n})\,{\mathrm{E}}\left\\{N_{t}(dx_{n})(N_{t}(dx_{2})\right.$
$\displaystyle-$
$\displaystyle\left.\delta_{x_{1}}(dx_{2}))\right.\\}\,\ell(dx_{1})-d\,{\mathrm{E}}\left\\{N_{t}(dx_{1})(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))\right\\}$
$\displaystyle-$ $\displaystyle
d_{N}\,\sum_{x_{n}\in\varphi_{t}}W(x_{1}-x_{n}){\mathrm{E}}\left\\{N_{t}(dx_{1})(N_{t}(dx_{n})\right.$
$\displaystyle-$
$\displaystyle\left.\delta_{x_{1}}(dx_{n}))(N_{t}(dx_{2})-\delta_{x_{1}}(dx_{2}))\right\\}.$
After dividing by $\ell(dx_{1})$ and $\ell(dx_{2})$, using the definition of
product densities (17) and taking the continuum limit in both space and time,
one arrives at the evolution equation for the second order product density
$\displaystyle\frac{1}{2}\,\frac{\partial}{\partial t}{m_{2}}(\xi_{1},t)$
$\displaystyle=$ $\displaystyle
b\int_{\Re^{2}}B(\xi_{2})\,{m_{2}}(\xi_{1}-\xi_{2},t)\,d\xi_{2}+b\,B(\xi_{1})\,{m_{1}}(t)-d\,{m_{2}}(\xi_{1},t)$
(68) $\displaystyle-$ $\displaystyle
d_{N}W(\xi_{1})\,{m_{2}}(\xi_{1},t)-d_{N}\int_{\Re^{2}}W(\xi_{2})\,{m_{3}}(\xi_{1},\xi_{2},t)\,d\xi_{2},$
where we see the dependence on the _third_ order product density in the last
integral
#### acknowledgements
M.R acknowledges the support granted by the International Institute for
Applied Systems Analysis (IIASA) to participate in the Young Scientist Summer
Program where part of this research was conducted during the summer of 2004.
M.R. is grateful to Richard Law who suggested to work on this problem and
generously shared his time and insights, Kenneth Lindsay who kindly shared his
probabilistic and simulation expertise, and fruitful discussions with Jonathan
Dushoff. The authors are also grateful with Benjamin Bolker, David Murrell and
David Grey who helped with details on the simulations of the point processes
and generously provided access to code and manuscripts. The support of Simon
A. Levin and Ioannis G. Kevrekidis is gratefully acknowledged.
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|
arxiv-papers
| 2012-02-27T23:40:47 |
2024-09-04T02:49:27.955057
|
{
"license": "Public Domain",
"authors": "Michael Raghib, Nicholas A. Hill and Ulf Dieckmann",
"submitter": "Michael Raghib",
"url": "https://arxiv.org/abs/1202.6092"
}
|
1202.6231
|
# Higgs at ILC in Universal Extra Dimensions
in Light of Recent LHC Data
Takuya Kakuda1 Kenji Nishiwaki2 Kin-ya Oda3 Naoya Okuda3 and Ryoutaro
Watanabe3
1- Department of Physics Niigata University
Niigata 950-2181 Japan
2- Harish-Chandra Research Institute
Chhatnag Road Jhusi Allahabad 211 019 India
3- Department of Physics Osaka University
Osaka 560-0043 Japan
###### Abstract
We present bounds on all the known universal extra dimension models from the
latest Higgs search data at the Large Hadron Collider, taking into account the
Kaluza-Klein (KK) loop effects on the dominant gluon-fusion production and on
the diphoton/digluon decay. The lower bound on the KK scale is from 500 GeV to
1 TeV depending on the model. We find that the Higgs production cross section
with subsequent diphoton decay can be enhanced by a factor 1.5 within this
experimental bound, with little dependence on the Higgs mass in between 115
GeV and 130 GeV. We also show that in such a case the Higgs decay branching
ratio into a diphoton final state can be suppressed by a factor 80%, which is
marginally observable at a high energy/luminosity option at the International
Linear Collider. The Higgs production cross section at a photon-photon
collider can also be suppressed by a similar factor 90%, being well within the
expected experimental reach.
## 1 Introduction
Higgs field is the last missing and the most important piece of the Standard
Model (SM) of elementary particles and interactions. Last year the Large
Hadron Collider (LHC) made a great achievement in Higgs searches. Now the SM
Higgs mass is highly constrained within a low mass range
$115.5\,\text{GeV}<M_{H}<127\,\text{GeV}$ or else is pushed up to a high mass
region $M_{H}>600\,\text{GeV}$ at the 95% CL [1, 2].
In particular the ATLAS experiment has observed an excess of events close to
$M_{H}=126\,\text{GeV}$ with a local significance $3.6\,\sigma$ above the
expected SM background without Higgs, though it becomes less significant
$2.3\,\sigma$ after taking into account the Look-Elsewhere Effect (LEE) [1].
On the other hand, the CMS experiment has observed the largest excess at 124
GeV with a local significance $3.1\,\sigma$ but reduces to $1.5\,\sigma$ after
taking the LEE into account over 110–600 GeV [2]. Note that the peak at ATLAS
is close to the CMS exclusion limit 127 GeV, but that the CMS local
significance at 126 GeV is still $\sim 2\,\sigma$ [2]. These peaks at ATLAS
and CMS are dominated by diphoton signals.111 Our analyses and statements
hereof are based on the results shown in the preliminary version presented on
the web in Refs. [1, 2].
An interesting observation is that the best fit value of the diphoton cross
section is enhanced from that of SM by factor $\sim 1.7$ and 2 for the peaks
at $M_{H}=124\,\text{GeV}$ (CMS [2]) and 126 GeV (ATLAS [1]), respectively.
For the latter, the enhancement needed for the total Higgs production cross
section is $\sim 1.5$ after taking into account all the related decay channels
(with the branching ratios being assumed to be the same as in the SM):
$H\to\gamma\gamma$, $H\to ZZ\to llll$ and $H\to WW\to l\nu l\nu$ [1].
The Universal Extra Dimension (UED) models assume that all the SM fields
propagate in the bulk of the compactified extra dimension(s). Currently known
UED models utilize compactifications on a one-dimensional orbifold
$S^{1}/Z_{2}$ (mUED), on two-dimensional orbifolds based on torus
$T^{2}/Z_{4}$ (T2Z4), $T^{2}/(Z_{2}\times Z_{2}^{\prime})$ (T2Z2Z2),
$T^{2}/Z_{2}$ (T2Z2), $RP^{2}$ (RP2), on a two-sphere based orbifold
$S^{2}/Z_{2}$ (S2Z2), and on two-dimensional manifolds, the projective sphere
(PS) and the sphere $S^{2}$ (S2); See [3, 4] for references.
We can list two virtues of the UED models (see e.g. [4] for references).
First, due to the compactification, there appears a tower of Kaluza-Klein (KK)
modes for each SM degree of freedom; Among these KK modes, the Lightest KK
Particle (LKP) is stable due to a symmetry of the compactified space and hence
becomes a good candidate for the dark matter. Second virtue is the explanation
of the number of generations to be three when there are two extra dimensions
in order to cancel the global gauge anomaly in six dimensions.
Further, the UED models allow a heavy Higgs. If the light Higgs is excluded in
the forthcoming LHC running and hence the Higgs turns out to be heavy in the
region $M_{H}>600\,\text{GeV}$, the SM with such a heavy Higgs is inconsistent
to the current electroweak precision data. In UED model the KK top loop
corrections may cure this discrepancy. However in this work, we pursue the
case for light Higgs mass and give a possible explanation for the above
mentioned enhancement of the Higgs production cross section.
Figure 1: 95% CL bounds from $H\to\gamma\gamma$ at ATLAS (red/orange dashed)
and at CMS (red/orange solid) and from $H\to WW\to l\nu l\nu$ at CMS with cut-
based (blue/cyan solid) and with multi-variate BDT (dotted) event selections.
The red and blue (orange and cyan) colors correspond to the maximum (minimum)
UV cutoff scale in 6D.
## 2 LHC bounds on UED models
In the LHC, the Higgs production is dominated by the gluon fusion process
$gg\to H$ induced by the top-quark loop. As a rule of thumb, one can expect
that loop-induced UED corrections are significant if a process is prohibited
at the tree level in the SM. The gluon fusion is such a process. The KK top
quarks make a correction to the Higgs production cross section as
$\displaystyle\hat{\sigma}_{gg\to H}^{\text{UED}}$
$\displaystyle={\pi^{2}\over 8M_{H}}\Gamma^{\text{UED}}_{H\to
gg}\,\delta(\hat{s}-M_{H}^{2}),$ (1) $\displaystyle\Gamma^{\text{UED}}_{H\to
gg}$ $\displaystyle=K{\alpha_{S}^{2}\over 8\pi^{3}}{M_{H}^{3}\over
v_{\text{EW}}^{2}}\left|J_{t}^{\text{SM}}+J_{t}^{\text{KK}}\right|^{2},$ (2)
where $K$ is the K-factor accounting for the higher order QCD corrections,
$\alpha_{S}$ is the fine structure constant for the QCD, $v_{\text{EW}}\simeq
246\,\text{GeV}$ is the electroweak scale, and explicit forms of the top and
KK-top loop functions $J_{t}^{\text{SM}}$ and $J_{t}^{\text{KK}}$,
respectively, are given in [3, 4]. As said above, the tree-level widths
$\Gamma_{H\to t\bar{t}}$, $\Gamma_{H\to b\bar{b}}$, $\Gamma_{H\to c\bar{c}}$,
$\Gamma_{H\to\tau\bar{\tau}}$, $\Gamma_{H\to WW}$, and $\Gamma_{H\to ZZ}$ are
not significantly modified from those in the SM by the KK loop corrections,
while the diphoton width becomes
$\displaystyle\Gamma^{\text{UED}}_{H\to\gamma\gamma}$
$\displaystyle={\alpha^{2}G_{F}M_{H}^{3}\over
8\sqrt{2}\pi^{3}}\left|J_{W}^{\text{SM}}+J_{W}^{\text{KK}}+{4\over
3}\left(J_{t}^{\text{SM}}+J_{t}^{\text{KK}}\right)\right|^{2},$ (3)
where $\alpha$ and $G_{F}$ are the fine-structure and Fermi constants,
respectively, and $J_{W}^{\text{SM}}$ ($J_{W}^{\text{KK}}$) are loop
corrections from SM-(KK-) gauge bosons [3]. Because of these additional
bosonic and fermionic loop correction, Higgs decay to $2\gamma$ receives a
nontrivial effect.
The diphoton and $WW$ experimental constraints [5, 7, 6] are put on the
following ratios, respectively,
$\displaystyle{\sigma^{\text{UED}}_{gg\to
H\to\gamma\gamma}\over\sigma^{\text{SM}}_{gg\to H\to\gamma\gamma}}$
$\displaystyle\simeq{\Gamma^{\text{UED}}_{H\to
gg}\Gamma^{\text{UED}}_{H\to\gamma\gamma}/\Gamma^{\text{UED}}_{H}\over\Gamma^{\text{SM}}_{H\to
gg}\Gamma^{\text{SM}}_{H\to\gamma\gamma}/\Gamma^{\text{SM}}_{H}},$ (4)
$\displaystyle{\sigma^{\text{UED}}_{gg\to H\to
WW}\over\sigma^{\text{SM}}_{gg\to H\to WW}}$
$\displaystyle\simeq{\Gamma^{\text{UED}}_{H\to
gg}/\Gamma^{\text{UED}}_{H}\over\Gamma^{\text{SM}}_{H\to
gg}/\Gamma^{\text{SM}}_{H}},$ (5)
where we have approximated $\Gamma^{\text{UED}}_{H\to
WW}\simeq\Gamma^{\text{SM}}_{H\to WW}$ and have taken into account the decay
modes into $t\bar{t}$, $b\bar{b}$, $c\bar{c}$, $\tau\bar{\tau}$, $gg$,
$\gamma\gamma$, $W^{+}W^{-}$ and $ZZ$ in the total width $\Gamma_{H}$.
In Fig. 1, we show 95% CL exclusion plots in $M_{\text{KK}}$ vs $M_{H}$ plane
from the $H\to\gamma\gamma$ modes at ATLAS [5] (red/orange dashed) and at CMS
[7] (red/orange solid) and from the $H\to WW$ mode at CMS [6] (blue/cyan),
where solid and dotted lines correspond to the cut-based and BDT event
selections for the $WW$ channel, respectively.222 As stated above, for all the
bounds, we have utilized the values shown in the preliminary version presented
on the web. We note that the newer CMS diphoton data set, which we have not
utilized, includes vector boson fusion (VBF) events that occurs at the tree
level in the SM and hence is not significantly enhanced by the UED loop
corrections. The red and blue (orange and cyan) colors correspond to the
maximum (minimum) UV cutoff scales in six dimensions; see [3, 4] for
details.333We can calculate the processes without UV cutoff dependence in five
dimensions. First we can see that the region $115\,\text{GeV}\lesssim
M_{H}\lesssim 127\,\text{GeV}$ is selected by the diphoton exclusion as in the
SM. The ATLAS diphoton exclusion around 121 GeV became strong due to a
statistical fluctuation. In the range $123\,\text{GeV}\lesssim M_{H}\lesssim
126\,\text{GeV}$, both ATLAS and CMS have an excess of events in the diphoton
channel and the bounds from $WW$ signals become stronger. We see that the
lower bound for the KK scale is about 500 GeV–1 TeV depending on the models in
this low Higgs mass region. The diphoton bounds do not exclude the low KK
scale $M_{\text{KK}}\lesssim 500\,\text{GeV}$ for the lower Higgs mass
$M_{H}\lesssim 123\,\text{GeV}$ in the case of RP2, PS and S2 models, in which
we have many low lying KK modes. This is because the KK top contribution
$J_{t}^{\text{UED}}$ cancels the dominant SM one $J_{W}^{\text{UED}}$ in that
region.444 In this parameter region, $J_{W}^{\text{SM}}\simeq 2$,
$J_{t}^{\text{SM}}\simeq-0.5$, and
$J_{W}^{\text{UED}}/J_{t}^{\text{UED}}\sim-0.4$. We can find a similar recent
study on mUED in [8].
Figure 2: Enhancement ratios of UED to SM at $M_{H}=125\,\text{GeV}$ for the
gluon-fusion Higgs production cross section $\sigma_{gg\to H}$ (dotted), for
the same with subsequent diphoton decay $\sigma_{gg\to H\to\gamma\gamma}$
(solid), and for the Higgs total decay width $\Gamma_{H}$ (dashed). The right
hand side of the vertical line is allowed by the CMS cut-based $H\to WW$ bound
given in Figure 1. Colors denote the same as in Figure 1.
In ATLAS, the best fit value for the ratio of the total Higgs production cross
section $\sigma_{gg\to H}/\sigma_{gg\to H}^{\text{SM}}$ is found to be $\sim
1.5$ around the observed excess of events at $M_{H}\simeq 126\,\text{GeV}$
[1]. In CMS, the best fit value for the ratio is $\sim 0.6$ (1.2) at
$M_{H}=126\,\text{GeV}$ (123–124 GeV). The preliminary version of Ref. [1]
reports that the diphoton ratio in Eq. (4) is $\sim 2$ at
$M_{H}=126\,\text{GeV}$. Let us examine whether this can be explained by the
UED models, keeping in mind the fact that this excess of the cross section
ratio is still only $\sim 1\sigma$ away from unity.
In Figure 2, we plot the enhancement factor for the total Higgs production
cross section due to the UED loop corrections (dotted), for the same with
subsequent diphoton decay (solid), and also for the total decay width for
comparison (dashed) as a function of the first KK mass $M_{\text{KK}}$. We
have chosen $M_{H}=125\,\text{GeV}$ while the result is insensitive to the
Higgs mass in the low mass region $M_{H}<130\,\text{GeV}$. Each vertical line
shows the lower bound for the first KK mass $M_{\text{KK}}$ whose left side is
excluded. Conventions on colors are the same as in Figure 1. We see that Higgs
cross section with subsequent diphoton decay can be enhanced by a factor $\sim
1.5$ within the current experimental constraint. Note however that the
diphoton ratio (solid) becomes smaller than the $WW$ ratio (dotted) in UED
models, in contrast to the observation at ATLAS, where the best fit values for
the former and latter are about 2 and 1.2 at the peak. Note that the $WW$
ratio is almost identical to the ratio for the total production cross section
$\sigma_{H}/\sigma_{H}^{\text{SM}}$ (dotted).
To summarize, the UED corrections become significant for the SM-loop induced
couplings $Hgg$ and $H\gamma\gamma$; The enhancement of the former can be seen
at LHC, even when multiplied by the reduction of the latter diphoton decay. In
the next section, let us see whether the latter reduction can be directly seen
at the International Linear Collider (ILC).
## 3 ILC and photon photon collider
Figure 3: Suppression ratios of UED to SM at $M_{H}=125\,\text{GeV}$ for the
Higgs branching ratio into diphoton $\text{BR}(H\to\gamma\gamma)$ (solid) and
for the Higgs production cross section at the photon-photon collider
$\sigma_{\gamma\gamma\to H}$ (dashed). Colors and vertical lines denote the
same as in Figure 2.
In Figure 3, we show the suppression ratio of UED to SM at
$M_{H}=125\,\text{GeV}$ for the Higgs branching ratio of diphoton decay
$\text{BR}(H\to\gamma\gamma)$ (solid) and for the Higgs production cross
section at the photon-photon collider $\sigma_{\gamma\gamma\to H}$ (dashed).
Colors indicate the same as in Figure 2. The Higgs decay branching ratio into
two photons is suppressed more than the corresponding decay width because the
former is divided by the total decay width that is enhanced by the decay into
gluons as shown by the dashed lines in Figure 2.
We see that the branching ratio (solid) can be suppressed by a factor $\sim
0.8$ within the current experimental bound. This is marginally accessible at
the ILC with integrated luminosity $500\,\text{fb}^{-1}$ at $500\,\text{GeV}$
whose expected precision for the $\text{BR}(H\to\gamma\gamma)$ is 23% for
$M_{H}=120\,\text{GeV}$ [9]. This precision is refined to 5.4% with luminosity
$1\,\text{ab}^{-1}$ at $1\,\text{TeV}$ for the same Higgs mass [10].
When we employ the photon photon collider option, $H\gamma\gamma$ coupling can
be measured more directly since it becomes the total production cross section
of the Higgs. From Figure 3, we see that the Higgs production cross section
(dashed) can be reduced by a factor $\sim 0.9$ in the allowed region to the
right of the vertical line. This is well within the reach for an integrated
photon-photon luminosity $410\,\text{fb}^{-1}$ at a linear $e^{+}e^{-}$
collider operated at $\sqrt{s}=210\,\text{GeV}$ which can measure
$\Gamma_{H\to\gamma\gamma}{\times}\text{BR}(H\to b\bar{b})$ with an accuracy
of 2.1% for $M_{H}=120\,\text{GeV}$ [11].
## 4 Summary
In UED models, the loop corrections from the KK-top and KK-gauge bosons modify
the $Hgg$ and $H\gamma\gamma$ couplings. Generally we have shown that the
former (latter) is enhanced (suppressed) from that in SM, with the former
effect dominating the latter.
We have obtained the 95% CL allowed region in the $M_{\text{KK}}$ vs $M_{H}$
parameter space for all the known UED models in the low mass region
$115\,\text{GeV}<M_{H}<130\,\text{GeV}$ in Figure 1. In this low Higgs mass
window, lower and upper bounds for the Higgs mass are given by the ATLAS and
CMS diphoton limits, respectively, whereas the lower bound for the KK scale is
put by the CMS limit from the $WW\to l\nu l\nu$ channel as
$M_{\text{KK}}\gtrsim 500\,\text{GeV}$–1 TeV.
We have also shown the suppression factor from the SM for
$\text{BR}(H\to\gamma\gamma)$ and $\Gamma_{H\to\gamma\gamma}$. We see that the
former can be suppressed by the factor 0.8 and that this is marginally
accessible at the ILC. The $H\gamma\gamma$ coupling itself can also be
suppressed by the factor 0.9 which is well within the reach for the photon
photon collider option.
## Acknowledgments
We thank Maria Krawczyk, Shinya Kanemura, and Howard Haber for useful comments
in the LCWS11.
## References
* [1] “Combined search for the Standard Model Higgs boson using up to 4.9 fb-1 of $pp$ collision data at $\sqrt{s}=7\text{TeV}$ with the ATLAS detector at the LHC,” Tech. Rep. ATLAS-CONF-2011-163; arXiv:1202.1408 [hep-ex].
* [2] S. Chatrchyan et al. [CMS Collaboration], “Combined results of searches for the Standard Model Higgs boson in $pp$ collisions at $\sqrt{s}=7\text{TeV}$,” arXiv:1202.1488 [hep-ex].
* [3] K. Nishiwaki, K. Oda, N. Okuda and R. Watanabe, “A Bound on Universal Extra Dimension Models from Up to $2\text{fb}^{-1}$ of LHC Data At 7TeV,” Phys. Lett. B 707 (2012) 506 [arXiv:1108.1764 [hep-ph]].
* [4] K. Nishiwaki, K. Oda, N. Okuda and R. Watanabe, “Heavy Higgs at Tevatron and LHC in Universal Extra Dimension Models,” arXiv:1108.1765 [hep-ph].
* [5] The ATLAS Collaboration, “Search for the Standard Model Higgs Boson in the diphoton decay channel with 4.9fb-1 of $pp$ collisions at $\sqrt{s}=7\text{TeV}$ with ATLAS,” arXiv:1202.1414 [hep-ex].
* [6] The CMS Collaboration, “Search for the Higgs Boson Decaying to W+W- in the Fully Leptonic Final State,” CMS-PAS-HIG-11-024, (December 2011).
* [7] The CMS Collaboration, “Search for a Higgs boson decaying into two photons in the CMS detector” CMS-PAS-HIG-11-030, (December 2011).
* [8] G. Bélanger et al., “Higgs Phenomenology of Minimal Universal Extra Dimensions,” arXiv:1201.5582 [hep-ph].
* [9] K. Desch [Higgs Working Group of the Extended ECFA/DESY Study], “Higgs Boson Precision Studies at a Linear Collider,” arXiv:hep-ph/0311092.
* [10] T. L. Barklow, “Higgs Coupling Measurements at a 1-Tev Linear Collider,” arXiv:hep-ph/0312268.
* [11] S. Heinemeyer et al., “Toward High Precision Higgs-Boson Measurements at the International Linear $e^{+}e^{-}$ Collider,” arXiv:hep-ph/0511332.
|
arxiv-papers
| 2012-02-28T14:15:16 |
2024-09-04T02:49:27.977541
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takuya Kakuda, Kenji Nishiwaki, Kin-ya Oda, Naoya Okuda and Ryoutaro\n Watanabe",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1202.6231"
}
|
1202.6251
|
The LHCb collaboration
# First evidence of direct $C\\!P$ violation in charmless two-body decays of
$B^{0}_{s}$ mesons
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
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Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, L. Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6,
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R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
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Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
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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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
###### Abstract
Using a data sample corresponding to an integrated luminosity of 0.35
$\mathrm{fb}^{-1}$ collected by LHCb in 2011, we report the first evidence of
$C\\!P$ violation in the decays of $B^{0}_{s}$ mesons to $K^{\pm}\pi^{\mp}$
pairs, $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)}$, with a significance of 3.3$\sigma$. Furthermore, we
report the most precise measurement of $C\\!P$ violation in the decays of
$B^{0}$ mesons to $K^{\pm}\pi^{\mp}$ pairs, $A_{C\\!P}(B^{0}\rightarrow
K\pi)=-0.088\pm 0.011\,\mathrm{(stat)}\pm 0.008\,\mathrm{(syst)}$, with a
significance exceeding 6$\sigma$.
###### pacs:
Valid PACS appear here
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | LHCb-PAPER-2011-029
| | CERN-PH-EP-2012-058
The violation of $C\\!P$ symmetry, _i.e._ the non-invariance of fundamental
forces under the combined action of the charge conjugation ($C$) and parity
($P$) transformations, is well established in the $K^{0}$ and $B^{0}$ meson
systems Christenson:1964fg ; Aubert:2001nu ; Abe:2001xe ; Nakamura:2010zzi .
Recent results from the LHCb collaboration have also provided evidence for
$C\\!P$ violation in the decays of $D^{0}$ mesons Aaij:2011in . Consequently,
there now remains only one neutral heavy meson system, the $B^{0}_{s}$, where
$C\\!P$ violation has not yet been seen. All current experimental measurements
of $C\\!P$ violation in the quark flavor sector are well described by the
Cabibbo-Kobayashi-Maskawa mechanism Cabibbo:1963yz ; Kobayashi:1973fv which
is embedded in the framework of the Standard Model (SM). However, it is
believed that the size of $C\\!P$ violation in the SM is not sufficient to
account for the asymmetry between matter and antimatter in the Universe
Hou:2008xd , hence additional sources of $C\\!P$ symmetry breaking are being
searched for as manifestations of physics beyond the SM.
In this Letter we report measurements of direct $C\\!P$ violating asymmetries
in $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays using data collected with the LHCb detector. The inclusion of charge-
conjugate modes is implied except in the asymmetry definitions. $C\\!P$
violation in charmless two-body $B$ decays could potentially reveal the
presence of physics beyond the SM Fleischer:1999pa ; Gronau:2000md ;
Lipkin:2005pb ; Fleischer:2007hj ; Fleischer:2010ib , and has been extensively
studied at the $B$ factories and at the Tevatron Aubert:2008sb ; Belle:2008zza
; Aaltonen:2011qt . The direct $C\\!P$ asymmetry in the $B^{0}_{(s)}$ decay
rate to the final state $f_{(s)}$, with $f=K^{+}\pi^{-}$ and
$f_{s}=K^{-}\pi^{+}$, is defined as
$A_{C\\!P}=\Phi\\!\left[\Gamma\\!\left(\overline{B}^{0}_{(s)}\rightarrow\bar{f}_{(s)}\right)\\!\\!,\,\Gamma\\!\left(B^{0}_{(s)}\rightarrow
f_{(s)}\right)\right]\\!\\!,$ (1)
where $\Phi[X,\,Y]=(X-Y)/(X+Y)$ and $\bar{f}_{(s)}$ denotes the charge-
conjugate of $f_{(s)}$.
LHCb is a forward spectrometer covering the pseudo-rapidity range $2<\eta<5$,
designed to perform flavor physics measurements at the LHC. A detailed
description of the detector can be found in Ref. Alves:2008zz . The analysis
is based on $pp$ collision data collected in the first half of 2011 at a
center-of-mass energy of $7\\!$ $\mathrm{\,Te\kern-1.00006ptV}$, corresponding
to an integrated luminosity of $0.35~{}\mathrm{fb}^{-1}$. The polarity of the
LHCb magnetic field is reversed from time to time in order to partially cancel
the effects of instrumental charge asymmetries, and about
$0.15~{}\mathrm{fb}^{-1}$ were acquired with one polarity and
$0.20~{}\mathrm{fb}^{-1}$ with the opposite polarity.
The LHCb trigger system comprises a hardware trigger followed by a High Level
Trigger (HLT) implemented in software. The hadronic hardware trigger selects
high transverse energy clusters in the hadronic calorimeter. A transverse
energy threshold of 3.5 $\mathrm{\,Ge\kern-1.00006ptV}$ has been adopted for
the data set under study. The HLT first selects events with at least one large
transverse momentum track characterized by a large impact parameter, and then
uses algorithms to reconstruct $D$ and $B$ meson decays. Most of the events
containing the decays under study have been acquired by means of a dedicated
two-body HLT selection. To discriminate between signal and background events,
this trigger selection imposes requirements on: the quality of the online-
reconstructed tracks ($\chi^{2}$ per degree of freedom), their transverse
momenta ($p_{\mathrm{T}}$) and their impact parameters ($d_{\mathrm{IP}}$,
defined as the distance between the reconstructed trajectory of the track and
the $pp$ collision vertex); the distance of closest approach of the decay
products of the $B$ meson candidate ($d_{\mathrm{CA}}$), its transverse
momentum ($p_{\mathrm{T}}^{B}$), its impact parameter ($d_{\mathrm{IP}}^{B}$)
and the decay time in its rest frame ($t_{\pi\pi}$, calculated assuming the
decay into $\pi^{+}\pi^{-}$). Only $B$ candidates within the $\pi\pi$
invariant mass range 4.7–5.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are
accepted. The $\pi\pi$ mass hypothesis is conventionally chosen to select all
charmless two-body $B$ decays using the same criteria.
Offline selection requirements are subsequently applied. Two sets of criteria
have been optimized with the aim of minimizing the expected uncertainty either
on $A_{C\\!P}(B^{0}\rightarrow K\pi)$ or on $A_{C\\!P}(B^{0}_{s}\rightarrow
K\pi)$. In addition to more selective requirements on the kinematic variables
already used in the HLT, two further requirements on the larger of the
transverse momenta and of the impact parameters of the daughter tracks are
applied. A summary of the two distinct sets of selection criteria is reported
in Table 1. In the case of $B^{0}_{s}\rightarrow K\pi$ decays a tighter
selection is needed because the probability for a $b$ quark to decay as
$B^{0}_{s}\rightarrow K\pi$ is about 14 times smaller than that to decay as
$B^{0}\rightarrow K\pi$ Aaltonen:2008hg , and consequently a stronger
rejection of combinatorial background is required. The two samples passing the
event selection are then subdivided into different final states using the
particle identification (PID) provided by the two ring-imaging Cherenkov
(RICH) detectors. Again two sets of PID selection criteria are applied: a
loose set optimized for the measurement of $A_{C\\!P}(B^{0}\rightarrow K\pi)$
and a tight set for that of $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$.
Table 1: Summary of selection criteria adopted for the measurement of $A_{C\\!P}(B^{0}\rightarrow K\pi)$ and $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. Variable | $A_{C\\!P}(B^{0}\rightarrow K\pi)$ | $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$
---|---|---
Track quality $\chi^{2}$/ndf | $<3$ | $<3$
Track $p_{\mathrm{T}}\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>1.1$ | $>1.2$
Track $d_{\mathrm{IP}}\,[\mathrm{mm}]$ | $>0.15$ | $>0.20$
$\mathrm{max}(p_{\mathrm{T}}^{K},\,p_{\mathrm{T}}^{\pi})\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>2.8$ | $>3.0$
$\mathrm{max}(d_{\mathrm{IP}}^{K},\,d_{\mathrm{IP}}^{\pi})\,[\mathrm{mm}]$ | $>0.3$ | $>0.4$
$d_{\mathrm{CA}}$ $[\mathrm{mm}]$ | $<0.08$ | $<0.08$
$p_{\mathrm{T}}^{B}\,[\textrm{Ge\kern-1.00006ptV\\!/}c]$ | $>2.2$ | $>2.4$
$d_{\mathrm{IP}}^{B}\,[\mathrm{mm}]$ | $<0.06$ | $<0.06$
$t_{\pi\pi}\,[\textrm{ps}]$ | $>0.9$ | $>1.5$
|
---|---
|
Figure 1: Invariant $K\pi$ mass spectra obtained using the event selection
adopted for the best sensitivity on (a, b) $A_{C\\!P}(B^{0}\rightarrow K\pi)$
and (c, d) $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. Plots (a) and (c) represent
the $K^{+}\pi^{-}$ invariant mass whereas plots (b) and (d) represent the
$K^{-}\pi^{+}$ invariant mass. The results of the unbinned maximum likelihood
fits are overlaid. The main components contributing to the fit model are also
shown.
To estimate the background from other two-body $B$ decays with a misidentified
pion or kaon (cross-feed background), the relative efficiencies of the RICH
PID selection criteria must be determined. The high production rate of charged
$D^{*}$ mesons at the LHC and the kinematic characteristics of the
$D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ decay chain make such events an
appropriate calibration sample for the PID of kaons and pions. In addition,
for calibrating the response of the RICH system for protons, a sample of
$\Lambda\rightarrow p\pi^{-}$ decays is used. PID information is not used to
select either sample, as the selection of pure final states can be realized by
means of kinematic criteria alone. The production and decay kinematics of the
$D^{0}\rightarrow K^{-}\pi^{+}$ and $\Lambda\rightarrow p\pi^{-}$ channels
differ from those of the $B$ decays under study. Since the RICH PID
information is momentum dependent, the distributions obtained from calibration
samples are reweighted according to the momentum distributions of $B$ daughter
tracks observed in data.
Unbinned maximum likelihood fits to the $K\pi$ mass spectra of the selected
events are performed. The $B^{0}\rightarrow K\pi$ and $B^{0}_{s}\rightarrow
K\pi$ signal components are described by single Gaussian functions convolved
with a function which describes the effect of final state radiation on the
mass lineshape Baracchini:2005wp . The background due to partially
reconstructed three-body $B$ decays is parameterized by means of an ARGUS
function Albrecht:1989ga convolved with a Gaussian resolution function. The
combinatorial background is modeled by an exponential and the shapes of the
cross-feed backgrounds, mainly due to $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays with one misidentified particle in
the final state, are obtained from Monte Carlo simulations. The
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ cross-
feed background yields are determined from fits to the $\pi^{+}\pi^{-}$ and
$K^{+}K^{-}$ mass spectra respectively, using events selected by the same
offline selection as the signal and taking into account the appropriate PID
efficiency factors. The $K^{+}\pi^{-}$ and $K^{-}\pi^{+}$ mass spectra for the
events passing the two offline selections are shown in Fig. 1.
From the two mass fits we determine respectively the signal yields
$N(B^{0}\rightarrow K\pi)=13\hskip 1.42262pt250\pm 150$ and
$N(B^{0}_{s}\rightarrow K\pi)=314\pm 27$, as well as the raw yield asymmetries
$A_{\mathrm{raw}}(B^{0}\rightarrow K\pi)=-0.095\pm 0.011$ and
$A_{\mathrm{raw}}(B^{0}_{s}\rightarrow K\pi)=0.28\pm 0.08$, where the
uncertainties are statistical only. In order to determine the $C\\!P$
asymmetries from the observed raw asymmetries, effects induced by the detector
acceptance and event reconstruction, as well as due to strong interactions of
final state particles with the detector material, need to be taken into
account. Furthermore, the possible presence of a
$B^{0}_{(s)}-\overline{B}^{0}_{(s)}$ production asymmetry must also be
considered. The $C\\!P$ asymmetry is related to the raw asymmetry by
$A_{C\\!P}=A_{\mathrm{raw}}-A_{\Delta}$, where the correction $A_{\Delta}$ is
defined as
$A_{\Delta}(B^{0}_{(s)}\rightarrow
K\pi)=\zeta_{d(s)}A_{\mathrm{D}}(K\pi)+\kappa_{d(s)}A_{\mathrm{P}}(B^{0}_{(s)}),$
(2)
where $\zeta_{d}=1$ and $\zeta_{s}=-1$, following the sign convention for $f$
and $f_{s}$ in Eq. (1). The instrumental asymmetry $A_{\mathrm{D}}(K\pi)$ is
given in terms of the detection efficiencies $\varepsilon_{\mathrm{D}}$ of the
charge-conjugate final states by
$A_{\mathrm{D}}(K\pi)=\Phi[\varepsilon_{\mathrm{D}}(K^{-}\pi^{+}),\,\varepsilon_{\mathrm{D}}(K^{+}\pi^{-})]$,
and the production asymmetry $A_{\mathrm{P}}(B^{0}_{(s)})$ is defined in terms
of the $\overline{B}^{0}_{(s)}$ and $B^{0}_{(s)}$ production rates,
$R(\overline{B}^{0}_{(s)})$ and $R(B^{0}_{(s)})$, as
$A_{\mathrm{P}}(B^{0}_{(s)})=\Phi[R(\overline{B}^{0}_{(s)}),\,R(B^{0}_{(s)})]$.
The factor $\kappa_{d(s)}$ takes into account dilution due to neutral
$B^{0}_{(s)}$ meson mixing, and is defined as
$\kappa_{d(s)}\\!=\\!\frac{\int_{0}^{\infty}\\!e^{-\Gamma_{d(s)}t}\\!\cos\\!\left(\Delta
m_{d(s)}t\right)\\!\varepsilon(B^{0}_{(s)}\\!\rightarrow\\!K\pi;\,t)\mathrm{d}t}{\int_{0}^{\infty}\\!e^{-\Gamma_{d(s)}t}\\!\cosh\\!\left(\frac{\Delta\Gamma_{d(s)}}{2}t\right)\\!\varepsilon(B^{0}_{(s)}\\!\rightarrow\\!K\pi;\,t)\mathrm{d}t},$
(3)
where $\varepsilon(B^{0}\rightarrow K\pi;\,t)$ and
$\varepsilon(B^{0}_{s}\rightarrow K\pi;\,t)$ are the acceptances as functions
of the decay time for the two reconstructed decays. To calculate $\kappa_{d}$
and $\kappa_{s}$ we assume that $\Delta\Gamma_{d}=0$ and we use the world
averages for $\Gamma_{d}$, $\Delta m_{d}$, $\Gamma_{s}$, $\Delta m_{s}$ and
$\Delta\Gamma_{s}$ Nakamura:2010zzi . The shapes of the acceptance functions
are parameterized using signal decay time distributions extracted from data.
We obtain $\kappa_{d}=0.303\pm 0.005$ and $\kappa_{s}=-0.033\pm 0.003$, where
the uncertainties are statistical only. In contrast to $\kappa_{d}$, the
factor $\kappa_{s}$ is small, owing to the large $B^{0}_{s}$ oscillation
frequency, thus leading to a negligible impact of a possible production
asymmetry of $B^{0}_{s}$ mesons on the corresponding $C\\!P$ asymmetry
measurement.
The instrumental charge asymmetry $A_{\mathrm{D}}(K\pi)$ can be expressed in
terms of two distinct contributions
$A_{\mathrm{D}}(K\pi)=A_{\mathrm{I}}(K\pi)+\alpha(K\pi)A_{\mathrm{R}}(K\pi)$,
where $A_{\mathrm{I}}(K\pi)$ is an asymmetry due to the different strong
interaction cross-sections with the detector material of $K^{+}\pi^{-}$ and
$K^{-}\pi^{+}$ final state particles, and $A_{\mathrm{R}}(K\pi)$ arises from
the possible presence of a reconstruction or detection asymmetry. The quantity
$A_{\mathrm{I}}(K\pi)$ does not change its value by reversing the magnetic
field, as the difference in the interaction lengths seen by the positive and
negative particles for opposite polarities is small. By contrast,
$A_{\mathrm{R}}(K\pi)$ changes its sign when the magnetic field polarity is
reversed. The factor $\alpha(K\pi)$ accounts for different signal yields in
the data sets with opposite polarities, due to the different values of the
corresponding integrated luminosities and to changing trigger conditions in
the course of the run. It is estimated by using the yields of the largest
decay mode, _i.e._ $B^{0}\rightarrow K\pi$, determined from the mass fits
applied to the two data sets separately. We obtain $\alpha(K\pi)=\Phi[N^{\rm
up}(B^{0}\rightarrow K\pi),\,N^{\rm down}(B^{0}\rightarrow K\pi)]=-0.202\pm
0.011$, where “up” and “down” denote the direction of the main component of
the dipole field.
The instrumental asymmetries for the final state $K\pi$ are measured from data
using large samples of tagged $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$
and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$ decays, and untagged
$D^{0}\rightarrow K^{-}\pi^{+}$ decays. The combination of the integrated raw
asymmetries of all these decay modes is necessary to disentangle the various
contributions to the raw asymmetries of each mode, notably including the
$K\pi$ instrumental asymmetry as well as that of the pion from the $D^{*+}$
decay, and the production asymmetries of the $D^{*+}$ and $D^{0}$ mesons. In
order to determine the raw asymmetry of the $D^{0}\rightarrow K\pi$ decay, a
maximum likelihood fit to the $K^{-}\pi^{+}$ and $K^{+}\pi^{-}$ mass spectra
is performed. For the decays $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$
and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$, we perform maximum
likelihood fits to the discriminating variable $\delta m=M_{D^{*}}-M_{D^{0}}$,
where $M_{D^{*}}$ and $M_{D^{0}}$ are the reconstructed $D^{*}$ and $D^{0}$
invariant masses respectively. Approximately 54 million $D^{0}\rightarrow
K^{-}\pi^{+}$ decays, 7.5 million $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and 1.1 million $D^{*+}\rightarrow
D^{0}(K^{-}K^{+})\pi^{+}$ decays are used. The mass distributions are shown in
Fig. 2 (a), (b) and (c). The $D^{0}\rightarrow K^{-}\pi^{+}$ signal component
is modeled as the sum of two Gaussian functions with common mean convolved
with a function accounting for final state radiation Baracchini:2005wp , on
top of an exponential combinatorial background. The $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$ and $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
signal components are modeled as the sum of two Gaussian functions convolved
with a function taking account of the asymmetric shape of the measured
distribution Aaij:2011in . The background is described by an empirical
function of the form $1-e^{-(\delta m-\delta m_{0})/\xi}$, where $\delta
m_{0}$ and $\xi$ are free parameters.
|
---|---
|
Figure 2: Distributions of the invariant mass or invariant mass difference of
(a) $D^{0}\rightarrow K^{-}\pi^{+}$, (b) $D^{*+}\rightarrow
D^{0}(K^{-}\pi^{+})\pi^{+}$, (c) $D^{*+}\rightarrow D^{0}(K^{-}K^{+})\pi^{+}$
and (d) $B^{0}\rightarrow J/\psi(\mu^{+}\mu^{-})K^{*0}(K^{+}\pi^{-})$. The
results of the maximum likelihood fits are overlaid.
Using the current world average of the integrated $C\\!P$ asymmetry for the
$D^{0}\rightarrow K^{-}K^{+}$ decay bib:hfagbase and neglecting $C\\!P$
violation in the Cabibbo-favored $D^{0}\rightarrow K^{-}\pi^{+}$ decay
Bianco:2003vb , from the raw yield asymmetries returned by the mass fits we
determine $A_{\mathrm{I}}(K\pi)=(-1.0\pm 0.2)\times 10^{-2}$ and
$A_{\mathrm{R}}(K\pi)=(-1.8\pm 0.2)\times 10^{-3}$, where the uncertainties
are statistical only.
The possible existence of a $B^{0}-\overline{B}^{0}$ production asymmetry is
studied by reconstructing a sample of $B^{0}\rightarrow J/\psi K^{*0}$ decays.
$C\\!P$ violation in $b\rightarrow c\bar{c}s$ transitions, which is predicted
in the SM to be at the $10^{-3}$ level Hou:2006du , is neglected. The raw
asymmetry $A_{\mathrm{raw}}(B^{0}\rightarrow J/\psi K^{*0})$ is determined
from an unbinned maximum likelihood fit to the
$J/\psi(\mu^{+}\mu^{-})K^{*0}(K^{+}\pi^{-})$ and
$J/\psi(\mu^{+}\mu^{-})\overline{K}^{*0}(K^{-}\pi^{+})$ mass spectra. The
signal mass peak is modeled as the sum of two Gaussian functions with common
mean, whereas the combinatorial background is modeled by an exponential. The
data sample contains approximately 25 400 $B^{0}\rightarrow J/\psi K^{*0}$
decays. The mass distribution is shown in Fig. 2 (d). To determine the
production asymmetry we need to correct for the presence of instrumental
asymmetries. Once the necessary corrections are applied, we obtain a value for
the $B^{0}$ production asymmetry $A_{\mathrm{P}}(B^{0})=0.010\pm 0.013$, where
the uncertainty is statistical only.
By using the instrumental and production asymmetries, the correction factor to
the raw asymmetry $A_{\Delta}(B^{0}\rightarrow K\pi)=-0.007\pm 0.006$ is
obtained. Since the $B^{0}_{s}$ meson has no valence quarks in common with
those of the incident protons, its production asymmetry is expected to be
smaller than for the $B^{0}$, an expectation that is supported by
hadronization models as discussed in Ref. Lambert:2009zz . Even conservatively
assuming a value of the production asymmetry equal to that for the $B^{0}$,
owing to the small value of $\kappa_{s}$ the effect of
$A_{\mathrm{P}}(B^{0}_{s})$ is negligible, and we find
$A_{\Delta}(B^{0}_{s}\rightarrow K\pi)=0.010\pm 0.002$.
The systematic uncertainties on the asymmetries fall into the following main
categories, related to: (a) PID calibration; (b) modeling of the signal and
background components in the maximum likelihood fits; and (c) instrumental and
production asymmetries. Knowledge of PID efficiencies is necessary in this
analysis to compute the number of cross-feed background events affecting the
mass fit of the $B^{0}\rightarrow K\pi$ and $B^{0}_{s}\rightarrow K\pi$ decay
channels. In order to estimate the impact of imperfect PID calibration, we
perform unbinned maximum likelihood fits after having altered the number of
cross-feed background events present in the relevant mass spectra according to
the systematic uncertainties affecting the PID efficiencies. An estimate of
the uncertainty due to possible imperfections in the description of the final
state radiation is determined by varying, over a wide range, the amount of
emitted radiation Baracchini:2005wp in the signal lineshape parameterization.
The possibility of an incorrect description of the core distribution in the
signal mass model is investigated by replacing the single Gaussian with the
sum of two Gaussian functions with a common mean. The impact of additional
three-body $B$ decays in the $K\pi$ spectrum, not accounted for in the
baseline fit — namely $B\rightarrow\pi\pi\pi$ where one pion is missed in the
reconstruction and another is misidentified as a kaon — is investigated. The
mass lineshape of this background component is determined from Monte Carlo
simulations, and then the fit is repeated after having modified the baseline
parameterization accordingly. For the modeling of the combinatorial background
component, the fit is repeated using a first-order polynomial. Finally, for
the case of the cross-feed backgrounds, two distinct systematic uncertainties
are estimated: one due to a relative bias in the mass scale of the simulated
distributions with respect to the signal distributions in data, and another
accounting for the difference in mass resolution between simulation and data.
All the shifts from the relevant baseline values are accounted for as
systematic uncertainties. Differences in the kinematic properties of $B$
decays with respect to the charm control samples, as well as different
triggers and offline selections, are taken into account by introducing a
systematic uncertainty on the values of the $A_{\Delta}$ corrections. This
uncertainty dominates the total systematic uncertainty related to the
instrumental and production asymmetries, and can be reduced in future
measurements with a better understanding of the dependence of such asymmetries
on the kinematics of selected signal and control samples. The systematic
uncertainties for $A_{C\\!P}(B^{0}\rightarrow K\pi)$ and
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ are summarized in Table 2.
Table 2: Summary of systematic uncertainties on $A_{C\\!P}(B^{0}\rightarrow
K\pi)$ and $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$. The categories (a), (b) and
(c) defined in the text are also indicated. The total systematic uncertainties
given in the last row are obtained by summing the individual contributions in
quadrature.
Systematic uncertainty | $A_{C\\!P}(B^{0}\rightarrow K\pi)$ | $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$
---|---|---
(a)PID calibration | 0.0012 | 0.001
(b)Final state radiation | 0.0026 | 0.010
(b)Signal model | 0.0004 | 0.005
(b)Combinatorial background | 0.0001 | 0.009
(b)3-body background | 0.0009 | 0.007
(b)Cross-feed background | 0.0011 | 0.008
(c)Instr. and prod. asym. ($A_{\Delta}$) | 0.0078 | 0.005
Total | 0.0084 | 0.019
In conclusion we obtain the following measurements of the $C\\!P$ asymmetries:
$A_{C\\!P}(B^{0}\rightarrow K\pi)=-0.088\pm 0.011\,\mathrm{(stat)}\pm
0.008\,\mathrm{(syst)}$
and
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)=0.27\pm 0.08\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)}.$
The result for $A_{C\\!P}(B^{0}\rightarrow K\pi)$ constitutes the most precise
measurement available to date. It is in good agreement with the current world
average provided by the Heavy Flavor Averaging Group
$A_{C\\!P}(B^{0}\rightarrow K\pi)=-0.098^{+0.012}_{-0.011}$ bib:hfagbase .
Dividing the central value of $A_{C\\!P}(B^{0}\rightarrow K\pi)$ by the sum in
quadrature of the statistical and systematic uncertainties, the significance
of the measured deviation from zero exceeds $6\sigma$, making this the first
observation (greater than 5$\sigma$) of $C\\!P$ violation in the $B$ meson
sector at a hadron collider. The same significance computed for
$A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is 3.3$\sigma$, therefore this is the
first evidence for $C\\!P$ violation in the decays of $B^{0}_{s}$ mesons. The
result for $A_{C\\!P}(B^{0}_{s}\rightarrow K\pi)$ is in agreement with the
only measurement previously available Aaltonen:2011qt .
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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arxiv-papers
| 2012-02-28T15:23:01 |
2024-09-04T02:49:27.987030
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, 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, S. Belogurov, K. Belous, I.\n Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R.\n Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S.\n Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, K. de\n Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff,\n J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, A.\n Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. 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, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi,\n S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli,\n C. Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew,\n J. Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy,\n L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J.\n Luisier, A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O.\n Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, S.\n Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan,\n B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, N. Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M.\n Otalora Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J.\n Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N. Patrick, C. Patrignani,\n C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, 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, J. Prisciandaro, V. Pugatch,\n A. Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel,\n I. Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues,\n P. Rodriguez Perez, G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J.\n Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P.\n Sail, B. Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina\n Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A.\n Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\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, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A.\n Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer,\n V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi,\n J.J. Velthuis, M. Veltri, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, R.\n Waldi, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams,\n M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K.\n Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Vincenzo Maria Vagnoni",
"url": "https://arxiv.org/abs/1202.6251"
}
|
1202.6267
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-2012-041 LHCb-PAPER-2011-042
Measurement of the ratio of branching fractions ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)$/${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$
The LHCb collaboration †††Authors are listed on the following pages.
The ratio of branching fractions of the radiative $B$ decays
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$ has
been measured using $0.37\,\mbox{\,fb}^{-1}$ of $pp$ collisions at a centre of
mass energy of $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$, collected by the
LHCb experiment. The value obtained is
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.12\pm
0.08^{+0.06}_{-0.04}\phantom{.}{}^{+0.09}_{-0.08},$
where the first uncertainty is statistical, the second systematic and the
third is associated to the ratio of fragmentation fractions $f_{s}/f_{d}$.
Using the world average for ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)=(4.33\pm 0.15)~{}\times 10^{-5}$, the branching fraction ${\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ is measured to be $(3.9\pm 0.5)\times
10^{-5}$, which is the most precise measurement to date.
Submitted to Physical Review D
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S.
Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, E. Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38,
S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30,
C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2,
M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44,
V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V.
Niess5, N. Nikitin29, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, K. Pal53,
J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M.
Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M.
Patel50, S.K. Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-
Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M.
Schiller39, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O.
Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A.
Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N.
Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P.
Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31, O. Shevchenko40, V.
Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49,
E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
In the Standard Model (SM) the decays $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ 111Charge conjugated modes are implicitly
included throughout the paper. proceed at leading order through
$b\\!\rightarrow s\gamma$ one-loop electromagnetic penguin transitions,
dominated by a virtual intermediate top quark coupling to a $W$ boson.
Extensions of the SM predict additional one-loop contributions that can
introduce sizeable effects on the dynamics of the transition [1, *Gershon:th-
null-tests:2006, *Mahmoudi:th-msugra:2006, *Altmannshofer:2011gn].
Radiative decays of the $B^{0}$ meson were first observed by the CLEO
collaboration in 1993 [5] through the decay mode $B\\!\rightarrow
K^{*}\gamma$. In 2007 the Belle collaboration reported the first observation
of the analogous decay in the $B^{0}_{s}$ sector,
$B^{0}_{s}\\!\rightarrow\phi\gamma$ [6]. The current world averages of the
branching fractions of $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ are $(4.33\pm 0.15)~{}\times 10^{-5}$ and
$(5.7^{+2.1}_{-1.8})~{}\times 10^{-5}$, respectively [7, 8,
*belle:exp-b2kstgamma:2004, *cleo:exp-excl-radiative-decays:1999]. These
results are in agreement with the latest SM theoretical predictions from NNLO
calculations using SCET [11], ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)=(4.3\pm 1.4)\times 10^{-5}$ and ${\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)=(4.3\pm 1.4)\times 10^{-5}$, which
suffer from large hadronic uncertainties. The ratio of experimental branching
fractions is measured to be ${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)/{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ = $0.7\pm 0.3$, in agreement with the
prediction of $1.0\pm 0.2$ [11].
This paper presents a measurement of ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)/{\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$ using a strategy
that ensures the cancellation of most of the systematic uncertainties
affecting the measurement of the individual branching fractions. The measured
ratio is used to determine ${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)$
assuming the world average value of ${\cal B}(B^{0}\\!\rightarrow
K^{*0}\gamma)$ [7].
## 2 The LHCb detector and dataset
The LHCb detector [12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about 4 Tm, and three stations of
silicon-strip detectors and straw drift-tubes placed downstream. The combined
tracking system has a momentum resolution $\Delta p/p$ that varies from 0.4%
at 5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at 100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP)
resolution of $20\,\mu$m for tracks with high transverse momentum. Charged
hadrons are identified using two ring-imaging Cherenkov (RICH) detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage running on a large
farm of commercial processors which applies a full event reconstruction.
The data used for this analysis correspond to $0.37\,\mbox{\,fb}^{-1}$ of $pp$
collisions collected in the first half of 2011 at the LHC with a centre of
mass energy of $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$.
$B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$
candidates are required to have triggered on the signal photon and vector
meson daughters, following a definite trigger path. The hardware level must
have been triggered by an ECAL candidate with $\mbox{$E_{\rm
T}$}>2.5\,\mathrm{\,Ge\kern-1.00006ptV}$. In the software trigger, the events
are selected when a track is reconstructed with IP $\chi^{2}>16$, and either
$\mbox{$p_{\rm T}$}>1.7\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ when the photon
has $\mbox{$E_{\rm T}$}>2.5\,\mathrm{\,Ge\kern-1.00006ptV}$ or $\mbox{$p_{\rm
T}$}>1.2\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ when the photon has
$\mbox{$E_{\rm T}$}>4.2\,\mathrm{\,Ge\kern-1.00006ptV}$. The selected track
must form a $K^{*0}$ or $\phi$ candidate when combined with an additional
track, and the invariant mass of the combination of the $K^{*0}(\phi)$
candidate and the photon candidate is requested to lie within a
$1\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ window around the nominal
$B^{0}(B^{0}_{s})$ mass.
Large samples of $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ Monte Carlo (MC) simulated events [13] are
used to optimize the signal selection and to parametrize the $B$ meson
invariant mass distribution. The $pp$ collisions are generated with Pythia 6.4
[14] and decays of hadronic particles are simulated using EvtGen [15] in which
final state radiation is generated using Photos [16]. The interaction of the
generated particles with the detector and its response are simulated using
Geant4 [17].
## 3 Event selection
The selection of both $B$ decays is designed to ensure the cancellation of
systematic uncertainties in the ratio of their efficiencies. The procedure and
requirements are kept as similar as possible: the $B^{0}(B^{0}_{s})$ mesons
are reconstructed from a selected $K^{*0}(\phi$), composed of oppositely
charged kaon-pion (kaon-kaon) pairs, combined with a photon.
The two tracks from the vector meson daughters are both required to have
$\mbox{$p_{\rm T}$}>500\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and to point
away from all $pp$ interaction vertices by requiring $\text{IP}\,\chi^{2}>25$.
The identification of the kaon and pion tracks is made by applying cuts to the
particle identification (PID) provided by the RICH system. The PID is based on
the comparison between two particle hypotheses, and it is represented by the
difference in logarithms of the likelihoods (DLL) between the two hypotheses.
Kaons are required to have $\mathrm{DLL}_{K\pi}$ $>5$ and DLL${}_{Kp}>2$,
while pions are required to have $\mathrm{DLL}_{K\pi}$ $<0$. With these cuts,
kaons (pions) coming from the studied channels are identified with a $\sim
70\,(83)\,\%$ efficiency for a $\sim 3\,(2)\,\%$ pion (kaon) contamination.
Two-track combinations are accepted as $K^{*0}(\phi)$ candidates if they form
a vertex with $\chi^{2}<9$ and their invariant mass lies within a $\pm
50\,(\pm 10)\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window of the
nominal $K^{*0}(\phi)$ mass. The resulting vector meson candidate is combined
with a photon of $\mbox{$E_{\rm T}$}>2.6\,\mathrm{\,Ge\kern-1.00006ptV}$.
Neutral and charged electromagnetic clusters in the ECAL are separated based
on their compatibility with extrapolated tracks [18] while photon and
$\pi^{0}$ deposits are identified on the basis of the shape of the
electromagnetic shower in the ECAL. The $B$ candidate invariant mass
resolution, dominated by the photon contribution, is about
$100\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the decays presented in
this paper.
The $B$ candidates are required to have an invariant mass within a $\pm
800\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around the corresponding
$B$ hadron mass, to have $\mbox{$p_{\rm
T}$}>3\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to point to a $pp$
interaction vertex by requiring $\text{IP}\,\chi^{2}<9$. The distribution of
the helicity angle $\theta_{\text{H}}$, defined as the angle between the
momentum of either of the daughters of the vector meson ($V$) and the momentum
of the $B$ candidate in the rest frame of the vector meson, is expected to
follow $\sin^{2}\theta_{\text{H}}$ for $B\\!\rightarrow V\gamma$, and
$\cos^{2}\theta_{\text{H}}$ for the $B\\!\rightarrow V\pi^{0}$ background.
Therefore, the helicity structure imposed by the signal decays is exploited to
remove $B\\!\rightarrow V\pi^{0}$ background, in which the neutral pion is
misidentified as a photon, by requiring that $|\cos\theta_{\text{H}}|<0.8$.
Background coming from partially reconstructed $b$-hadron decays is rejected
by requiring vertex isolation: the $\chi^{2}$ of the $B$ vertex must increase
by more than half a unit when adding any other track in the event.
## 4 Determination of the ratio of branching fractions
The ratio of the branching fractions is calculated from the number of signal
candidates in the $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ channels,
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=\frac{N_{B^{0}\\!\rightarrow
K^{*0}\gamma}}{N_{B^{0}_{s}\\!\rightarrow\phi\gamma}}\times\frac{{\cal
B}(\phi\rightarrow K^{+}K^{-})}{{\cal B}(K^{*0}\rightarrow
K^{+}\pi^{-})}\times\frac{f_{s}}{f_{d}}\times\frac{\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}}{\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}},$ (1)
where $N$ corresponds to the observed number of signal candidates (yield),
${\cal B}(\phi\rightarrow K^{+}K^{-})$ and ${\cal B}(K^{*0}\rightarrow
K^{+}\pi^{-})$ are the visible branching fractions of the vector mesons,
$f_{s}/f_{d}$ is the ratio of the $B^{0}$ and $B^{0}_{s}$ hadronization
fractions in $pp$ collisions at $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$,
and
$\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}/\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}$ is the ratio of efficiencies for the two decays. This latter
ratio is split into contributions coming from the acceptance
($r_{\text{acc}}$), the reconstruction and selection requirements
($r_{\text{reco}}$), the PID requirements ($r_{\text{PID}}$), and the trigger
requirements ($r_{\text{trig}}$) :
$\frac{\epsilon_{B^{0}_{s}\\!\rightarrow\phi\gamma}}{\epsilon_{B^{0}\\!\rightarrow
K^{*0}\gamma}}=r_{\text{acc}}\times r_{\text{reco}}\times r_{\text{PID}}\times
r_{\text{trig}}.$ (2)
The PID efficiency ratio is measured from data to be $r_{\text{PID}}=0.787\pm
0.010\,\text{(stat)}$, by means of a calibration procedure using pure samples
of kaons and pions from $D^{*\pm}\\!\rightarrow D^{0}(K^{+}\pi^{-})\pi^{\pm}$
decays selected utilizing purely kinematic criteria. The other efficiency
ratios have been extracted using simulated events. The acceptance efficiency
ratio, $r_{\text{acc}}=1.094\pm 0.004\,\text{(stat)}$, exceeds unity because
of the correlated acceptance of the kaons due to the limited phase space in
the $\phi\\!\rightarrow K^{+}K^{-}$ decay. These phase-space constraints also
cause the $\phi$ vertex to have a worse spatial resolution than the $K^{*0}$
vertex. This affects the $B^{0}_{s}\\!\rightarrow\phi\gamma$ selection
efficiency through the IP $\chi^{2}$ and vertex isolation cuts while the
common track cut $\mbox{$p_{\rm
T}$}>500\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ is less efficient on the
softer pion from the $K^{*0}$ decay. Both effects almost compensate and the
reconstruction and selection efficiency ratio is found to be
$r_{\text{reco}}=0.949\pm 0.006\,\text{(stat)}$, where the main systematic
uncertainties in the numerator and denominator cancel since the kinematic
selections are mostly identical for both decays. The trigger efficiency ratio
$r_{\text{trig}}=1.057\pm 0.008\,\text{(stat)}$ has been computed taking into
account the contributions from the different trigger configurations during the
data taking period.
The yields of the two channels are extracted from a simultaneous unbinned
maximum likelihood fit to the invariant mass distributions of the data.
Signals are described using a Crystal Ball function [19], with the tail
parameters fixed to their values extracted from MC simulation and the mass
difference between the $B^{0}$ and $B^{0}_{s}$ signals fixed [20]. The width
of the signal peak is left as a free parameter. Combinatorial background is
parametrized by an exponential function with a different decay constant for
each channel. The results of the fit are shown in Fig. 1. The number of events
obtained for $B^{0}\\!\rightarrow K^{*0}\gamma$ and
$B^{0}_{s}\\!\rightarrow\phi\gamma$ are $1685\pm 52$ and $239\pm 19$, with a
signal over background ratio of $S/B=3.1\pm 0.4$ and $3.7\pm 1.3$ in a $\pm
3\sigma$ window, respectively.
Figure 1: Result of the fit for the $B^{0}\\!\rightarrow K^{*0}\gamma$ (left)
and $B^{0}_{s}\\!\rightarrow\phi\gamma$ (right). The black points represent
the data and the fit result is represented as a solid line. The signal is
fitted with a Crystal Ball function (light dashed line) and the background is
described as an exponential (dark dashed line). Below each invariant mass
plot, the Poisson $\chi^{2}$ residuals [21] are shown.
Several potential sources of peaking background have been studied:
$B^{0}_{(s)}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ and $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}\pi^{0}$, where the two photons from the $\pi^{0}$ can be merged
into a single cluster and misidentified as a single photon,
$\Lambda_{b}^{0}\\!\rightarrow\Lambda^{*0}(Kp)\,\gamma$, where the proton can
be misidentified as a pion or a kaon, and the irreducible
$B^{0}_{s}\\!\rightarrow K^{*0}\gamma$. Their invariant mass distributions and
selection efficiencies have been evaluated from simulated events and the
number of predicted background events is determined and subtracted from the
signal yield.
$B$ decays in which one of the decay products has not been reconstructed, such
as $B\\!\rightarrow(K^{*0}\pi^{0})X$, tend to accumulate towards lower values
in the invariant mass distribution but can contaminate the signal peak.
However, their contributions have not been included in the fit, and the
correction to the fitted signal yield has been quantified by means of a
statistical study. The mass distribution of the partially reconstructed $B$
decays is first extracted from a sample of simulated events and the
corresponding shape has been added to the fit with a free amplitude. The fit
is then repeated many times varying the shape parameters and the amplitude of
the partially reconstructed component within their uncertainties. The
correction to be applied to the signal yield and its uncertainty at a $95\%$
confidence level are determined from the obtained distribution of the signal
yield variation.
The effects of the cross-feed between the two channels, i.e.
$B^{0}\\!\rightarrow K^{*0}\gamma$ signal misidentified as
$B^{0}_{s}\\!\rightarrow\phi\gamma$ and vice-versa, as well as the presence of
multiple $B$ candidates per event, have also been computed using simulation.
The statistical uncertainty due to finite MC sample size is taken as the
uncertainty in these corrections.
Table 1 summarizes all the corrections applied to the fitted signal yields, as
well as the corresponding uncertainties, for each source of background.
Table 1: Correction factors and corresponding uncertainties affecting the signal yields, in percent, induced by peaking backgrounds, partially reconstructed backgrounds, signal cross-feed and multiple candidates. The total uncertainty is obtained by summing the individual contributions in quadrature. | $B^{0}\\!\rightarrow K^{*0}\gamma$ | $B^{0}_{s}\\!\rightarrow\phi\gamma$ | Ratio
---|---|---|---
Contribution | Corr. | Error | Corr. | Error | Corr. | Error
$B^{0}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ | $-1.3$ | $\pm 0.4$ | — | $<0.1$ | $-1.3$ | $\pm 0.4$
$B^{0}_{s}\\!\rightarrow K^{+}\pi^{-}\pi^{0}$ | $-0.5$ | $\pm 0.5$ | — | $<0.1$ | $-0.5$ | $\pm 0.5$
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}\pi^{0}$ | — | $<0.1$ | $-1.3$ | $\pm 1.3$ | $+1.3$ | $\pm 1.3$
$\Lambda_{b}^{0}\\!\rightarrow\Lambda^{*0}\gamma$ | $-0.7$ | $\pm 0.2$ | $-0.3$ | $\pm 0.2$ | $-0.4$ | $\pm 0.3$
$B^{0}_{s}\\!\rightarrow K^{*0}\gamma$ | $-0.8$ | $\pm 0.4$ | — | — | $-0.8$ | $\pm 0.4$
Partially reconstructed $B$ | $+0.04$ | ${}^{+3.1}_{-0.2}$ | $+4.5$ | ${}^{+1.3}_{-2.9}$ | $-4.5$ | ${}^{+4.2}_{-1.3}$
$\phi\gamma/K^{*0}\gamma$ cross-feed | $-0.4$ | $\pm 0.2$ | — | $<0.1$ | $-0.4$ | $\pm 0.2$
Multiple candidates | $-0.5$ | $\pm 0.2$ | $-0.3$ | $\pm 0.2$ | $-0.2$ | $\pm 0.3$
Total | $-4.2$ | ${}^{+3.2}_{-0.9}$ | $+2.6$ | ${}^{+1.9}_{-3.2}$ | $-6.8$ | ${}^{+4.5}_{-2.0}$
The ratio of branching fractions from Eq. 1 is calculated using the fitted
yields of the signal corrected for the backgrounds, the values of the visible
branching fractions [20], the LHCb measurement of $f_{s}/f_{d}$ [22, 23], and
the values of the efficiency ratios described above. The result is
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.12\pm 0.08\mathrm{(stat)}.$
## 5 Systematic uncertainties
The limited size of the MC sample used in the calculation of $r_{\text{acc}}$,
$r_{\text{reco}}$, and $r_{\text{trig}}$ induces a systematic uncertainty in
the ratio of branching fractions. In addition, $r_{\text{acc}}$ is affected by
uncertainties in the hadron reconstruction efficiency, arising from
differences in the interaction of pions and kaons with the detector and the
uncertainties in the description of the material of the detector. Differences
in the mass window size of the vector mesons, combined with small differences
in the position of the $K^{*0}(\phi)$ mass peaks between data and MC, produce
a systematic uncertainty in $r_{\text{reco}}$ which has been evaluated by
moving the centre of the mass window to the value found in data. The
reliability of the simulation to describe the $\text{IP}\,\chi^{2}$ of the
tracks and the $B$ vertex isolation has been propagated into an uncertainty
for $r_{\text{reco}}$. For this, the MC sample has been reweighted to
reproduce the background-subtracted distributions from data, obtained by
applying the sPlot technique [24] to separate signal and background
components, using the invariant mass of the $B$ candidate as the discriminant
variable. No further systematic errors are associated with the use of MC
simulation, since kinematic properties of the decays are known to be well
modelled. Systematic uncertainties associated with the photon are negligible
due to the fact that its reconstruction in both decays is identical.
The systematic uncertainty associated with the PID calibration method has been
evaluated using MC simulation. The statistical error due to the size of the
kaon and pion calibration samples has also been propagated to
$r_{\text{PID}}$.
The systematic effect introduced by applying a $B$ mass window cut of $\pm
800\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ has been evaluated by repeating
the fit procedure with a tighter $B$ mass window reduced to $\pm
600\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Table 2 summarizes all sources of systematic uncertainty, including the
background contributions detailed in Table 1. The uncertainty on the ratio of
efficiency-corrected yields is obtained by combining the individual sources in
quadrature. The uncertainty on the ratio $f_{s}/f_{d}$ is given as a separate
source of uncertainty.
Table 2: Summary of contributions to the relative systematic uncertainty on the ratio of branching fractions. Note that $f_{s}/f_{d}$ is quoted as a separate systematic uncertainty. Source | Uncertainty (%)
---|---
Acceptance ($r_{\text{acc}}$) | $\pm 0.3$
Selection ($r_{\text{reco}}$) | $\pm 1.4$
PID efficiencies ($r_{\text{PID}}$) | $\pm 2.7$
Trigger ($r_{\text{trig}}$) | $\pm 0.8$
$B$ mass window | $\pm 0.9$
Background | ${}^{+4.5}_{-2.0}$
Visible fraction of vector mesons | $\pm 1.0$
Quadratic sum of above | ${}^{+5.4}_{-3.3}$
$f_{s}/f_{d}$ | ${}^{+7.9}_{-7.5}$
Besides $f_{s}/f_{d}$, the dominant source of systematic uncertainty is the
imperfect modelling of the backgrounds due to partially reconstructed $B$
decays. This specific uncertainty is expected to be reduced when more data are
available.
## 6 Results and conclusions
In $0.37\,\mbox{\,fb}^{-1}$ of $pp$ collisions at a centre of mass energy of
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ the ratio of branching fractions
of $B^{0}\\!\rightarrow K^{*0}\gamma$ and $B^{0}_{s}\\!\rightarrow\phi\gamma$
decays has been measured to be
$\frac{{\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)}{{\cal
B}(B^{0}_{s}\\!\rightarrow\phi\gamma)}=1.12\pm
0.08\mathrm{(stat)}\phantom{.}^{+0.06}_{-0.04}\mathrm{(syst)}\phantom{.}^{+0.09}_{-0.08}(f_{s}/f_{d})$
in good agreement with the theoretical prediction of $1.0\pm 0.2$ [11].
Using ${\cal B}(B^{0}\\!\rightarrow K^{*0}\gamma)=(4.33\pm 0.15)~{}\times
10^{-5}$ [7], one obtains
${\cal B}(B^{0}_{s}\\!\rightarrow\phi\gamma)=(3.9\pm 0.5)\times 10^{-5}$
(statistical and systematic errors combined), which agrees with the previous
experimental value. This is the most precise measurement of the
$B^{0}_{s}\\!\rightarrow\phi\gamma$ branching fraction to date.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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|
arxiv-papers
| 2012-02-28T16:05:38 |
2024-09-04T02:49:27.996840
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S.\n Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw,\n S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi,\n A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, K. de\n Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff,\n J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.A.\n Cowan, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, S. De\n Capua, M. De Cian, F. De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone,\n D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista,\n F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn,\n B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March,\n H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, J. Magnin, 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, M. Meissner, M. Merk, J. Merkel, R. Messi,\n S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil,\n D. Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R.\n Muresan, B. Muryn, B. Muster, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada,\n R. Nandakumar, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, A.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, K. Pal, J. Palacios,\n A. Palano, 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, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, K. Sobczak, F.J.P. Soler, A.\n Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V.K. Subbiah, 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, N. Torr,\n E. Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D.R.\n Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang,\n W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Albert Puig",
"url": "https://arxiv.org/abs/1202.6267"
}
|
1202.6351
|
# Numerical Study of the Properties of the Central Moment Lattice Boltzmann
Method
Yang Ning yning@uwyo.edu Department of Mechanical Engineering, University of
Wyoming, Laramie, WY 82071
Kannan N. Premnath knandhap@uwyo.edu Department of Mechanical Engineering,
University of Wyoming, Laramie, WY 82071
###### Abstract
Central moment lattice Boltzmann method (LBM) is one of the more recent
developments among the lattice kinetic schemes for computational fluid
dynamics. A key element in this approach is the use of _central_ moments to
specify collision process and forcing, and thereby naturally maintaining
Galilean invariance, an important characteristic of fluid flows. When the
different central moments are relaxed at different rates like in a standard
multiple relaxation time (MRT) formulation based on _raw_ moments, it is
endowed with a number of desirable physical and numerical features. Since the
collision operator exhibits a cascaded structure, this approach is also known
as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a
systematic study of its numerical properties, such as accuracy, grid
convergence and stability for well defined canonical problems is lacking and
the present work is intended to fulfill this need. We perform a quantitative
study of the performance of the cascaded LBM for a set of benchmark problems
of differing complexity, viz., Poiseuille flow, decaying Taylor-Green vortex
flow and lid-driven cavity flow. We first establish its grid convergence and
demonstrate second order accuracy under diffusive scaling for both the
velocity field and its derivatives, i.e. components of the strain rate tensor,
as well. The method is shown to quantitatively reproduce steady/unsteady
analytical solutions or other numerical results with excellent accuracy. The
cascaded MRT LBM based on central moments is found to be of similar accuracy
when compared with the standard MRT LBM based on raw moments, when detailed
comparison of the flow fields are made, with both well reproducing even small
scale vortical features. Numerical experiments further demonstrate that the
central moment MRT LBM results in significant stability improvements when
compared with certain existing collision models at moderate additional
computational cost.
###### pacs:
47.11.Qr,05.20.Dd,47.27.-i
††preprint: PREPRINT
## I Introduction
Early developments in the area of computational fluid dynamics (CFD) have
focused on the solution of the classical discretizations of the continuum
description of fluid motion. During the last two decades, there has been much
interest and effort in the development of schemes that derive their basis on a
more smaller scale picture involving particle motion, which may be classified
as mesoscopic methods. One of the most promising of such approaches is the
lattice Boltzmann method (LBM) Chen and Doolen (1998); Succi (2001); Luo et
al. (2010). Based on kinetic theory, it involves the solution of the lattice
Boltzmann equation (LBE), which specifies the evolution of the particle
populations along discrete directions, which comprise the lattice. This
evolution involves a Lagrangian free streaming process along such lattice
links and a local collision step specified as a relaxation process. Various
elements involved in these two simple steps are constructed based on symmetry
considerations, while obeying certain conservation constraints, in such a way
that they recover the dynamics of fluid flow in the near incompressible limit.
The resulting scheme has a number of desirable features. These include the
ability to naturally represent complex fluid physics such as multiphase and
multicomponent flows based on kinetic theory, amenability to parallelization
due to the locality of the method and representation of flow through complex
geometries. Furthermore, due to the exact conservation in the streaming step
and machine round-off conservation in the collision process, it has
considerably low numerical dissipation for a second-order numerical scheme
Ubertini et al. (2010). Due to such competitive advantages, the LBM has found
applications in the simulation of a wide range of fluid flow problems Chen and
Doolen (1998); Succi (2001); Luo et al. (2010).
Since the LBM is usually developed by means of a bottom-up strategy, there is
certain level of flexibility in the construction of its various elements to
recover the macroscopic fluid motion. In particular, the choice of a suitable
collision model can have profound influence on the fidelity as well as the
stability of the approach. As such, the construction of the collision step has
been the subject of considerable attention since the inception of the LBM. The
simplest among these is the so-called single-relaxation-time (SRT) model Chen
et al. (1992); Qian et al. (1992), which is based on the Bhatnagar-Gross-Krook
(BGK) approximation Bhatnagar et al. (1954). While it is popular, it has
limitations in the representation of certain flow problems and is generally
prone to numerical instability, particularly at high Reynolds numbers. A major
development to address these aspects is the moment approach d‘Humières (1992),
which has been constructed based on multiple relaxation times (MRT) in
particular to significantly improve the numerical stability Lallemand and Luo
(2000). While it is related to its precursor involving a more general
relaxation approximation Higuera and Jiménez (1989); Higuera et al. (1989),
the characteristic difference being that it performs collision in an
orthogonal moment space leading to an efficient and flexible numerical scheme.
This moment approach, which is designated as the standard MRT formulation in
this paper, has recently been studied and compared with some of the other
collision models in detail Luo et al. (2011). A simpler version that is
intermediate between the SRT and MRT model is the so-called two-relaxation-
time (TRT) model Ginzburg (2005), in which the moments of even and odd orders
are relaxed to their equilibrium at different rates. This, along with the MRT
model, can be adjusted such that it results in a minimization of undesirable
discrete kinetic effects near walls. Another significant development is the
so-called entropic LBM Karlin et al. (1999). It involves an equilibria, which
is based on a constrained minimization of a Lyapunov-type functional. By
modulating the collision process through enforcing entropy involution locally,
this approach aims to maintain non-linear stability. This approach has
resulted in a number of simplified variants recently Asinari and Karlin
(2009); Karlin et al. (2011).
An important physical feature of the fluid motion is that their description be
independent of any inertial frame of reference (e.g. Pope (2000)). This
invariance property, which is termed as the Galilean invariance, should be
satisfied by any model or numerical scheme for its general applicability.
Furthermore, it has recently been shown that stabilization of classical
schemes for compressible flow can be achieved when they are specifically
constructed to respect this physical property Scovazzi (2007a, b); Hughes et
al. (2010). Keeping these general notions in mind, Galilean invariance can be
naturally prescribed in the LBM when its various elements are represented in
terms of the _central_ moments, i.e. moments obtained by shifting the particle
velocity by the local fluid velocity. That is, any dynamical changes due to
the collision process and impressed forces can be represented in terms of
suitable variations of a set of such central moments. In particular, a
collision model based on the relaxation of central moments was constructed
recently Geier et al. (2006). The model exhibits a cascaded structure, which
was later shown to be equivalent to considering a generalized equilibrium in
the lattice or rest frame of reference Asinari (2008). These central moments
can be relaxed at different rates during collision leading to a cascaded MRT
or central moment MRT formulation, whereas by contrast the standard MRT
formulation considers _raw_ moments. A systematic derivation of this approach
by including the effect of impressed forces based on central moments was
presented in Premnath and Banerjee (2009). This leads to considering
generalized sources, analogous to the generalized equilibrium in the rest
frame of reference. They also presented a detailed Chapman-Enskog analysis of
the cascaded MRT LBM for its consistency with the macroscopic fluid dynamical
equations of motion. This approach was further extended to various lattice
models in three-dimensions in Premnath and Banerjee (2011), in the cylindrical
coordinate system for axisymmetric flows in Premnath and Ning (2012) and for
accounting of non-equilibrium effects in Premnath and Banerjee (2012).
Prior work on the cascaded LBM as discussed above have focused mainly on
method developments or their mathematical analysis, with little attention
towards their numerics except for few validation cases. In particular, a
detailed numerical study of the properties of the cascaded LBM for established
benchmark problems and also their performance against other LBM approaches is
lacking. The focus of the present work is intended to fill this gap by
presenting a systematic study of the numerical properties of the cascaded LBM,
viz., grid convergence, accuracy and stability for various canonical problems
of differing complexity in terms of flow features and temporal evolution.
Establishing the reliability and merits of the method in quantitative terms
could provide confidence in their extension and applications to various
complex flow problems of interest. To study the numerics of the cascaded LBM,
we consider the Poiseuille flow, decaying Taylor-Green vortex flow, and lid-
driven cavity flow, for which either analytical solutions or detailed prior
numerical results are available for comparison. Much of the literature on the
LBM with other collision models on grid convergence studies have focused only
on those for the velocity field. In this work, we present numerical results on
the grid convergence of the cascaded LBM for the velocity field as well as its
derivatives, i.e. the strain rate tensor. Furthermore, an advantage of the
kinetic schemes such as the LBM is that the strain rate tensor can be computed
locally in terms of non-equilibrium moments. In this work, we also present a
direct comparison of the results obtained using the non-equilibrium moments of
the cascaded LBM with those involving the finite differencing of the velocity
field at various locations for the lid-driven cavity flow problem to assess
their quantitative accuracy. It may be noted that a detailed comparison study
of the SRT and the standard MRT models have recently been performed in Luo et
al. (2011). Thus, in this work, we present a quantitative accuracy comparison
between the standard MRT LBM and the cascaded or central moment MRT LBM for
the lid-driven cavity flow. Finally, we will discuss the numerical stability
performance of the various LBM schemes for the above benchmark problem.
The paper is organized as follows. Section II presents the details of the
particular version of the cascaded MRT LBM used in this work. In Sec. III, the
results of the grid convergence study of the cascaded MRT LBM together with
the raw moment based standard MRT LBM for the three benchmark problems are
discussed. Subsequently, the quantitative accuracy of the cascaded LBM is
demonstrated by making detailed comparison with either analytical or other
numerical solutions for the above problems in Sec. IV. In Sec. V, numerical
stability test results are presented for the lid-driven cavity flow using the
SRT LBM, standard MRT LBM and cascaded MRT LBM. Summary and conclusions of
this work are given in Sec. VI.
## II Cascaded Lattice Boltzmann Method
We will now discuss the main features of the cascaded LBM. Similar to the
standard MRT LBM, the cascaded MRT LBM also performs collisions in moment
space, but these moments are obtained by shifting the particle velocity by the
local fluid velocity, i.e. using central moments. As a result, the approach
can naturally maintain Galilean invariance. Central moment relaxation process
was specified in Geier et al. (2006), which was re-interpreted by considering
generalized equilibrium in Asinari (2008). Its detailed mathematical
consistency analysis in a MRT formulation with forcing was carried out in
Premnath and Banerjee (2009). The computations of the cascaded LBM are
actually performed after transforming the central moments into raw moments by
means of a binomial formula. In this work, the specific formulation of the
cascaded LBM given in Premnath and Banerjee (2009), whose details are somewhat
different from that given in Geier et al. (2006), is used. This is briefly
discussed in what follows.
In this work, the standard two-dimensional, nine velocity (D2Q9) lattice is
employed. We consider the usual bra-ket notations in the description of the
method as it provides a convenient representation. That is, we consider the
depiction of vectors as $\langle\phi|$ and $|\phi\rangle$, where
$\langle\phi|$ represents a row vector of $\phi$ of any state in the
corresponding direction $(\phi_{0},\phi_{1},\phi_{2},\cdots,\phi_{8})$ and
$|\phi\rangle$ represents a column vector
$(\phi_{0},\phi_{1},\phi_{2},\cdots,\phi_{8})^{T}$. The inner product
$\sum^{8}_{\alpha=0}\phi_{\alpha}\varphi_{\alpha}$ is then denoted by
$\langle\phi|\varphi\rangle$. As the cascaded LBM is a moment approach, we
need a set of nine linearly independent moment basis vectors for its
specification. The (raw) moments of the distribution function $f_{\alpha}$ of
different orders can be defined as $\sum^{8}_{\alpha=0}e^{m}_{\alpha
x}e^{n}_{\alpha y}f_{\alpha}$. Here, $\alpha$ is the discrete particle
direction, and $m$ and $n$ are integers. Thus, a set of nine linearly
independent nonorthogonal basis vectors obtained using the monomials
$e^{m}_{\alpha x}e^{n}_{\alpha y}$ in an ascending order can be written as
$\begin{split}&|\rho\rangle=||\vec{e}_{\alpha}|^{0}\rangle=(1,1,1,1,1,1,1,1,1)^{T},\\\
&|e_{\alpha x}\rangle=(0,1,0,-1,0,1,-1,-1,1)^{T},\\\ &|e_{\alpha
y}\rangle=(0,0,1,0,-1,1,1,-1,-1)^{T},\\\ &|e^{2}_{\alpha x}+e^{2}_{\alpha
y}\rangle=(0,1,1,1,1,2,2,2,2)^{T},\\\ \end{split}$
$\begin{split}&|e^{2}_{\alpha x}-e^{2}_{\alpha
y}\rangle=(0,1,-1,1,-1,0,0,0,0)^{T},\\\ &|e_{\alpha x}e_{\alpha
y}\rangle=(0,0,0,0,0,1,-1,1,-1)^{T},\\\ &|e^{2}_{\alpha x}e_{\alpha
y}\rangle=(0,0,0,0,0,1,1,-1,-1)^{T},\\\ &|e_{\alpha x}e^{2}_{\alpha
y}\rangle=(0,0,0,0,0,1,-1,-1,1)^{T},\\\ &|e^{2}_{\alpha x}e^{2}_{\alpha
y}\rangle=(0,0,0,0,0,1,1,1,1)^{T}.\end{split}$ (1)
This can be transformed by means of the Gram-Schmidt procedure into an
equivalent set of _orthogonal_ basis vectors, which provides a computationally
more efficient and convenient setting for the description of the method. As a
result, we have the following orthogonal set Premnath and Banerjee (2009):
$\begin{split}&|K_{0}\rangle=|\rho\rangle,\\\ &|K_{1}\rangle=|e_{\alpha
x}\rangle,\\\ &|K_{2}\rangle=|e_{\alpha y}\rangle,\\\
&|K_{3}\rangle=3|e^{2}_{\alpha x}+e^{2}_{\alpha y}\rangle-4|\rho\rangle,\\\
&|K_{4}\rangle=|e^{2}_{\alpha x}-e^{2}_{\alpha y}\rangle,\\\
&|K_{5}\rangle=|e_{\alpha x}e_{\alpha y}\rangle,\\\
&|K_{6}\rangle=-3|e^{2}_{\alpha x}e_{\alpha y}\rangle+2|e_{\alpha
y}\rangle,\\\ &|K_{7}\rangle=-3|e_{\alpha x}e^{2}_{\alpha
y}\rangle+2|e_{\alpha x}\rangle,\\\ &|K_{8}\rangle=9|e^{2}_{\alpha
x}e^{2}_{\alpha y}\rangle-6|e^{2}_{\alpha x}+e^{2}_{\alpha
y}\rangle+4|\rho\rangle.\end{split}$ (2)
Collecting the above set of vectors as a matrix $\mathcal{K}$, it immediately
follows that $\mathcal{K}\mathcal{K}^{T}$ is a diagonal matrix, owing to
orthogonality. This orthogonal matrix $\mathcal{K}$ can be written in
component form as
$\begin{split}\mathcal{K}&=\bigl{[}|K_{0}\rangle,|K_{1}\rangle,|K_{2}\rangle,|K_{3}\rangle,|K_{4}\rangle,|K_{5}\rangle,|K_{6}\rangle,|K_{7}\rangle,|K_{8}\rangle)\bigr{]}\\\
&=\begin{bmatrix}1&0&0&-4&0&0&0&0&4\\\ 1&1&0&-1&1&0&0&2&-2\\\
1&0&1&-1&-1&0&2&0&-2\\\ 1&-1&0&-1&1&0&0&-2&-2\\\ 1&0&-1&-1&-1&0&-2&0&-2\\\
1&1&1&2&0&1&-1&-1&1\\\ 1&-1&1&2&0&-1&-1&1&1\\\ 1&-1&-1&2&0&1&1&1&1\\\
1&1&-1&2&0&-1&1&-1&1.\end{bmatrix}\end{split}$ (3)
To specify the collision step and forcing, we need the central moments of the
local equilibrium and sources, which can be obtained as follows. First, the
local Maxwell-Boltzmann distribution function in continuous particle velocity
space $(\xi_{x},\xi_{y})$ is written as $f^{\mathcal{M}}\equiv
f^{\mathcal{M}}(\rho,\vec{u},\xi_{x},\xi_{y})=\frac{\rho}{2\pi
c_{s}^{2}}\exp{\left[-\frac{(\vec{\xi}-\vec{u})^{2}}{2c_{s}^{2}}\right]}$,
where $c_{s}$ is the speed of sound. Typically, $c_{s}^{2}=1/3$. Based on
this, the continuous central moments of the equilibrium of order $(m+n)$ can
be defined as
$\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\mathcal{M}}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}$,
which yields
$\begin{split}|\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}}\rangle&=(\widehat{\Pi}^{\mathcal{M}}_{0},\widehat{\Pi}^{\mathcal{M}}_{x},\widehat{\Pi}^{\mathcal{M}}_{y},\widehat{\Pi}^{\mathcal{M}}_{xx},\widehat{\Pi}^{\mathcal{M}}_{yy},\widehat{\Pi}^{\mathcal{M}}_{xy},\widehat{\Pi}^{\mathcal{M}}_{xxy},\widehat{\Pi}^{\mathcal{M}}_{xyy},\widehat{\Pi}^{\mathcal{M}}_{xxyy})^{T},\\\
&=(\rho,0,0,c_{s}^{2}\rho,c_{s}^{2}\rho,0,0,0,c_{s}^{4}\rho)^{T}.\end{split}$
(4)
Considering that the impressed forces only influence the fluid momentum, the
central moments of the sources of order $(m+n)$ due to a force field
$(F_{x},F_{y})$ defined by
$\widehat{\Gamma}^{\mathcal{F}}_{x^{m}y^{n}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Delta
f^{\mathcal{F}}(\xi_{x}-u_{x})^{m}(\xi_{y}-u_{y})^{n}d\xi_{x}d\xi_{y}$, where
$\Delta f^{\mathcal{F}}$ is the change in the distribution function due to
force fields, can be simply written as Premnath and Banerjee (2009)
$\begin{split}|\widehat{\Gamma}^{\mathcal{F}}_{x^{m}y^{n}}\rangle&=(\widehat{\Gamma}^{\mathcal{F}}_{0},\widehat{\Gamma}^{\mathcal{F}}_{x},\widehat{\Gamma}^{\mathcal{F}}_{y},\widehat{\Gamma}^{\mathcal{F}}_{xx},\widehat{\Gamma}^{\mathcal{F}}_{yy},\widehat{\Gamma}^{\mathcal{F}}_{xy},\widehat{\Gamma}^{\mathcal{F}}_{xxy},\widehat{\Gamma}^{\mathcal{F}}_{xyy},\widehat{\Gamma}^{\mathcal{F}}_{xxyy})^{T},\\\
&=(0,F_{x},F_{y},0,0,0,0,0,0)^{T}.\end{split}$ (5)
Based on the above continuous central moments, the elements of the cascaded
LBE can be formulated. Using the trepezoidal rule representation of the source
term, the cascaded LBE can be written as Premnath and Banerjee (2009)
$f_{\alpha}(\vec{x}+\vec{e}_{\alpha}{\delta_{t}},t+\delta_{t})=f_{\alpha}(\vec{x},t)+\Omega^{\mathcal{C}}_{\alpha(\vec{x},t)}+\frac{1}{2}\bigl{[}S_{\alpha(\vec{x},t)}+S_{\alpha(\vec{x}+\vec{e}_{\alpha},t+\delta_{t})}\bigr{]}.$
(6)
Here, the collision term $\Omega^{\mathcal{C}}_{\alpha}$ can be represented as
$\Omega^{\mathcal{C}}_{\alpha}\equiv\Omega^{\mathcal{C}}_{\alpha}(\mathbf{f},\bf{\widehat{g}})=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}$,
where $\mathbf{f}$ $\equiv|f_{\alpha}\rangle=(f_{0},f_{1},\cdots,f_{8})^{T}$
is the vector of distribution functions and $\mathbf{\widehat{g}}$
$\equiv|\widehat{g}_{\alpha}\rangle=(\widehat{g}_{0},\widehat{g}_{1},\cdots,\widehat{g}_{8})^{T}$
is the vector of unknown collision kernel to be obtained later. Owing to the
cascaded nature of the central moment based approach, it satisfies the
following functional relation
$\widehat{g}_{\alpha}\equiv\widehat{g}_{\alpha}(\mathbf{f},\widehat{g}_{\beta}),\
\ \ \ \ \beta=0,1,\cdots,\alpha-1$. The discrete form of the source term
$S_{\alpha}$ in the cascaded LBE given above represents the influence of the
force field $(F_{x},F_{y})$ in the velocity space and is defined as
$\mathbf{S}\equiv|S_{\alpha}\rangle=(S_{0},S_{1},S_{2},\cdots,S_{8})^{T}$.
Noting that Eq. (6) is semi-implicit, by using the standard variable
transformation $\overline{f}=f_{\alpha}-\frac{1}{2}S_{\alpha}$, its
implicitness can be effectively removed. This yields
$\overline{f}_{\alpha}(\vec{x}+\vec{e}_{\alpha}{\delta_{t}},t+\delta_{t})=\overline{f}_{\alpha}(\vec{x},t)+\Omega^{\mathcal{C}}_{\alpha(\vec{x},t)}+S_{\alpha(\vec{x},t)}.$
(7)
The derivation of the collision term, i.e. the collision kernel
$\mathbf{\widehat{g}}$ and the source term $\mathbf{S}$ involves matching the
_discrete_ central moments and the _continuous_ central moments of equilibria
and sources, which are specified above, of all orders supported by the lattice
set. We designate this step as the _Galilean invariance matching principle_.
First, the discrete central moments of the distribution functions and sources
of order $(m+n)$ can be defined, respectively, as
$\widehat{\kappa}_{x^{m}y^{n}}=\langle(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}|f_{\alpha}\rangle$ and
$\widehat{\sigma}_{x^{m}y^{n}}=\langle(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}|S_{\alpha}\rangle$. Also, in terms of the transformed
distribution functions we define
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\langle(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}|\overline{f}_{\alpha}\rangle$, which
satisfies
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}=\widehat{\kappa}_{x^{m}y^{n}}-\frac{1}{2}\widehat{\sigma}_{x^{m}y^{n}}$,
and similarly for the local equilibria
$\widehat{\overline{\kappa}}^{eq}_{x^{m}y^{n}}=\langle(e_{\alpha
x}-u_{x})^{m}(e_{\alpha y}-u_{y})^{n}|\overline{f}^{eq}_{\alpha}\rangle$.
Then, the Galilean invariance matching principle reads
$\displaystyle\widehat{\kappa}^{eq}_{x^{m}y^{n}}=\widehat{\Pi}^{\mathcal{M}}_{x^{m}y^{n}},$
(8)
$\displaystyle\widehat{\sigma}_{x^{m}y^{n}}=\widehat{\Gamma}^{\mathcal{F}}_{x^{m}y^{n}}.$
(9)
This immediately specifies the various discrete central moments. Hence, we get
$\begin{split}\hskip
42.67912pt|\widehat{\kappa}^{eq}_{x^{m}y^{n}}\rangle&=(\widehat{\kappa}^{eq}_{0},\widehat{\kappa}^{eq}_{x},\widehat{\kappa}^{eq}_{y},\widehat{\kappa}^{eq}_{xx},\widehat{\kappa}^{eq}_{yy},\widehat{\kappa}^{eq}_{xy},\widehat{\kappa}^{eq}_{xxy},\widehat{\kappa}^{eq}_{xyy},\widehat{\kappa}^{eq}_{xxyy})^{T}\\\
&=(\rho,0,0,c_{s}^{2}\rho,c_{s}^{2}\rho,0,0,0,c_{s}^{4}\rho)^{T},\end{split}$
(10)
$\begin{split}|\widehat{\sigma}_{x^{m}y^{n}}\rangle&=(\widehat{\sigma}_{0},\widehat{\sigma}_{x},\widehat{\sigma}_{y},\widehat{\sigma}_{xx},\widehat{\sigma}_{yy},\widehat{\sigma}_{xy},\widehat{\sigma}_{xxy},\widehat{\sigma}_{xyy},\widehat{\sigma}_{xxyy})^{T}\\\
&=(0,F_{x},F_{y},0,0,0,0,0,0)^{T},\end{split}$ (11)
and
$\begin{split}|\widehat{\overline{\kappa}}^{eq}_{x^{m}y^{n}}\rangle=&(\widehat{\overline{\kappa}}^{eq}_{0},\widehat{\overline{\kappa}}^{eq}_{x},\widehat{\overline{\kappa}}^{eq}_{y},\widehat{\overline{\kappa}}^{eq}_{xx},\widehat{\overline{\kappa}}^{eq}_{yy},\widehat{\overline{\kappa}}^{eq}_{xy},\widehat{\overline{\kappa}}^{eq}_{xxy},\widehat{\overline{\kappa}}^{eq}_{xyy},\widehat{\overline{\kappa}}^{eq}_{xxyy})^{T},\\\
=&(\rho,-\frac{1}{2}F_{x},-\frac{1}{2}F_{y},c_{s}^{2}\rho,c_{s}^{2}\rho,0,0,0,c_{s}^{4}\rho)^{T}.\end{split}$
(12)
The next important step is to transform all the above discrete central moments
in terms of raw moments, which can be readily accomplished by means of the
following binomial formula: $\langle(e_{\alpha x}-u_{x})^{m}(e_{\alpha
y}-u_{y})^{n}|\varphi\rangle=\langle e_{\alpha x}^{m}e_{\alpha
y}^{n}|\varphi\rangle+\bigl{\langle}e_{\alpha
x}^{m}\bigl{[}\sum_{j=1}^{n}C^{n}_{j}e^{n-j}_{\alpha
y}(-1)^{j}u^{j}_{y}\bigr{]}|\varphi\bigr{\rangle}+\bigl{\langle}e_{\alpha
y}^{m}\bigl{[}\sum_{i=1}^{m}C^{m}_{i}e^{m-i}_{\alpha
x}(-1)^{i}u^{i}_{x}\bigr{]}|\varphi\bigr{\rangle}+\bigl{\langle}\bigl{[}\sum_{i=1}^{m}C^{m}_{i}e^{m-i}_{\alpha
x}(-1)^{i}u^{i}_{x}\bigr{]}\bigl{[}\sum_{j=1}^{n}C^{n}_{j}e^{n-j}_{\alpha
y}(-1)^{j}u^{j}_{y}\bigr{]}|\varphi\bigr{\rangle}$, where
$C^{p}_{q}=p!/\bigl{(}q!(p-q)!)$. Thus, we obtain the following discrete raw
moments of sources $\widehat{\sigma}_{x^{m}y^{n}}^{{}^{\prime}}$ as
$\begin{split}&\widehat{\sigma}_{0}^{{}^{\prime}}=\langle
S_{\alpha}|\rho\rangle=0,\\\ &\widehat{\sigma}_{x}^{{}^{\prime}}=\langle
S_{\alpha}|e_{\alpha x}\rangle=F_{x},\\\
&\widehat{\sigma}_{y}^{{}^{\prime}}=\langle S_{\alpha}|e_{\alpha
y}\rangle=F_{y},\\\ &\widehat{\sigma}_{xx}^{{}^{\prime}}=\langle
S_{\alpha}|e_{\alpha x}^{2}\rangle=2F_{x}u_{x},\\\
&\widehat{\sigma}_{yy}^{{}^{\prime}}=\langle S_{\alpha}|e_{\alpha
y}^{2}\rangle=2F_{y}u_{y},\\\ &\widehat{\sigma}_{xy}^{{}^{\prime}}=\langle
S_{\alpha}|e_{\alpha x}e_{\alpha y}\rangle=F_{x}u_{y}+F_{y}u_{x},\\\
&\widehat{\sigma}_{xxy}^{{}^{\prime}}=\langle S_{\alpha}|e_{\alpha
x}^{2}e_{\alpha y}\rangle=F_{y}u_{x}^{2}+2F_{x}u_{x}u_{y},\\\
&\widehat{\sigma}_{xyy}^{{}^{\prime}}=\langle S_{\alpha}|e_{\alpha x}e_{\alpha
y}^{2}\rangle=F_{x}u_{y}^{2}+2F_{y}u_{y}u_{x},\end{split}$ (13)
$\begin{split}&\widehat{\sigma}_{xxyy}^{{}^{\prime}}=\langle
S_{\alpha}|e_{\alpha x}^{2}e_{\alpha
y}^{2}\rangle=2F_{x}u_{x}u_{y}^{2}+2F_{y}u_{y}u_{x}^{2}.\end{split}$
Based on the above, we now obtain the source terms projected to the orthogonal
moment basis vectors, i.e. $\braket{K_{\beta}}{S_{\alpha}}$,
$\beta=0,1,2,\ldots,8$. This intermediate step is needed to obtain the source
terms in the velocity space. It immediately follows that
$\displaystyle\widehat{m}^{s}_{0}=\braket{K_{0}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\widehat{m}^{s}_{1}=\braket{K_{1}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{x},$
$\displaystyle\widehat{m}^{s}_{2}=\braket{K_{2}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle F_{y},$
$\displaystyle\widehat{m}^{s}_{3}=\braket{K_{3}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 6(F_{x}u_{x}+F_{y}u_{y}),$
$\displaystyle\widehat{m}^{s}_{4}=\braket{K_{4}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle 2(F_{x}u_{x}-F_{y}u_{y}),$
$\displaystyle\widehat{m}^{s}_{5}=\braket{K_{5}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(F_{x}u_{y}+F_{y}u_{x}),$
$\displaystyle\widehat{m}^{s}_{6}=\braket{K_{6}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(2-3u_{x}^{2})F_{y}-6F_{x}u_{x}u_{y},$
$\displaystyle\widehat{m}^{s}_{7}=\braket{K_{7}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle(2-3u_{y}^{2})F_{x}-6F_{y}u_{y}u_{x},$
$\displaystyle\widehat{m}^{s}_{8}=\braket{K_{8}}{S_{\alpha}}$ $\displaystyle=$
$\displaystyle
6\left[(3u_{y}^{2}-2)F_{x}u_{x}+(3u_{x}^{2}-2)F_{y}u_{y}\right].$
Equivalently, this can be written in matrix form as
$\mathcal{K}^{T}\mathbf{S}=(\mathcal{K}\cdot\mathbf{S})_{\alpha}=(\braket{K_{0}}{S_{\alpha}},\braket{K_{1}}{S_{\alpha}},\braket{K_{2}}{S_{\alpha}},\ldots,\braket{K_{8}}{S_{\alpha}})=(\widehat{m}^{s}_{0},\widehat{m}^{s}_{1},\widehat{m}^{s}_{2},\ldots,\widehat{m}^{s}_{8})^{T}\equiv\ket{\widehat{m}^{s}_{\alpha}}$.
By exploiting the orthogonal property of $\mathcal{K}$, i.e.
$\mathcal{K}^{-1}=\mathcal{K}^{T}\cdot D^{-1}$, where the diagonal matrix is
$D=\mbox{diag}(\braket{K_{0}}{K_{0}},\braket{K_{1}}{K_{1}},\braket{K_{2}}{K_{2}},\ldots,\braket{K_{8}}{K_{8}})$,
we exactly invert the above to obtain the source terms in velocity space
$S_{\alpha}$ as
$\begin{split}S_{0}=&\frac{1}{9}\bigl{(}-m_{3}^{s}+m_{8}^{s}\bigr{)},\\\
S_{1}=&\frac{1}{36}\bigl{(}6m_{1}^{s}-m_{3}^{s}+9m_{4}^{s}+6m_{7}^{s}-2m_{8}^{s}\bigr{)},\\\
S_{2}=&\frac{1}{36}\bigl{(}6m_{2}^{s}-m_{3}^{s}-9m_{4}^{s}+6m_{6}^{s}-2m_{8}^{s}\bigr{)},\\\
S_{3}=&\frac{1}{36}\bigl{(}-6m_{1}^{s}-m_{3}^{s}+9m_{4}^{s}-6m_{7}^{s}-2m_{8}^{s}\bigr{)},\\\
S_{4}=&\frac{1}{36}\bigl{(}-6m_{2}^{s}-m_{3}^{s}-9m_{4}^{s}-6m_{6}^{s}-2m_{8}^{s}\bigr{)},\\\
S_{5}=&\frac{1}{36}\bigl{(}6m_{1}^{s}+6m_{2}^{s}+2m_{3}^{s}+9m_{5}^{s}-3m_{6}^{s}-3m_{7}^{s}+m_{8}^{s}\bigr{)},\\\
S_{6}=&\frac{1}{36}\bigl{(}-6m_{1}^{s}+6m_{2}^{s}+2m_{3}^{s}-9m_{5}^{s}-3m_{6}^{s}+3m_{7}^{s}+m_{8}^{s}\bigr{)},\\\
S_{7}=&\frac{1}{36}\bigl{(}-6m_{1}^{s}-6m_{2}^{s}+2m_{3}^{s}+9m_{5}^{s}+3m_{6}^{s}+3m_{7}^{s}+m_{8}^{s}\bigr{)},\\\
S_{8}=&\frac{1}{36}\bigl{(}6m_{1}^{s}-6m_{2}^{s}+2m_{3}^{s}-9m_{5}^{s}+3m_{6}^{s}-3m_{7}^{s}+m_{8}^{s}\bigr{)}.\end{split}$
(14)
The discrete raw moments of the transformed distribution functions
$\widehat{\overline{\kappa}}_{x^{m}y^{n}}^{{}^{\prime}}$, which will be needed
in the evaluation of the collision kernel, can be conveniently written as
follows:
$\begin{split}\widehat{\overline{\kappa}}_{0}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|\rho\rangle&=\rho,\\\
\widehat{\overline{\kappa}}_{x}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}\rangle&=\rho u_{x}-\frac{1}{2}F_{x},\\\
\widehat{\overline{\kappa}}_{y}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
y}\rangle&=\rho u_{y}-\frac{1}{2}F_{y},\\\
\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}^{2}\rangle&=\left(\sum_{\alpha}^{\\{1,3,5,6,7,8\\}}\right)\otimes\overline{f}_{\alpha},\\\
\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
y}^{2}\rangle&=\left(\sum_{\alpha}^{\\{2,4,5,6,7,8\\}}\right)\otimes\overline{f}_{\alpha},\\\
\end{split}$ (15)
$\begin{split}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}e_{\alpha
y}\rangle&=\left(\sum_{\alpha}^{\\{5,7\\}}-\sum_{\alpha}^{\\{6,8\\}}\right)\otimes\overline{f}_{\alpha},\\\
\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}^{2}e_{\alpha
y}\rangle&=\left(\sum_{\alpha}^{\\{5,6\\}}-\sum_{\alpha}^{\\{7,8\\}}\right)\otimes\overline{f}_{\alpha},\\\
\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}e_{\alpha
y}^{2}\rangle&=\left(\sum_{\alpha}^{\\{5,8\\}}-\sum_{\alpha}^{\\{6,7\\}}\right)\otimes\overline{f}_{\alpha},\\\
\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}=\langle\overline{f}_{\alpha}|e_{\alpha
x}^{2}e_{\alpha
y}^{2}\rangle&=\left(\sum_{\alpha}^{\\{5,6,7,8\\}}\right)\otimes\overline{f}_{\alpha}.\end{split}$
where we have used
$\left(a\sum_{\alpha}^{A}+b\sum_{\beta}^{B}+\cdots\right)\otimes\overline{f}_{\alpha}=a(\overline{f}_{\alpha_{1}}+\overline{f}_{\alpha_{2}}+\overline{f}_{\alpha_{3}}+\cdots)+b(\overline{f}_{\beta_{1}}+\overline{f}_{\beta_{2}}+\overline{f}_{\beta_{3}}+\cdots)+\cdots$,
with $A=\\{\alpha_{1},\alpha_{2},\alpha_{3},\cdots\\}$,
$B=\\{\beta_{1},\beta_{2},\beta_{3},\cdots\\},\cdots$, as a compact summation
operator for ease of presentation. Furthermore, the raw moments of the
collision kernels
$\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{m}e_{\alpha y}^{n}=\sum_{\beta}\langle K_{\beta}|e_{\alpha x}^{m}e_{\alpha
y}^{n}\rangle\widehat{g}_{\beta}$ are needed in its construction. Collision
invariants of conserved moments imply
$\widehat{g}_{0}=\widehat{g}_{1}=\widehat{g}_{2}=0$. Exploiting the orthogonal
property of the matrix $\mathcal{K}$, the non-conserved moments of
$\widehat{g}_{\beta}$ at higher orders, i.e. $\beta=3,4,\cdots,8$ can be
obtained as follows Premnath and Banerjee (2009):
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}=\sum_{\beta}\braket{K_{\beta}}{\rho}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}+2\widehat{g}_{4},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha y}^{2}}\widehat{g}_{\beta}$
$\displaystyle=$ $\displaystyle 6\widehat{g}_{3}-2\widehat{g}_{4},$ (16)
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle 4\widehat{g}_{5},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}^{2}e_{\alpha
y}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{6},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha x}e_{\alpha
y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle-4\widehat{g}_{7},$
$\displaystyle\sum_{\alpha}(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}e_{\alpha
x}^{2}e_{\alpha y}^{2}=\sum_{\beta}\braket{K_{\beta}}{e_{\alpha
x}^{2}e_{\alpha y}^{2}}\widehat{g}_{\beta}$ $\displaystyle=$ $\displaystyle
8\widehat{g}_{3}+4\widehat{g}_{8}.$
Using the above, the collision kernel $\widehat{g}_{\beta}$ of the cascaded
collision operator
$\Omega^{\mathcal{C}}_{\alpha}\equiv\Omega^{\mathcal{C}}_{\alpha}(\mathbf{f},\bf{\widehat{g}})=(\mathcal{K}\cdot\mathbf{\widehat{g}})_{\alpha}$
can be obtained as follows. Starting from the lowest order central moments
that are non-collisional invariants (i.e. $\widehat{\overline{\kappa}}_{xx}$
and higher), they are successively set equal to their local attractors based
on the transformed equilibria. This step provides tentative expressions for
$\widehat{g}_{\alpha}$ based on the equilibrium assumption. This is then
modified to allow for relaxation process during collision. That is, they are
multiplied with corresponding relaxation parameters Geier et al. (2006). In
this step, care needs to be exercised to multiply the relaxation parameters
only with those terms that are not yet in post-collision states (i.e. terms
not involving $\widehat{g}_{\beta},\beta=0,1,2,\ldots,\alpha-1$) for a given
$\widehat{g}_{\alpha}$. See Premnath and Banerjee (2009) for various details
involved in this procedure. Here, we summarize the final expressions of the
non-conserved collision kernels, which are given as follows:
$\displaystyle\widehat{g}_{3}$ $\displaystyle=$
$\displaystyle\frac{\omega_{3}}{12}\left\\{\frac{2}{3}\rho+\rho(u_{x}^{2}+u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}+\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}+\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(17) $\displaystyle\widehat{g}_{4}$ $\displaystyle=$
$\displaystyle\frac{\omega_{4}}{4}\left\\{\rho(u_{x}^{2}-u_{y}^{2})-(\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}})-\frac{1}{2}(\widehat{\sigma}_{xx}^{{}^{\prime}}-\widehat{\sigma}_{yy}^{{}^{\prime}})\right\\},$
(18) $\displaystyle\widehat{g}_{5}$ $\displaystyle=$
$\displaystyle\frac{\omega_{5}}{4}\left\\{\rho
u_{x}u_{y}-\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xy}^{{}^{\prime}}\right\\},$
(19) $\displaystyle\widehat{g}_{6}$ $\displaystyle=$
$\displaystyle\frac{\omega_{6}}{4}\left\\{2\rho
u_{x}^{2}u_{y}+\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{y}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xxy}\right\\}-\frac{1}{2}u_{y}(3\widehat{g}_{3}+\widehat{g}_{4})$
(20) $\displaystyle-2u_{x}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{7}$
$\displaystyle=$ $\displaystyle\frac{\omega_{7}}{4}\left\\{2\rho
u_{x}u_{y}^{2}+\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}-u_{x}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}-\frac{1}{2}\widehat{\sigma}_{xyy}\right\\}-\frac{1}{2}u_{x}(3\widehat{g}_{3}-\widehat{g}_{4})$
(21) $\displaystyle-2u_{y}\widehat{g}_{5},$ $\displaystyle\widehat{g}_{8}$
$\displaystyle=$
$\displaystyle\frac{\omega_{8}}{4}\left\\{\frac{1}{9}\rho+3\rho
u_{x}^{2}u_{y}^{2}-\left[\widehat{\overline{\kappa}}_{xxyy}^{{}^{\prime}}-2u_{x}\widehat{\overline{\kappa}}_{xyy}^{{}^{\prime}}-2u_{y}\widehat{\overline{\kappa}}_{xxy}^{{}^{\prime}}+u_{x}^{2}\widehat{\overline{\kappa}}_{yy}^{{}^{\prime}}+u_{y}^{2}\widehat{\overline{\kappa}}_{xx}^{{}^{\prime}}\right.\right.$
(22)
$\displaystyle\left.\left.+4u_{x}u_{y}\widehat{\overline{\kappa}}_{xy}^{{}^{\prime}}\right]-\frac{1}{2}\widehat{\sigma}_{xxyy}^{{}^{\prime}}\right\\}-2\widehat{g}_{3}-\frac{1}{2}u_{y}^{2}(3\widehat{g}_{3}+\widehat{g}_{4})-\frac{1}{2}u_{x}^{2}(3\widehat{g}_{3}-\widehat{g}_{4})$
$\displaystyle-4u_{x}u_{y}\widehat{g}_{5}-2u_{y}\widehat{g}_{6}-2u_{x}\widehat{g}_{7}.$
In the above, $\omega_{\beta}$, where $\beta=3,4,5,\ldots,8$, are the
relaxation parameters, satisfying the usual bounds $0<\omega_{\beta}<2$. When
a Chapman-Enskog expansion Chapman and Cowling (1964) is applied to the
cascaded LBM, it can be shown to recover the Navier-Stokes equations with the
relaxation parameters $\omega_{3}=\omega^{\chi}$ and
$\omega_{4}=\omega_{5}=\omega^{\nu}$ controlling the fbulk and shear
viscosities, respectively (e.g.,
$\nu=c_{s}^{2}\left(\frac{1}{\omega^{\nu}}-\frac{1}{2}\right)$) Premnath and
Banerjee (2009). The rest of the parameters can be adjusted independently
improve numerical stability. In this work,
$\omega_{4}=\omega_{5}=\frac{1}{\tau}$ is selected based on the specified
kinematic viscosity, while the rest of the relaxation parameters are set to
$1$.
The cascaded LBE can now be re-written in the form of the usual stream-and-
collide procedure, leading to the following two steps:
$\displaystyle\widetilde{\overline{f}}_{\alpha}(\vec{x},t)=\overline{f}_{\alpha}(\vec{x},t)+\Omega^{\mathcal{C}}_{\alpha(\vec{x},t)}+S_{\alpha(\vec{x},t)},$
(23)
$\displaystyle\overline{f}_{\alpha}(\vec{x}+\vec{e}_{\alpha},t+\delta_{t})=\widetilde{\overline{f}}_{\alpha}(\vec{x},t),$
(24)
where the symbol “tilde” ($\sim$) in the above equations refers to the post-
collision state of the distribution function. Expanding the collision term in
the first step, the components of the post-collision distribution function can
be explicitly written as
$\displaystyle\widetilde{\overline{f}}_{0}$ $\displaystyle=$
$\displaystyle\overline{f}_{0}+\left[\widehat{g}_{0}-4(\widehat{g}_{3}-\widehat{g}_{8})\right]+S_{0},$
$\displaystyle\widetilde{\overline{f}}_{1}$ $\displaystyle=$
$\displaystyle\overline{f}_{1}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}+2(\widehat{g}_{7}-\widehat{g}_{8})\right]+S_{1},$
$\displaystyle\widetilde{\overline{f}}_{2}$ $\displaystyle=$
$\displaystyle\overline{f}_{2}+\left[\widehat{g}_{0}+\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}+2(\widehat{g}_{6}-\widehat{g}_{8})\right]+S_{2},$
$\displaystyle\widetilde{\overline{f}}_{3}$ $\displaystyle=$
$\displaystyle\overline{f}_{3}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{3}+\widehat{g}_{4}-2(\widehat{g}_{7}+\widehat{g}_{8})\right]+S_{3},$
$\displaystyle\widetilde{\overline{f}}_{4}$ $\displaystyle=$
$\displaystyle\overline{f}_{4}+\left[\widehat{g}_{0}-\widehat{g}_{2}-\widehat{g}_{3}-\widehat{g}_{4}-2(\widehat{g}_{6}+\widehat{g}_{8})\right]+S_{4},$
(25) $\displaystyle\widetilde{\overline{f}}_{5}$ $\displaystyle=$
$\displaystyle\overline{f}_{5}+\left[\widehat{g}_{0}+\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}-\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{5},$
$\displaystyle\widetilde{\overline{f}}_{6}$ $\displaystyle=$
$\displaystyle\overline{f}_{6}+\left[\widehat{g}_{0}-\widehat{g}_{1}+\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}-\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{6},$
$\displaystyle\widetilde{\overline{f}}_{7}$ $\displaystyle=$
$\displaystyle\overline{f}_{7}+\left[\widehat{g}_{0}-\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}+\widehat{g}_{5}+\widehat{g}_{6}+\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{7},$
$\displaystyle\widetilde{\overline{f}}_{8}$ $\displaystyle=$
$\displaystyle\overline{f}_{8}+\left[\widehat{g}_{0}+\widehat{g}_{1}-\widehat{g}_{2}+2\widehat{g}_{3}-\widehat{g}_{5}+\widehat{g}_{6}-\widehat{g}_{7}+\widehat{g}_{8}\right]+S_{8}.$
The hydrodynamic fields, i.e. the fluid density and the velocity then follow
from taking the zeroth and first moments of the distribution function,
yielding
$\displaystyle\rho=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}=\langle\overline{f}_{\alpha}|\rho\rangle,$
(26) $\displaystyle\rho u_{i}=$
$\displaystyle\sum_{\alpha=0}^{8}\overline{f}_{\alpha}e_{\alpha
i}+\frac{1}{2}F_{i}=\langle\overline{f}_{\alpha}|e_{\alpha
i}\rangle+\frac{1}{2}F_{i},i=x,y,$ (27)
and the pressure $p$ satisfies $p=c_{s}^{2}\rho$. A particularly useful
feature of kinetic schemes such as the cascaded LBM is that the strain-rate
tensor can be computed _locally_ from a knowledge of the non-equilibrium
moments. In fact, this can be shown by means of the Chapman-Enskog analysis,
which was performed on the cascaded LBE in Premnath and Banerjee (2009).
Setting the components of the momentum as $j_{x}=\rho u_{x}$ and $j_{y}=\rho
u_{y}$, such an analysis shows Premnath and Banerjee (2009)
$\displaystyle\widehat{f_{3}}^{(neq)}=$
$\displaystyle-\frac{2}{3\omega_{3}}\bigl{(}\partial_{x}j_{x}+\partial_{y}j_{y}\bigr{)},$
(28) $\displaystyle\widehat{f_{4}}^{(neq)}=$
$\displaystyle-\frac{2}{3\omega_{4}}\bigl{(}\partial_{x}j_{x}-\partial_{y}j_{y}\bigr{)},$
(29) $\displaystyle\widehat{f_{5}}^{(neq)}=$
$\displaystyle-\frac{1}{3\omega_{5}}\bigl{(}\partial_{x}j_{y}+\partial_{y}j_{x}\bigr{)},$
(30)
where
$\widehat{f}_{\beta}^{(neq)}\approx\widehat{f}_{\beta}-\widehat{f}^{eq}_{\beta}$
are the non-equilibrium raw moments. Specifically,
$\widehat{f}_{3}=\widehat{\kappa}_{xx}^{{}^{\prime}}+\widehat{\kappa}_{yy}^{{}^{\prime}}$,
$\widehat{f}_{4}=\widehat{\kappa}_{xx}^{{}^{\prime}}-\widehat{\kappa}_{yy}^{{}^{\prime}}$,
and $\widehat{f}_{5}=\widehat{\kappa}_{xy}^{{}^{\prime}}$, whose equilibria
are $\widehat{f}_{3}^{eq}=2/3\rho+\rho(u_{x}^{2}+u_{y}^{2})$,
$\widehat{f}_{4}^{eq}=\rho(u_{x}^{2}-u_{y}^{2})$, and
$\widehat{f}_{5}^{eq}=\rho u_{x}u_{y}$, respectively Premnath and Banerjee
(2009). It thus follows that
$\displaystyle\partial_{x}j_{x}=$
$\displaystyle-\frac{3\omega_{3}}{2}\biggl{[}\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}e^{2}_{\alpha
x}-\left(\frac{1}{3}\rho+\rho u_{x}^{2}\right)\bigg{]},$ (31)
$\displaystyle\partial_{y}j_{y}=$
$\displaystyle-\frac{3\omega_{4}}{2}\biggl{[}\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}e^{2}_{\alpha
y}-\left(\frac{1}{3}\rho+\rho u_{y}^{2}\right)\bigg{]},$ (32)
$\displaystyle\partial_{x}j_{y}+\partial_{y}j_{x}=$
$\displaystyle-3\omega_{5}\biggl{[}\displaystyle\sum_{\alpha=0}^{8}f_{\alpha}e_{\alpha
x}e_{\alpha y}-\rho u_{x}u_{y}\bigg{]}.$ (33)
These specific expressions will be exploited in the numerical study of the
cascaded LBM in the remainder of this paper. In the sections that follow, we
will present the results obtained with the cascaded LBM for a set of benchmark
problems to assess its numerical properties in terms of grid convergence,
accuracy and stability.
## III Grid Convergence Study on the Benchmark Problems
We first perform a numerical study involving grid convergence for canonical
flows including a steady 2D Poiseuille flow, a time-dependent 2D decaying
Taylor-Green vortex flow, and a 2D lid-driven cavity flow characterized by
various complex features. In the various figures presented in this section,
the symbols represent the computed solution using the cascaded MRT LBM, the
thin solid lines are the resulting slopes representing changes in the relative
errors as the grid resolution increases, and the thick solid lines are the
ideal slopes corresponding to second-order accuracy. In this work, a
_diffusive scaling_ is applied to perform the convergence tests Junk et al.
(2005). According to this scaling, the errors due to compressibility effects
decrease at the same rate as the errors due to grid discretization thus
prescribing a consistent limit process to represent incompressible flow. That
is, the velocity scales in the same proportion as the length scales.
Equivalently, this means that the ratio of the Mach number and the grid
Knudsen number remains constant for different grid resolutions, _i.e._ $Ma/Kn$
= constant.
### III.1 2D Poiseuille Flow
The 2D Poiseuille flow is first considered. The flow is between two parallel
plates of infinite length in the streamwise direction subjected to a constant
body force. A periodic boundary condition is applied at the inlet and the
outlet and a no-slip boundary condition at the solid boundaries by employing
the standard half-way bounce back approach. The grid convergence is
established by considering the following resolutions consisting of $3\times
24,3\times 36,\ldots,3\times 192$ lattice nodes under diffusive scaling. The
relaxation time for shear modes is set to $\tau=0.55$ that specifies
$\omega_{4}$ and $\omega_{5}$. The rest of relaxation parameters are set to
unity. The flow is driven by a constant body force with the components $F_{x}$
specified to yield desired condition (see below) and $F_{y}=0$. This classical
flow problem has the well known parabolic profile as the analytical solution
given by $u(y)=u_{max}(1-y^{2}/L^{2})$, where
$u_{max}=\frac{F_{x}L^{2}}{2\nu}$ is the maximum velocity occurring midway
between the plates, $\nu$ is the kinematic viscosity related the to relaxation
time $\tau$ as given in the previous section, and $L$ denotes the half-width
between the plates. Figure 1 illustrates the relative global errors between
the computed solutions obtained using the cascaded MRT LBM and the analytical
solutions for such flow at different Reynolds numbers of $100$, $200$ and
$400$. The relative global error, which quantifies the difference between the
computed and analytical solutions, is defined as
$\text{Relative
Error}=\frac{\sum_{i}||(u_{c,i}-u_{a,i})||}{\sum_{i}||u_{a,i}||},$ (34)
where $u_{c,i}$ and $u_{a,i}$ are the computed and the analytical solutions,
respectively, and a standard Euclidean norm is used in the above measurements.
It is seen that the relative errors have slopes of almost equal to $2.00$,
which tells that the cascaded MRT LBM is well-posed second-order accurate for
this problem. In addition, the relative errors are seen to slightly increases
as the Reynolds number increases.
Figure 1: Grid convergence of the cascaded MRT LBM for the velocity field in a
2D Poiseuille flow with constant body force under diffusive scaling.
### III.2 2D Decaying Taylor-Green Vortex Flow
The second problem considered is the decaying Taylor-Green vortex Taylor
(1923), which is a 2D unsteady flow induced by a prescribed initial vortex
distribution and decaying due to fluid viscosity. The fluid domain is a square
of side $2\pi$ with no inflow/outflow and wall boundaries. The initial
condition is set to be periodic array of vortices in both x and y directions
as follows
$\displaystyle u(x,y,0)=$ $\displaystyle-u_{0}\cos(kx)\sin(ky),$ (35)
$\displaystyle v(x,y,0)=$ $\displaystyle+u_{0}\sin(kx)\cos(ky),$ (36)
$\displaystyle p(x,y,0)=$ $\displaystyle
p_{0}\biggl{[}1-\frac{u_{0}^{2}}{4c_{s}^{2}}\bigl{(}\cos(2kx)+\cos(2ky)\bigr{)}\biggr{]},$
(37)
where $k=\frac{2\pi}{N}$ is the wavenumber, $u_{0}$ and $p_{0}$ are the
initial values for velocity and pressure, respectively. Here, $N$ is the
number of grid nodes in each direction. The temporal evolution has the
characteristic time scale given by $T=\frac{1}{2k^{2}\nu}$. Since there is no
external energy supplied and because of the presence of fluid viscosity, the
velocity field will decay with time due to fluid viscous dissipation. There
exists an analytical solution for this problem which is a solution of the
Navier-Stokes equations in a periodic domain and given by
$\displaystyle u(x,y,t)=$ $\displaystyle-u_{0}\cos(kx)\sin(ky)e^{-2k^{2}\nu
t},$ (38) $\displaystyle v(x,y,t)=$
$\displaystyle+u_{0}\sin(kx)\cos(ky)e^{-2k^{2}\nu t},$ (39) $\displaystyle
p(x,y,t)=$ $\displaystyle
p_{0}-\frac{u_{0}^{2}}{4}\biggl{[}\cos(2kx)+\cos(2ky)\biggr{]}e^{-4k^{2}\nu
t}.$ (40)
Furthermore, the components of the strain rate tensor also satisfy the
following explicit analytical solution:
$\displaystyle S_{xx}=$ $\displaystyle\frac{\partial u}{\partial
x}=ku_{0}\sin(kx)\sin(ky)e^{-2\nu k^{2}t}$ (41) $\displaystyle S_{yy}=$
$\displaystyle\frac{\partial u}{\partial y}=-S_{xx}$ (42) $\displaystyle
S_{xy}=$ $\displaystyle\frac{1}{2}\biggl{(}\frac{\partial u}{\partial
y}+\frac{\partial v}{\partial x}\biggr{)}=0$ (43)
In this test, the Reynolds number of the flow is set to
$Re=\frac{u_{0}l}{\nu}=14.4$, where $l=2\pi$ is the length of the domain. A
periodic boundary condition is applied to all the sides of the domain. We
consider the following parameters in our grid convergence study: $\tau=0.55$,
$k=1,2$ and $u_{0}=0.01$. Applying the diffusive scaling, we obtain the
relative global errors between the computed and the analytical solutions for
the grid resolutions of $24\times 24$, $48\times 48$, $96\times 96$,
$192\times 192$ for a representative time $t=30.1T$. In Fig. 2 shown are the
relative errors for the u-velocity component, which have the slopes of $1.99$
and $1.98$ for the wavenumbers $k=1$ and $k=2$, respectively. Figure 3 shows
the relative errors for the only independent strain rate tensor component
$S_{xx}$ with the slopes of $1.99$ and $1.98$ as well for the above two
wavenumbers.
Figure 2: Grid convergence of the cascaded MRT LBM for the velocity field in a
2D Taylor-Green vortex flow with $k=1$ and $k=2$. Figure 3: Grid convergence
of the cascaded MRT LBM for the strain rate in a 2D Taylor-Green vortex flow
with $k=1$ and $k=2$.
Thus, it is evident that the cascaded MRT LBM is second-order accurate not
only for the velocity field, but also for the components of the strain rates
as well. This finding is consistent with a recent study with the SRT LBM for
this problem Kruger et al. (2010).
### III.3 2D Lid-driven Cavity Flow
Finally, the 2D lid-driven cavity flow is considered, whose geometric
simplicity is contrasted by various complex flow features. It is generally
considered a standard benchmark test for CFD methods and has been a subject of
many investigations using a variety of methods (see e.g. Ghia et al. (1982);
Schreiber and Keller (1983); Vanka (1986); Erturk et al. (2005); Bruneau and
Saad (2006)). Grid convergence for this problem has been studied using
different collision models (SRT and standard MRT) for the LBM by various
researchers (e.g. Luo et al. (2011)). In this section, the aim is to analyze
the grid convergence and an estimation of the order of accuracy of the
cascaded MRT LBM for this flow problem. More detailed accuracy investigation
of the various flow features will be carried out in the next section. While
the geometry is simple from the boundary condition implementation point of
view, the flow contains singular points and becomes very complicated in terms
of flow structures, particularly as the Reynolds number increases (see e.g.
Erturk et al. (2005) for a review). A schematic of the arrangement of the
boundaries in a 2D lid-cavity flow is shown in Fig. 4.
Figure 4: Illustration of the geometry of a lid-driven cavity flow.
Fluid is enclosed inside a square cavity of length, $L$, and is set into
motion by the moving upper wall that has a constant velocity $U_{o}$. The side
and the bottom walls are considered to be stationary, which allows to
implement a simple half-way bounce-back boundary condition on them. However,
because the upper wall is in constant motion, a momentum correction needs to
be added Lallemand and Luo (2003) into the regular bounce-back scheme for the
upper boundary. This is implemented as
$f_{\alpha}(i,N_{y}-1)=\widetilde{f}_{\overline{\alpha}}(i,N_{y}-1)+6\rho
w_{\alpha}e_{\alpha y}U_{p}$, where
$\widetilde{f}_{\overline{\alpha}}(i,N_{y}-1)$ is the post-collision
distribution function, for $\alpha=4,7,8$, with $\overline{\alpha}=2,5,6$ as
the opposite directions of $\alpha$, and $w_{\alpha}$ is the weighting factor
Lallemand and Luo (2003). Ghia _et al._ Ghia et al. (1982) have
systematically studied this problem in much detail by employing a vorticity-
stream function formulation of the 2D incompressible Navier-Stokes equations,
which is solved by a multigrid method. Some of their numerical results have
been used for making accuracy comparisons in this work which will be discussed
in a later section. Because of the lack of analytical solutions, the computed
solutions obtained by a relatively very fine grid resolution, i.e. with _i.e._
$801\times 801$, are treated as the approximate benchmark or reference
(“analytical”) solutions. Not only is the convergence of velocity fields
tested, but also the grid convergence of the components of the strain rate
tensor is considered. It may be noted that the study involving the latter
quantity has not so far received enough attention for this problem using the
LBM.
The components of the velocity field and the strain rate tensor at the
centerlines of the cavity in both vertical and horizontal directions are
computed for a given Reynolds number once the solutions converge to steady
state. The solutions are considered to reach steady state convergence when the
relative global errors is small than $10^{-15}$. Again, diffusive scaling is
employed to set the parameters for different grid resolutions consisting of
$13\times 13$, $19\times 19$, $25\times 25$, $31\times 31$, $37\times 37$,
$49\times 49$, $61\times 61$, $85\times 85$, $97\times 97$ and $121\times 121$
nodes. Figure 5 shows the grid convergence of the U-component of the velocity
field at a Reynolds number of $100$. It is found that the best fit slopes are
$2.11$ and $2.19$ along the vertical and the horizontal centerlines,
respectively, for the U-velocity. Likewise, the slopes are $2.18$ and $2.11$
respectively along the vertical and the horizontal centerlines for the
V-velocity as shown in Fig. 6.
Figure 5: Grid convergence of the cascaded MRT LBM for the U-velocity
component in a 2D lid-driven cavity flow for $Re=100$. Figure 6: Grid
convergence of the cascaded MRT LBM for the V-velocity component in a 2D lid-
driven cavity flow for $Re=100$.
For the normal strain rate tensor component $\frac{\partial v}{\partial y}$ ,
the slopes are found to be $1.81$ and $1.95$ respectively for the vertical and
the horizontal centerlines, which is shown in Fig. 7. Furthermore, it is seen
that along the vertical and horiztonal centerlines, the strain rate tensor
component $\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$ has
the slopes of $2.12$ and $2.07$, respectively, for grid convergence (see Fig.
8). One reason why the slopes are either somewhat higher or lower than $2$,
rather than very close to the ideal value as seen with the other two problems
discussed before, is that the reference solution for obtaining the relative
error is taken to be that of the numerical solution with the very fine grid.
This is often the practice as the “analytical” solution does not exist for
this problem.
Figure 7: Grid convergence of the cascaded MRT LBM for the strain rate tensor
component $\frac{\partial v}{\partial x}$ in a 2D lid-driven cavity flow for
$Re=100$. Figure 8: Grid convergence of the cascaded MRT LBM for the strain
rate tensor component $\frac{\partial u}{\partial y}+\frac{\partial
v}{\partial x}$ in a 2D lid-driven cavity flow for $Re=100$.
Overall, it is seen that the cascaded MRT LBM gives a very respectable second
order accuracy for a variety of flows, including the relatively simple
Poiseuille flow and decaying Taylor-Green vortex flow, and for relatively
complex flows such as the lid-driven cavity flow. The method is found to be
second order accurate not only for the velocity field, but also for their
derivatives for the above problems.
## IV Accuracy Studies on the Benchmark Problems
Let us now make a more detailed comparison of the accuracy of the solutions
computed using the cascaded LBM with prior results involving either analytical
or other numerical solution for the flow fields of the three benchmark
problems considered in the previous section.
### IV.1 2D Poiseuille Flow
Figure 9 shows a comparison of the velocity profiles of the 2D Poiseuille flow
between the results obtained using the cascaded LBM and the parabolic
analytical solution at a constant Reynolds number of $200$ with constant
relaxation time $\tau=0.515$ for different grid resolutions in the wall normal
direction starting from $26$ to $401$. Here, diffusive scaling is employed in
the selection of parameters. That is, as the resolution is doubled, the
maximum flow velocity or the Mach number is decreased by a factor of $2$. The
results are in excellent agreement with the analytical solution, in which the
maximum relative error is less than $0.22$ percent.
Figure 9: Comparison of the velocity profiles in a 2D Poiseuille flow for
$Re=200$ at different grid resolutions $N$ in the wall normal direction with a
constant relaxation time $\tau=0.55$.
### IV.2 2D Decaying Taylor-Green Vortex Flow
Using the same set of parameters specified for this time-dependent problem in
the previous section, we now compare the computed $U-$ and $V-$ velocity
components along the vertical and horizontal centerlines, respectively, with
the corresponding analytical solutions (Eq. (38)-(39)) at three different
representative instants. Figures 11 and 11 show such a comparison of the
velocity components at times $t=6.55T$, $13.10T$ and $25.20T$, where the
characteristic time $T$ is defined in the previous section, reflecting the
decaying of the initial vortex distribution. It is evident that the cascaded
MRT LBM is in excellent agreement with the analytical solution at all times
shown.
Figure 10: Comparison of the U-velocity component in a decaying Taylor-Green
vortex flow for $Re=14.4$ at three different non-dimensional times $T$:
$t=6.55T,13.10T$ and $26.20T$.
Figure 11: Comparison of the V-velocity component in a decaying Taylor-Green
vortex flow for $Re=14.4$ at three different non-dimensional times $T$:
$t=6.55T,13.10T$ and $26.20T$.
### IV.3 2D Lid-driven Cavity Flow
Let us now consider more detailed features of the lid-driven cavity flow
problem discussed in the last section at various Reynolds numbers in order to
make quantitative comparisons. Figures 13 and 13 show the U- and V- components
of the velocity, respectively, along the centerlines of the square cavity at
Reynolds numbers of $100$, $400$, $1000$, $3200$, $5000$, and $7500$ obtained
using the cascaded MRT LBM along with the previous numerical data presented by
Ghia _et al_ Ghia et al. (1982). The cascaded MRT LBM results corresponding to
the finest grid considered earlier, i.e. for the $401\times 401$ grid
resolution are chosen to make comparison. In these figures, the solid lines
represent the computed results obtained by the cascaded MRT LBM, and the
symbols are the prior data provided by Ghia _et al_ Ghia et al. (1982). The
velocities are normalized by the lid velocity $U_{0}$. Very good agreement is
seen for all the Reynolds numbers considered.
Figure 12: Comparison of the $U$\- component of the velocity field along the
vertical centerline of the cavity flow at various Reynolds numbers:
$Re=100,400,1000,3200,5000$ and $7500$. Lines – cascaded MRT LBM and symbols –
data by Ghia _et al_ Ghia et al. (1982).
Figure 13: Comparison of the $V$\- component of the velocity field along the
horizontal centerline of the cavity flow at various Reynolds numbers:
$Re=100,400,1000,3200,5000$ and $7500$. Lines – cascaded MRT LBM and symbols –
data by Ghia _et al_ Ghia et al. (1982).
In a previous work, it was established that the standard MRT LBM based on raw
moments is superior when compared with the SRT LBM for the computation of lid-
driven cavity flow Luo et al. (2011). Hence, it would be sufficient to make a
direct comparison between the cascaded MRT LBM based on central moments and
the standard MRT LBM for various flow characteristics of this problem. First,
in order to provide a global characteristics of the flow field, it would be
interesting to compare the streamlines in the cavity at various Reynolds
numbers. It is known that at a certain Reynolds number above $7500$, the flow
field becomes unsteady and we restrict such comparisons for stationary state
solutions only. Hence, Fig. 14 shows the computed streamlines at Reynolds
numbers of $100$, $400$, $1000$, $5000$ and $7500$ using both the above
methods. The streamlines computed by both these approaches are plotted side-
by-side for comparison. It is found that the streamlines appear to be
remarkably very similar with both the raw moment and central moment based
approaches. At Reynolds numbers of $100$, $400$ and $1000$, a major vortex
appears around the geometric center of the cavity with two minor vortices
around the lower corners. Since the lid is driven from left to right, the
major vortex circulates in a clockwise direction and the two minor vortices
circulate in a counter-clockwise direction. At Reynolds numbers of $3200$ and
$5000$, in addition to the vortices that exist with the lower Reynolds number
cases, there appears another minor vortex on the left upper corner, which
circulates in a counter-clockwise direction. When the Reynolds number
increases further to $7500$, a fourth minor vortex is found on the right lower
corner, which circulates in a clockwise direction. All the above flow features
correspond to steady states. Furthermore, in order to provide a more detailed
comparison, we present various secondary vortices that appear in the cavity at
$Re=7500$ in Fig. 15. Again, remarkable similarity between the cascaded MRT
LBM and the standard MRT LBM is found for these more detailed secondary flow
structures.
(a) Cascaded MRT $Re=100$
(b) Standard MRT $Re=100$
(c) Cascaded MRT $Re=400$
(d) Standard MRT $Re=400$
(e) Cascaded MRT $Re=1000$
(f) Standard MRT $Re=1000$
(g) Cascaded MRT $Re=3200$
(h) Standard MRT $Re=3200$
(i) Cascaded MRT $Re=5000$
(j) Standard MRT $Re=5000$
(k) Cascaded MRT $Re=7500$
(l) Standard MRT $Re=7500$
Figure 14: Comparison of the streamlines in a 2D lid-driven cavity flow at
different Reynolds numbers computed with cascaded (central moment) MRT LBM and
standard (raw moment) MRT LBM: $Re=100,400,1000,3200,5000$ and $7500$.
Solutions obtained using $201^{2}$ grids with both methods.
(a) Cascaded MRT Top
(b) Standard MRT Top
(c) Cascaded MRT Bottomleft
(d) Standard MRT Bottomleft
(e) Cascaded MRT Bottomright
(f) Standard MRT Top
Figure 15: Comparison of the streamlines of the secondary vortices in a 2D
lid-driven cavity flow at $Re=7500$ computed with cascaded (central moment)
MRT LBM and standard (raw moment) MRT LBM.
In order to provide a more quantitative perspective, Fig. 16 illustrates a
comparison of the center of the primary vortex location in the cavity flow at
different Reynolds numbers ($Re=100,400,1000,3200,5000$, and $7500$) between
the cascaded and standard MRT LBM as well as the data by Ghia _et al_ Ghia et
al. (1982).
Figure 16: Comparison of the Cartesian coordinates of the location of the
center of the primary vortex in a lid-driven cavity flow at different Reynolds
numbers.
From the earlier streamline plots, it can be observed that the location of the
primary vortex moves towards the geometric center of the cavity as the
Reynolds number increases. The computed results using the cascaded MRT LBM and
the standard MRT LBM are in excellent agreement (within $0.014$ percent) with
each other for all Reynolds numbers. In addition, they are both in very good
agreement with the data by Ghia _et al_ Ghia et al. (1982) to within $0.50$
percent for all Reynolds numbers. These quantitative results for the primary
vortex locations are enumerated in Table 1.
Table 1: Comparison of the location of the primary vortex in a lid-driven cavity flow at different Reynolds numbers. $Re$ | Cascaded MRT LBM | Standard MRT LBM | Ghia _et al_(1982) Ghia et al. (1982)
---|---|---|---
100 | $(0.61482,0.73543)$ | $(0.61467,0.73524)$ | $(0.61720,0.73440)$
400 | $(0.55380,0.60514)$ | $(0.55380,0.60514)$ | $(0.55470,0.60550)$
1000 | $(0.53070,0.56512)$ | $(0.53070,0.56512)$ | $(0.53130,0.56250)$
3200 | $(0.51778,0.54027)$ | $(0.51777,0.54028)$ | $(0.51650,0.54690)$
5000 | $(0.51499,0.53522)$ | $(0.51497,0.53524)$ | $(0.51150,0.53520)$
7500 | $(0.51299,0.53186)$ | $(0.51298,0.53188)$ | $(0.51170,0.53220)$
In addition, Table 2 presents a comparison between the above two methods and
the prior numerical data for the location of secondary vortices at different
Reynolds numbers. Again, both the cascaded MRT LBM and the standard MRT LBM
are in excellent quantitative agreement for the location of these detailed
secondary vortical structures with the data by Ghia _et al_ Ghia et al.
(1982).
Table 2: Comparison of the location of various secondary vortices in a lid-
driven cavity flow at differnt Reynolds numbers. First Secondary Vortex
---
| $Re$ | Cascaded MRT LBM | Standard MRT LBM | Ghia _et al_(1982) Ghia et al. (1982)
Top | 100 | NA | NA | NA
400 | NA | NA | NA
1000 | NA | NA | NA
3200 | $(0.0547,0.8976)$ | $(0.0546,0.8973)$ | $(0.0547,0.8984)$
5000 | $(0.0644,0.9081)$ | $(0.0641,0.9076)$ | $(0.0625,0.9102)$
7500 | $(0.0676,0.9102)$ | $(0.0677,0.9099)$ | $(0.0664,0.9141)$
Bottom Left | 100 | $(0.0387,0.0387)$ | $(0.0373,0.0373)$ | $(0.0313,0.0391)$
400 | $(0.0533,0.0493)$ | $(0.0530,0.0494)$ | $(0.0508,0.0469)$
1000 | $(0.0842,0.0791)$ | $(0.0842,0.0791)$ | $(0.0859,0.0781)$
3200 | $(0.0821,0.1207)$ | $(0.0821,0.1207)$ | $(0.0859,0.1094)$
5000 | $(0.0740,0.1378)$ | $(0.0740,0.1378)$ | $(0.0703,0.1367)$
7500 | $(0.0654,0.1536)$ | $(0.0654,0.1536)$ | $(0.0645,0.1504)$
Bottom Right | 100 | $(0.9383,0.0658)$ | $(0.9386,0.0654)$ | $(0.9453,0.0625)$
400 | $(0.8833,0.1243)$ | $(0.883,0.1243)$ | $(0.8906,0.1250)$
1000 | $(0.8631,0.1128)$ | $(0.8631,0.1128)$ | $(0.8594,0.1094)$
3200 | $(0.8229,0.0853)$ | $(0.8229,0.0852)$ | $(0.8125,0.0859)$
5000 | $(0.8037,0.0739)$ | $(0.8037,0.0739)$ | $(0.8086,0.0742)$
7500 | $(0.7892,0.0663)$ | $(0.7893,0.0663)$ | $(0.7813,0.0625)$
Second Secondary Vortex
Bottom Left | 100 | NA | NA | NA
400 | NA | NA | NA
1000 | NA | NA | NA
3200 | $(0.0075,0.0075)$ | $(0.0073,0.0073)$ | $(0.0078,0.0078)$
5000 | $(0.0075,0.0075)$ | $(0.0074,0.0074)$ | $(0.0117,0.0078)$
7500 | $(0.0125,0.0125)$ | $(0.0115,0.0115)$ | $(0.0117,0.0117)$
Bottom Right | 100 | NA | NA | NA
400 | $(0.9926,0.0075)$ | NA | $(0.9922,0.0078)$
1000 | $(0.9923,0.0076)$ | $(0.9928,0.0073)$ | $(0.9922,0.0078)$
3200 | $(0.9875,0.0113)$ | $(0.9885,0.0115)$ | $(0.9844,0.0078)$
5000 | $(0.9775,0.0200)$ | $(0.9771,0.0193)$ | $(0.9805,0.0195)$
7500 | $(0.9508,0.0429)$ | $(0.9509,0.0429)$ | $(0.9492,0.0430)$
Third Secondary Vortex
Bottom Right | 100 | NA | NA | NA
400 | NA | NA | NA
1000 | NA | NA | NA
3200 | NA | NA | NA
5000 | NA | NA | NA
7500 | $(0.9964,0.0037)$ | NA | $(0.9961,0.0039)$
Another useful global characteristic for comparison is the vorticity contours
in the cavity at different Reynolds numbers. Figure 17 shows the vorticity
contours computed using both the standard MRT LBM and the cascaded MRT LBM at
three different Reynolds numbers ($Re=100,400$, and $1000$). As Reynolds
number increases, the vorticity contours become denser and denser approaching
the boundary walls. Overall, the vorticity distribution is found to be very
similar using both the methods for all the Reynolds numbers considered thus
corraborating the earlier results.
(a) Cascaded MRT $Re=100$
(b) Standard MRT $Re=100$
(c) Cascaded MRT $Re=400$
(d) Standard MRT $Re=400$
(e) Cascaded MRT $Re=1000$
(f) Standard MRT $Re=1000$
Figure 17: Comparison of the vorticity contours in a 2D lid-driven cavity flow
at different Reynolds numbers computed with cascaded (central moment) MRT LBM
and standard (raw moment) MRT LBM: $Re=100,400$ and $1000$.
As discussed earlier, one of the useful features of kinetic schemes such as
the cascaded MRT LBM is that the components of the strain rate tensor can be
obtained locally from the components of the non-equilibrium moments of the
distribution function (see Eqs. (31)-(33)). The cavity flow being a shear
driven problem generally has all the components of the strain rate tensor non-
zero, and whose magnitudes can dramatically change with the Reynolds number.
Hence, this problem provides a good test for the evalution of the accuracy of
the computation of strain rate tensor by kinetic theory considerations, i.e.
using non-equilibrium moments (Eqs. (31)-(33)). For the sake of comparison, we
will make use of the standard second-order central differencing of the
velocity field to obtain the usual direct estimation of the strain rate tensor
components. In this regard, flow at two different Reynolds numbers are
considered ($Re=100$ and $1000$) and the components of the strain rate tensor
are obtained at five different locations within the cavity using the above two
methods, which are enumerated in Table 3. As the Reynolds number is increased
from $100$ to $1000$, the magnitudes of the strain rate tensor change
significantly, which are quite well captured by the kinetic approach. Indeed,
remarkably the local computation using the non-equilibrium moments are in very
good agreement with the finite-difference estimation at various locations in
the cavity for both the Reynolds numbers, with the maximum difference within 2
percent. This further demonstrates the numerical fidelity of the approach. In
particular, such non-equilibrium moments based approach for the strain rate
components can be used in the subgrid scale models for large eddy simulation
of turbulent flows using the cascaded MRT LBM.
Table 3: Comparison of the components of the strain rate tensor computed using
the local non-equilibrium moments (Eqs. (31)-(33)) and the finite-differencing
(second-order central) of the velocity field with the cascaded MRT LBM at five
different locations within the cavity for two different Reynolds numbers
($Re=100$ and $1000$). $Re=100$
---
| | Location | Non-eqm. Moments | Finite Difference | Difference
$\frac{\partial v}{\partial y}$ | A | $(\frac{L}{4},\frac{L}{2})$ | $2.416\times 10^{-4}$ | $2.415\times 10^{-4}$ | 0.044$\%$
B | $(\frac{L}{2},\frac{L}{4})$ | $1.711\times 10^{-5}$ | $1.721\times 10^{-5}$ | 0.613$\%$
C | $(\frac{L}{2},\frac{L}{2})$ | $1.850\times 10^{-4}$ | $1.848\times 10^{-4}$ | 0.102$\%$
D | $(\frac{L}{2},\frac{3L}{4})$ | $-9.020\times 10^{-5}$ | $-9.025\times 10^{-5}$ | 0.057$\%$
E | $(\frac{3L}{4},\frac{L}{2})$ | $-3.541\times 10^{-4}$ | $-3.536\times 10^{-4}$ | 0.125$\%$
$Re=100$
$\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$ | A | $(\frac{L}{4},\frac{L}{2})$ | $3.516\times 10^{-5}$ | $3.526\times 10^{-5}$ | 0.300$\%$
B | $(\frac{L}{2},\frac{L}{4})$ | $-4.344\times 10^{-4}$ | $-4.342\times 10^{-4}$ | 0.045$\%$
C | $(\frac{L}{2},\frac{L}{2})$ | $-3.368\times 10^{-4}$ | $-3.363\times 10^{-4}$ | 0.135$\%$
D | $(\frac{L}{2},\frac{3L}{4})$ | $4.599\times 10^{-4}$ | $4.603\times 10^{-4}$ | 0.093$\%$
E | $(\frac{3L}{4},\frac{L}{2})$ | $-5.290\times 10^{-4}$ | $-5.280\times 10^{-4}$ | 0.198$\%$
$Re=1000$
$\frac{\partial v}{\partial y}$ | A | $(\frac{L}{4},\frac{L}{2})$ | $4.220\times 10^{-5}$ | $4.217\times 10^{-5}$ | 0.050$\%$
B | $(\frac{L}{2},\frac{L}{4})$ | $2.196\times 10^{-5}$ | $2.209\times 10^{-5}$ | 0.596$\%$
C | $(\frac{L}{2},\frac{L}{2})$ | $5.017\times 10^{-5}$ | $5.017\times 10^{-5}$ | 0.008$\%$
D | $(\frac{L}{2},\frac{3L}{4})$ | $2.370\times 10^{-5}$ | $2.372\times 10^{-5}$ | 0.073$\%$
E | $(\frac{3L}{4},\frac{L}{2})$ | $6.397\times 10^{-5}$ | $6.446\times 10^{-5}$ | 0.750$\%$
$Re=1000$
$\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}$ | A | $(\frac{L}{4},\frac{L}{2})$ | $-1.344\times 10^{-4}$ | $-1.334\times 10^{-4}$ | 0.782$\%$
B | $(\frac{L}{2},\frac{L}{4})$ | $8.137\times 10^{-5}$ | $7.984\times 10^{-5}$ | 1.914$\%$
C | $(\frac{L}{2},\frac{L}{2})$ | $-3.980\times 10^{-5}$ | $-3.979\times 10^{-5}$ | 0.021$\%$
D | $(\frac{L}{2},\frac{3L}{4})$ | $2.291\times 10^{-4}$ | $2.289\times 10^{-4}$ | 0.049$\%$
E | $(\frac{3L}{4},\frac{L}{2})$ | $-1.688\times 10^{-4}$ | $-1.699\times 10^{-4}$ | 0.626$\%$
## V Numerical Stability Studies on the Benchmark Problems
We will now discuss the results of numerical stability studies. Among the
three benchmark problems discussed earlier, the lid-driven cavity flow
presents the most stringent test since it is a fully 2D problem with
boundaries containing singularity and the flow is shear driven. In fact, such
a cavity flow problem was considered in detail to determine stability regimes
of the SRT and the standard MRT collision models in a recent work Luo et al.
(2011). Earlier, its three-dimensional counterpart was also considered from
this viewpoint Premnath et al. (2009). These studies have demonstrated the
superiority of the use of multiple relaxation times in providing controlled
additional numerical dissipation to enhance numerical stability on either
coarser grids or at high Reynolds numbers when compared with the single
relaxation time models. Hence, it is appropriate to consider the 2D lid-driven
cavity flow to establish the stability regime of the cascaded MRT LBM in the
context of other collision models. We now make a direct comparison of the
maximum threshold Reynolds number for numerical stability of the SRT LBM, the
standard MRT LBM and the cascaded MRT LBM for this problem. With the cascaded
MRT LBM, the relaxation parameters $\omega_{4}=\omega_{5}=1/\tau$ are selected
based on the specified kinematic viscosity, while the rest of relaxation
parameters are set to unity for simplicity. For each approach, for a given
grid resolution, the lid velocity was fixed and the relaxation time $\tau$ was
decreased gradually until the computation became unstable.
Figure 18 shows the maximum Reynolds number ($Re=U_{0}L/\nu$) that could be
attained for each method before the computations became unstable, i.e. when
the relative global error increases rapidly or becomes exponentially large as
the simulation progresses. Results are provided for different grid resolutions
for these three approaches. It is clear that the cascaded MRT computations can
reach Reynolds numbers that are about $2$ or $3$ times higher than that of the
standard MRT approach and the standard MRT computations can reach Reynolds
numbers that are $3$ or $4$ times higher than that of the SRT approach. The
latter results are consistent with prior findings Luo et al. (2011); Premnath
et al. (2009). Relaxation of different _central_ moments at different rates
provides a controlled additional numerical dissipation to maintain numerical
stability. That is, maintaining frame invariance in conjunction with the use
of multiple relaxation times further promotes the stability of the method. It
may be noted that stabilization of certain classical methods have been
achieved by constructing discretization operators that enforce Galilean
invariance Scovazzi (2007a, b); Hughes et al. (2010). Hence, it may be
expected that explicitly incorporating an invariance property could aid with
other standard mechanisms of stabilization of the LBM. As carried out in Luo
et al. (2011), we also perform an alternate stability test with the three
approaches on a chosen coarse grid for this problem. In this test, the grid
resolution is fixed at a relatively coarse resolution of $26\times 26$, and
then viscosity $\nu$ (or equivalently $\tau$) is also set for all the three
approaches. We then intend to find the maximum lid velocity which can maintain
the stability of computations for $50,000$ time steps Luo et al. (2011).
Figure 19 shows how the three methods behave for this test. It is seen that
the parameter regime or the maximum lid velocity for stability is considerably
higher with the cascaded MRT LBM when compared with the other approaches. This
further establishes the merits of the use of multiple relaxation times for
central moment relaxation. Often, the stability of the CFD methods are
characterized in terms of the grid or cell Reynolds number given by
$Re_{c}=U_{0}\Delta x/\nu$ (e.g. Wesseling (2000)). Thus, we also present the
maximum cell Reynolds number for stability of the three approaches for this
problem in Table 4, which demonstrates the advantages of the cascaded MRT LBM.
Figure 18: Comparison of the maximum Reynolds number for numerical stability of different methods for simulation of the lid-driven cavity flow. Figure 19: Alternative stability test to determine the maximum threshold lid velocity for different methods for a chosen coarse resolution ($26\times 26$). Table 4: Comparison of the maximum cell Reynolds number ($Re_{c}=U_{0}\Delta x/\nu$) for numerical stability of different methods for simulation of the lid-driven cavity flow problem. Grid Resolution | SRT LBM | Standard MRT LBM | Cascaded MRT LBM
---|---|---|---
$101\times 101$ | $14.14$ | $59.40$ | $148.50$
$201\times 201$ | $14.21$ | $62.18$ | $165.83$
$401\times 401$ | $14.25$ | $62.34$ | $199.50$
Another important aspect is the computational cost. As shown previously, the
cascaded MRT approach can be more stable with similar accuracy compared with
the standard MRT for the lid-driven cavity flow. But if it is much more
expensive for numerical computations than the standard MRT, its advantages
will not be very useful. In this regard, we fully exploit all the optimization
strategies that could be used with a moment approach, such as those specified
in d‘Humières et al. (2002) for the cascaded MRT LBM. It is found that for the
2D lid-driven cavity flow problem, the cascaded MRT LBM takes about $11.6\%$
longer than the standard MRT LBM, which is acceptable in view of the
significant advantages in terms of numerical stability. It should be pointed
out that these results pertain only to 2D problems. Additional work is
required in three-dimensions to optimize the computational cost of the
cascaded MRT LBM and also to optimize its relaxation parameters by means of a
linear Fourier analysis.
## VI Summary and Conclusions
Galilean invariance is one of the main physical attributes in the description
of the fluid motion. This is naturally achieved by considering dynamical
changes in terms of central moments in kinetic schemes, as was done in the
recently introduced cascaded LBM. Enforcing frame invariance is generally
expected to have a positive influence on numerical stability as seen in some
recent work with other classical schemes. The use of multiple relaxation times
(MRT) in the central moment or cascaded LBM brings in the various flexibility
associated with the standard MRT LBM based on raw moments. In particular, the
relaxation of different central moments at different rates introduces
additional dissipation as in the raw moment based approach, which can lead to
enhanced stability.
In this paper, we discussed our results from systematic numerical studies on
grid convergence, accuracy, and stability of the cascaded MRT LBM. We have
chosen three commonly used 2D benchmark problems including the Poiseuille
flow, the decaying Taylor-Green vortex flow, and the lid-driven cavity flow.
In the grid convergence tests, the cascaded MRT approach has been found to be
second order accurate under diffusive scaling for all the benchmark problems
considered. These results are shown to hold not only for the velocity field,
but also for the components of the strain rate tensors. Furthermore,
comparisons of the numerical accuracy of the cascaded MRT LBM were made with
other collision models and also with prior analytical or numerical results
based on the solution of the Navier-Stokes equations. These demonstrated that
the cascaded MRT LBM is in excellent agreement with the prior results for all
the canonical problems considered. In particular, the detailed flow structures
for the more complex lid-driven cavity flow predicted by the cascaded MRT LBM
are in very good quantitative agreement with the standard MRT LBM. In
addition, the utility and the accuracy of the use of non-equilibrium moments
with the cascaded MRT LBM for the computation of the components of the strain
rate tensor is demonstrated. Finally, stability tests on a 2D lid-driven
cavity flow problem was carried out, which showed substantial improvements in
numerical stability of the cascaded MRT LBM, with higher threshold Reynolds
numbers, when compared to other models. With the use of proper optimization
strategies, the 2D cascaded MRT LBM was found to be only about $10\%$ to
$20\%$ more expensive when compared to the standard MRT LBM in terms of
computational time.
Future work could include further development of more optimized formulations
of the three-dimensional cascaded LBM based on central moments with a view to
maintain computational efficiency and their applications to unsteady
multiscale problems such as turbulence. Optimization of the relaxation
parameters by a linear Fourier analysis to introduce adequate additional
dissipation for enhanced numerical stability while maintaining necessary
physics with this approach is also desired.
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|
arxiv-papers
| 2012-02-28T20:37:42 |
2024-09-04T02:49:28.013653
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Ning and Kannan N. Premnath",
"submitter": "Kannan Premnath",
"url": "https://arxiv.org/abs/1202.6351"
}
|
1202.6354
|
∎
11institutetext: Bernhard Haslhofer 22institutetext: Cornell University,
Department of Information Science
301 College Avenue, Ithaca, NY 14850, USA
22email: bernhard.haslhofer@cornell.edu 33institutetext: Robert Sanderson and
Herbert van de Sompel 44institutetext: Los Alamos National Laboratory
Los Alamos, NM 87544, USA
44email: rsanderson,herbertv@lanl.gov 55institutetext: Rainer Simon
66institutetext: Austrian Institute of Technology
Donau-City-Str. 1, A-1220 Vienna, Austria
66email: rainer.simon@ait.ac.at
# Open Annotations on Multimedia Web Resources
Bernhard Haslhofer Robert Sanderson Rainer Simon Herbert van de Sompel
(Received: / Accepted: date)
###### Abstract
Many Web portals allow users to associate additional information with existing
multimedia resources such as images, audio, and video. However, these portals
are usually closed systems and user-generated annotations are almost always
kept locked up and remain 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. We present the current state of the Open Annotation
Model, explain our design rationale, and describe how the model can represent
user annotations on multimedia Web resources. Applying this model in Web
portals and devices, which support user annotations, should allow clients to
easily publish and consume, thus exchange annotations on multimedia Web
resources via common Web standards.
###### Keywords:
Annotations Web Linked Data
††journal: Multimedia Tools and Applications
## 1 Introduction
Youtube and Flickr are examples of large-scale Web portals that allow users to
annotate multimedia resources by adding textual notes and comments to images
or videos. Figure 1 shows an
image111http://www.flickr.com/photos/library_of_congress/3175009412/ from the
Flickr Commons collection contributed by the Library of Congress. Flickr users
added several annotations to specific image segments, one of them telling us
that this picture shows a cathedral in Bergen.
Figure 1: An annotation example on Flickr.
Annotations describe resources with additional information, which is valuable
to other users, who are searching and browsing resource collections. They are
also important for underlying information systems, which can exploit the high-
level descriptive semantics of annotations in combination with automatically
extracted low-level features, such as image size and color, to implement
search and retrieval over multimedia resources. Taking the previous example,
users who are searching for “Bergen” or even “Bergen Cathedral” will now find
this particular image in Flickr, because some user provided this descriptive
information in textual form.
Annotations are also becoming an increasingly important component in scholarly
cyber-infrastructures (cf. Bradley:2008kx ), which are often realized as Web
systems. Therefore, 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.
Instead, _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 Heath:2011uq 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., Hunter2009 ) annotating an image or video. Things become
more complex when a particular paragraph in an HTML document annotates a
segment (cf., Hausenblas:LDOW09 ) 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 an association created between one
_body_ resource and other _target_ resources, where the body must be somehow
_about_ the target.
Annotea Kahan:2001vn already defines a specification for publishing
annotations on the Web 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 Koivunen:2006s introduced
additional types of annotations, such as _bookmark_ and _topic_. Schroeter and
Hunter Schroeter:uq 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.
Haslhofer:2009ve 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 guidelines. However, these extensions were developed
separately from each other and inherit some of the above-mentioned Annotea
shortcomings.
In this article we describe the _Open Annotation Model_ 222Open Annotation
Model: Beta Data Model Guide http://www.openannotation.org/spec/beta/, which
is currently being developed in an international collaboration. It applies a
Web- and resource-centric view on annotations and defines a modular
architecture, which has a simple base line model in its core. The model also
provides means to address segments in multimedia resources either by encoding
segment information using the Media Fragment URI specification or by
introducing custom segment constraints for more complex annotation use cases.
By allowing fixity and timestamp information on the resources involved in an
annotation, it also takes into account the ephemeral nature of Web resources.
Pulling together the functionalities provided by various, partly independent
Annotea extensions is a major goal of this effort.
This article extends our work previously published in Haslhofer:2011fk by the
following: it explains the design rationale that lead to the specification of
the current annotation model and also the technical aspects of the model in
more detail. It also contains an updated and extended related work section.
## 2 Design Rationale
In this section we outline the design rationale that drives the specification
of the Open Annotation Model. We give examples illustrating common
requirements we found in several real-world annotation use cases (e.g.,
Sanderson:2011a ; Verspoor:2005kx ; Simon:2011vn ) and describe the reasons
behind our design decisions. From each design decision we derived a set of
guiding design principles, which are reflected in the current Open Annotation
Model design.
### 2.1 Annotations are qualified associations between resources
Textual notes or tags on images, as in the Flickr example in Figure 1, occur
frequently on the Web and are the simplest examples for Web annotations. In
Open Annotation Model terms, these are _annotations_ that have a textual
_body_ and an image resource as _target_. However, in many scenarios the
prevailing view that annotation bodies are textual is insufficient. Figure 2
shows an annotation in which the body is not a textual note, but a video,
which itself is an addressable Web resource identified by a URI. In order to
cover such cases, we must abstract from purely textual annotation bodies and
model them as resources that can be of any media type.
From a conceptual point of view, an annotation is an association between two
resources, the _body_ and the _target_. However, in most cases this
association needs to be qualified and addressable in some way. Indicating the
creator and creation date of a resource, or replying to existing annotations,
are frequently occurring scenarios that require an annotation to be expressed
as a first-class entity. Web annotations are then instances of an annotation
and relate together the body and target resources that are involved in the
annotation association. With this approach we follow a common design pattern,
which is also known as Qualified Relation in Linked Data Dodds:uq or
Association Class in UML.
Figure 2: A Youtube video annotating an image on the Web.
In contrast to existing tagging models (see KimEtAl2008 ) the annotation
_creator_ is not a mandatory core model entity. We believe that adopters of
the Open Annotation Model already have user models in place or rely on open
Web identities (e.g., a Google or Facebook account) and are most likely not
willing to map their user models against another user model, which is
specified as part of an annotation model. However, we encourage adopters to
relate annotations with existing Web resources, which identify users.
From these considerations we derive the following guiding principles:
* •
All core entities (annotation, body, target) must be resources.
* •
Annotations must allow for both body and target of any media type.
* •
Annotations, bodies, and targets can have different authorship.
### 2.2 Annotations involve parts of multimedia resources
Our introductory Flickr example illustrates how annotations can target
specific segments in Web resources. In Figure 2 we gave an example that
involves multimedia resources: a video as body, and an image as target. There
is strong dependency between these requirements because a resource’s media
type affects the way segments need to be addressed. Annotating an area in an
image requires a different segment representation than those that target
segments in the spatial and temporal dimensions of a video. Similarly,
addressing text segments in a PDF document differs from addressing text in
plain text or HTML documents. Since we abstract from purely textual annotation
bodies, we must take into account that both the body and the target of an
annotation can be resource segments. We could, for instance, refine our
previous example by saying that some sequence within the video annotates a
certain image segment.
The problem of addressing media segments, also known as fragment
identification, is well known and will be explained in more detail in Section
6.3. Since the Open Annotation Model will be implemented in Web environments
and all resources involved in an annotation are Web resources identified by
URIs, it should reuse the fragment identification mechanisms that are already
defined as part of the Architecture of the World Wide Web Jacobs2004 and
extensions thereof, such as fragment construction rules for specific media
types. However, many annotation use cases require more complex segment
representations such as polygon regions in images, which cannot be expressed
with available standards. Therefore, our guiding principles with respect to
addressing segments in multimedia resources are:
* •
Annotations must support resource segment addressing both on body and target
resources.
* •
Preferably this should be done with (media) fragment URIs, but extensibility
must be provided for cases in which the use of URIs for segment addressing is
not possible.
### 2.3 Annotation resources are ephemeral Web resources
Previously we argued that all core entities of the Open Annotation Model must
be first class Web resources, which are identified by URIs, if possible HTTP
URIs. The great benefit of this approach is that existing technologies and
solutions that can be applied for Web resources (e.g., mime-types, fragments,
access control, etc.) also work for resources involved in a Web annotation
without the necessity to include these aspects in an annotation specification.
However, there is one big problem we inherit from the Web architecture and
which is severe in the context of annotations: URI-addressable Web resources
are ephemeral, which means that the representations obtained by dereferencing
their URIs may change over time.
The annotation example in Figure 3 shows a Twitter tweet annotating the CNN
web site and illustrates the ephemerality problem: the tweet refers to the CNN
page at a certain point in time and might be misinterpreted when the CNN main
web site changes. Measures should be provided that can help in avoiding
misinterpretations of annotations, including the expression of timestamps and
fixity information for body and target resources.
Figure 3: A Twitter tweet annotating the CNN website.
### 2.4 Annotations should be interoperable
The focus of the Open Annotation Model is on sharing annotations on scholarly
resources and therefore the model should have sufficient richness of
expression to satisfy scholars’ needs. However, in order to maximize the
likelihood of adoption, the model should also be an interoperability framework
readily applicable in other domains. One possibility to achieve this is to
follow a modular and extensible modeling approach, with a generally applicable
baseline model and domain- or media-type specific extensions. An annotation
client should implement at least the baseline model and, if possible, provide
fallback behavior for annotations that contain domain-specific extensions the
client might not be aware of.
In order to increase the likelihood of adoption, and in alignment with the
goal of sharing annotations, no client-server protocol for publishing,
updating, or deleting annotations will be specified. Rather, the
specifications will take a perspective whereby clients publish annotations to
the Web and make them discoverable using common Web approaches. Such an
approach does not require a preferred annotation server for a client, yet it
does not preclude one either.
Our guiding principles to achieve annotation interoperability and widespread
adoption are:
* •
The Open Annotation Model should have a simple but expressive baseline model
defining top-level classes/entities and properties/relationships.
* •
The baseline model should be extensible.
* •
Annotation protocols are out of scope.
## 3 The Baseline Model
The Open Annotation data model draws from various extensions of Annotea to
form 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/beta/,
is still under development. In the following, we describe the major technical
building blocks of the Open Annotation data model and use some simple examples
to illustrate how to apply the model in practice. For further examples, which
also cover different use cases, such as the annotation of medieval
manuscripts, we refer to the online documentation.
Following its predecessors, the Open Annotation Model, shown in Figure 4, has
three primary classes of resources. In all cases below, the oac namespace
prefix expands to http://www.openannotation.org/ns/.
Figure 4: Open Annotation baseline data model.
* •
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 either one body per annotation, or an annotation without any body,
but not annotations with multiple bodies.
* •
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 is an RDF 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.
Figure 5: Additional properties and relationships in the Open Annotation
baseline model.
An annotation is a Web resource, which is identified by an HTTP URI and
returns an RDF document when being dereferenced. As with any RDF data,
additional properties and relationships can be associated with any of the
resources. It is recommended that an annotation has a timestamp of when the
annotation relationship was created (dcterms:created) and a reference to the
agent that created it (dcterms:creator). Resources referenced by additional
relationships may themselves have additional properties and relationships.
Figure 5 gives an example of recommended and other possible (e.g., dc:title)
properties and relationships that can be added to the Open Annotation baseline
model. The set of properties and relationship in this example is by no means
exhaustive. Properties and relationships from other vocabularies may also be
used. It is also important to note that the creator and created timestamp of
each of the three resource types above may be different. An annotation might
refer to an annotation body created by a third party, perhaps from before the
Open Annotation specification was published, and a target created by yet
another party.
Similarly, there may be additional subclasses of oac:Annotation that further
specify restrictions on the meaning of the annotation, such as a _reply_ ; an
annotation where the single target is itself an annotation. This allows
chaining of annotations into a threaded discussion model.
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, it 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.
Figure 6: An example annotation with inline body.
The Open Annotation Model allows the inclusion of information directly in the
annotation document by adding the representation of a resource inline within
the RDF document using the _Content in RDF_ specification Koch:2009uq from
the W3C. The example annotation in Figure 6 shows how to express this: the
representation is the object of the cnt:chars predicate, and its character
encoding the object of the cnt:characterEncoding predicate. Further classes
from this specification include Base64 encoded resources and XML encoded
resources.
## 4 Annotating Media Segments
Most of the use cases, which have been explored before specifying the model,
involved comments that were about a segment of a resource, rather than the
entire resource identified by a URI. The data model allows two different
methods of identifying and describing the region of interest of a resource;
either using a _fragment URI_ , or a more expressive _constraint_ resource. It
is clearly recommended to use fragment URIs whenever possible, because this
method relies on normative specifications, which brings interoperability with
other applications. Only if there are no appropriate URI fragment
specifications available, the creators should define their own constraints.
### 4.1 Describing segments with fragment URIs
A fragment URI normally identifies a part of a resource, and the method for
constructing and interpreting these URIs is dependent on the media type of the
resource. In general, fragment URIs are created by appending a fragment that
describes the section of interest to the URI of the full resource, separated
by a ’#’ character (see Berners-Lee:2005uq ).
There are two main sources for existing fragment URI specifications, which can
both identify and describe how to discover a segment of interest within a
resource. The first is the set of Mime Type specification RFCs from the IETF.
This includes X/HTML (RFC 2854/3236), XML (RFC 3023), PDF (RFC 3778) and Plain
Text (RFC 5147). The second is W3C Media URIs specification
fragmentsURI:2011ab , which is defined at a broader level to cover images,
video and audio resources, regardless of the exact format. The following
examples show how to apply these specifications to define media-type specific
URIs that describe a certain resource segment:
* •
http://www.example.net/foo.html#namedSection identifies the section named as
“namedSection” in an HTML document.
* •
http://www.example.net/foo.pdf#page=10&viewrect=20,100,50,60 identifies a
rectangle starting at 20 pixels in from the left, and 100 down from the top,
with a width of 50 and a height of 60 in a PDF document.
* •
http://www.example.net/foo.png#xywh=160,120,320,240 identifies a 320 by 240
box, starting at x=160 and y=120 in an image.
* •
http://www.example.net/foo.mpg#t=npt:10,20 identifies a sequence starting just
before the 10th second, and ending just before the 20th in a video.
We recommend that when a definition exists for how to construct a fragment URI
for a particular document format, and such a fragment would accurately
describe the section of interest for an annotation, then this technique should
be used. It is recommended to also use dcterms:isPartOf with the full resource
as the object, in order to make the annotation more easily discoverable.
Figure 7 shows an example in which a tweet annotates a rectangular section in
an image, which in turn is identified and described by a media fragment URI.
Figure 7: Annotating media segments using a fragment URI.
### 4.2 Describing segments via constraint resources
There are many situations when segments cannot be described with fragment
URIs, but it is still desirable to be able to annotate a segment of a
resource. For example, a non-rectangular section of an image, or a segment of
a resource with a format or media type that is not covered by either fragment
specification, such as a 3-dimensional model or a dataset. To handle these
situations, we introduce an oac:Constraint resource that describes the segment
of interest using an appropriate standard, and a oac:ConstrainedTarget
resource that identifies the segment of interest. This _constrained target_ is
the object of the oac:hasTarget predicate of the oac:Annotation, and
subsequently oac:constrains the full target resource. Figure 8 shows how
constrained targets extend the Open Annotation baseline model.
Figure 8: Constraint annotation targets in the Open Annotation Model.
The nature of the constraint description will be dependent on the type of the
resource for which the segment is being conveyed. It is then up to the
annotation client to interpret the segment description with respect to the
full resource. Figure 9 shows an example in which an area within an image is
described by an SVG path element. The document containing the SVG
specification is identified by a dereferencable HTTP URI and a specialized
type oac:SvgConstraint in combination with a dc:format property informs the
client about the type of constraint it needs to deal with.
Figure 9: Annotating media segments using an SVG constraint.
Alternatively, it is also possible to include the constraint information
inline within the annotation document using the same technique as used for
including the body. The oac:Constraint is given a URN (normally a urn:UUID)
and then the constraint information is included as the value of the cnt:chars
property. The requirements for doing this are the same as for including the
oac:Body inline within the annotation document. For more complex use cases it
also possible to express constraints in RDF and to apply constraints also on
the body of an annotation.
One goal of the ongoing Open Annotation demonstrator activities is to collect
real-world constraint definitions from various use cases and to specify them
in the context of the OA model. We hope that this also serves as as feedback
loop for possible enhancements or additional URI fragment specifications.
## 5 Robust Annotations over 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 Open Annotation Model introduces three ways to express
temporal context. The manner in which these three types of annotations 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 oac:Annotation. For
example, if Alice recurrently publishes a tweet 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. Figure 10 shows how Uniform Time Annotations can be represented using
the Open Annotation Model.
Figure 10: A Uniform Time Annotation example.
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:WebTimeConstraint node, which is a specialization of
oac:Constraint, for both body and target. If, in the aforementioned example,
Alice would have the habit to publish a 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 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 Van-de-
Sompel:2009fk ; Sompel:2010fk , which is a framework that proposes a simple
extension of HTTP in order to connect the original and archived resources, can
be applied for recreating annotations. 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) Sanderson:2010fk .
## 6 Related Work
In this section we give an overview of existing work in the area of Web
annotations. After summarizing general works about annotations and annotation
interoperability, we analyze the features of existing Web annotation models
and compare them with those of the Open Annotation Model.
### 6.1 Annotations and annotation interoperability
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. Agosti:2007uq provide a comprehensive study on the
contours and complexity of annotations. A representative discussion on how
annotations can be used in various scholarly disciplines is given by Bradley
Bradley:2008kx . He describes how annotations can support interpretation
development by collecting notes, classifying resources, and identifying novel
relationships between resources.
The different forms and functions that annotations can take are analyzed by
Marshall Marshall:2000kx . She distinguishes between _formal_ and _informal_
annotations, whereby formal annotations follow structural standards and
informal ones are unstructured. Furthermore, Marshall divides into _implicit_
annotations that are intended for sharing and _explicit_ annotations of
personal nature, often interpretable only by the original creator. Further
divisions defined by Marshall with regard to the function of an annotation
include _permanent_ vs. _transient_ , _annotation as writing_ vs. _annotation
as reading_ , _extensive_ vs. _intensive_ , _published_ vs. _private_ and
_institutional_ vs. _workgroup_ vs. _individual_. The difference between
personal and public annotations in a digital environment is further
investigated in a study by Marshall and Brush Marshall2004 . They derive
design implications for annotation systems, e.g. regarding find and filtering
requirements, and user interface strategies for processing and sharing
annotations.
A taxonomy of annotation types and marking symbols used by readers of
scholarly documents is presented by Qayyum Qayyum2008 . His taxonomy is
derived from the results of a user study conducted with students reading
research articles in a private as well as a collaborative digital setting. A
related recent effort is presented by Blustein et al. Blustein2011 . In their
field study, conducted over the course of three years, they identify six
purposes for scholarly annotation: _interpretation_ , _problem-working_ ,
_tracing progress_ , _procedural annotations_ , _place marking and aiding
memory_ and _incidental markings_.
### 6.2 Web annotation models
The idea of publishing user annotations on the Web is not new. Annotea
Kahan:2001vn was specified more than a decade ago and defines a data model
and protocol for uploading, downloading, and modifying annotations. Since the
Web has changed over time and now also comprises non-document Web resources,
the Annotea model soon became insufficient for many annotation use cases, as
we explained at the beginning of this article. Annotea extensions, such as Co-
Annotea Hunter:2008ab or LEMO Haslhofer:2009ve , were developed to deal with
the Annotea shortcomings and to take into account emerging architectural
styles, such as RESTful Web Services, or Linked Open Data. Recent Web
annotation model specification efforts include the M3O ontology
Saathoff:2010vn , the Annotation Ontology Ciccarese2011 , and the Open
Annotation Model, which we presented in this article.
Figure 11: Feature analysis of existing Web annotation models.
In Figure 11 we present the results of a feature comparison we performed
across the previously mentioned models. The models are timely ordered by their
publication year, and the feature selection is based on the requirements and
use case descriptions we found in the model documentations. Although we cannot
generalize from this representative set of annotation models and features, we
can observe hat annotation models have continually been adapted to emerging
standards, needs, and architectures: Linked Open Data is increasingly adopted
for publishing and sharing annotations, multimedia resources can now be
annotated also by other multimedia resources, extensibility has become a key
requirement, security and provenance are being outsourced to other models, and
standard segment identification mechanisms are being combined with custom
solutions (context, fragment, selectors) to capture complex domain-specific
needs.
The Annotation Ontology, which is an open ontology for the annotation of
scientific documents on the Web, is technically very similar to the Open
Annotation Model. However, they differ e.g., in terms of how fragments are
being expressed: by representing constraints and constraint targets as first-
class resources the Open Annotation Model supports direct addressing of
fragments, thus enabling use cases where different users annotate the same
fragment, or search scenarios where annotations are retrieved by fragment.
Furthermore, the Open Annotation Model supports structured annotation bodies
and allows to overlay semantic statements pertaining to one or more annotation
targets, which offers potentially more flexibility, e.g., for use cases of
entity and entity-relation extraction in scientific literature.
A related strand of research concerns models for _social tagging_ of Web
resources. Hunter Hunter2009 describes tags as “ _a subclass of annotations
that comprise simple, unstructured labels or keywords assigned to digital
resources to describe and classify the digital resource_ ”. A comparison of
tagging ontologies is presented by Kim et al. KimEtAl2008 . They survey the
state of the art in tagging models and identify three building blocks common
to existing tagging models: _taggers_ , the _tags_ themselves and the
_resources_ being tagged.
Semantic annotations features can also be found in multimedia metadata
frameworks such as MPEG-7 and multimedia metadata ontologies such as COMM
Arndt:2007uq or the recent _W3C Ontology for Media Resources_
333http://www.w3.org/TR/mediaont-10/#example3 specification, which provides a
core metadata vocabulary for media resources on the Web. It defines two
metadata properties that can be used for the textual description of a media
resource (fragment) or for relating RDF files or named graphs to a media
resource. Other ontologies were designed to embed annotations directly into
the multimedia content representation. The M3O Ontology Saathoff:2010vn , for
instance, allows the integration of annotations with SMIL and SVG documents.
On the contrary, the Open Annotation Model is more in line with the previously
discussed Web Annotation models. It treats annotations as first class Web
resources, which can exist independently from the content or metadata
representations of media objects. This design choice is motivated by a set of
scholarly use cases, which require that multimedia content objects can be
annotated any time after the content production and metadata extraction
process.
### 6.3 Media segment identification
Early related work on the issue of describing segments in multimedia resources
can be traced back to research on linking in hypermedia documents (cf.
Hardman:1994zr ). For describing segments using a non-URI based mechanism one
can use MPEG-7 Shape Descriptors (cf. Nack:1999ly ) or terms defined in a
dedicated multimedia ontology. SVG svg:2003bh and MPEG-21 ISO/IEC:2006qf
introduced XPointer-based URI fragment definitions for linking to segments in
multimedia resources. The Temporal URI specification Pfe07 addresses a
temporal segment in a time-based media resource through a defined URL query
parameter (’t=’). YouTube supports similar direct linking to a particular
point in time in a video using a fragment URI. The Media Fragments URI
Specification VanDeursen:2010 is a W3C Working Draft that introduces a
standard, URI-based approach for addressing temporal, spatial and track sub-
parts of any non-textual media content, thus making audiovisual media segments
first class citizens on the Web Hausenblas:LDOW09 .
### 6.4 Robustness of Web resources
The ephemeral nature of Web resources and methods to deal with that problem
have been studied from the early years of the Web on. Phelps and Wilensky
Phelps:2000ys proposed to decorate hyperlinks with lexical signatures to re-
find disappeared web resources. Recent works include Klein et al. Klein:2011kx
, who proposed to compute lexical signatures from the link neighborhood of a
Web page, and Morishima et al. Morishima:2009vn who describe a method to fix
broken links when link targets have moved. The problem has also been realized
in the Linked Data context and solutions like DSNotify Popitsch:2011zr were
proposed to re-find resources by their representations.
## 7 Summary and Future Directions
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
Open Annotation Model, which originates from activities in the Open Annotation
Collaboration and aims at building an interoperable environment for publishing
annotations on the Web.
At the time of this writing, the Open Annotation Model is still in beta stage
and is currently implemented in several demonstration projects covering use
cases ranging from annotating medieval manuscripts, over annotating online
maps, to annotating online video segments. As part of these demonstration
projects client APIs are being developed for various programming environments
to ease model adoption for developers. We expect to obtain user and developer
feedback from these projects, which should further refine the Open Annotation
Model.
Pursuing better integration of the proposed segment identification approach
with the W3C Media Fragment URI specification is on our research agenda.
However, this requires an extension mechanism in that specification, which is
not within the scope of the responsible working group at the moment. Also the
question of how to model multiple annotation targets and how to interpret this
information correctly, is currently being discussed. Finally, as a first
outcome of the demonstration projects, we observed that a common set of use
cases is annotating data with other data, rather than with information
intended for human consumption. This raises the question of how to model _Data
Annotations_ in an interoperable way.
As a final result, we expect a data model, which provides an interoperable
method of expressing annotations such that they can easily be shared between
platforms, with sufficient richness of expression to satisfy also complex
annotation scenarios.
###### Acknowledgements.
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 Program. The
development of OAC is funded by the Andrew W. Mellon foundation.
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|
arxiv-papers
| 2012-02-28T20:46:54 |
2024-09-04T02:49:28.025422
|
{
"license": "Public Domain",
"authors": "Bernhard Haslhofer, Robert Sanderson, Rainer Simon, Herbert van de\n Sompel",
"submitter": "Bernhard Haslhofer",
"url": "https://arxiv.org/abs/1202.6354"
}
|
1202.6409
|
# Classification of poset-block spaces admitting MacWilliams-type identity
Jerry Anderson Pinheiro and Marcelo Firer J. A. Pinheiro is with
IMECC–UNICAMP, State University of Campinas, CEP 13083-859, Campinas, SP,
Brazil (e-mail: jerryapinheiro@gmail.com).M. Firer is with IMECC–UNICAMP,
State University of Campinas, CEP 13083-859, Campinas, SP, Brazil (e-mail:
mfirer@ime.unicamp.br).
###### Abstract
In this work we prove that a poset-block space admits a MacWilliams-type
identity if and only if the poset is hierarchical and at any level of the
poset, all the blocks have the same dimension. When the poset-block admits the
MacWilliams-type identity we explicit the relation between the weight
enumerators of a code and its dual.
###### Index Terms:
Poset-block codes, MacWilliams identity, weight distribution, MacWilliams-type
identity.
## I Introduction
Due to both the interest in generalizing classic problems in coding theory and
to applications in cryptography, experimental designs and high-dimensional
numerical integration (see for example [1] and [2]), by the mid 1990s
researches began to study codes considering metrics others than the usual
Hamming metric over $\mathbb{F}_{q}^{n}$. Among those families of metrics are
the poset metrics [3] and the block metrics [2]. Much of the classical theory
has been generalized to codes in spaces endowed with a poset metric, as can be
seen, for example, in [4], [5], [6] and [7].
In 2008 Firer et al [8] presented the family of metrics called poset-block
that generalizes all the previous ones. In this work we generalize to poset-
block spaces the characterization given in [5] for poset-metric spaces of
poset-block metrics admitting MacWilliams-type identity.
Let $[m]:=\\{1,2,\cdots,m\\}$ be a finite set. If $\preccurlyeq$ is a partial
order relation in $[m]$, we say $P:=([m],\preccurlyeq)$ is a poset and denote
by $\preccurlyeq_{P}$ the order in $P$. An ideal in a poset is a nonempty
subset $I\subset[m]$ such that, for $i\in I$ and $j\in[m]$, if
$j\preccurlyeq_{P}i$ then $j\in I$. Given $A\subset[m]$, we denote by $\langle
A\rangle_{P}$ the smaller ideal of $P$ containing $A$. If $A=\\{i\\}$, we will
denote by $\langle i\rangle_{P}$ the ideal $\langle\\{i\\}\rangle_{P}$. A
chain in a poset $P$ is a subset of $[m]$ such that every two elements are
comparable.
Let $\mathbb{F}_{q}$ be a finite field and $\mathbb{F}_{q}^{n}$ the vector
space of $n$-tuples over $\mathbb{F}_{q}$. Given $m\in[n]$, $P$ a poset over
$[m]$ and $\pi:[m]\rightarrow\mathbb{N}$ a map such that
$n=\sum_{i=1}^{m}\pi(i)$, we say that $\pi$ is a labeling of the poset $P$ and
that the pair $(P,\pi)$ is a poset-block structure over $[m]$.
We denote $k_{i}=\pi(i)$, and consider the vector space over $\mathbb{F}_{q}$
$V:=\mathbb{F}_{q}^{k_{1}}\times\mathbb{F}_{q}^{k_{2}}\times\cdots\times\mathbb{F}_{q}^{k_{m}},$
isomorphic to $\mathbb{F}_{q}^{n}$. Given $u\in\mathbb{F}_{q}^{n}$, there is a
unique decomposition $u=(u_{1},\cdots,u_{m})$ with
$u_{i}\in\mathbb{F}_{q}^{k_{i}}$, $i\in[m]$. The $\pi$-support and the
$(P,\pi)$-weight of $u$ are defined respectively as
$supp_{\pi}(u):=\\{i\in[m]:u_{i}\neq 0\in\mathbb{F}_{q}^{k_{i}}\\}$
and
$w_{(P,\pi)}(u):=|\langle supp_{\pi}(u)\rangle_{P}|,$
where $|.|$ denotes the cardinality of the given set. For
$u,v\in\mathbb{F}_{q}^{n}$,
$d_{(P,\pi)}(u,v):=w_{(P,\pi)}(u-v)$
defines a metric over $\mathbb{F}_{q}^{n}$ called poset-block metric, or just
$(P,\pi)$-distance between $u$ and $v$.
We note that when $\pi(i)=1$ for every $i\in[m]$ the $(P,\pi)$-distance is
usual poset distance introduced in [3], while imposing $P$ to be a trivial
poset ($i\preccurlyeq j\iff i=j$) turns the $(P,\pi)$-distance into the block
distance defined in [2]. Interweaving the poset and the block structures opens
a wide range of possibilities for searching for codes with interesting metric
characteristics, such as perfect codes, since poset and block metrics have
opposite effects on distances: while enlarging the relations on a poset
enlarges the distances (hence “shrinks” metric balls), enlarging the blocks
diminishes distances (hence “blows” metric balls).
Concerned with MacWilliams-type identities, dual posets play a crucial role:
###### Definition 1
Given a poset $P$ over $[m]$, the dual poset is the poset $\overline{P}$
defined by the relations
$i\preccurlyeq_{P}j\iff j\preccurlyeq_{\overline{P}}i$
for every $i,j\in[m]$. The pair $(\overline{P},\pi)$ is called the dual poset-
block.
Given $j\in[m]$, the rank of $j$, denoted by $h_{P}(j)$, is
$h_{P}(j):=max\\{|C|:C\subset\langle j\rangle_{P}\ \mbox{and}\ C\ \mbox{is a
chain}\\}.$
The height $h(P)$ of $P$ is the maximal rank of the elements of $[m]$. The
$i$-level of $P$ is $\Gamma_{P}^{i}:=\\{j\in[m]:h_{P}(j)=i\\}$. We define
$b_{i}=\sum_{j\in\Gamma_{P}^{i}}k_{j}$ as the sum of the dimensions of the
blocks associated by $\pi$ to the $i$-level of $P$, and we call it the
dimension of $\Gamma_{P}^{i}$.
A poset-block $(P,\pi)$ is said to be hierarchical if given
$j_{1}\in\Gamma_{P}^{i}$ we have that $j_{1}\preccurlyeq_{P}j$ for all
$j\in\Gamma_{P}^{i+1}$. Defining a hierarchical poset on $[m]$ is equivalent
to choosing an ordered partition of $[m]$ (the partition defined by the
different levels), thus it is a quite large set of posets (or poset metrics)
including, as a particular case, the block structures presented in [2] when
the poset structure is trivial ($h(P)=1$), the Niederreiter-Rosenbloom-
Tsfasman metric (see [9]) with a unique chain when $h(P)=m$ and the block
structure is trivial ($k_{i}=1$ for every $i\in[m]$) and the usual Hamming
structure when both the poset and the block structures are trivial.
Given a poset-block $(P,\pi)$ over $[m]$ such that $|\Gamma_{P}^{i}|=m_{i}$,
let $\sigma$ be a permutation of $[m]$ such that
$\\{\sigma^{-1}(r_{i}+1),\cdots,\sigma^{-1}(r_{i}+m_{i})\\}=\Gamma_{P}^{i}$
where $r_{i}=m_{1}+\cdots+m_{i-1}$ and $m_{0}=0$. We let $P_{1}$ be the poset
induced by $\sigma$, ie, the poset in which
$\sigma(j_{1})\preccurlyeq_{P_{1}}\sigma(j_{2})$ if
$j_{1}\preccurlyeq_{P}j_{2}$. Obviously, $P_{1}$ and $P$ are isomorphic
posets. If we put $\pi_{1}(i)=\pi(\sigma^{-1}(i))=k_{i}^{\prime}$, then the
map
$g:(\mathbb{F}_{q}^{k_{1}}\times\cdots\times\mathbb{F}_{q}^{k_{m}},d_{(P,\pi)})\rightarrow(\mathbb{F}_{q}^{k_{1}^{\prime}}\times\cdots\times\mathbb{F}_{q}^{k_{m}^{\prime}},d_{(P_{1},\pi_{1})})$
$\ \ \ \ \ \ \ \
(v_{1},\cdots,v_{m})\mapsto(v_{\sigma(1)},\cdots,v_{\sigma(m)})$
is, by construction, a linear isometry. Hence, up to a linear isometry, we can
and will assume that $\Gamma_{P}^{i}=\\{r_{i}+1,\cdots,r_{i}+m_{i}\\}$, and in
this case we say $(P,\pi)$ has a natural labeling. Hence, given
$u\in\mathbb{F}_{q}^{n}$ we may decompose it as
$u=\sum_{i=1}^{h(P)}\sum_{j=1}^{m_{i}}\sum_{l=1}^{k_{(r_{i}+j)}}u_{r_{i}+j}^{l}e_{s(i,j,l)}$
where $u_{r_{i}+j}^{l}\in\mathbb{F}_{q}$ are scalars and
$\left\\{e_{s(i,j,l)}:1\leqslant l\leqslant k_{(r_{i}+j)},1\leqslant
j\leqslant m_{i},1\leqslant i\leqslant h(P)\right\\}$
is the usual basis of $\mathbb{F}_{q}^{n}$, with
$s(i,j,l)=l+\sum_{t=0}^{r_{i}+j-1}k_{t}$ and $k_{0}=0$.
A $[n,k,\delta]_{q}$ linear $(P,\pi)$-code is a $k$-dimensional subspace
$\mathcal{C}\subset\mathbb{F}_{q}^{n}$ where $\mathbb{F}_{q}^{n}$ is equipped
with the poset-block metric $d_{(P,\pi)}$ and
$\delta=min\\{w_{(P,\pi)}(v):0\neq v\in\mathcal{C}\\}$
is the $(P,\pi)$-minimum distance of $\mathcal{C}$.
###### Definition 2
Let $\mathcal{C}$ be a linear $(P,\pi)$-code. Its dual code is defined as
$\mathcal{C}^{\perp}=\\{x\in\mathbb{F}_{q}^{n}:x\cdot u=0\ \forall\
u\in\mathcal{C}\\}$
where $x\cdot u$ is the usual formal inner product. We remark that
$\mathcal{C}^{\perp}$ is an $(n-k)$-dimensional linear code. Along this work,
$\mathcal{C}^{\perp}$ is considered to be a linear $(\overline{P},\pi)$-code
with parameters $[n,n-k]_{q}$ and we denote by $\delta^{\perp}$ its minimal
distance (according to the $(\overline{P},\pi)$-metric).
Given a linear $(P,\pi)$-code $\mathcal{C}$, the $(P,\pi)$-weight enumerator
of $\mathcal{C}$ is the polynomial
$W_{\mathcal{C},(P,\pi)}(x)=\sum_{u\in\mathcal{C}}x^{w_{(P,\pi)}(u)}=\sum_{i=0}^{m}A_{i,(P,\pi)}(\mathcal{C})x^{i},$
where $A_{i,(P,\pi)}(\mathcal{C})=|\\{u\in\mathcal{C}:w_{(P,\pi)}(u)=i\\}|$.
When no confusion may arise, we will use a simplified notation for those
coefficients: $A_{i}=A_{i,(P,\pi)}(\mathcal{C})$ and
$\overline{A}_{i}=A_{i,(\overline{P},\pi)}(\mathcal{C}^{\perp})$.
Note that
$V:=\mathbb{F}_{q}^{b_{1}}\times\cdots\times\mathbb{F}_{q}^{b_{t}}$
is a vector space over $\mathbb{F}_{q}$ isomorphic to $\mathbb{F}_{q}^{n}$, so
that given $u\in\mathbb{F}_{q}^{n}$ we can write $u=(u^{1},\cdots,u^{t})$
where $u^{i}\in\mathbb{F}_{q}^{b_{i}}$ and
$u^{i}=(u_{r_{i}+1},\cdots,u_{r_{i}+m_{i}})$ is such that
$u_{r_{i}+j}\in\mathbb{F}_{q}^{k_{(r_{i}+j)}}$.
If $P$ is a poset with $t$ levels, the leveled $(P,\pi)$-weight enumerator of
$\mathcal{C}$ is the formal expression
$W_{\mathcal{C},(P,\pi)}(x;y_{0},\cdots,y_{t}):=\sum_{u\in\mathcal{C}}x^{w_{(P,\pi)}(u)}y_{s_{P}(u)},$
where $s_{P}(u)=max\\{i:u^{i}\in\mathbb{F}_{q}^{b_{i}}\backslash\\{0\\}\\}$
and $s_{P}(0)=0$. This definition is similar to the one used in [5] in the
classification of poset metrics that admits MacWilliams-type identity, ie, the
case where the block structure is trivial. It is clear that
$W_{\mathcal{C},(P,\pi)}(x)=W_{\mathcal{C},(P,\pi)}(x;1,\cdots,1)$.
###### Definition 3
We say that a poset-block $(P,\pi)$ admits a MacWilliams-type identity (MW-I)
if the $(\overline{P},\pi)$-weight enumerator of $\mathcal{C}^{\perp}$ is
uniquely determined by the $(P,\pi)$-weight enumerator of $\mathcal{C}$ for
every linear $(P,\pi)$-code $\mathcal{C}$.
MacWilliams-type identities in the context of poset codes have interested
researchers (see [4], [10] and [11]) since they establish a relation between
important invariants of a high information rate code with those of a low
dimension code, that are much easier to compute. In 2005, Kim and Oh [5]
proved that a poset space admits a MW-I if and only if the poset is
hierarchical. In this work we extend this result to the instances that
remained open: the instance of poset-block (and block metrics as a particular
case).
## II MacWilliams-type identity in $(P,\pi)$ spaces
The example below shows that the condition established in [5] is not
sufficient to ensure MacWilliams-type identity in $(P,\pi)$ spaces.
###### Example 1
Let $P=\\{1,2,3\\}$ be the hierarchical poset with partial order defined by
the relations $1\preccurlyeq_{P}2$ and $1\preccurlyeq_{P}3$ so that the dual
poset $\overline{P}$ is defined by the relations
$2\preccurlyeq_{\overline{P}}1$ and $3\preccurlyeq_{\overline{P}}1$. Define
$\pi:[3]\rightarrow\mathbb{N}$ by $\pi(1)=1$, $\pi(2)=1$ and $\pi(3)=2$. Then,
direct computations shows that the linear codes
$\mathcal{C}_{1}=\\{(0,0,0,0),(0,0,1,0)\\}$
and
$\mathcal{C}_{2}=\\{(0,0,0,0),(0,1,0,0)\\}$
over $\mathbb{F}_{2}^{4}$ has the same $(P,\pi)$-weight enumerator:
$W_{\mathcal{C}_{1},(P,\pi)}(x)=1+x^{2}=W_{\mathcal{C}_{2},(P,\pi)}(x).$
However,
$W_{\mathcal{C}_{1}^{\perp},(\overline{P},\pi)}(x)=1+2x+x^{2}+4x^{3}$
and
$W_{\mathcal{C}_{2}^{\perp},(\overline{P},\pi)}(x)=1+3x+4x^{3},$
so that MW-I does not hold.
### II-A Necessary condition for MacWilliams-type identity
Let $(P,\pi)$ be a poset-block in $[m]$ with $t$ levels such that
$|\Gamma_{P}^{i}|=m_{i}$ for $i\in[t]$. The three lemmas below are the
equivalent, for the poset-block case, of Lemmas (2.1)–(2.4) in [5]. Despite
the fact their proofs for poset-block being more delicate than in the case of
posets (where the blocks are trivial), they are quite similar.
###### Lemma 1
Given $u\in\mathbb{F}_{q}^{n}$ then
$w_{(\overline{P},\pi)}(u)=m\Leftrightarrow
supp_{\pi}(u)\supset\Gamma_{P}^{1}$. Furthermore, if $u$ satisfies
$supp_{\pi}(u)\subset\Gamma_{P}^{1}$, we have that
$q^{n-b_{1}}\ \mbox{\large\textbar}\ |\\{v\in\mathbb{F}_{q}^{n}:u\cdot
v=0\mbox{ and }w_{(\overline{P},\pi)}(v)=m\\}|.$
where $a\mbox{\large\textbar}b$ means $a$ divides $b$ and $b_{1}$ is the
dimension of $\Gamma_{P}^{1}$.
###### Proof:
The first affirmation is evident. Let $u\in\mathbb{F}_{q}^{n}$ such that
$supp_{\pi}(u)\subset\Gamma_{P}^{1}$. Without loss of generality we can assume
that $\Gamma_{P}^{1}=[m_{1}]$ and $u=(u_{1},\cdots,u_{i},0,\cdots,0)$ where
$i\leqslant m_{1}$ and $u_{j}\in\mathbb{F}_{q}^{k_{j}}\backslash\\{0\\}$ for
all $j\in[i]$. Set
$\displaystyle
A:=\\{(v_{1},\cdots,v_{i}):v_{j}\in\mathbb{F}_{q}^{k_{j}}\backslash\\{0$
$\displaystyle\\}\ \forall\ j\in[i]\mbox{ and }$ $\displaystyle u_{1}\cdot
v_{1}+\cdots+u_{i}\cdot v_{i}=0\\}.$
In each $\mathbb{F}_{q}^{k_{j}}$ space we have $q^{k_{j}}-1$ non null vectors,
then we have $\prod_{j=i+1}^{m_{1}}(q^{k_{j}}-1)$ possibilities of vectors in
the blocks associated to elements of the subset $\\{i+1,\cdots,m_{1}\\}$ of
$[m]$, since we do not impose restrictions in the $m-m_{1}$ remaining blocks,
by first claim it follows that
$\displaystyle|\\{v\in\mathbb{F}_{q}^{n}:u\cdot v=0\mbox{ and
}w_{(\overline{P},\pi)}(v)$ $\displaystyle=m\\}|=$ $\displaystyle
q^{n-b_{1}}|A|\prod_{j=i+1}^{m_{1}}(q^{k_{j}}-1).$
∎
###### Lemma 2
If a poset-block $(P,\pi)$ admits a MW-I, then $j\preceq_{P}i$ for every
$i\in\Gamma_{P}^{2}$ and $j\in\Gamma_{P}^{1}$.
###### Proof:
Assuming $\Gamma_{P}^{2}\neq\emptyset$, it follows that $m>m_{1}$. Suppose
there is $i\in\Gamma_{P}^{2}$ that is not comparable to some
$j\in\Gamma_{P}^{1}$, that is, such that $|\langle
i\rangle_{P}|<1+|\Gamma_{P}^{1}|$. In this instance there are
$u,v\in\mathbb{F}_{q}^{n}$ such that $supp_{\pi}(u)=\\{i\\}$,
$supp_{\pi}(v)\subset\Gamma_{P}^{1}$ and $|\langle
supp_{\pi}(u)\rangle_{P}|=|\langle supp_{\pi}(v)\rangle_{P}|$. Without loss of
generality we can admit that $u=e_{s(2,1,1)}$. If $\mathcal{C}_{u}$ and
$\mathcal{C}_{v}$ are two one-dimensional linear $(P,\pi)$-codes generated by
$u$ and $v$ respectively, then $\mathcal{C}_{u}$ and $\mathcal{C}_{v}$ have
same $(P,\pi)$-weight enumerator. Assuming the MW-I in $(P,\pi)$,
$\mathcal{C}_{u}^{\perp}$ and $\mathcal{C}_{v}^{\perp}$ must have the same
$(\overline{P},\pi)$-weight enumerator. If $x\in\mathcal{C}_{u}^{\perp}$ then
$x_{r_{2}+1}^{1}=0$. Furthermore, by Lemma 1 $w_{(\overline{P},\pi)}(x)=m$ if
and only if $\Gamma_{P}^{1}\subset supp_{\pi}(x)$, so that
$\displaystyle|\\{x\in\mathcal{C}_{u}^{\perp}:$ $\displaystyle
w_{(\overline{P},\pi)}(x)=m\\}|=$ $\displaystyle\ \ \ \
|\\{x\in\mathbb{F}_{q}^{n}:x_{r_{2}+1}^{1}=0\mbox{ and }\Gamma_{P}^{1}\subset
supp_{\pi}(x)\\}|.$
Set
$A:=\\{x_{i}\in\mathbb{F}_{q}^{k_{i}}:x_{r_{2}+1}^{1}=0\\}$
and
$B:=\\{(x_{1},\cdots,x_{m}):x_{j}\neq 0\ \forall\ j\in[m_{1}]\mbox{ and
}x_{i}=0\\}.$
Since $i\notin\Gamma_{P}^{1}$, $|A|=q^{k_{i}-1}$ and
$|B|=q^{n-k_{i}-b_{1}}\prod_{j=1}^{m_{1}}(q^{k_{j}}-1)$, it follows that
$\displaystyle|\\{x\in\mathcal{C}_{u}^{\perp}:w_{(\overline{P},\pi)}(x)=m\\}|=$
$\displaystyle|B||A|=$ $\displaystyle
q^{n-b_{1}-1}\prod_{j=1}^{m_{1}}(q^{k_{j}}-1).$ (1)
On the other hand
$\displaystyle\\{x\in\mathcal{C}_{v}^{\perp}:$ $\displaystyle
w_{(\overline{P},\pi)}(x)=m\\}=$ $\displaystyle\ \
\\{x\in\mathbb{F}_{q}^{n}:x\cdot v=0\mbox{ and
}w_{(\overline{P},\pi)}(x)=m\\},$ (2)
hence, by Lemma 1 and by Equations (1) and (2) it follows that
$q\ \mbox{\large\textbar}\prod_{j=1}^{m_{1}}(q^{k_{j}}-1),$
a contradiction because $q$ is power of a prime. Therefore $|\langle
i\rangle_{P}|=1+|\Gamma_{P}^{1}|$, ie, $j\preceq_{P}i$ for all
$j\in\Gamma_{P}^{1}$. ∎
Let $P^{j}=P\backslash\cup_{i=1}^{j}\Gamma_{P}^{i}$. Consider on $P^{j}$ the
order induced by $P$ and let
$\pi^{j}=\pi|_{[m]\backslash\cup_{i=1}^{j}\Gamma_{P}^{i}}$ be the restriction
of $\pi$ to $[m]\backslash\cup_{i=1}^{j}\Gamma_{P}^{i}$.
###### Lemma 3
If a poset-block $(P,\pi)$ admits the MW-I, then the poset-block
$(P^{1},\pi^{1})$ also admits.
###### Proof:
If $m=m_{1}$ we have that $[m]\backslash\Gamma_{P}^{1}=\emptyset$ and there is
nothing to be proved. Let us assume that $m>m_{1}$ and let
$\mathcal{C}_{1}^{\prime}$ and $\mathcal{C}_{2}^{\prime}$ be linear
$(P^{1},\pi^{1})$-codes with length $n-b_{1}$ and same
$(P^{1},\pi^{1})$-weight enumerator. For $i=1,2$, let
$\mathcal{C}_{i}:=\mathbb{F}_{q}^{b_{1}}\oplus\mathcal{C}_{i}^{\prime}=\\{(u,v):u\in\mathbb{F}_{q}^{b_{1}}\mbox{
and }v\in\mathcal{C}_{i}^{\prime}\\}$
be linear $(P,\pi)$-codes with length $n$ and same $(P,\pi)$-weight
enumerator. Since $(P,\pi)$ admits MW-I, $\mathcal{C}_{1}^{\perp}$ and
$\mathcal{C}_{2}^{\perp}$ have the same $(\overline{P},\pi)$-weight
enumerator. Furthermore, the dual codes $\mathcal{C}_{1}^{\perp}$ and
$\mathcal{C}_{2}^{\perp}$ can be described as
$\displaystyle\mathcal{C}_{i}^{\perp}=\\{(u,v)\in\mathbb{F}_{q}^{b_{1}}\times\mathbb{F}_{q}^{n-b_{1}}:(u,v)$
$\displaystyle\cdot(a,b)=0$ $\displaystyle\forall\
a\in\mathbb{F}_{q}^{b_{1}}\mbox{ and }b\in\mathcal{C}_{i}^{\prime}\\}.$
Being $b\in\mathcal{C}_{i}^{\prime}$ the null code-word of
$\mathcal{C}_{i}^{\prime}$, by definition of $\mathcal{C}_{i}^{\perp}$ it
follows that $u$ is the null element of $\mathbb{F}_{q}^{b_{1}}$, hence
$\mathcal{C}_{i}^{\perp}=\\{(u,v):u=0\in\mathbb{F}_{q}^{b_{1}}\mbox{ and
}v\in\mathcal{C}_{i}^{\prime\perp}\\}.$
Therefore, by puncturing the codes $\mathcal{C}_{1}^{\perp}$ and
$\mathcal{C}_{2}^{\perp}$ in the first $b_{1}$ coordinates, it follows that
$\mathcal{C}_{1}^{\prime\perp}$ and $\mathcal{C}_{2}^{\prime\perp}$ have the
same $(P^{1},\pi^{1})$-weight enumerator. ∎
By induction, using Lemmas 2 and 3 we have the following necessary condition
for a poset-block $(P,\pi)$ to admit a MW-I.
###### Proposition 1
If $(P,\pi)$ admits the MW-I, then $P$ is a hierarchical poset.
By Example 1 we can conclude that the previous condition is not sufficient to
assure an MW-I and the following is also necessary:
###### Proposition 2
Suppose that $(P,\pi)$ admits a MW-I. Then, $\pi(j_{1})=\pi(j_{2})$ for all
$j_{1},j_{2}\in\Gamma_{P}^{i}$ and every $1\leqslant i\leqslant h(P)$, ie,
blocks at the same level have the same dimension.
###### Proof:
Given $i\in[h(P)]$ consider $j_{1},j_{2}\in\Gamma_{P}^{i}$ and assume
$\pi(j_{1})\leqslant\pi(j_{2})$. Let $\mathcal{C}_{u}$ and $\mathcal{C}_{v}$
be the one-dimensional linear $(P,\pi)$-codes with length $n$ generated by
$u=e_{s(i,j_{1}-r_{i},1)}$ and $v=e_{s(i,j_{2}-r_{i},1)}$ respectively, where
$r_{i}=m_{1}+\cdots+m_{i-1}$. By Proposition 1 the poset $P$ is hierarchical,
and since there are
$(q^{k_{j_{1}}-1}-1)+\displaystyle\sum_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{i}}{j\neq
j_{1}}}(q^{k_{j}}-1)$
elements in $\mathcal{C}_{u}^{\perp}$ with support contained in a unique block
at the $i$-level of $P$, then
$\displaystyle
A_{m_{i+1}+\cdots+m_{t}+1,(\overline{P},\pi)}(\mathcal{C}_{u}^{\perp})=$
$\displaystyle(q^{k_{j_{1}}-1}-1)\prod_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{l}}{i<l\leqslant
t}}q^{k_{j}}$
$\displaystyle+\displaystyle\sum_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{i}}{j\neq
j_{1}}}(q^{k_{j}}-1)\prod_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{l}}{i<l\leqslant
t}}q^{k_{j}}$
since, when considering the dual poset $\overline{P}$, there are no
restrictions on the coordinates in the blocks belonging to levels higher (in
$P$) than $i$. In a similar way we find that
$\displaystyle
A_{m_{i+1}+\cdots+m_{t}+1,(\overline{P},\pi)}(\mathcal{C}_{v}^{\perp})=$
$\displaystyle(q^{k_{j_{2}}-1}-1)\prod_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{l}}{i<l\leqslant
t}}q^{k_{j}}$
$\displaystyle+\displaystyle\sum_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{i}}{j\neq
j_{2}}}(q^{k_{j}}-1)\prod_{\genfrac{}{}{0.0pt}{}{j\in\Gamma_{P}^{l}}{i<l\leqslant
t}}q^{k_{j}}.$
Assuming that $(P,\pi)$ admits a MW-I it follows that
$A_{m_{i+1}+\cdots+m_{t}+1,(\overline{P},\pi)}(\mathcal{C}_{u}^{\perp})=A_{m_{i+1}+\cdots+m_{t}+1,(\overline{P},\pi)}(\mathcal{C}_{v}^{\perp}),$
ie, $\pi(j_{1})=\pi(j_{2})$. ∎
From the two previous propositions it follows that:
###### Theorem 1
If $(P,\pi)$ admits a MacWilliams-type identity then $P$ is a hierarchical
poset and blocks at the same level have the same dimension.
Figure 1: Diagram of a typical hierarchical poset-block with blocks of equal
dimension at each level.
### II-B Sufficient condition for MacWilliams-type identity
In this section we will prove that the conditions found to be necessary will
also be sufficient. Let $(P,\pi)$ be a hierarchical poset-block over $[m]$
with $t$ levels such that $|\Gamma_{P}^{i}|=m_{i}$, with $i\in[t]$. As before,
we let $m_{0}=0$ and $r_{i}=m_{1}+\cdots+m_{i-1}$. We can assume without loss
of generality that $\Gamma_{P}^{i}=\\{r_{i}+1,\cdots,r_{i}+m_{i}\\}$. Let
$d_{i}=\pi(r_{i}+j)$ for every $j\in[m_{i}]$, ie, blocks at the same level
have the same dimension. Under this condition the dimension of the $i$-level
is given by
$b_{i}=\sum_{j=1}^{m_{i}}\pi(r_{i}+j)=m_{i}d_{i}.$
We note that $n=b_{1}+\cdots+b_{t}$ and $m=m_{1}+\cdots+m_{t}$. Given
$i\in\\{0,1,\cdots,t\\}$, set
* •
$\widehat{b_{i}}=n-(b_{1}+\cdots+b_{i})$;
* •
$\widehat{m_{i}}=m-(m_{1}+\cdots+m_{i})$ and
* •
$\widetilde{u^{i+1}}=(u^{i+1},\cdots,u^{t})\in\mathbb{F}_{q}^{\widehat{b_{i}}}$.
With this definitions we have that
$w_{(\overline{P},\pi)}(u)=\widehat{m_{i}}+w_{\pi_{i}}(u^{i})$ where
$w_{\pi_{i}}(u^{i})$ is the $(\Gamma_{P}^{i},\pi|_{\Gamma_{P}^{i}})$-weight of
$u^{i}$, the block weight as introduced in [2]. Given a linear $(P,\pi)$-code
$\mathcal{C}$, the set
$\mathcal{C}_{i}=\\{u\in\mathcal{C}:\widetilde{u^{i+1}}=0\\}$
is a subcode of $\mathcal{C}$ that can be decomposed as
$\mathcal{C}_{i}=\mathcal{C}_{i}^{0}\sqcup\mathcal{C}_{i}^{1}$ where
$\mathcal{C}_{i}^{0}=\\{u\in\mathcal{C}_{i}:u^{i}=0\\}\ \mbox{ and }\
\mathcal{C}_{i}^{1}=\\{u\in\mathcal{C}_{i}:u^{i}\neq 0\\}.$
Given $i\in[t]$, the weight enumerator of the $i$-level of $P$ is defined as
$LW_{\mathcal{C},(P,\pi)}^{(i)}(x):=\sum_{j=1}^{m_{i}}A_{r_{i}+j}x^{r_{i}+j}.$
(3)
The coefficients of this polynomial represent the weight distribution of code-
words such that its support contains elements in the $i$-level and do not
contain elements that are above the $i$-level. If we define
$LW_{\mathcal{C},(P,\pi)}^{(0)}(x)=A_{0}$, it is clear that
$W_{\mathcal{C},(P,\pi)}(x;y_{0},y_{1},\cdots,y_{t})=\sum_{i=0}^{t}LW_{\mathcal{C},(P,\pi)}^{(i)}(x)y_{i}.$
(4)
If for each $i\in[t]$ we have that $y_{j}=1$ for $j\leqslant i$ and $y_{j}=0$
for all $j>i$, then the leveled $(P,\pi)$-weight enumerator of $\mathcal{C}$
coincides with the $(P,\pi)$-weight enumerator of $\mathcal{C}_{i}$, hence,
$\displaystyle
W_{\mathcal{C}_{i},(P,\pi)}(x)-W_{\mathcal{C}_{i-1},(P,\pi)}(x)$
$\displaystyle=LW_{\mathcal{C},(P,\pi)}^{(i)}(x)$
$\displaystyle=\sum_{u\in\mathcal{C}_{i}^{1}}x^{w_{(P,\pi)}(u)}.$ (5)
We introduce now some concepts related to additive characters, that will be
used in the proof in a way similar to what was done first by MacWilliams [12]
in the classical case and later in the poset case (see [4], [5] and [11]).
###### Definition 4
An additive character $\chi$ in $\mathbb{F}_{q}$ is an homomorphism of the
additive group $\mathbb{F}_{q}$ into the multiplicative group of complex
numbers with norm $1$. If $\chi\equiv 1$, we say that $\chi$ is the trivial
additive character.
###### Lemma 4
Let $\chi$ be a non trivial additive character of $\mathbb{F}_{q}$ and
$\alpha$ a fix element of $\mathbb{F}_{q}^{j}$. Then
$\sum_{\beta\in\mathbb{F}_{q}^{j}}\chi(\alpha\cdot\beta)=\left\\{\begin{array}[]{lc}q^{j},&\mbox{if
}\alpha\mbox{ is null}\\\ 0,&\mbox{otherwise}\\\ \end{array}\right.$
###### Lemma 5
Let $\chi$ be a non trivial additive character of $\mathbb{F}_{q}$. For any
linear code $\mathcal{C}\subset\mathbb{F}_{q}^{n}$
$\sum_{v\in\mathcal{C}}\chi(u\cdot
v)=\left\\{\begin{array}[]{lcl}0,&\mbox{if}&u\in\mathbb{F}_{q}^{n}\backslash\mathcal{C}^{\perp}\\\
|\mathcal{C}|,&\mbox{if}&u\in\mathcal{C}^{\perp}\\\ \end{array}\right.$
###### Definition 5
(Hadamard Transform) Let $f$ be a complex function defined in
$\mathbb{F}_{q}^{n}$. The Hadamard transform of $f$ is
$\widehat{f}(u)=\sum_{v\in\mathbb{F}_{q}^{n}}\chi(u\cdot v)f(v).$
The proof of the following lemma may be found in [13].
###### Lemma 6
(Discrete Poisson Summation Formula) Let
$\mathcal{C}\subset\mathbb{F}_{q}^{n}$ be a linear code and $f$ a complex
function defined on $\mathbb{F}_{q}^{n}$. Then
$\sum_{v\in\mathcal{C}^{\perp}}f(v)=\frac{1}{|\mathcal{C}|}\sum_{u\in\mathcal{C}}\widehat{f}(u).$
(6)
In case both the block and the poset structures are trivial (the Hamming
case), the use of the discrete Poisson summation formula to establish the
MacWilliams identity is simple: just consider $f(u)=x^{w_{H}(u)}$ and apply
the discrete Poisson summation formula to the Hadamard transform
$\widehat{f}(u)=(1+(q-1)x)^{n-w_{H}(u)}(1-x)^{w_{H}(u)}$ (as in [12]). If
$f(u)=x^{w_{(\overline{P},\pi)}(u)}z_{s_{\overline{P}}(u)}$, where
$s_{\overline{P}}(u)=min\\{i:u^{i}\in\mathbb{F}_{q}^{b_{i}}\backslash\\{0\\}\\}$
and $s_{\overline{P}}(0)=t+1$, then
$\sum_{u\in\mathcal{C}^{\perp}}f(u)=W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x;z_{t+1},\cdots,z_{1}).$
(7)
Therefore we will extend this result determining the Hadamard transform of the
function $f(u)=x^{w_{(\overline{P},\pi)}(u)}z_{s_{\overline{P}}(u)}$.
Given $i\in\\{0,\cdots,t\\}$, we set
$B_{i}=\\{u\in\mathbb{F}_{q}^{n}:u^{j}=0\ \forall\ 1\leqslant j\leqslant
i\text{ and }u^{i+1}\neq 0\\}$
and then
$\displaystyle\widehat{f}(u)$
$\displaystyle=\sum_{v\in\mathbb{F}_{q}^{n}}\chi(u\cdot v)f(v)$
$\displaystyle=\sum_{i=0}^{t}\sum_{v\in B_{i}}\chi(u\cdot v)f(v)$
$\displaystyle=\sum_{i=0}^{t}\sum_{v\in B_{i}}\chi(u\cdot
v)x^{w_{(\overline{P},\pi)}(v)}z_{s_{\overline{P}}(v)}.$
Defining $S_{i}(u)=\sum_{v\in B_{i}}\chi(u\cdot
v)x^{w_{(\overline{P},\pi)}(v)}z_{s_{\overline{P}}(v)}$, since $B_{t}=\\{0\\}$
it follows that
$\widehat{f}(u)=z_{t+1}+\sum_{i=1}^{t}S_{i-1}(u).$ (8)
The proof of the sufficiency condition will be done with the aid of four
lemmas that allow us to determine $\sum_{u\in\mathcal{C}}\widehat{f}(u)$ as a
function of the leveled weight enumerator of $\mathcal{C}$. From Equation (8)
and assuming that the poset is hierarchical and that blocks at the same level
has the same dimension, we get the following four lemmas.
###### Lemma 7
To $i\in[t]$, denote $\gamma_{i}=(q^{d_{i}}-1)$, then for all
$u\in\mathbb{F}_{q}^{n}$ we have that
$S_{i-1}(u)=z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}[(1-x)^{w_{\pi_{i}}(u^{i})}(1+\gamma_{i}x)^{m_{i}-w_{\pi_{i}}(u^{i})}-1]$
if $\widetilde{u^{i+1}}$ is a null vector and $S_{i-1}(u)=0$ if
$\widetilde{u^{i+1}}$ is not a null vector.
###### Proof:
Since $P$ is a hierarchical poset, if $v\in B_{i-1}$, then
$w_{(\overline{P},\pi)}(v)=\widehat{m_{i}}+w_{\pi_{i}}(v^{i})$, and we denote
$v=(v^{1},\cdots,v^{i},\widetilde{v^{i+1}})$. By definition of $S_{i-1}(u)$
and since a character is an additive homomorphism, we have that
$\displaystyle
S_{i-1}(u)=z_{i}x^{\widehat{m_{i}}}\sum_{\widetilde{v^{i+1}}\in\mathbb{F}_{q}^{\widehat{b_{i}}}}$
$\displaystyle\chi(\widetilde{u^{i+1}}\cdot\widetilde{v^{i+1}})\times$
$\displaystyle\sum_{v^{i}\in\mathbb{F}_{q}^{b_{i}}\backslash\\{0\\}}\chi(u^{i}\cdot
v^{i})x^{w_{\pi_{i}}(v^{i})}.$ (9)
By Lemma 4
$\sum_{\widetilde{v^{i+1}}\in\mathbb{F}_{q}^{\widehat{b_{i}}}}\chi(\widetilde{u^{i+1}}\cdot\widetilde{v^{i+1}})=\left\\{\begin{array}[]{lc}q^{\widehat{b_{i}}},&\mbox{if
}\widetilde{u^{i+1}}\mbox{ is null}\\\ 0,&\mbox{otherwise}\\\
\end{array}\right.$ (10)
Being $r_{i}=m_{1}+\cdots+m_{i-1}$ and $\chi$ a non trivial additive
character, since $v_{r_{i}+j}\in\mathbb{F}_{q}^{d_{i}}$ for every
$j\in\\{1,\cdots,m_{i}\\}$ and
$w_{\pi_{i}}(v^{i})=\sum_{j=1}^{m_{i}}\delta(v_{r_{i}+j})$ where $\delta(u)$
is the Kronecker function (it returns $1$ if $u$ is not null and $0$
otherwise), it follows that
$\displaystyle\sum_{v^{i}\in\mathbb{F}_{q}^{b_{i}}}\chi(u^{i}\cdot v^{i})$
$\displaystyle x^{w_{\pi_{i}}(v^{i})}=$ $\displaystyle\ \ \ \ \
\prod_{j=1}^{m_{i}}\sum_{v_{r_{i}+j}\in\mathbb{F}_{q}^{d_{i}}}\chi(u_{r_{i}+j}\cdot
v_{r_{i}+j})x^{\delta(v_{r_{i}+j})}.$
Therefore, if $u_{r_{i}+j}$ is a null vector, then
$\sum_{v_{r_{i}+j}\in\mathbb{F}_{q}^{d_{i}}}\chi(u_{r_{i}+j}\cdot
v_{r_{i}+j})x^{\delta(v_{r_{i}+j})}=1+\gamma_{i}x.$
If $u_{r_{i}+j}$ is not a null vector, since
$u_{r_{i}+j}\notin(\mathbb{F}_{q}^{d_{i}})^{\perp}$, then by Lemma 5
$\displaystyle\sum_{v_{r_{i}+j}\in\mathbb{F}_{q}^{d_{i}}}\chi($ $\displaystyle
u_{r_{i}+j}\cdot v_{r_{i}+j})x^{\delta(v_{r_{i}+j})}=$ $\displaystyle
1+x\sum_{v_{r_{i}+j}\in\mathbb{F}_{q}^{d_{i}}\backslash\\{0\\}}\chi(u_{r_{i}+j}\cdot
v_{r_{i}+j})=1-x,$
hence
$\displaystyle\sum_{v^{i}\in\mathbb{F}_{q}^{b_{i}}\backslash\\{0\\}}\chi(u^{i}$
$\displaystyle\cdot v^{i})x^{w_{\pi_{i}}(v^{i})}=$
$\displaystyle(1-x)^{w_{\pi_{i}}(u^{i})}(1+\gamma_{i}x)^{m_{i}-w_{\pi_{i}}(u^{i})}-1.$
(11)
The result follows from Equations (II-B), (10) and (11). ∎
###### Lemma 8
Given $i\in[t]$, define
$Q_{i}(x):=\frac{1-x}{1+\gamma_{i}x},$
$a_{i}(x):=q^{\widehat{b_{i}}}\left(\frac{1+\gamma_{i}x}{x}\right)^{m-\widehat{m_{i}}}(1-x)^{\widehat{m_{i-1}}}$
and
$c_{i}(x):=x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left(\frac{1-x}{Q_{i}(x)}\right)^{m_{i}},$
where $\gamma_{i}=q^{d_{i}}-1$. Then,
$\displaystyle\sum_{u\in\mathcal{C}}\widehat{f}(u)=$
$\displaystyle|\mathcal{C}|z_{t+1}$
$\displaystyle+\left(\frac{x}{1-x}\right)^{m}\sum_{i=1}^{t}a_{i}(x)z_{i}LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))$
$\displaystyle+\sum_{i=1}^{t}z_{i}c_{i}(x)|\mathcal{C}_{i-1}|-\sum_{i=1}^{t}z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}|\mathcal{C}_{i}|.$
(12)
###### Proof:
If $u\notin\mathcal{C}_{i}$ then $\widetilde{u^{i+1}}$ is not a null vector
and by Lemma 7 we find that
$\displaystyle\sum_{u\in\mathcal{C}}$ $\displaystyle S_{i-1}(u)=$
$\displaystyle\sum_{u\in\mathcal{C}_{i}}S_{i-1}(u)+\sum_{u\in\mathcal{C}\backslash\mathcal{C}_{i}}S_{i-1}(u)=$
$\displaystyle
z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left[\left(\frac{1-x}{Q_{i}(x)}\right)^{m_{i}}\sum_{u\in\mathcal{C}_{i}}Q_{i}(x)^{w_{\pi_{i}}(u^{i})}-|\mathcal{C}_{i}|\right].$
(13)
If $u\in\mathcal{C}_{i}^{1}$, then
$w_{(P,\pi)}(u)=w_{\pi_{i}}(u^{i})+(m-\widehat{m_{i-1}})$, and if
$u\in\mathcal{C}_{i}^{0}$ we have $w_{\pi_{i}}(u^{i})=0$. Since
$\mathcal{C}_{i-1}=\mathcal{C}_{i}^{0}$, then
$|\mathcal{C}_{i-1}|=|\mathcal{C}_{i}^{0}|$ and hence
$\displaystyle\sum_{u\in\mathcal{C}_{i}}Q_{i}(x$
$\displaystyle)^{w_{\pi_{i}}(u^{i})}=$
$\displaystyle\sum_{u\in\mathcal{C}_{i}^{1}}Q_{i}(x)^{w_{\pi_{i}}(u^{i})}+\sum_{u\in\mathcal{C}_{i}^{0}}Q_{i}(x)^{w_{\pi_{i}}(u^{i})}=$
$\displaystyle\frac{1}{Q_{i}(x)^{m-\widehat{m_{i-1}}}}\sum_{u\in\mathcal{C}_{i}^{1}}Q_{i}(x)^{w_{(P,\pi)}(u)}+|\mathcal{C}_{i-1}|.$
(14)
Since $m-\widehat{m_{i+1}}+m_{i}=m-\widehat{m_{i}}$ and by Equation (3) we
have that
$\sum_{u\in\mathcal{C}_{i}^{1}}Q_{i}(x)^{w_{(P,\pi)}(u)}=LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))$,
by replacing Equation (14) into (13) it follows that
$\displaystyle\sum_{u\in\mathcal{C}}S_{i-1}(u)$
$\displaystyle=\left(\frac{x}{1-x}\right)^{m}q^{\widehat{b_{i}}}\left(\frac{1+\gamma_{i}x}{x}\right)^{m-\widehat{m_{i}}}\times$
$\displaystyle(1-x)^{\widehat{m_{i-1}}}z_{i}LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))+$
$\displaystyle+z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left[\left(\frac{1-x}{Q_{i}(x)}\right)^{m_{i}}|\mathcal{C}_{i-1}|-|\mathcal{C}_{i}|\right].$
(15)
By Identity (8), $\widehat{f}(u)=z_{t+1}+\sum_{i=1}^{t}S_{i-1}(u)$, then by
Equation (15)
$\displaystyle\sum_{u\in\mathcal{C}}$
$\displaystyle\widehat{f}(u)=|\mathcal{C}|z_{t+1}+\sum_{i=1}^{t}\sum_{u\in\mathcal{C}}S_{i-1}(u)$
$\displaystyle=|\mathcal{C}|z_{t+1}+\left(\frac{x}{1-x}\right)^{m}\sum_{i=1}^{t}a_{i}(x)z_{i}LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))$
$\displaystyle\ \ \
+\sum_{i=1}^{t}z_{i}c_{i}(x)|\mathcal{C}_{i-1}|-\sum_{i=1}^{t}z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}|\mathcal{C}_{i}|.$
∎
In the definition of $W_{\mathcal{C},(P,\pi)}(x;y_{0},\cdots,y_{t})$, the
$y_{i}^{\prime}s$ were considered as formal symbols. In two next lemmas we
consider specific situations that will determine the weight enumerator in the
stated conditions.
###### Lemma 9
Let
$g_{j}=\left\\{\begin{array}[]{ll}\sum_{i=j+1}^{t}c_{i}(x)z_{i},&\mbox{if
}0\leqslant j\leqslant t-1\\\ 0,&\mbox{if }j=t\end{array}.\right.$
Then
$\sum_{i=1}^{t}z_{i}c_{i}(x)|\mathcal{C}_{i-1}|=W_{\mathcal{C},(P,\pi)}(1;g_{0},\cdots,g_{t}).$
###### Proof:
Since $r_{i}=m_{1}+\cdots+m_{i-1}$ and
$|\mathcal{C}_{i}|=A_{0}+A_{1}+\cdots+A_{r_{i}+m_{i}}=\sum_{j=0}^{i}LW_{\mathcal{C},(P,\pi)}^{(j)}(1),$
(16)
then
$\displaystyle\sum_{i=1}^{t}z_{i}c_{i}($ $\displaystyle
x)|\mathcal{C}_{i-1}|=$ $\displaystyle
A_{0}(c_{1}(x)z_{1}+c_{1}(x)z_{2}+\cdots+c_{t}(x)z_{t})$
$\displaystyle+(A_{1}+\cdots+A_{m_{1}})(c_{2}(x)z_{2}+\cdots+c_{t}(x)z_{t})$
$\displaystyle+\cdots+$
$\displaystyle+(A_{m_{1}+\cdots+m_{t-2}+1}+\cdots+A_{m_{1}+\cdots+m_{t-1}})c_{t}(x)z_{t}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{t}LW_{\mathcal{C},(P,\pi)}^{(i)}(1)g_{i}$
hence the result follows from Identity (4). ∎
The proof of the next lemma is omitted since it follows the same steps as in
the proof of Lemma 9.
###### Lemma 10
Let
$h_{j}=\left\\{\begin{array}[]{ll}\sum_{i=j}^{t}z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}},&\mbox{if
}1\leqslant j\leqslant t\\\
\sum_{i=1}^{t}z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}},&\mbox{if
}j=0\end{array}.\right.$
Then
$\sum_{i=1}^{t}z_{i}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}|\mathcal{C}_{i}|=W_{\mathcal{C},(P,\pi)}(1;h_{0},\cdots,h_{t}).$
Before we proceed to prove the next theorem we recall we are assuming the
following collection of conditions and notations:
* •
$(P,\pi)$ a poset-block over $[m]$ with $t$ levels;
* •
$P$ is hierarchical;
* •
$r_{i}=m_{1}+\cdots+m_{i-1}$;
* •
$\Gamma_{P}^{i}=\\{r_{i}+1,\cdots,r_{i}+m_{i}\\}$;
* •
$d_{i}=\pi(r_{i}+j)$ for every $j\in\\{1,\cdots,m_{i}\\}$;
* •
$b_{i}=m_{i}d_{i}$ is such that $\sum_{i=1}^{t}b_{i}=n$.
Now we can prove that necessary conditions stated in Theorem 1 are also
sufficient to have a MW-I.
###### Theorem 2
Under the conditions above stated, the poset-block $(P,\pi)$ admits a
MacWilliams-type identity.
###### Proof:
By (6) and (7) we have that
$W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x;z_{t+1},\cdots,z_{1})=\frac{1}{|\mathcal{C}|}\sum_{u\in\mathcal{C}}\widehat{f}(u).$
(17)
Considering Equation (4) we have that
$a_{i}(x)z_{i}LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))=W_{\mathcal{C},(P,\pi)}(Q_{i}(x);y_{0},\cdots,y_{t}),$
for every $i\in\\{1,\cdots,t\\}$, where $a_{i}(x)z_{i}=y_{i}$ and $y_{j}=0$
for every $j\neq i$. Substituting the identities obtained in Lemma 9 and Lemma
10 into Equation (8) it follows that
$\displaystyle|\mathcal{C}|$ $\displaystyle
W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x;z_{t+1},\cdots,z_{1})=|\mathcal{C}|z_{t+1}$
$\displaystyle\ \ \
+\left(\frac{x}{1-x}\right)^{m}W_{\mathcal{C},(P,\pi)}(Q_{1}(x);0,a_{1}(x)z_{1},0,\cdots,0)$
$\displaystyle\ \ \
+\left(\frac{x}{1-x}\right)^{m}W_{\mathcal{C},(P,\pi)}(Q_{2}(x);0,0,a_{2}(x)z_{2},0,\cdots,0)$
$\displaystyle\ \ \
+\cdots+\left(\frac{x}{1-x}\right)^{m}W_{\mathcal{C},(P,\pi)}(Q_{t}(x);0,\cdots,0,a_{t}(x)z_{t})$
$\displaystyle\ \ \
+W_{\mathcal{C},(P,\pi)}(1;g_{0},\cdots,g_{t})-W_{\mathcal{C},(P,\pi)}(1;h_{0},\cdots,h_{t}).$
On the left side of the above equality we have the leveled weight enumerator
of $\mathcal{C}^{\perp}$ (the dual code of $\mathcal{C}$). On the right side
we have an expression that depends not on the code itself but only on the
leveled weight enumerator of $\mathcal{C}$. Hence, if $\mathcal{C}_{1}$ is a
linear $(P,\pi)$-code that has the same $(P,\pi)$-polynomial as $\mathcal{C}$,
since $W_{\mathcal{C}_{1}^{\perp},(\overline{P},\pi)}(x;1,\cdots,1)$ is the
$(\overline{P},\pi)$-polynomial of $\mathcal{C}_{1}^{\perp}$, it follows that
$W_{\mathcal{C}_{1}^{\perp},(\overline{P},\pi)}(x;1,\cdots,1)=W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x;1,\cdots,1),$
ie, the $(\overline{P},\pi)$-polynomial of $\mathcal{C}^{\perp}$ is uniquely
determined by $(P,\pi)$-polynomial of $\mathcal{C}$ for every code
$\mathcal{C}$, hence the poset-block structure admits a MW-I. ∎
### II-C Relationship between Weight Distributions
In this section, we will use the same conditions and notations stated before
Theorem 2 in the previous section. For every $k\in\\{0,\cdots,n\\}$, let
$P_{k}^{\gamma_{i}}(x:n)=\sum_{l=0}^{k}(-1)^{l}\gamma_{i}^{k-l}\genfrac{(}{)}{0.0pt}{}{x}{l}\genfrac{(}{)}{0.0pt}{}{n-x}{k-l}$
be the Krawtchouk polynomial whose generator function is given by
$(1+\gamma_{i}z)^{n-x}(1-z)^{x}=\sum_{k=0}^{\infty}P_{k}^{\gamma_{i}}(x:n)z^{k}.$
(18)
If $x\in\\{0,\cdots,n\\}$, we can switch the upper limit of summation by $n$.
This generator functions arise naturally when we are setting a relationship
between the $(P,\pi)$-polynomial coefficients of $\mathcal{C}$ and the
$(\overline{P},\pi)$-polynomial coefficients of $\mathcal{C}^{\perp}$ (for
details about the Krawtchouk polynomials in coding theory, see [12]).
###### Lemma 11
Let $(P,\pi)$ be a poset-block over $[m]$ that admits MW-I and $\mathcal{C}$ a
linear $(P,\pi)$-code with length $n$. Then
$\displaystyle|$
$\displaystyle\mathcal{C}|W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x)=|\mathcal{C}|+$
$\displaystyle\sum_{i=1}^{t}q^{\widehat{b_{i}}}x^{\widehat{m_{i}}}\left[\sum_{k=1}^{m_{i}}\left(a_{k}(j:m_{i})+\binom{m_{i}}{k}\gamma_{i}^{k}|\mathcal{C}_{i-1}|\right)x^{k}\right]$
(19)
where
$a_{k}(j:m_{i})=\sum_{j=1}^{m_{i}}A_{r_{i}+j}P_{k}^{\gamma_{i}}(j:m_{i})$.
###### Proof:
Set
$\displaystyle E_{1}$ $\displaystyle(x)=$
$\displaystyle\sum_{i=1}^{t}q^{\widehat{b_{i}}}\left(\frac{1+\gamma_{i}x}{x}\right)^{m-\widehat{m_{i}}}(1-x)^{\widehat{m_{i-1}}}LW_{\mathcal{C},(P,\pi)}^{(i)}(Q_{i}(x))$
and
$E_{2}(x)=\sum_{i=1}^{t}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left(\left(\frac{1-x}{Q_{i}(x)}\right)^{m_{i}}|\mathcal{C}_{i-1}|-|\mathcal{C}_{i}|\right).$
Putting $z_{1}=\cdots=z_{t+1}=1$ and replacing (8) in (17), it follows that
$W_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x)=\
1+F_{1}(x)+\frac{1}{|\mathcal{C}|}E_{2}(x)$ (20)
where $F_{1}(x)=\frac{1}{|\mathcal{C}|}\frac{x^{m}}{(1-x)^{m}}E_{1}(x)$. Using
the Identity (3) in $E_{1}(x)$ and recalling that $r_{i}=m-\widehat{m_{i-1}}$
and $\widehat{m_{i}}-\widehat{m_{i-1}}=m_{i}$, it follows that
$\displaystyle E_{1}(x)$
$\displaystyle=\sum_{i=1}^{t}q^{\widehat{b_{i}}}\left(\frac{1+\gamma_{i}x}{x}\right)^{m-\widehat{m_{i}}}(1-x)^{\widehat{m_{i-1}}}\times$
$\displaystyle\ \ \ \ \ \ \ \
\sum_{j=1}^{m_{i}}A_{r_{i}+j}\left(\frac{1-x}{1+\gamma_{i}x}\right)^{r_{i}+j}$
$\displaystyle=\sum_{i=1}^{t}\frac{q^{\widehat{b_{i}}}}{x^{m-\widehat{m_{i}}}}\sum_{j=1}^{m_{i}}A_{r_{i}+j}(1+\gamma_{i}x)^{m_{i}-j}(1-x)^{m+j},$
and therefore
$\displaystyle F_{1}(x)$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{i=1}^{t}q^{\widehat{b_{i}}}x^{\widehat{m_{i}}}\sum_{j=1}^{m_{i}}A_{r_{i}+j}(1+\gamma_{i}x)^{m_{i}-j}(1-x)^{j}$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{i=1}^{t}q^{\widehat{b_{i}}}x^{\widehat{m_{i}}}\sum_{j=1}^{m_{i}}A_{r_{i}+j}\sum_{k=0}^{m_{i}}P_{k}^{\gamma_{i}}(j:m_{i})x^{k}$
(21)
where the second equality follows from (18). Hence if
$a_{k}(j:m_{i})=\sum_{j=1}^{m_{i}}A_{r_{i}+j}P_{k}^{\gamma_{i}}(j:m_{i}),$
since $P_{0}^{\gamma_{i}}=1$, and then
$a_{0}(j:m_{i})=|\mathcal{C}_{i}|-|\mathcal{C}_{i-1}|$ by (16). Therefore
$\displaystyle|\mathcal{C}|$ $\displaystyle F_{1}(x)=$
$\displaystyle\sum_{i=1}^{t}q^{\widehat{b_{i}}}x^{\widehat{m_{i}}}\sum_{k=0}^{m_{i}}\left(\sum_{j=1}^{m_{i}}A_{r_{i}+j}P_{k}^{\gamma_{i}}(j:m_{i})\right)x^{k}$
$\displaystyle=\sum_{i=1}^{t}q^{\widehat{b_{i}}}x^{\widehat{m_{i}}}\left(|\mathcal{C}_{i}|-|\mathcal{C}_{i-1}|+\sum_{k=1}^{m_{i}}a_{k}(j:m_{i})x^{k}\right).$
(22)
From Newton’s binomial theorem we have that
$\displaystyle E_{2}(x)=$
$\displaystyle\sum_{i=1}^{t}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left[\left(1+\sum_{k=1}^{m_{i}}\binom{m_{i}}{k}\gamma_{i}^{k}x^{k}\right)|\mathcal{C}_{i-1}|-|\mathcal{C}_{i}|\right]$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{t}x^{\widehat{m_{i}}}q^{\widehat{b_{i}}}\left(|\mathcal{C}_{i-1}|-|\mathcal{C}_{i}|+\sum_{k=1}^{m_{i}}\binom{m_{i}}{k}\gamma_{i}^{k}|\mathcal{C}_{i-1}|x^{k}\right)$
(23)
and the result follows from (20), (22) and (23). ∎
In the conditions stated in Lemma (11) we have that
$\displaystyle W$ ${}_{\mathcal{C}^{\perp},(\overline{P},\pi)}(x)=$
$\displaystyle\overline{A}_{0}+(\overline{A}_{1}x+\cdots+\overline{A}_{m_{t}}x^{m_{t}})$
$\displaystyle+(\overline{A}_{m_{t}+1}x+\cdots+\overline{A}_{m_{t}+m_{t-1}}x^{m_{t-1}})x^{m_{t}}$
$\displaystyle+\cdots+$
$\displaystyle+(\overline{A}_{m_{t}+\cdots+m_{2}+1}x+\cdots+\overline{A}_{m_{t}+\cdots+m_{1}}x^{m_{1}})x^{m_{t}+\cdots+m_{2}}$
$\displaystyle=1+\sum_{i=1}^{t}x^{\widehat{m_{i}}}\sum_{k=1}^{m_{i}}\overline{A}_{\widehat{m_{i}}+k}x^{k}$
(24)
therefore from (19) and (24) follows the next theorem, that characterizes the
weight distribution of $\mathcal{C}^{\perp}$ in terms of the distribution of
$\mathcal{C}$.
###### Theorem 3
Let $(P,\pi)$ be a hierarchical poset-block over $[m]$ with $t$ levels
satisfying MW-I and $\mathcal{C}$ a linear $(P,\pi)$-code with length $n$ over
$\mathbb{F}_{q}$. Being $\gamma_{i}=(q^{d_{i}}-1)$ and $b_{j}$ the dimension
of $\Gamma_{P}^{j}$, for any given $i\in[t]$ and $k\in[m_{i}]$ we have that
$\displaystyle\overline{A}_{\widehat{m_{i}}+k}=$
$\displaystyle\frac{q^{\widehat{b_{i}}}}{|\mathcal{C}|}\sum_{j=1}^{m_{i}}\left(A_{r_{i}+j}P_{k}^{\gamma_{i}}(j:m_{i})\right)$
$\displaystyle+\frac{q^{\widehat{b_{i}}}}{|\mathcal{C}|}\binom{m_{i}}{k}\gamma_{i}^{k}\sum_{j=0}^{r_{i}}A_{j}.$
We remark that when we consider a trivial structure of blocks, $b_{j}=m_{j}$
and $d_{j}=1$ for all $j\in[t]$, then we have the result obtained in Theorem
4.4 from [5]. On the other hand, when considering a trivial poset structure
(an antichain poset where none of elements are comparable), then $t=1$ and
$m=m_{1}$, hence given $k\in[m_{1}]$ we have that
$\overline{A}_{k}=\frac{1}{|\mathcal{C}|}\sum_{j=0}^{m}A_{j}P_{k}^{\gamma_{1}}(j:m).$
## Acknowledgment
The first author is currently pursuing the M.Sc. degree in the Institute of
Mathematics, Statistics and Scientific Computing–UNICAMP. He was supported by
CAPES. The second author was partially supported by FAPESP, grant
$2007$/$56052$–$8$. A partial and initial version of this work was will appear
in the Proceedings of ITW 2011.
## References
* [1] H. Niederreiter, “A combinatorial problem for vector spaces over finite fields,” _Discrete Mathematics_ , vol. 96, pp. 221–228, 1991.
* [2] K. Feng, L. Xu, and F. J. Hickernell, “Linear error-block codes,” _Finite Fields and Their Applications_ , vol. 12, pp. 638–652, 2006.
* [3] R. A. Brualdi, J. Graves, and K. Lawrence, “Codes with a poset metric,” _Discrete Mathematics_ , vol. 147, pp. 57–72, 1995.
* [4] D. S. Kim and D. C. Kim, “Character sums and MacWilliams identities,” _Discrete Mathematics_ , vol. 287, pp. 155–160, 2004.
* [5] H. K. Kim and D. Y. Oh, “A classification of posets admitting the MacWilliams identity,” _IEEE Transactions on Information Theory_ , vol. 51, no. 4, pp. 1424–1431, Apr. 2005.
* [6] S. Ling and F. Özbudak, “Constructions and bounds on linear error-block codes,” _Des. Codes Cryptogr._ , vol. 45, pp. 297–316, 2007.
* [7] L. Panek, M. Firer, H. K. Kim, and J. Y. Hyun, “Groups of linear isometries on poset structures,” _Discrete Mathematics_ , vol. 308, pp. 4116–4123, 2008\.
* [8] M. M. S. Alves, L. Paneck, and M. Firer, “Error-block codes and poset metrics,” _Advances in Mathematics of Communications_ , vol. 2, no. 1, pp. 95–111, 2008.
* [9] M. Y. Rosembloom and M. A. Tsfasman, “Codes for m-metric,” _Problems of Information Transmission_ , vol. 33, no. 1, pp. 45–52, 1997.
* [10] J. N. Gutiérrez and H. Tapia-Recillas, “A MacWilliams identity for poset-codes,” _Congr. Numer_ , vol. 133, pp. 63–73, 1998.
* [11] D. S. Kim and J. G. Lee, “A MacWilliams-type identity for linear codes on weak order,” _Discrete Mathematics_ , vol. 262, pp. 181–194, 2003.
* [12] F. J. MacWilliams and N. J. Sloane, _The Theory of Error-Correcting Codes_. Amsterdam, The Netherlands: North-Holland, 1977.
* [13] R. Lidl and H. Niederreiter, _Finite Fields_ , 2nd ed., ser. Encyclopedia of Mathematics and its Applications. Cambridge, U.K.: Cambridge University Press, 1997, no. 20.
Marcelo Firer received the B.Sc. and M.Sc. degrees in 1989 and 1991
respectively, from State University of Campinas, Brazil, and the Ph.D. degree
from the Hebrew University of Jerusalem, in 1997, all in Mathematics. He is
currently an Associate Professor of the State University of Campinas. His
research interest includes coding theory, action of groups, semigroups and
Tits buildings.
---
Jerry A. Pinheiro receives the B.Sc. in 2006 in Computer Science and in 2008
in Mathematics, from Higher Education Center of Foz do Iguaçu and Western
Parana State University respectively. He is currently pursuing the M.Sc.
degree in the Institute of Mathematics, Statistics and Scientific Computing of
the State University of Campinas. His current research interest includes
poset-block codes and error-correcting codes.
---
|
arxiv-papers
| 2012-02-28T23:24:38 |
2024-09-04T02:49:28.039125
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jerry Anderson Pinheiro and Marcelo Firer",
"submitter": "Jerry Pinheiro",
"url": "https://arxiv.org/abs/1202.6409"
}
|
1202.6428
|
# Inverse Spin Hall Effect in Ferromagnetic Metal with Rashba Spin Orbit
Coupling
M.-J. Xing Computational Nanoelectronics and Nano-device Laboratory,
Electronic and Computer Engineering Department, National University of
Singapore, Singapore, 117576 State Key Laboratory for Advanced Metals and
Materials, School of Materials Science and Engineering, University of Science
and Technology Beijing, China, 100083 M. B. A. Jalil elembaj@nus.edu.sg
Computational Nanoelectronics and Nano-device Laboratory, Electronic and
Computer Engineering Department, National University of Singapore, Singapore,
117576 Information Storage Materials Laboratory, Electronic and Computer
Engineering Department, National University of Singapore, Singapore, 117576
Seng Ghee Tan Computational Nanoelectronics and Nano-device Laboratory,
Electronic and Computer Engineering Department, National University of
Singapore, Singapore, 117576 Data Storage Institute, Agency for Science,
Technology and Research (A*STAR). DSI Building, 5 Engineering Drive 1,
Singapore, 117608 Y. Jiang State Key Laboratory for Advanced Metals and
Materials, School of Materials Science and Engineering, University of Science
and Technology Beijing, China, 100083
###### Abstract
We report an intrinsic form of the inverse spin Hall effect (ISHE) in
ferromagnetic (FM) metal with Rashba spin orbit coupling (RSOC), which is
driven by a normal charge current. Unlike the conventional form, the ISHE can
be induced without the need for spin current injection from an external
source. Our theoretical results show that Hall voltage is generated when the
FM moment is perpendicular to the ferromagnetic layer. The polarity of the
Hall voltage is reversed upon switching the FM moment to the opposite
direction, thus promising a useful readback mechanism for memory or logic
applications.
## I Introduction
Recent research showed that Rashba spin orbit coupling (RSOC) at the surfaces
of metals can be enhanced by the presence of heavy atoms Gambardella ; Ast
and/or surface oxidation LaShell in adjacent layers. This interfacial
enhancement enables significant RSOC effect to be manifested in ferromagnetic
metals with small or moderate atomic number at room temperature Miron ,
whereas previously, strong RSOC effect is confined only to semiconductor
heterostructures. In this paper, we choose a typical ferromagnetic (FM) metal
(Co) as the central conducting layer, sandwiched between an oxide and a Pt
layer, the latter supplying the heavy atoms [see Fig. 1(a)]. The electron
accumulation which develops in the FM layer in the presence of a charge
(unpolarized) current is theoretically evaluated via the non-equilibrium
Green’s function (NEGF) method in the ballistic limit. In the presence of
$s$-$d$ coupling, the incoming charge current becomes polarized by the FM
moments in the central region. When this intrinsic polarization of current is
coupled to the RSOC, an inverse spin Hall effect (ISHE) saitoh ; Hankiewicz
will be induced. Thus, a Hall voltage is generated without the need for spin
injection from an external spin polarizing layer. By contrast, in previous
works, the ISHE is experimentally realized by injecting spin polarized current
Valenzuela ; Zhang from an external FM electrode, or by the inflow of pure
spin current Hankiewicz ; Saitoh1 ; Kimura ; Xing ; Li ; Ando , generated
externally e.g. via spin pumping or non-local spin accumulation. In this work,
we show theoretically that Hall voltage can be generated when the FM moment in
the central region is oriented perpendicular to the plane, which persists at
room temperature. Furthermore, the generated Hall voltage can be reversed
symmetrically when the FM moment is switched to the opposite direction. Thus,
the charge current-induced ISHE signal can be used to detect the polarity of
the FM moment, and potentially serve as a read-back mechanism in memory
applications.
Figure 1: (a) Schematic diagram of the proposed FM moment detector utilizing
the ISHE phenomenon. (b) Lattice discretization of the device for tight-
binding NEGF calculation (top view). The Hall voltage $V_{t}$ is the potential
difference between the two strip electrodes running along the top and bottom
edges of the central region. The covered area schematically shows the edge
width which will be used for the Hall voltage calculation.
## II Model Hamiltonian and Theory
The schematic diagram of the FM moment detector is shown in Fig. 1(a); the
central region comprises of a triple-layer structure for the enhancement of
RSOC within the FM (Co) layer Miron . Charge accumulation within the Co layer
is calculated via tight-binding NEGF method Xing . To perform the tight-
binding calculation, the central region of the device is discretized into
$(M\times N)$ lattice of points [see Fig. 1(b)]. The conduction electrons
within the Co layer experiences the RSOC effect and the $s$-$d$ exchange
interaction with the local FM moments $\textbf{M}(\theta,\phi)$. Thus, the
Hamiltonian of the central region can be expressed as
$H_{C}=H_{K}+H_{M}+H_{Rso}$, where $H_{K}$ is the kinetic term, $H_{M}$ the
$s$-$d$ coupling term, and $H_{Rso}$ the RSOC term. The Hamiltonian can be
expressed as Rashba :
$\displaystyle H_{K}$ $\displaystyle=$
$\displaystyle\sum_{mn\sigma}[4td^{\dagger}_{mn\sigma}d_{mn\sigma}$ (1)
$\displaystyle-$ $\displaystyle
t(d^{\dagger}_{m+1,n\sigma}d_{mn\sigma}+d^{\dagger}_{m,n+1\sigma}d_{mn\sigma}+\mathrm{h.c.})],$
$\displaystyle H_{M}$ $\displaystyle=$
$\displaystyle\sum_{mn\sigma}\mathrm{sgn}[\sigma]M\cos(\theta)d^{\dagger}_{mn\sigma}d_{mn\sigma}$
(2)
$\displaystyle+M\sin(\theta)e^{i\sigma\phi}d^{\dagger}_{mn\sigma}d_{mn\bar{\sigma}},$
$\displaystyle H_{Rso}$ $\displaystyle=$
$\displaystyle\sum_{mn\sigma\sigma^{\prime}}-it_{so}[(d^{\dagger}_{m+1,n}d_{mn}-d^{\dagger}_{m-1,n}d_{mn})\otimes\hat{\sigma}_{y}$
(3)
$\displaystyle-(d^{\dagger}_{m,n+1}d_{mn}-d^{\dagger}_{m,n-1}d_{mn})\otimes\hat{\sigma}_{x}],$
where $M$ denotes the $s$-$d$ coupling strength, $t_{so}=\frac{\alpha}{2a}$
denotes the RSOC strength. Similarly, the Hamiltonian of the normal metal (NM)
leads, and the coupling energy between the leads and central region can be
expressed as:
$\displaystyle H_{L(R)}$ $\displaystyle=$
$\displaystyle\sum_{mn\sigma}[4ta^{\dagger}_{mn\sigma}a_{mn\sigma}$ (4)
$\displaystyle-$ $\displaystyle
t(a^{\dagger}_{m+1,n\sigma}a_{mn\sigma}+a^{\dagger}_{m,n+1\sigma}a_{mn\sigma}+\mathrm{h.c.})].$
$\displaystyle H_{T}$ $\displaystyle=$
$\displaystyle\sum_{n\sigma}[t^{\prime}_{L}a^{\dagger}_{0n\sigma}d_{1n\sigma}+t^{\prime}_{R}a^{\dagger}_{M+1,n\sigma}d_{Mn\sigma}+\mathrm{h.c.}].$
(5)
From the eigenvalue equation of the total Hamiltonian and the definition of
retarded Green’s function, one can obtain an equation:
$(E-H_{mn}+i\eta)G^{n,n}_{m,m}(\sigma\sigma)=I$. From this relation, one can
obtain a series of linear equations involving $G^{n,n}_{m,m}(\sigma\sigma)$ by
considering each spatial point $(m,n)$. For instance, within the central
region, i.e. $1<m<M$, one obtains:
$\displaystyle I$ $\displaystyle=$ $\displaystyle[E-4t-\sigma
M\cos(\theta)]G^{n,n}_{m,m}(\sigma\sigma)-e^{i\sigma\phi}M\sin(\theta)G^{n,n}_{m,m}(\bar{\sigma}\sigma)+t[G^{n,n}_{m-1,m}(\sigma\sigma)+G^{n,n}_{m+1,m}(\sigma\sigma)$
(6) $\displaystyle+$ $\displaystyle
G^{n-1,n}_{m,m}(\sigma\sigma)+G^{n+1,n}_{m,m}(\sigma\sigma)]+t_{so}[\sigma
G^{n,n}_{m+1,m}(\bar{\sigma}\sigma)-iG^{n+1,n}_{m,m}(\bar{\sigma}\sigma)-\sigma
G^{n,n}_{m-1,m}(\bar{\sigma}\sigma)+iG^{n-1,n}_{m,m}(\bar{\sigma}\sigma)].$
Collectively, all these equations can be expressed in matrix form:
$(E[I]-[H])[G]^{r}=I$. The infinitely large matrix $[H]$ consists of sub-
matrices denoting the Hamiltonian of the central region ($[H_{C}]$) and the
coupling coefficients between the two leads and the central region separately
($[\tau_{L/R,C}]$). Following standard procedures in the tight-binding method,
one then obtains:
$(E[I]-[H_{C}]-[\Sigma]^{r}_{L}-[\Sigma]^{r}_{R})[G_{C}]^{r}=I,$ (7)
in which the non-zero terms of the self energy are:
$[\Sigma^{n,n^{\prime}}_{m,m}]^{r}_{L(R)}=[\tau]^{n,n}_{m,0(M+1)}[g^{r}]^{n,n^{\prime}}_{0(M+1),0(M+1)}[\tau]^{n^{\prime},n^{\prime}}_{0(M+1),m}$.
The retarded Green’s functions of the isolated left(right) lead
$[g^{r}]^{n,n^{\prime}}_{0(M+1),0(M+1)}$ can be expressed as (for the left
lead):
$[g^{r}]^{n,n^{\prime}}_{0,0}=-\frac{1}{t}\sum_{i}\chi_{i}(p_{n})e^{ik_{i}a}\chi_{i}(p_{n^{\prime}}).$
(8)
In the above, $k_{i}$ is the wave vector along the semi-infinite longitudinal
direction, $\chi_{i}(p_{n})$ is the $\tilde{i}$th eigenfunction in the
transverse dimension at site $(0,n)$ in the lead, which can be expressed as:
$\chi_{i}(p_{n})=\sqrt{\frac{2}{N+1}\sin{\frac{i\pi n}{N+1}}}.$ (9)
The retarded Green’s function of the central region can then be solved by
inverting Eq. (7). Thus one can express the lesser Green’s function $[G]^{<}$
via the Langreth formula: $[G]^{<}=[G]^{r}[\Sigma]^{<}[G]^{a}$, in which
$[\Sigma]^{<}=\sum_{\mu=L,R}([\Sigma]^{a}_{\mu}-[\Sigma]^{r}_{\mu})f_{\mu}$,
with $f_{\mu}$ being the Fermi distribution function within lead $\mu$. The
total charge accumulation for a given cell at lattice coordinate $(m,n)$ with
an area of $a^{2}$ is given by:
$\tilde{\rho}_{m,n}=-\frac{ie}{2\pi}\int^{\infty}_{-\infty}Tr[G]^{<}_{mn,mn}(E)dE,$
(10)
where the trace is over the spin degree of freedom. The surface charge density
$\rho_{m,n}$ is then given by $\rho_{m,n}=\frac{\tilde{\rho}_{m,n}}{a^{2}}$.
The Hall voltage at longitudinal position $x=m$ ($V_{tm}$) is given, up to a
proportionality constant, by the difference in the surface charge density
between the top and bottom edges corresponding to $x=m$, i.e.
$V_{tm}\propto\Delta\rho_{m}=\rho_{tm}-\rho_{bm}$. In calculating the charge
densities $\rho_{tm}$ and $\rho_{bm}$, we consider a finite width of each
edge. The values of $\rho_{tm}$ and $\rho_{bm}$ are averaged over some number
of rows ($W$) adjacent to the top and bottom edges, such that $Wa\approx$ 0.3
nm. Thus, the surface charge density difference along $x$-direction is given
by:
$\Delta\rho_{m}=\frac{\sum^{W}_{n=1}\rho_{m,N-n+1}-\rho_{m,n}}{W}.$ (11)
In practice, the Hall voltage $V_{t}$ is given by the potential difference
between the two electrodes. We assume that each electrode runs along the
entire length of the edges (i.e., from $m=1$ to $M$). Thus, computationally,
the Hall voltage $V_{t}$ is given by the surface charge density difference
averaged over the longitudinal dimension, i.e.,
$\Delta\rho_{av}=\frac{\sum^{M}_{m=1}\sum^{W}_{n=1}\rho_{m,N-n+1}-\rho_{m,n}}{M\times
W}.$ (12)
## III Results and Discussion
The following parameters are assumed in the numerical calculations: (i) The
device is modeled at room temperature ($T=300$ K); the Fermi energy of the
central FM layer is set to $7.38$ eV, which is a typical value for Co
Wawrzyniaka . (ii) The lattice cell dimension is set to $a=0.045$ nm, which is
significantly smaller than the Fermi wavelength $(a\sim\lambda/10)$, so that
the lattice Green’s function model can simulate a continuum system to a good
approximation. (iii) The coupling strength is
$t=\frac{\hbar^{2}}{2ma^{2}}=18.69$ eV, while for simplicity, the coupling
between the lead and the central region is set to $t_{L/R}=0.8t$. (iv) The
RSOC strength in Eq. (3) is given by $t_{so}=\frac{\alpha}{2a}$. For a typical
FM RSOC material, $\alpha$ lies between $4\times 10^{-11}$ and $3\times
10^{-10}$ eVm Ast ; Henk ; Krupin , which translates to a range of coupling
parameter values of $0.4<t_{so}<3.3$ eV. (v) The $s$-$d$ exchange energy is
set to $|M|=0.85$ eV Wakoh . (vi) The electrochemical potentials of the two
leads are set to $\mu_{L}=-\mu_{R}=2$ eV. (vii) Finally, the central region is
discretized into a square lattice of $(M\times N)=(200\times 100)$ of unit
cells. This corresponds to an actual dimensions of (9 nm$\times$ 4.5 nm) for
the central region.
Figure 2: (a) Distribution of $\Delta\rho_{m}$ along the longitudinal
$x$-axis. The FM moments in the central region are oriented along the $\pm x$
(blue), $\pm y$ (red) and $\pm z$ (black) directions. The corresponding
electron density distributions are schematically depicted in (b), (c) and (d),
respectively. Fig (e) shows a possible electrode configuration to detect the
Hall voltage when FM moment is along $\pm x$ directions. The following
parameter values are assumed: RSOC strength of $t_{so}=3$ eV, bias voltage of
$V=4$ eV, and $s$-$d$ coupling strength of $M=0.85$ eV. The central region is
discretized into a lattice of $(200\times 100)$ unit cells.
We first calculate the transverse charge density difference of
$\Delta\rho_{m}$ as a function of the longitudinal position $x$, when the FM
moments in the Co layer are separately oriented along $\pm x$, $\pm y$ and
$\pm z$ directions. The results are plotted in Figure 2(a). Our results show
that when the FM moments are in the $y$-direction, the spatial charge
distribution $\rho_{m,n}$ is symmetric about the central longitudinal axis of
$n=(N+1)/2$, resulting in $\Delta\rho_{m}=0$. When the FM moments are switched
to $-y$ direction, the charge distribution $\rho_{m,n}$ remains symmetric
about the central longitudinal axis, and hence $\Delta\rho_{m}$ is still $0$
[see Fig. 2(b) for a schematic representation]. Thus, when the FM moments of
the Co layer are aligned along $\pm y$, no Hall voltage would be observed. By
contrast, when the FM moments are oriented along the $x$-direction,
$\Delta\rho_{m}$ is symmetric about the central point
$(m,n)=((M+1)/2,(N+1)/2)$ [see Figs. 2(a) and (c)]. Furthermore, the sign of
$\Delta\rho_{m}$ is reversed when the FM moments are switched to the $-x$
direction. However, when averaged along the entire edge, the surface charge
density difference will be $\Delta\rho_{av}=0$ due to the point symmetry.
However, if the Hall electrodes extend to only half the entire length of the
top and bottom edges [as shown schematically in Fig. 2(e)], a finite Hall
voltage can still be detected.
Of greater interest is the case where the FM moments are along the out-of-
plane $z$-direction. The charge density difference $\Delta\rho_{m}$ is
symmetric about the central vertical axis, i.e. $m=(M+1)/2$. Thus, a finite
$\Delta\rho_{av}$, i.e., a Hall voltage $V_{t}$ is generated [see Figs. 2(a)
and (d)]. Since only charge current but not spin current is injected into the
system, the above can be regarded as a charge current-induced ISHE in FM metal
with RSOC.
Figure 3: The spatial distribution of $\rho_{m,n}$ (in unit of
$\frac{e}{m^{2}}$). The longitudinal and transverse dimensions are expressed
in unit of the lattice constant $a$. The $s$-$d$ coupling strength is $M=0.85$
eV, the RSOC strength is $t_{so}=3$ eV, the bias voltage is $V=4$ eV, the
central region is a $(200\times 100)$ lattice ($9$ nm $\times 4.5$ nm).
Figure 4: The detailed distribution of $\rho_{tm}$ and $\rho_{bm}$ (in unit of
$\frac{e}{m^{2}}$) at (a) the top, and (b) bottom edges of Fig. 3. The
longitudinal and transverse dimensions are expressed in unit of the lattice
constant $a$.
In the following, we will investigate this charge current-driven ISHE in
greater detail. The spatial distribution of $\rho_{m,n}$ is plotted over the
central region when the FM moments are in the $+z$-direction [see Fig. 3]. For
clarity, the detailed distribution of charge densities $\rho_{tm}$ and
$\rho_{bm}$ are shown in Figs. 4(a) and (b). It is observed that the surface
charge density is larger along the top edge, which will result in a finite
Hall voltage or ISHE effect.
Figure 5: (a) The oscillatory increase of $\Delta\rho_{av}$ with the RSOC
strength $t_{so}$, for different edge width $W$ and FM moment orientations
(solid lines for $z$, dashed lines for $-z$ ). The central region is
discretized into a $(200\times 100)$ lattice, with dimensions (9 nm$\times$4.5
nm). The $s$-$d$ coupling strength is $M=0.85$ eV, while the bias voltage is
$V=4$ V. (b) The schematic diagram of the spin electron distribution due to
Yang-Mills-like Lorentz force. The number of Hall deflection pairs increases
with increasing RSOC strength.
Fig. 5(a) shows the dependence of $\Delta\rho_{av}$ on the RSOC strength
$t_{so}$ for different edge width $W$. $\Delta\rho_{av}$, and hence the Hall
voltage $V_{t}$ across the central region, show an increasing trend with the
RSOC strength $t_{so}$, but in an oscillatory manner. The physics underlying
this oscillatory increase can be understood in terms of the Yang-Mills-like
Lorentz force arising from the RSOC gauge Tan ; Shen . Here, the FM moments
$M$ merely play the role of sustaining a vertical spin polarization of current
but do not contribute directly to the Lorentz force. The Lorentz force leads
to the transverse separation of electrons of opposite spins [shown
schematically in Fig. 5(b)], which we term as “Hall deflection pair”. Since
there are unequal number of charges on the two transverse sides, each Hall
deflection pair will contribute to a charge Hall voltage. For a fixed $M$, an
increase in the RSOC strength results in an increasing number of Hall
deflection pairs along the length of the device, as shown in Fig. 5(b). The
charge imbalance in a Hall deflection pair coupled with the increase in the
number of such pairs with the RSOC strength provide a heuristic explanation of
the oscillatory increase of $\Delta\rho_{av}$ and the Hall voltage with the
RSOC strength. The sign of $\Delta\rho_{av}$ can be reversed by switching the
orientation of FM moment between $\pm z$. Furthermore, $\Delta\rho_{av}$ is
not sensitive to the definition of edge, i.e. the general trend remains
unchanged for the range of edge width $W$ considered in our calculation.
Previous work on RSOC in semiconductors has shown that when the surface charge
density difference is in the order of $10^{12}e/\mathrm{m}^{2}$, the generated
Hall voltage will be large enough for detection (0.1 mV) Li . The electron
density in our metallic FM RSOC device will be much higher than that in
semiconductors, so that $\Delta\rho_{av}$ could attain a value of the order of
$10^{16}e/\mathrm{m}^{2}$ and generate a sizable Hall voltage of $V_{t}\approx
1V$. By selecting optimal $t_{so}$ which corresponds to the peak Hall voltage
values (see Fig. 5), we conjecture that a reasonably large $\Delta\rho_{av}$,
hence $V_{t}$ can be measured when the FM moments are oriented along the out-
of-plane $z$ axis. The charge density difference $\Delta\rho_{av}$ switches in
sign upon reversal of the FM moments to the $-z$ direction. The resulting
large difference in the Hall voltage corresponding to the two FM orientations
($\pm z$) suggests that the ISHE can be utilized in a metal FM RSOC system for
the sensitive detection of the FM moment orientation.
In summary,we have investigated the inverse spin Hall effect (ISHE) which is
induced by the combination of RSOC and $s$-$d$ coupling to the FM moments. A
Hall voltage is generated when the FM moments are oriented in the
perpendicular-to-plane direction. The Hall voltage increases in an oscillating
manner with the RSOC strength $t_{so}$. The polarity of the Hall voltage is
reversed when the FM moment is switched to the opposite direction. This
property suggests the utility of the ISHE in FM metals with strong RSOC effect
for the detection of the FM moment direction, e.g., as a possible memory
readback mechanism.
###### Acknowledgements.
This work was supported by the ASTAR SERC Grant No. 092 101 0060
(R-398-000-061-331) and the NSFC Grant No. 50831002, 51071022\.
## References
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|
arxiv-papers
| 2012-02-29T03:09:51 |
2024-09-04T02:49:28.048642
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.-J. Xing, M. B. A. Jalil, Seng Ghee Tan and Y. Jiang",
"submitter": "Mingjun Xing",
"url": "https://arxiv.org/abs/1202.6428"
}
|
1202.6478
|
# A 95 GHz Class I Methanol Maser Survey Toward A Sample of GLIMPSE Point
Sources Associated with BGPS Clumps
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 , Jin-Hua He44affiliation: National
Astronomical Observatories/Yunnan Observatory, Chinese Academy of Sciences,
Kunming 650011, China , Ye Xu55affiliation: Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210008, China , 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: , 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: , Tao An11affiliation:
Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
chenxi@shao.ac.cn 22affiliation: Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, China , Yan Sun55affiliation: Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210008, China , Bing-Gang
Ju 55affiliation: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210008, China
###### Abstract
We report a survey with the Purple Mountain Observatory (PMO) 13.7-m radio
telescope for class I methanol masers from the 95 GHz ($8_{0}$ – $7_{1}$A+)
transition. The 214 target sources were selected by combining information from
both the _Spitzer_ GLIMPSE and 1.1 mm BGPS survey catalogs. The observed
sources satisfy both the GLIMPSE mid-IR criteria of [3.6]-[4.5]$>$1.3,
[3.6]-[5.8]$>$2.5, [3.6]-[8.0]$>$2.5 and 8.0 $\mu$m magnitude less than 10,
and also have an associated 1.1 mm BGPS source. Class I methanol maser
emission was detected in 63 sources, corresponding to a detection rate of 29%
for this survey. For the majority of detections (43), this is the first
identification of a class I methanol maser associated with these sources. We
show that the intensity of the class I methanol maser emission is not closely
related to mid-IR intensity or the colors of the GLIMPSE point sources,
however, it is closely correlated with properties (mass and beam-averaged
column density) of the BGPS sources. Comparison of measures of star formation
activity for the BGPS sources with and without class I methanol masers
indicate that the sources with class I methanol masers usually have higher
column density and larger flux density than those without them. Our results
predict that the criteria $log(S_{int})\leq-38.0+1.72log(N_{H_{2}}^{beam})$
and $log(N_{H_{2}}^{beam})\geq 22.1$, which utilizes both the integrated flux
density (Sint) and beam-averaged column density ($N_{H_{2}}^{beam}$) of the
BGPS sources, are very efficient for selecting sources likely to have an
associated class I methanol maser. Our expectation is that searches using
these criteria will detect 90% of the predicted number of class I methanol
masers from the full BGPS catalog ($\sim$ 1000), and do so with a high
detection efficiency ($\sim$75%).
masers – stars:formation – ISM: molecules – radio lines: ISM – infrared: ISM
## 1 Introduction
Methanol masers from a number of transitions are common in active star forming
regions (SFRs) and have been empirically classified into two categories (class
I and class II). Initial studies found that strong emission from the two
classes are preferentially found towards different star formation regions
(Batrla et al. 1988; Menten 1991). Class I methanol masers (e.g. the $7_{0}$ –
$6_{1}$ A+ and $8_{0}$ – $7_{1}$ A+ at 44 and 95 GHz respectively) are
typically observed in multiple locations across the star-forming region spread
over an area up to a parsec in extent (e.g. Plambeck & Menten 1990; Kurtz et
al. 2004; Voronkov et al. 2006; Cyganowski et al. 2009). In contrast class II
methanol masers (e.g. the $5_{1}$ – $6_{0}$ A+ and $2_{0}$ – $3_{-1}$ E at 6.7
and 12.2 GHz respectively) are often associated with ultracompact (UC) Hii
regions, infrared sources and OH masers and reside close to (within
1$\arcsec$) a high-mass young stellar objects (YSO) (e.g., Caswell et al.
2010). See Müller et al. (2004) for accurate rest frequencies and other basic
data on methanol maser transitions. The empirical classification and
observational findings were supported by early theoretical models of methanol
masers which suggest that the pumping mechanism of class I masers is dominated
by collisions with molecular hydrogen, in contrast to class II masers which
are pumped by external far-infrared radiation (Cragg et al. 1992). More recent
modelling has found that in some cases weak class II methanol masers can be
associated with strong class I masers and vice versa (e.g. Voronkov et al.
2005), i.e., bright masers of different classes can not reside in the same
volume of gas. High spatial resolution observations (e.g. Cyganowski et al.
2009) suggest that where both masers are seen in the same vicinity, while the
two types of masers are not co-spatial on arcsecond scales, they are often
driven by the same young stellar object.
A number of surveys have been performed for class II methanol masers
especially at 6.7 GHz, resulting in the detection of $\sim$ 900 class II maser
sources in the Galaxy to date (e.g., the surveys summarized in the compilation
of Pestalozzi et al. 2005 and the recent searches of Ellingsen 2007, Pandian
et al. 2007, Xu et al. 2008, 2009, Green et al. 2009, 2010, 2012 and Caswell
et al. 2010, 2011). Class I methanol masers are less well studied than class
II masers, but have recently become the focus of more intense research (e.g.,
Sarma & Momjian 2009, 2011; Fontani et al. 2010; Kalenskii et al. 2010;
Voronkov et al. 2010a, b, 2011; Chambers et al. 2011; Chen et al. 2011; Fish
et al. 2011; Pihlström et al. 2011). Early studies of class I masers include
only a small number of large surveys (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).
Recently, some surveys have been done at other transitions of class I
methanol, e.g. 9.9 GHz by Voronkov et al. (2010a) and a new class I methanol
maser transition at 23.4 GHz has been discovered by Voronkov et al. (2011).
Interferometric observations show that the class I methanol masers at
different transitions (e.g., 36 GHz and 44 GHz) usually have similar larger-
scale spatial distributions, but are rarely found to produce a maser at the
same site (e.g., Fish et al. 2011). Surveys have revealed that class I
methanol maser (unlike class II masers) are associated not only with high-mass
star formation, but also lower mass counterparts (Kalenskii et al. 2010).
Recently Chen et al. (2009) demonstrated that a new sample of massive young
stellar object (MYSO) candidates associated with ongoing outflows (known as
extended green objects or EGOs and identified from the _Spitzer_ GLIMPSE
survey by Cyganowski et al. (2008)), provide another productive target for
class I maser searches. On the basis of their statistical analysis Chen et al.
predicted a detection rate of 67% for class I masers toward EGOs. A follow-up
systemic survey towards a complete EGO sample (192 sources) with the Australia
Telescope National Facility (ATNF) Mopra 22-m radio telescope resulted in the
detection of 105 new 95 GHz class I methanol masers (Chen et al. 2011). The
majority of these detections (92) are newly-identified class I methanol maser
sources, thus demonstrating that there is a high detection rate (55%) of class
I methanol masers toward EGOs. This search, combined with previous
observations increased the number of known class I methanol masers to
$\sim$300\. Chambers et al. (2011) obtained an apparently contradictory result
for a similar search, achieving a relatively low detection rate (8/31=26%) of
class I methanol maser at 44 GHz towards 4.5 $\mu$m emission sources. The low
detection rate in this survey may be because Chambers et al. have targeted
sources with relatively less extended green emission than the EGOs identified
by Cyganowski et al. (2008).
The generally held view of class I methanol masers is that they trace regions
of mildly shocked gas, where the methanol abundance is significantly enhanced
and the gas is heated and compressed providing more frequent collisions.
Voronkov et al. (2010a) suggested that the shocks which produce class I
methanol masers may be driven into molecular clouds not only by outflows (it
is worth noting that a high-velocity feature from a class I methanol maser
associated with outflow parallel to the line of sight has been detected in the
EGO source G309.38-0.13 by Voronkov et al. 2010b), but also from expanding Hii
regions. Based on the results of their analysis of GLIMPSE properties and the
findings of Voronkov et al., Chen et al. (2011) suggest that class I methanol
masers may arise at two distinct two-evolutionary phases during the high-mass
star formation process: they may appear as one of the first signatures of
massive star formation associated with young outflows, and also that they can
be re-activated at a later evolutionary stage associated with OH masers and
Hii regions.
Further searches for class I methanol masers are very important for our
understanding of the range of environments and circumstances in which they
arise. Ellingsen (2006) developed criteria for targeting class II methanol
maser searches using GLIMPSE point source colors. He suggested that targeted
searches toward GLIMPSE point sources with [3.6]-[4.5] $>$ 1.3 and an 8.0
$\mu$m magnitude less than 10 will detect more than 80% of all class II
methanol maser sources with an efficiency of greater than 10% (although the
actual efficiency obtained from the only follow-up search reported to date is
much lower (Ellingsen 2007)). In comparison, the mid-IR color analysis of
GLIMPSE point sources toward EGOs undertaken by Chen et al. (2011) shows that
the color-color region occupied by the GLIMPSE point sources towards EGOs
which are, and are not, associated with class I methanol masers are very
similar, and mostly located within color ranges -0.6$<$[5.8]-[8.0]$<$1.4 and
0.5$<$[3.6]-[4.5]$<$4.0. This suggests that the GLIMPSE point source colors
may not be a very sensitive diagnostic for constructing a sample to search for
class I methanol masers. Despite the significant overlap in the color space
occupied by EGOs with and without an associated class I methanol, Chen et al.
(2011) do find the detection rate of class I methanol masers is higher in
those sources with redder GLIMPSE point source colors. Therefore the reddest
GLIMPSE point sources may provide a reasonable sample for searching for class
I methanol masers with a relatively high detection efficiency. One point to
note is that the implication of a relatively high detection efficiency for
class I methanol masers for the redder GLIMPSE point sources is based on the
EGO sample. The GLIMPSE point sources associated with EGOs are believed to be
MYSOs with ongoing outflows, and the EGOs themselves have a high detection
rate of class I methanol masers. Therefore a relatively high detection
efficiency of class I methanol masers is not unexpected in these redder
GLIMPSE point sources associated with EGOs. Further searches for class I
methanol masers toward non-EGO associated GLIMPSE point sources is necessary
to more reliably characterise the mid-IR characteristics of class I methanol
maser sources. The mid-IR colors of some other astrophysical objects, (e.g.
AGB stars) also are located within a similar color-color region as that found
for class I methanol masers (Robitaille et al. 2008). Thus finding additional
measures by which GLIMPSE point sources associated with active star formation
can be distinguished from other objects with similar mid-IR colors is an
important step required for such searches.
Recently the Bolocam Galactic Plane Survey (BGPS) has detected 1.1 mm thermal
dust emission from thousands of regions of dense gas, many of which are
closely associated with star formation. The typical H2 column density of BGPS
sources is $\sim 10^{22}$ cm -2, the typical mass a few hundred M⊙, and the
typical size a parsec (Aguirre et al. 2011; Dunham et al. 2011a, b). So the
BGPS is identifying high column density regions and is a sensitive tracer of
massive clumps, in contrast to signposts such as class II methanol maser
emission, which are only present once a YSO has formed. Dunham et al. (2011a)
found that approximately half the BGPS sources contain at least one GLIMPSE
source (within the area where both BGPS and GLIMPSE surveys overlap). Chen et
al. (2011) found that the detection rate of class I methanol masers is
significantly higher towards those EGOs with an associated BGPS source
(35/54=65%) than for those without (1/9=11%), or in the complete EGO sample
(55%). Dunham et al. (2011a) also found that EGOs are frequently associated
with BGPS sources. Of the 84 EGOs within the BGPS survey area, 79 are
associated with BGPS sources. All of the above factors suggest that the BGPS
may be a useful supplement to the GLIMPSE point source catalog in constructing
a reliable and efficient targeted sample for class I methanol masers.
In this paper, we report the results of a 95 GHz class I methanol maser survey
towards a sample of GLIMPSE point sources with associated 1.1 mm BGPS sources
which has been undertaken with the Purple Mountain Observatory (PMO) 13.7 m
radio telescope. In Section 2 we describe the sample and observations, in
Section 3 we present the results of the survey, analysis and discussion is
given in Section 4, followed by a summary of the important results in Section
5.
## 2 Source selection and Observations
### 2.1 Source selection
We used the released catalogs from the GLIMPSE survey (version 2.0) and the
BGPS (version 1.0.1) to construct a target sample for our class I methanol
maser search. The properties of the two surveys are summarized below. The BGPS
111See http://irsa.opac.caltech.edu/data/BOLOCAM-GPS is a 1.1mm continuum
survey of 170 square degrees of the Galactic Plane in the northern hemisphere
with the Bolocam instrument (Glenn et al. 2003; Haig et al. 2004), employed on
the Caltech Submillimeter Observatory (CSO). Two distinct portions were
included in the survey: a blind survey of the inner Galaxy region spanning
$-10\arcdeg<l<90\arcdeg$ where $|b|\leq 0.5\arcdeg$ everywhere, except for
$1.0\arcdeg$ cross-cuts at $l=3\arcdeg$, $15\arcdeg$, $30\arcdeg$, and
$31\arcdeg$ where $|b|\leq 1.5\arcdeg$, and a targeted survey towards known
star formation regions in several outer Galaxy regions, including Cygnus-X
($70\arcdeg<l<90.5\arcdeg$, $|b|\leq 1.5\arcdeg$), the Perseus Arm ($l\sim
111\arcdeg$, $b=0\arcdeg$), the W3/4/5 region ($l\sim 135\arcdeg$, $b\sim
0.5\arcdeg$), IC1396 ($l\sim 99\arcdeg$, $b\sim 3.5\arcdeg$) and the Gemini
OB1 molecular cloud ($l\sim 190\arcdeg$, $b\sim 0.5\arcdeg$). The survey
detected approximately 8400 sources with an rms noise level in the maps
ranging from 30 to 60 mJy beam-1. The details of the survey methods and data
reduction are described in Aguirre et al. (2011), and the source extraction
algorithm and catalog (v1.0 BGPS data) are described in Rosolowsky et al.
(2010). The effective FWHM beam size of the BGPS is 33$\arcsec$, corresponding
to a solid angle of $2.9\times 10^{-8}$ steradians, which is equivalent to a
tophat function with a 40′′ diameter ($\Omega=2.95\times 10^{-8}$). Thus the
BGPS catalog presents aperture flux densities within a 40$\arcsec$ diameter
aperture ($S_{40\arcsec}$), corresponding to the flux density within one beam.
The BGPS catalog also provides an integrated flux density ($S_{int}$), which
is the sum of all pixels within a radius ($R$ also given in the catalog) of
the BGPS source. Dunham et al. (2010) suggested that a correction factor of
1.5 must be applied to the Rosolowsky et al. BGPS catalog flux densities. This
factor is based on a comparison of BGPS data with previous 1.2 mm data
acquired with the MAMBO and SIMBA instruments (Aguirre et al. 2011). In this
paper, we have also applied this correction factor to the flux densities in
the Rosolowsky et al. BGPS catalog. The BGPS catalog includes the coordinates
of both a geometric centroid and of the peak of the 1.1 mm emission. We have
used the peak positions for the dust continuum emission (rather than centroid
positions) in our analysis.
The Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE)
222http://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ is a legacy science
program of the _Spitzer Space Telescope_ in a number of mid-infrared
wavelength bands at 3.6, 4.5, 5.8, and 8.0 $\mu$m using the Infrared Array
Camera (IRAC; Benjamin et al. 2003; Churchwell et al. 2009). The survey
resolution is better than 2$\arcsec$ in all wavelength bands. The survey
catalogs for GLIMPSE I and II have been released. The GLIMPSE I survey covers
$10\leq|l|\leq 65\arcdeg$ with $|b|\leq 1\arcdeg$, and the GLIMPSE II survey
covers the region of $|l|\leq 10\arcdeg$ with $|b|\leq 1\arcdeg$ for
$|l|>5\arcdeg$, $|b|\leq 1.5\arcdeg$ for $2\arcdeg<|l|\leq 5\arcdeg$, and
$|b|\leq 2\arcdeg$ for $|l|\leq 2\arcdeg$. The data products include both
highly reliable point source catalogs, and less reliable but more complete
point source archives. In our analysis, we have used only the highly reliable
point source catalogs from the GLIMPSE I and II surveys. The total area of the
GLIMPSE I and II surveys is 274 square degrees. The overlap region between the
BGPS and the two GLIMPSE surveys is $-10\arcdeg<l<65\arcdeg$, $|b|<0.5\arcdeg$
and $|b|<1.0\arcdeg$ at $l=3\arcdeg$, $15\arcdeg$, $30\arcdeg$, and
$31\arcdeg$. We have used data from the overlap region to compile a sample of
target sources to search for class I methanol masers.
The target sample was constructed by applying the following criteria: (1) A
GLIMPSE point source with [3.6]-[4.5]$>$1.3, [3.6]-[5.8]$>$2.5,
[3.6]-[8.0]$>$2.5 and an 8.0 $\mu$m magnitude less than 10; (2) the point
sources meeting this mid-IR criterion must have a 1.1 mm BGPS counterpart
within $15\arcsec$ (half the beam size of the BGPS survey); (3) the source
must be at a declination greater than $-25\arcdeg$ (so as to be accessible to
the PMO 13.7-m telescope); (4) the separation of each target source from all
other target sources must be greater than half the beam size of the PMO 13.7-m
telescope at 95 GHz (30′′) (where this is not the case the source with
stronger 4.5 $\mu$m emission is retained in the sample). The mid-IR criteria
for selecting the initial sample of GLIMPSE point sources are based on the
observed colors of known class I and class II methanol masers (see Figures 15,
16 and 18 of Ellingsen (2006)). Although Chen et al. (2011) found that some
class I methanol masers are associated with GLIMPSE point sources with less-
red colors ([3.6]-[4.5]$\sim$0.5), the detection rates are highest for redder
colors and so these criteria should be more efficient. When cross-matching the
GLIMPSE point sources and the BGPS sources we only considered the separation
between the GLIMPSE point source position and the BGPS peak position. We did
not consider the measured size of the BGPS source, even though this method may
miss some true associations between GLIMPSE and BGPS sources.
Within the BGPS-GLIMPSE overlap regions a total of $\sim$420 GLIMPSE point
sources satisfied the four criteria outlined above. Of these a total of 214
(approximately half) were randomly picked as the initial target sample for our
95 GHz class I methanol maser survey with the PMO 13.7-m telescope. Table 1
lists the target sample source parameters including the mid-IR magnitudes of
the GLIMPSE point sources and the main parameters of the BGPS sources
(including the BGPS ID number) extracted from the relevant catalogs used in
this study. The separation between the GLIMPSE point source and the BGPS
source range from 0.3′′ to 14.7$\arcsec$ with a mean of 7.3$\arcsec$. A
histogram of the separations is shown in Figure 1.
### 2.2 Observations
The observations of the $8_{0}$ – $7_{1}$ A+ (95.1964630 GHz) class I methanol
maser transition were made using the PMO 13.7 m telescope in Delingha, China
during 2011 March – April. We used the position of the 1.1 mm BGPS source peak
emission rather than GLIMPSE point source as the target position for the
observations. The positions of the target sources in Equatorial Coordinates
(J2000) are given in Table 2. A new cryogenically cooled 9-beam SIS receiver
($3\times 3$ with a separation of 174$\arcsec$ between the centers of adjacent
beams) was used for the observations. This receiver operates in the 80–115 GHz
band and the central beam of the 9 beam receiver was pointed at the target
position. The system temperature for the observations was in the range 105–140
K, depending on the weather conditions and the atmospheric absorption $\tau$
was typically 0.15 – 0.2. A Fast Fourier Transform Spectrometer (FFTS) with
16384 spectral channels across a bandwidth of 1 GHz (corresponding to a
velocity range of $\sim$ 3000 km s-1) was available for each beam during the
observations. This gives an effective velocity resolution of 0.19 km s-1 for
the 95 GHz class I methanol masers. However we only searched for maser
emission over the velocity range from -200 to 200 km s-1 to cover the range of
observed molecular gas in the Milky Way. Each source was observed in a
position-switching mode with off positions offset 10$\arcmin$ in right
ascension. The pointing rms was better than 5$\arcsec$. The standard chopper
wheel calibration technique was applied to measure an antenna temperature,
T${}_{A}^{*}$ corrected for atmospheric absorption. The FWHM beam size of the
telescope is approximately 53$\arcsec$ at this frequency with a main beam
efficiency $\eta_{mb}$ of 46%. The antenna efficiency is 42%, thus resulting
in a factor of 45 Jy K-1 for conversion of antenna temperature into flux
density. The initial observations had an on-source integration time of 10 mins
for each of the 214 targeted sources yielding a T${}_{A}^{*}$ 1$\sigma$ noise
level of about 20 mK (corresponding to about 1.0 Jy) for each beam after
Hanning smoothing of the spectra. Then, depending on the intensity of any
detected emission we observed for an additional 10-20 minutes (on-source) to
improve the signal-to-noise (SNR) of the final spectra. This yielded a typical
rms noise level of 15 – 20 mK in the T${}_{A}^{*}$ scale (corresponding to 0.7
– 1.0 Jy) after Hanning smoothing. The corresponding rms noise
($\sigma_{rms}$) for each target source is summarized in Table 2.
The spectral data were reduced and analyzed with the GILDAS/CLASS package.
Although data from all 9-beams were recorded, the locations of the 8 offset
beams rotate with changing azimuth/elevation during the observation, thus only
data from the central beam which was placed on the target position is valid.
We only focus on the data from the central beam in this work. As part of the
processing a low-order polynomial baseline fitting and subtraction, and
Hanning smoothing were performed for the averaged spectra. Usually the 95 GHz
methanol spectra do not have a particularly Gaussian profile, possibly because
the spectra consist of multiple maser features within a similar velocity
range. However, to characterize the spectral characteristics of the emission
we have performed Gaussian fitting of each feature for each detected source.
## 3 Results
### 3.1 Class I methanol maser detection
95 GHz class I methanol emission above 3 $\sigma_{rms}$ was detected toward 63
of the 214 targeted sources, corresponding to a detection rate of 29% for this
survey. The spectra of the 63 detected class I methanol sources are shown in
Figure 2. The detected objects are listed, along with the parameters of
Gaussian fits to their 95 GHz spectral features in Table 3. The flux densities
of the detected emission derived from the Gaussian fits range from $\sim$ 0.6
to 43.4 Jy (corresponding to main beam temperatures TBM $\sim$ 0.03 to 2.1 K).
The flux densities obtained from integrating the emission over all spectral
features for each source are also given in Table 3 and range from 3 to 136 Jy
km s-1, with a mean of 24 Jy km s-1. The measured FWHM of individual spectral
features derived from Gaussian fitting are in the range 0.18 – 11.5 km s-1
with a mean of 2.1 km s-1. The spectra of the class I methanol emission in
most sources usually include one or more narrow spectral features (typical
line width $<$1 km s-1 seen in Table 2) which are clearly maser emission (see
Figure 2), but the same spectra often also contain broader emission features
(typical line width $>$1 km s-1 seen in Table 2). The pattern of class I
methanol transitions containing both strong narrow spectral features and
weaker broader emission has been seen in all previous single-dish surveys
(e.g. Ellingsen 2005 and Chen et al. 2011), and their nature was discussed in
detail by Chen et al. (2011). There are 13 sources which show a single broad
Gaussian profile with a width of $>$ $\sim$2 km s-1 (sources N20, N29, N32,
N41, N78, N94, N99, N102, N117, N148, N154, N164 and N194; see Figure 2). At
present our single-dish observations can not distinguish from their
characteristics whether these broader emission sources are maser or thermal.
For the purposes of our subsequent analysis we have assumed that some of the
detected 95 GHz emission in these single broad line sources arises also from
masers, recognising that future interferometric observations are required to
determine whether or not this is correct. One point to note is that even if
these single broad line sources are found to be purely thermal, the number
(only 13) of these sources is too small to affect most of the statistical
conclusions drawn in Section 4.
### 3.2 Comparison with previous detections
Among the 63 detected 95 GHz methanol sources, 20 have previously been
detected as class I methanol masers in one or more transitions. The previous
class I maser observations of these 20 sources are summarized in Table 4,
including information as to which transitions have been detected. Table 4
shows that 12 of these sources (all of them are EGOs) have previously been
detected in the 95 GHz transition, 11 of them by Chen et al. (2011) (Mopra EGO
survey) and the other one from the survey of Val’tts et al. (2000). Twelve
sources were also detected in the 44 GHz transition, including 6 EGOs detected
by Cyganowski et al. (2009) and Slysh et al. (1994) as well as 6 sources from
other surveys. Therefore 51 new 95 GHz class I methanol maser sources have
been found in this survey, of which 43 are newly-identified as class I
methanol maser sources. One source (source number N22 in our survey) was
detected as a class I methanol maser at 44 GHz, but undetected at 95 GHz in a
survey with the Nobeyama 45-m telescope by Fontani et al. (2010). While in our
observations we detected emission in the 95 GHz transition with a peak flux
density of $\sim$12 Jy. We have compared the targeted positions for this
source in the two surveys, and found that there is an angular separation of
$\sim 18\arcsec$ between the targeted positions used. If the 95 GHz methanol
maser emission detected in our PMO-13.7 m observations is located close to our
targeted position, the non-detection with the Nobeyama 45-m telescope may be
due to the relatively smaller beam size at 95 GHz ($18\arcsec$) which may not
have covered the maser emission region in this case.
We have compared the spectra of the 11 sources which were detected in both the
EGO-based Mopra survey (Chen et al. 2011), and in the current PMO 13.7-m
survey. The two spectra overlayed are shown in Figure 3 and it can be clearly
seen that the line profile and velocity range of each source are similar in
both surveys. The observed emission intensities are consistent in 4 sources
(N43, N73, N76 and N83), but are different in other 7 sources. Usually the
emission detected in the current PMO survey is (1.5 – 2 times) stronger than
that in the previous Mopra survey (except for one source N97 with stronger
emission detected in the Mopra survey). In addition to flux density
calibration uncertainties between the two telescopes, the following factors
may cause the observed difference in the detected emission intensity between
the two surveys: 1) different target positions were adopted in the two
surveys; 2) the different beam sizes of the telescopes used in the two surveys
cover different regions; 3) intrinsic intensity variability in the class I
methanol maser emission between the two epochs. We have compared the targeted
positions used in the two surveys, and found that the angular separation
typically ranges from 1$\arcsec$ to 10 $\arcsec$ in both the sources with and
without a significant difference in the observed intensity, thus it does not
seem that case 1 is the major factor in explaining the differences. Case 2 is
plausible if the maser emission is extended to spatial scales comparable to,
or larger than the Mopra beam (36$\arcsec$ at 95 GHz), in which case the PMO
would detect additional maser emission outside the Mopra beam. This is
consistent with the observed results in the two surveys for most sources, as
stronger emission was detected by the PMO, but one source (N97) shows the
opposite trend with stronger emission detected by the Mopra rather than the
PMO. In this case one of possibilities is that there is intrinsic intensity
variability in this source, although we can not characterise the nature of the
variations with only two epochs of data collected using different telescopes.
Moreover the exact coordinates of this source are unknown, so it is possible
that both in the Mopra and PMO observations are at an offset position, in
which case even a fairly small difference in the telescope pointing of about
10$\arcsec$ can lead to a higher intensity observed with a narrower beam
(Mopra) than with a broader beam (PMO), provided that Mopra was pointing more
directly towards the source. Variations in the intensity of 6.7 GHz class II
methanol masers have been detected with timescales on the order of days to
years (e.g. Goedhart et al. 2004; Ellingsen 2007; Goedhart et al. 2009;
Szymczak et al. 2011) and some sources have been found to exhibit periodic
variability (e.g. Goedhart et al. 2009; Szymczak et al. 2011). Intensity
variation in class I methanol masers has also been reported in a few sources
(e.g. Kurtz et al. 2004; Pratap et al. 2007), but to date there are no
systematic observations of class I methanol maser variability. It will be
necessary to perform multi-epoch observations with accurate calibration to
determine the characteristics of the intensity variations in class I methanol
masers.
### 3.3 Distance and luminosity of class I methanol masers
The distance and the distance-dependent integrated maser luminosity for each
of the 63 detected methanol maser sources are given in Table 5. We used the
Galactic rotation model of Reid et al. (2009), with the Galactic constants set
to, R⊙= 8.4 kpc and $\Theta_{\odot}$= 254 km s-1 to estimate the distances.
Since class I methanol maser emission is generally observed to lie close to
the VLSR as measured from the thermal gas (e.g. Cyganowski et al. 2009), the
velocity of the brightest feature in the 95 GHz maser spectrum was used in the
distance calculation. All Galactic rotation models suffer from ambiguity
(known as kinematic distance ambiguity) for sources which lie within the solar
circle. With the exception of the velocity associated with the tangent point,
there are two distances (referred to as a near and far distance), either side
of the tangent point, which will produce the same line-of-sight velocity. All
of the sources with 95 GHz methanol masers detected in our survey fall within
the solar circle. Where present, an association between the detected class I
methanol maser source and an infrared dark cloud (IRDC) may allow us to
resolve the distance ambiguity. IRDCs are believed to represent sites where
the earliest stages of massive star formation are present (e.g. Egan et al.
1998; Carey et al. 1998, 2000; Simon et al. 2006a, 2006b). They are observed
in absorption against the diffuse infrared background especially at 8.0
$\mu$m, and hence the identification of IRDCs is greatly biased toward nearby
sources (and hence the near kinematic distance), where they will show greater
contrast against the diffuse IR background (see Jackson et al. 2008). We have
cross-matched the 63 detected 95 GHz methanol masers with the catalog of IRDCs
seen in the _Spitzer_ GLIMPSE images (Peretto & Fuller 2009), and we have
undertaken visual inspection of the GLIMPSE 8 $\mu$m images for those sources
with $|l|<10\arcdeg$ (which are not included in Peretto & Fuller catalog). The
information as to whether the class I methanol maser detections are associated
with IRDCs or not is summarized in Column (8) of Table 5. We found 33 of 63
maser sources for which the associated BGPS sources are spatially coincident
and structurally similar to IRDCs. We have assumed that these 33 sources are
at the near kinematic distance. The remaining 30 class I maser sources are
associated with BGPS which are not coincident with IRDCs, and for these we
have adopted the far kinematic distance.
To examine how reasonable (or otherwise) the above distance assumptions are,
we have cross-checked our distance determinations for a subsample of class I
maser sources for which the distance ambiguity has been resolved in other
studies. Some of our detected class I maser sources have a class II methanol
maser association (see Section 4.3 for the identification of the class II
maser associations), and some of these have had the distance ambiguity more
directly resolved using HI self-absorption (HISA) from the Southern Galactic
Plane Survey (SGPS) or the VLA Galactic Plane Survey (VGPS) by Green &
McClure-Griffiths (2011). We found that 9 of the 10 sources with IRDC
associations (which we assume to be at the near distance) are assigned the
near distance by Green & McClure-Griffiths (2011), and 5 out of the 6 sources
without IRDCs (which we assume to be at the far distances) are assigned to be
at the far distance by their work. We have marked these sources with a “G” in
Table 5, and adopted the distances from their work in our analysis for these
sources. Moreover some of our detected class I maser sources are included in
the sample of BGPS sources studied with molecular lines (e.g., NH3, HCO+ and
N2H+) by Dunham et al. (2011b) and Schlingman et al. (2011). There are 16
sources (marked by “S” in Table 5) which are contained in the BGPS sample with
distances determined in Table 5 of Schlingman et al. (2011). Among them 14
sources have distance solutions determine from Galactic rotation (the other
two sources N194 and N210 have no reliable distance estimations from the
Galactic rotation; see below), and our distance determinations with the IRDC
method for them (including 13 sources with IRDC associations at near distance,
and 1 source without IRDC associations at far distance) are consistent with
that determined in Schlingman et al. (2011). The 9 sources (marked by “D” in
Table 5) are included in the BGPS sample with distances determined in Table 6
of Dunham et al. (2011b). Comparing their distances with those estimated from
our analysis on the basis of the presence or otherwise of an IRDC (7 sources
with and 2 sources without IRDCs, respectively) are also generally consistent
with those estimated by Dunham et al. (2011b). In addition, one point to note
is that the identification of an IRDC depends on the presence of a bright 8
$\mu$m infrared background, so a source at the near distance without a
significant infrared background might be not identified as IRDC. Therefore for
those sources without an identified IRDC, the distance may be less certain and
biased toward large distances. The reliability of the distance determinations
for our sources without IRDC associations could potentially be improved
through additional HISA investigations, however, Dunham et al. (2011b) find
that HISA is unlikely to be present for BGPS sources without an associated
IRDC. They find that for 215 BGPS sources without IRDC identifications listed
in Table 6 of Dunham et al. (2011b), only 26 present a definite HISA features.
Hence, we have not undertaken any additional HISA determinations beyond those
already available in the literature, as the available cross-checks show that
our assumption of the near and far kinematic distances for sources with and
without IRDCs respectively appear reasonable. The accuracy of this
discriminator for kinematic distances can’t be accurately assessed with such a
small sample, however, if our results are representative then it is at
$\sim$90%. In some cases the Galactic rotation model is not able to provide a
reliable distance estimate and for these sources (sources N33, N194 and N210
in Table 1) we have adopted a distance of 4 kpc for source N33 (which has an
IRDC association), and that determined by Schlingman et al. (2011) for the
other two sources N194 and N210.
Based on the estimated distances, the integrated luminosity of 95 GHz methanol
maser, Lm can be calculated from Lm=4$\pi$$\cdot$D2$\cdot$S${}_{int}^{m}$,
where $D$ is the estimated distance and S${}_{int}^{m}$ is the integrated flux
density of the 95 GHz emission. This assumes that maser emission is isotropic,
which is known to be false, however, in the absence of any information on the
beaming angle of the maser emission, nor our alignment with respect to it,
this is the only feasible approach that can be undertaken.
## 4 Analysis and Discussion
### 4.1 Mid-IR characteristics of GLIMPSE point sources
Analysis of the mid-IR colors of GLIMPSE point sources associated with EGOs
with and without class I methanol maser detections has been performed by Chen
et al. (2011). No significant difference in the mid-IR colors was found
between the GLIMPSE point sources with and without class I methanol masers in
the EGO sample (see Figure 5 of their work). We have performed the same
analysis for our observing sample to further investigate the mid-IR
characteristics of the GLIMPSE point sources which are, and are not,
associated with class I methanol masers. Although the detected class I maser
sources in our PMO survey include 12 EGOs which were considered in the color
analysis by Chen et al. (2011), the remaining 51 newly-discovered 95 GHz class
I methanol maser sources (which includes 3 EGO associated sources previously
only detected in the 44 GHz transition) allows us to explore in a more
unbiased manner, the mid-IR characteristics of GLIMPSE point sources
associated class I methanol masers.
A number of color-color diagrams were constructed to compare the mid-IR colors
of the GLIMPSE point sources with and without an associated class I methanol
maser detection in our survey. In Figure 4 we plot three color-color diagrams
([3.6]-[4.5] vs. [5.8]-[8.0]; [3.6]-[5.8] vs. [3.6]-[8.0] and [3.6]-[4.5] vs.
[4.5]-[8.0]) using different symbols for the sources which are, and which are
not associated with class I methanol masers. There are 63 members of the group
associated with class I methanol masers and 151 members of the group which are
not associated with class I methanol masers. This figure shows that there are
no clear differences in the mid-IR colors between those sources in our sample
which are associated with a class I maser, and those which are not, consistent
with the findings from the EGO-based sample of Chen et al. (2011). There are
15 sources in total associated with known EGOs in our observing sample (see
Table 4). The color regions occupied by the sources at evolutionary Stages I,
II and III, (derived from the 2D radiative transfer model of Robitaille et al.
(2006)), are marked on the [3.6]-[4.5] vs. [5.8]-[8.0] color diagram of Figure
4 (left panel). We found that most (187/214) sources in our observed sample
fall in the region occupied by the youngest protostars (Stage I), with the
remaining 27 sources found in the upper-left of the color-color diagram,
outside the Stage I evolutionary region. Chen et al. (2011) have discussed
these redder GLIMPSE sources which lie outside the Stage I color region in
detail. They may be deeply reddened sources (with reddening vector
A${}_{v}\sim$80; a typical reddening vector of Av$=$20 derived from the
Indebetouw et al. (2005) extinction law is shown in Figure 4 to demonstrate
the reddening effect), MYSOs with an extremely high mass envelope, or caused
by emission mechanisms such as H2 or PAH line emission which were not included
in the Robitaille et al. (2006) models. One of the most likely explanations is
that they have excess 4.5 $\mu$m emission from shocked H2 in particularly
strong/active outflows, which in turn readily produces class I maser emission.
This is supported by the high detection rate of class I methanol masers
towards these redder sources seen in both the current observations (17/27=63%
in this survey), and the EGO survey of Chen et al. (2011) (a detection rate of
75%). We discuss possible dependence of the detection rate of class I methanol
masers with the colors or magnitudes of GLIMPSE point sources in greater
detail in Section 4.4. For the redder GLIMPSE point sources (outside the Stage
I region), 8 sources with an associated class I methanol maser are also
associated with EGOs (marked by red triangles in Figure 4), which means the
other 9 sources with an associated class I masers are not associated with an
EGO, although 3 of them are associated with known MYSOs (sources N22, N90 and
N101; see Table 4).
We have undertaken a detailed analysis of possible correlations between the
class I methanol maser emission and the associated GLIMPSE point sources.
Figure 5 (left panel) shows a log-log plot of the integrated luminosity of the
class I methanol masers versus the luminosity of the GLIMPSE point sources in
the 4.5 $\mu$m band. The distance to the source listed in Table 5 was used to
calculate the luminosity for both the class I maser and the GLIMPSE point
source (see the discussion of distance assignment in Section 3.3). A linear
regression analysis for this distribution was undertaken, and the line of best
fit obtained is plotted in the figure. Our analysis suggests that there is a
statistically significant positive slope in the distribution, but with a weak
correlation (the best fit shows a slope of 0.41 with a statistically
significant p-value of 10-4 which allows us to reject zero slope in the data,
and a low correlation coefficient of 0.47). Such a correlation seems
reasonable if the 4.5 $\mu$m emission is believed to be enhanced by shocks,
which are also thought to be responsible for the class I methanol maser
emission. On the other hand, this correlation may be simply a consequence of
the correlation between the class I methanol maser and central source
luminosity, which has been obtained by, e.g., Bae et al. (2011) for the 44 GHz
masers. However, our determination of the distances using the presence or
absence of an IRDC to resolve the distance ambiguity will introduce
unpredictable uncertainties as discussed in Section 3. To eliminate the
possible effects of distance dependencies in our investigations we compared
mid-IR color [3.6]-[4.5] with [3.6]-log(Sm), where Sm is the integrated flux
density of the class I methanol maser (a plot of this is shown in the right-
hand panel of Figure 5). This plot shows no significant correlation between
the “colors”, with the linear regression analysis giving a slope of 0.6, a
non-significant p-value of 0.10 and a small correlation coefficient (0.22).
One possible reason for weak or non-significant correlation between them is
that the GLIMPSE point sources which have been identified as being associated
with the class I methanol masers may not be the true driving sources. Within
the large field-of-view covered by the PMO beam size (52$\arcsec$), there will
always be a number of GLIMPSE point sources, and from the present observations
with this resolution we can not determine which one is the driving source of
the class I methanol maser. Our assumption that the GLIMPSE point source which
satisfies the mid-IR color criteria for the class I maser search in our survey
is the driving source is almost certainly wrong in some cases, indeed some
driving sources of class I methanol maser are likely not present in the
GLIMPSE point source catalog due to saturation, the presence of bright diffuse
emission, or intrinsically extended morphology in the IRAC bands (e.g. from
extended PAH emission or extended H2 emission in shocked gas (see Robitaille
et al. 2008, Povich et al. 2009, and Povich & Whitney 2010)). On the other
hand, if the GLIMPSE point sources do correspond to the true driving sources
of the class I methanol masers, the lack of significant correlations between
the maser and GLIMPSE mid-IR colors suggests that the excitation of the class
I methanol masers are not directly related to the mechanism responsible for
the mid-IR emission. This view is supported by the fact that class I methanol
maser spots are often distribute over large angular and spatial scales
(usually of the order of 10$\arcsec$), and are excited in shocked regions
(e.g. Cyganowski et al. 2009), whereas the GLIMPSE point sources emission
reflects the thermal dust or molecular environments within a smaller region
around the protostar. Moreover, the 4.5 $\mu$m emission may still be
dominanted by the thermal dust emission from the driving protostar, rather
than the molecular gas (such as H2 or CO) excited by shocks, thus masking any
relationship between the class I methanol maser properties and the 4.5 $\mu$m
intensity.
### 4.2 Relationships between class I methanol masers and BGPS sources
A close correlation between GLIMPSE point sources with an associated class I
methanol masers and the presence of a 1.1 mm BGPS sources was first noted in
the EGO-based survey of Chen et al. (2011), the analysis of which motivated
the investigations undertaken here. Chen et al. showed that the luminosity of
the class I methanol masers in the EGO sample strongly depends on the
properties (including both the mass and volume density) of the associated 1.1
mm dust clump: the more massive and denser the clump, the stronger the class I
methanol emission. Here we perform a similar analysis to Chen et al. (2011) on
a sample of class I methanol masers which combines GLIMPSE point sources and
BGPS sources to investigate the relationship between the dust clumps and the
maser emission in a wider sample of sources.
Based on the assumption that the 1.1 mm emission from the BGPS source arises
from optically thin dust, we can calculate the associated gas mass using the
equation:
$M_{gas}=\frac{S_{int}D^{2}}{\kappa_{d}B_{\nu}(T_{dust})R_{d}},$ (1)
where $S_{int}$ is the 1.1 mm integrated flux density of the BGPS source, $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 at
temperature $T_{dust}$, and $R_{d}$ is the dust-to-gas mass ratio. Here we
have used $\kappa_{d}$$=$1.14 cm2 g-1 for 1.1 mm (Ossenkopf & Henning 1994)
and a dust-to-gas ratio ($R_{d}$) of 1:100 in our calculations and
$B_{\nu}(T_{dust})$ was calculated for an assumed dust temperature of 20 K.
The average H2 column density ($N_{H_{2}}$) and volume density ($n_{H_{2}}$)
of each dust clump were then derived from its mass and radius ($R$), assuming
a spherical geometry and a mean mass per particle of $\mu=2.37$ mH. The
parameters of the 1.1 mm continuum integrated flux density, $S_{int}$ and 1.1
mm source radius, $R$ were obtained from the BGPS catalog (Rosolowsky et al.
2010) for the 214 sources in our sample and the values are listed in Table 1.
We applied a correction factor of 1.5 to the Rosolowsky et al. BGPS catalog
flux densities (which are also listed in Column (11) of Table 1 of our work)
to derive the gas masses for the 63 BGPS sources with an associated class I
methanol maser detection. For the two sources (sources N39 and N143) which are
unresolved with the BGPS beam, we were not able to determine their gas column
and volume densities due to the absence of the size of the BGPS source. The
derived masses and gas densities for the 1.1 mm dust clumps with an associated
class I methanol maser are given in Table 5. As stated in Section 3.3, there
is a small number of detected class I maser sources which are also included in
the sample of BGPS sources investigated by Dunham et al. (2011b) or Schlingman
et al. (2011). Comparing the physical parameters (gas mass and volume/column
density) derived for those BGPS sources which are in common with the two
previous studies, we find that they are consistent with each other (usually
similar but not identical).
A log-log plot of the luminosity of the class I methanol maser versus the
derived gas mass (left panel) and H2 volume density (right panel) of the
associated 1.1 mm BGPS source is shown in Figure 6. From this figure it can be
seen that there is significant positive correlation between the class I maser
luminosity and the gas mass of the BGPS source, while a very weak negative
correlation exists between the class I maser luminosity and the H2 volume
density. Linear regression analysis for both distributions (the corresponding
best fit lines are overlaid in each panel of Figure 6) find a statistically
significant (p-value of 8.1E-13) linear dependence with a slope of 0.81
existing between the maser luminosity and the gas mass (the slope has a
standard error of 0.07 and a correlation coefficient of 0.84). In contrast,
there is no statistically significant correlation (p-value of 0.10) between
the class I methanol maser luminosity and the gas density (the fit has a slope
of -0.25 and a small correlation coefficient of 0.22). The statistically-
significant positive correlation between class I maser luminosity and BGPS
source mass obtained in this study is similar to that measured in the EGO-
based sample of Chen et al. (2011). Chen et al. (2011) also found a weak but
statistically significant positive correlation between the class I maser
luminosity and the gas volume density in the EGO sample, however, no
statistically significant or a very weak negative correlation is observed in
our larger and more diverse sample.
We also carried out an investigation of the dependence between the BGPS beam-
averaged gas column density and the class I methanol maser integrated flux
density (both of which are independent of the assumed distance to the source).
The H2 column density per beam can be estimated by
$N_{H_{2}}^{beam}=\frac{S_{40^{\prime\prime}}}{\Omega_{beam}\mu\kappa_{d}B_{\nu}(T_{dust})R_{d}},$
(2)
where S${}_{40^{\prime\prime}}$ is the 1.1 mm flux density within an aperture
with a diameter of 40$\arcsec$, $\Omega_{beam}$ is the solid angle of the
beam, $\mu$ is the mean mass per particle, $\kappa_{d}$ is the mass absorption
coefficient per unit mass of dust, $B_{\nu}(T_{dust})$ is the Planck function
at temperature $T_{dust}=20$ K, and $R_{d}$ is the dust-to-gas mass ratio, as
described above. S${}_{40^{\prime\prime}}$ was adopted as the measure of the
flux within a beam since a top-hat function with a 20$\arcsec$ radius has the
same solid angle as a Gaussian beam with an FWHM of 33$\arcsec$ (see also
Section 2.1). In addition to a flux correction factor of 1.5 (see above), an
aperture correction of 1.46 should be applied to flux density
S${}_{40^{\prime\prime}}$ (which is given in Column (10) of Table 1 in our
work) to account for power outside the 40$\arcsec$ aperture due to the
sidelobes of the CSO beam (Aguirre et al. 2011) in the calculation of beam-
averaged column density. Since this property is independent of the distance to
the source, we can derive it for all BGPS sources in our sample and we have
listed it for each source in Column (12) of Table 1. The results are shown as
a log-log plot in Figure 7 which demonstrates that there is a statistically
significant positive correlation between the beam-averaged gas column density
of BGPS sources and the integrated flux density of class I methanol masers
(S${}_{int}^{m}$). We have performed a linear regression analysis for this
distribution and obtain a best fit linear equation of:
$log(S_{int}^{m})=0.75[0.10]log(N_{H_{2}}^{beam})-15.94[2.28]$ (3)
with a correlation coefficient of 0.69 and p-value of 3.25E-10. This
relationship between the class I maser flux density and the beam-averaged gas
column is important for refining future class I methanol maser surveys based
on BGPS sources because it is independent of distance and other intrinsic
physical parameters of the sources. For example, toward nearby low-mass star-
forming regions a threshold column density of 123 M⊙ pc-2 (corresponding to
$6.5\times 10^{21}$ cm-2) has been observed (Lada et al. 2010; Heiderman et
al. 2010), and substituting this into the above relationship we can estimate a
lower limit of 2.6 Jy km s-1 for the integrated flux density of 95 GHz class I
methanol masers. The lowest class I maser integrated flux density from our
observations is only slightly higher ($\sim$ 3.0 Jy km s-1), which suggests
that we are likely to have detected significant part the 95 GHz class I maser
sources in the observed sample.
### 4.3 Star formation activity associated with methanol masers
The star formation activity of the BGPS sources was characterized by Dunham et
al. (2011a), through the properties of mid-IR sources along a line of sight
coincident with the BGPS sources. They divided the BGPS sources into four
groups representing increasing probability of the associated mid-IR sources
indicating star formation activity. The sources with the highest probability
of star formation activity are classified as group 3 and include BGPS sources
matched with EGOs or Red MSX Survey (RMS; Hoare et al. 2004; Urquhart et al.
2008) sources. The lowest probability group (group 0) includes BGPS sources
which were not matched with any mid-IR sources and are considered to be
“starless” in their work. Groups 1 and 2 represent BGPS sources matched with
GLIMPSE red sources cataloged by Robitaille et al. (2008), or a deeper list of
GLIMPSE red sources created by Dunham et al. (2011a). Overall they found that
the mid-IR emission associated with BGPS sources with a high probability of
star formation activity (group 3) are typically extended with large skirts of
emission, while the low probability sources (group 1) are more compact, with
weak emission. In this section, we explore the star formation activity in the
sources with and without methanol maser associations using the parameters of
the BGPS sources.
Histograms of BGPS source parameters (beam-averaged H2 column density
$N_{H_{2}}^{beam}$, integrated flux density $S_{int}$ and radius R) for those
sources with and without an associated class I methanol maser are presented in
Figure 8. Unfortunately we are not able to compare any intrinsic physical
parameters such as mass, source size in pc etc between the two groups, due to
the absence of a distance estimate for the sources without an associated class
I methanol maser. For each distribution in Figure 8, the upper and lower
panels correspond to the BGPS sources with and without a class I maser
detection, respectively. It can be clearly seen that the distributions differ
significantly between BGPS sources with an associated class I maser and those
without for the beam-averaged H2 column density and the integrated flux
density of BGPS sources (see left-hand and middle panels). In contrast there
is no significant difference in the observed distribution of the radius of the
BGPS sources for the two samples (see right-hand panel). The basic statistical
parameters such as mean, median, standard deviation, for each of these
distributions are summarized in Table 6\. The mean logarithm of the beam-
averaged column density $N_{H_{2}}^{beam}$ is 21.9 [cm-2] for the sources with
no class I methanol maser detections, but 22.7 [cm-2] for the sample of
sources with an associated class I methanol maser (a difference of
approximately 3 standard deviations). While the mean logarithm of the BGPS
integrated flux density is 0.0 [Jy] in sources without an associated class I
masers, but greater at 0.7 [Jy] in sources with class I masers (a difference
of approximately 2 standard deviations). However, a t-test finds that the
difference in the mean of each of the distributions for these two properties
is statistically significant for the two groups. The distributions of radii
are not significantly different between the two groups, each having a mean of
around 50$\arcsec$ and a large range (mostly distributed between 20 and
100$\arcsec$). The beam-averaged column density for the BGPS sources without
an associated class I maser ranges between 21.4 [cm-2] $\leq
log(N_{H_{2}}^{beam})\leq$ 22.7 [cm-2], whereas for sources with an associated
class I masers the range is 21.9 [cm-2] $\leq log(N_{H_{2}}^{beam})\leq$ 23.8
[cm-2]. Similarly the range of the logarithm of integrated flux density is
from -0.9 to 1.1 [Jy] for BGPS sources without an associated class I masers,
but from -0.6 to 1.5 [Jy] for those with a class I masers. The large
overlapping range in the integrated flux density distribution of the two
groups suggests that the beam-averaged column density is the most efficacious
parameter for selecting BGPS sources likely to be associated with a class I
methanol maser.
Comparing the distribution of the beam-averaged H2 column density for the four
star formation activity groups described by Dunham et al. (2011a; Figure 12),
with that for the class I methanol maser sample we can see that it is similar
to that shown for group 3\. While the distribution for the sources without an
associated class I methanol maser is similar to that seen for group 0 and
group 1 by Dunham et al. Comparing distributions of BGPS source flux for our
samples with Dunham et al. (2011a), those without class I masers appear to
agree well with their group 1. As described above, group 3 contains the
sources with the highest probability of star formation activity include BGPS
sources matched with EGOs or RMS sources, while group 0 represents those with
the lowest probability, including BGPS sources without any associated mid-IR
source (referred as “starless”). Since all of our target BGPS sources have an
associated GLIMPSE point source we would expect that the distributions we
observe should differ from those seen for group 0 sources, which have no
associated mid-IR source. The group 1 category sources include at least one IR
object which may be an AGB star catalogued by Robitaille et al.(2008) or a
deeper GLIMPSE red source from the list of Dunham et al. The BGPS sources in
our sample with an associated class I masers (63 in total) includes 15 EGOs
(which are classified into group 3 by Dunham et al.), however, the relatively
small number of EGOs can not dominant the BGPS parameter distributions
observed for this group. The remaining 48 sources must also have a similar
BGPS parameter distribution to that observed for group 3. Since class I
methanol maser emission is only known to be found towards active star
formation regions, the similar distribution of the BGPS properties seen in the
class I maser sources and the group 3 sources supports the speculation of
Dunham et al. (2011a) that group 3 sources are those with the highest
probability of star formation activity. The BGPS sources in our target sample
without an associated class I methanol maser, correspond to group 1 in the
Dunham et al. classification (which have a lower probability of star formation
activity), and these may be regions which are either too young, or have too
low gas density, or too weak outflows to excite class I maser emission.
Comparing our observing sample with the GLIMPSE red source catalog complied by
Robitaille et al. (2008), we found that there are 95 sources are common in the
two data sets (including 22 sources with class I masers and 73 sources without
class I masers). Using the criteria of Robitaille et al. to separate AGB stars
and YSOs, 8 of 22 sources for which we have detected an associated class I
masers are classified as extreme AGB stars with high mass-loss rates and
therefore significant circumstellar dust. However, since class I methanol
masers appear to only be associated with star formation, this suggests that
there may be a relatively high mis-classification rate for the extreme AGB
sources using the Robitaille et al. criteria. We found that only 9 of 73 BGPS
sources from our sample which are not associated with class I masers are
classified as AGB stars. This also supports the hypothesis that the sources
without class I masers may be objects at early stages of star formation,
rather than AGB stars.
Analysis of the properties of 1.1 mm BGPS associated with EGOs by Chen et al.
(2011) showed that those which are associated with class I methanol masers,
but not class II methanol masers have a lower mass/density of dust clump than
those which are associated with both class I and II methanol masers. We have
cross-matched the 63 sources with an associated class I methanol maser
detected in our survey with the catalog of 6.7 GHz class II methanol masers
(usually better than 1′′) from the Parkes Methanol Multibeam (MMB) blind
survey published to date (Caswell et al. 2010; Green et al. 2010 ; Caswell et
al. 2011 ; Green et al. 2012), or from the ATCA observations of Caswell
(2009). The MMB masers positions have been measured to high positional
accuracy (better than 1$\arcsec$) and the observations have a sensitivity of
about 0.2 Jy ($3\sigma$ from the subsequent ATCA observations). The MMB survey
published to date covers the region $186\arcdeg<l<20\arcdeg$ with
$|b|<2\arcdeg$. Thus the overlap region between the class I methanol maser
sources detected in our survey and the class II methanol masers in the MMB
survey is $0\arcdeg<l<20\arcdeg$ with $|b|<0.5\arcdeg$. The MMB survey data
from the overlap region allow us to identify the associations between the two
classes of methanol masers, and in particular to identify those class I
methanol maser sources without an associated class II masers. Whether the
class I methanol maser detected in this survey is associated with a class II
maser or not is summarized in Column (9) of Table 5. Thirty three of the 63
class I methanol masers in our sample lie in the MMB overlap region and
Caswell (2009) data set, and of these 20 have an associated class II maser and
13 do not. Histograms of the beam-averaged H2 column density and flux density
of BGPS for the sources associated with only class I methanol masers compared
to those associated with both classes of methanol maser are shown in Figure 9.
Although the sample sizes for the two groups are relatively small, they still
allow us to investigate whether the BGPS properties discriminate between the
two groups. The statistical parameters for each distribution are summarized in
Table 6. From Figure 9 and the statistical parameters we can see that there is
a trend for the sources associated both methanol maser classes to have higher
BGPS flux densities and column densities than the sources associated with only
class I masers. The mean column density and flux density of the associated
BGPS sources are marked with a dashed line in the corresponding histogram, and
are significantly larger for the sample of sources associated with both
classes of methanol maser. A t-test shows that the difference in the mean of
the two group distributions for the two BGPS properties is statistically
significant.
It is important to note that the sample size used in the current analysis is
small. A larger sample is required to more thoroughly investigate the star
formation activity and physical properties of the regions with associated
class I and II methanol masers. In addition, our assumption of a dust
temperature (Tdust) of 20 K for all sources in our analysis will affect the
physical parameters such as mass and column/volume density derived from the
BGPS data. For example, a dust temperature of 7.2 K for the BGPS sources with
an associated class I methanol maser and a dust temperature of 20K for those
without would result in distributions of the beam-averaged column density for
the two samples having the same mean. However, the mean gas kinetic
temperature derived from the NH3 observations for group 3 sources (those
similar to the class I maser group) was 22.7 K (Dunham et al. 2011b), much
higher than the 7.2 K required to give the distributions the same mean.
Furthermore, since the BGPS sources without an associated class I maser are
similar to group 1 of Dunham et al., for which the mean temperature from NH3
observations is 14.6 K (Dunham et al. 2011b), the expectation is that more
accurate temperature estimates for individual BGPS sources would produce a
greater difference in the distributions of the physical properties derived
from BGPS data, rather than reducing it.
### 4.4 Detection rates
In this section, we compared the detection rates of class I methanol masers
with the cataloged parameters of the associated GLIMPSE point sources and 1.1
mm BGPS sources with the aim of developing more efficient criteria for future
targeted class I methanol maser searches.
Figure 10 presents a histogram showing the detection rate of class I methanol
masers as a function of the 4.5 $\mu$m magnitude (left panel) and [3.6]-[4.5]
color (right panel) of the associated GLIMPSE point sources. It can be clearly
seen that the detection rate for class I masers increases (from 0.1 to 0.5) as
the 4.5 $\mu$m magnitude decreases (i.e. with increasing 4.5 $\mu$m flux
density). In contrast the detection rate for class I methanol masers shows no
significant variation for [3.6]-[4.5] color $<$ 3.2, being $\sim 0.2-0.3$,
however for [3.6]-[4.5]$>$3.2 it is much higher (0.8–1.0). Recalling the
discussion in Section 4.1, these results are consistent there being no
significant differences between the mid-IR colors of the sources with and
without an associated class I methanol maser, however, there is a higher
detection rate for class I methanol masers towards GLIMPSE point sources with
the most extreme red range for [3.6]-[4.5] color. Chen et al. (2011) have
suggested that these redder sources may correspond to higher mass, high
luminous YSOs. The correlation between the detection rate of class I methanol
masers and the 4.5 $\mu$m magnitude (or flux density) of the associated
GLIMPSE point source suggests that the outflows or shocks are stronger or more
active for those with more intense 4.5 $\mu$m emission which thus are more
likely to produce maser emission. Apart from correlation between the emission
intensities of class I methanol masers and the GLIMPSE 4.5 $\mu$m band (Figure
5, left), there is clearly an increased probability of the presence of a
methanol maser for sources with stronger 4.5 $\mu$m emission. This may be
because although strong 4.5 $\mu$m emission is a good indicator of the
presence of shocks (and hence the possibility of a class I maser), the
intensity of the maser may depend more strongly on other physical factors such
as the gas mass and column density of the parent clouds. We also note that
while sources with [4.5]$<$8.0 have a higher detection rate for class I
methanol masers (0.4 – 0.5), they were typically classified as extreme AGB
stars by Robitaille et al. (2008). At present all class I methanol masers are
thought to be associated with star formation, which suggests that there is a
high mis-classified rate for the extreme AGB star population in Robitaille et
al. (2008) and that many of these sources correspond to luminous YSOs.
The detection rates of class I methanol masers as a function of the BPGS
cataloged parameters (beam-averaged H2 column density and integrated flux
density) are shown in Figure 11. The BGPS radius parameter is not included in
the analysis because as discussed in section 4.2, the radius is the least
useful BGPS parameter in terms of its ability to select BGPS sources with a
higher likelihood of having associated class I maser emission. From this
figure, we can clearly see that the detection rates for class I methanol
masers significantly increase with increasing values of the BGPS source
parameters. To allow a more detailed comparison the number and rate of
detection for class I masers in each bin for each BGPS parameter in our
observed sample and the full BGPS catalog are summarized in Table 7. This
shows that the detection rate for this sample is 100% for sources with the
highest beam-averaged column density (larger than 23.0 [cm-2] in logarithm)
and BGPS integrated flux density (larger than 1.2 [Jy] in logarithm). We note
that if there are thermal sources among the objects detected at 95 GHz, they
may be preferentially associated with BGPS sources with the highest column
densities. As the number of these BGPS sources in the high column density bins
is small, even a few sources can potentially distort the statistics. To test
for this we excluded the 13 (potentially thermal) sources with a single broad
line profile (as identified in Section 3.1), from the class I methanol maser
detection sample and from the total sample and redid our analysis. Our re-
analysis excluding potential thermal sources showed a similar trend to that
seen in Figure 11. In fact, among the 13 broad line profile sources, only 3
are located in the high column density bins ($>10^{23}$ cm-2) with 100%
probability of a 95 GHz maser detection. Thus the possible thermal 95 GHz
sources do not precisely correspond to BGPS sources with the highest column
densities. The rate (3/9) of the possible thermal sources to the class I
methanol detection sources in the high column density bins is relatively low,
thus the possible thermal sources do not significantly distort the statistics.
Moreover, as stated in Section 3.1, we can not exclude the possibility that
the emission from weak maser features contributes to the broad line profiles.
Our analysis using all 95 GHz detections does not exclude any possible maser
sources for a future survey toward a larger BGPS sample (see below). Even if
we assume that all the broad line profile sources are totally thermal, the
rate of real maser sources would still be very high (50/63=80%) in any sample
derived on the basis of all 95 GHz detections.
For class I methanol maser surveys with a single dish with a beam size of
around an arcminute, it seems that the millimetre continuum emission on
similar scales (e.g. the BGPS sources) can provide a better targeting criteria
than the arcsecond-scale mid-IR emission (e.g. GLIMPSE point sources). Our
earlier discussion shows that the class I methanol maser emission intensity is
not closely related to the mid-IR emission of GLIMPSE points sources, but does
depend on the mass and beam-averaged column density of the associated BGPS
sources, also suggesting that BGPS properties are likely to provide a better
basis for constructing samples for further class I methanol maser searches. We
also undertook binomial generalized linear modeling (GLM) for the class I
maser presence and absence using both the GLIMPSE point source and BGPS
properties, similar to that undertaken for water masers by Breen et al. (2007)
and Breen & Ellingsen (2011). This investigation showed that the BGPS source
properties are a much stronger predictor of the likelihood that a particular
source will host a maser, than are the mid-IR properties, consistent with the
investigations outlined above. As the results of the binomial GLM are less
readily interpreted than the more direct correlation investigations in
sections 4.1, 4.2 and 4.3, and don’t reveal any significant new information we
do not discuss them further here.
To more efficiently search for class I methanol masers using the BGPS sources,
we can combine the two BGPS properties of beam-averaged column density and
integrated flux density to develop better criterion for future searches. In
Figure 12 (left panel) we plot a log-log distribution comparing the BGPS flux
density versus BGPS beam-average column density from the current observations
using different symbols for the sources with and without class I methanol
maser detections (including also possible thermal sources). This clearly shows
that there is a significant discrimination between sources with class I masers
(marked by red circles) and those without class I masers (marked by blue
triangles). From inspection of this plot we have defined a region wherein most
(90%) of class I methanol maser detected in our current survey are placed,
constructed with red lines in the plot. The defined region can be expressed as
follows:
$log(S_{int})\leq-38.0+1.72log(N_{H_{2}}^{beam}),and\
log(N_{H_{2}}^{beam})\geq 22.1,$ (4)
We can then extrapolate the identified class I methanol maser region to the
full BGPS sample to estimate the likely number of class I methanol masers. The
distribution of all BGPS sources with the class I maser region overlaid is
present in the right panel of Figure 13. In total, approximate 1200 sources
are located within the defined class I maser region. If we extrapolate the
results of this study we would predict that we can detect 90% of all the
expected ($\sim 1000$; see Table 7) class I methanol masers associated with
BGPS sources and that the detection efficiency would be about 75% towards the
sources within the defined region. Since the above estimates are based on all
95 GHz detections (including possible thermal sources), the number of real
maser source detections may be at 80% (at worst) of the above predictions (see
discussion above).
However, a search for class I methanol maser towards an unbiased sample of
BGPS sources is required to clarify all possible sample selection effects and
to reliably estimate the true number of methanol masers associated with the
full BGPS catalog. Since the BGPS only covers around half of the inner Galaxy,
the total number of class I methanol masers in the Galaxy would be expected to
be at least double the number associated with BGPS sources, suggesting that
class I methanol masers may be significantly more numerous in the Galaxy than
are class II methanol masers.
## 5 Summary
Using the PMO 13.7-m radio telescope, we have performed a search for 95 GHz
class I methanol masers toward a sample selected from a combination of the
mid-IR _spitzer_ GLIMPSE and 1.1 mm CSO BGPS surveys. A total of 214 sources
were selected as the observing sample, and these satisfy the GLIMPSE mid-IR
criteria of [3.6]-[4.5]$>$1.3, [3.6]-[5.8]$>$2.5, [3.6]-[8.0]$>$2.5 and 8.0
$\mu$m magnitude less than 10, and are also associated with a 1.1 mm BGPS
source. 95 GHz class I methanol maser emission was detected toward 63 sources,
of these 51 are new 95 GHz class I methanol maser sources, and 43 have no
previously observed class I methanol maser activity. Thus a detection rate of
$\sim$29% was observed for class I methanol masers in the conjunct sample of
GLIMPSE and BGPS surveys from our single-dish survey. We also find that the
sensitivity of survey exceeds the theoretical detection limit derived from the
observed dependence between the integrated class I maser emission and the BGPS
beam-averaged column density.
Analysis of the mid-IR colors of GLIMPSE point sources in our observing sample
indicates that the color-color region occupied by those sources with and
without an associated class I methanol maser are not significantly different.
However, the detection rate of class I methanol masers is higher towards those
GLIMPSE point sources with redder mid-IR colors. The mid-IR characteristics
the GLIMPSE sources associated with class I methanol masers in the current
sample is very similar to that derived in our earlier EGO-selected sample. We
find that the class I methanol maser intensity is not closely related to
either the mid-IR emission intensity nor the color of the associated GLIMPSE
point sources. However, the maser emission is well correlated with the gas
mass derived for the BGPS sources. Comparison of the properties of BGPS
sources with and without an associated methanol maser shows that those with an
associated class I methanol maser usually have higher beam-averaged H2 column
density and larger BGPS flux density than those without an associated maser.
A series of investigations of the detection rates of class I methanol masers
as a function of GLIMPSE mid-IR and BGPS properties were undertaken, with the
aim of developing more efficient criteria for future targeted class I methanol
maser searches. Although the detection rates of class I methanol masers appear
to some extent to be dependent on the mid-IR properties of GLIMPSE point
sources (such as 4.5 $\mu$m magnitude and [3.6]-[4.5] color), tighter
correlations are observed between the class I methanol maser detection rate
and the BGPS source properties. This suggests that the BGPS catalog could
provide much more efficient target samples for future class I methanol maser
searches. Based on the observed relationship between the detection rate of
class I methanol maser and the BGPS beam-averaged H2 column density, we
estimate that approximately 1000 (of $\sim$8400) BGPS sources may have an
associated class I methanol maser. We identify a region in the distribution of
BGPS beam-average column density versus BGPS integrated flux density
(satisfying $log(S_{int})\leq-38.0+1.72log(N_{H_{2}}^{beam})$, and
$log(N_{H_{2}}^{beam})\geq 22.1$), towards which we we expect to find 90% of
all ($\sim 1000$) class I methanol masers with a high detection efficiency
($\sim$75%).
We thank an anonymous referee for their helpful comments which have improved
this paper. We are grateful to the staff of Qinghai Station of Purple Mountain
Observatory for their assistance in the observation. This research has made
use of 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, and made use of information from the BGPS survey
database at http://irsa.ipac.caltech.edu/data/BOLOCAM-GPS/. This work is
partly supported by China Ministry of Science and Technology under State Key
Development Program for Basic Research (2012CB821800), the National Natural
Science Foundation of China (grants 10621303, 10625314, 10803017, 10821302,
11073041, 11073054, 11133008, 11173046), the CAS/SAFEA International
Partnership Program for Creative Research Teams, and Key Laboratory for Radio
Astronomy, CAS. .
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Table 1: Sample parameters
| GLIMPSE Point Source | | BGPS Source
---|---|---|---
Number | Name | 3.6 $\mu$m | 4.5 $\mu$m | 5.8 $\mu$m | 8.0 $\mu$m | | ID | Name | R | S${}_{40^{\prime\prime}}$ | Sint | $N_{H_{2}}^{beam}$
| | (mag) | (mag) | (mag) | (mag) | | | | (′′) | (Jy) | (Jy) | (1022 cm-2)
(1) | (2) | (3) | (4) | (5) | (6) | | (7) | (8) | (9) | (10) | (11) | (12)
1 | G004.7808$+$00.0429 | 12.12(0.04) | 10.09(0.04) | 8.96(0.03) | 8.19(0.04) | | 1051 | G004.783$+$00.043 | 27.96 | 0.175(0.041) | 0.422(0.081) | 1.05
2 | G004.8020$+$00.1880 | 13.30(0.06) | 11.06(0.05) | 9.99(0.04) | 9.56(0.05) | | 1053 | G004.803$+$00.191 | 25.94 | 0.094(0.038) | 0.211(0.065) | 0.57
3 | G004.9676$+$00.0500 | 13.00(0.05) | 10.83(0.06) | 9.07(0.03) | 7.91(0.02) | | 1066 | G004.967$+$00.047 | – | 0.096(0.041) | 0.166(0.060) | 0.58
4 | G005.0424$-$00.0977 | 13.31(0.07) | 11.09(0.08) | 9.57(0.04) | 8.47(0.03) | | 1071 | G005.043$-$00.099 | 57.77 | 0.157(0.041) | 0.553(0.110) | 0.95
5 | G005.3294$-$00.0949 | 14.04(0.11) | 11.42(0.07) | 9.81(0.04) | 8.74(0.03) | | 1084 | G005.333$-$00.093 | – | 0.087(0.039) | 0.198(0.063) | 0.52
6 | G005.3608$+$00.0179 | 12.39(0.08) | 10.21(0.05) | 8.76(0.03) | 7.82(0.03) | | 1087 | G005.361$+$00.017 | 46.14 | 0.281(0.041) | 0.925(0.119) | 1.69
7 | G005.3706$+$00.3179 | 9.55(0.09) | 7.45(0.06) | 6.00(0.02) | 5.37(0.03) | | 1090 | G005.373$+$00.319 | – | 0.145(0.036) | 0.230(0.054) | 0.87
8 | G005.8418$-$00.3756 | 13.34(0.09) | 10.99(0.06) | 9.17(0.04) | 8.01(0.04) | | 1131 | G005.841$-$00.379 | – | 0.107(0.058) | 0.203(0.091) | 0.64
9 | G006.0560$-$00.0319 | 13.55(0.15) | 10.50(0.09) | 8.69(0.04) | 7.70(0.04) | | 1164 | G006.057$-$00.029 | – | 0.116(0.043) | 0.233(0.069) | 0.70
10 | G006.4042$-$00.0413 | 14.21(0.17) | 10.98(0.07) | 9.43(0.04) | 8.71(0.03) | | 1203 | G006.407$-$00.039 | 5.13 | 0.126(0.039) | 0.246(0.062) | 0.76
11 | G006.9221$-$00.2513 | 10.33(0.06) | 8.24(0.06) | 6.74(0.03) | 5.72(0.02) | | 1251 | G006.923$-$00.251 | 36.42 | 0.441(0.059) | 1.453(0.164) | 2.66
12 | G007.0097$-$00.2542 | 12.49(0.13) | 10.01(0.08) | 8.71(0.09) | 7.68(0.28) | | 1259 | G007.013$-$00.253 | 36.39 | 0.327(0.042) | 0.956(0.114) | 1.97
13 | G007.3350$-$00.5666 | 12.13(0.09) | 9.94(0.06) | 8.67(0.04) | 7.87(0.02) | | 1289 | G007.335$-$00.567 | 37.71 | 0.455(0.059) | 1.366(0.159) | 2.74
14 | G008.2032$+$00.1916 | 15.42(0.34) | 11.89(0.06) | 10.13(0.04) | 9.30(0.04) | | 1341 | G008.206$+$00.192 | 41.18 | 0.222(0.063) | 0.691(0.155) | 1.34
15 | G008.2761$+$00.5124 | 12.95(0.14) | 11.21(0.14) | 9.99(0.06) | 9.12(0.06) | | 1346 | G008.274$+$00.512 | – | 0.159(0.066) | 0.243(0.088) | 0.96
16 | G008.3264$-$00.0932 | 11.40(0.04) | 9.88(0.04) | 8.85(0.03) | 8.22(0.04) | | 1352 | G008.326$-$00.096 | – | 0.237(0.060) | 0.349(0.092) | 1.43
17 | G008.4200$-$00.2710 | 13.68(0.08) | 11.52(0.08) | 10.20(0.07) | 9.72(0.20) | | 1360 | G008.422$-$00.274 | 65.48 | 0.616(0.073) | 3.268(0.309) | 3.71
18 | G008.4404$-$00.1689 | 10.58(0.05) | 8.93(0.07) | 7.64(0.04) | 6.96(0.03) | | 1361 | G008.440$-$00.168 | – | 0.140(0.064) | 0.211(0.085) | 0.84
19 | G008.4522$-$00.2885 | 14.17(0.08) | 11.72(0.06) | 10.19(0.04) | 9.57(0.05) | | 1362 | G008.454$-$00.290 | 16.12 | 0.217(0.065) | 0.478(0.120) | 1.31
20 | G008.4602$-$00.2231 | 12.68(0.07) | 11.25(0.07) | 9.98(0.05) | 9.37(0.04) | | 1363 | G008.458$-$00.224 | 32.57 | 0.367(0.062) | 1.016(0.150) | 2.21
21 | G008.7082$-$00.4162 | 14.85(0.11) | 11.97(0.06) | 10.65(0.06) | 9.94(0.05) | | 1380 | G008.710$-$00.414 | 43.96 | 0.558(0.072) | 2.237(0.214) | 3.36
22 | G008.8315$-$00.0278 | 11.93(0.35) | 9.73(0.20) | 9.10(0.07) | 9.02(0.05) | | 1395 | G008.832$-$00.028 | 70.29 | 0.913(0.081) | 3.865(0.340) | 5.50
23 | G008.9560$+$00.1823 | 13.32(0.07) | 10.75(0.08) | 9.42(0.03) | 8.33(0.02) | | 1405 | G008.956$+$00.186 | 43.89 | 0.186(0.057) | 0.685(0.142) | 1.12
24 | G009.0285$-$00.3086 | 14.00(0.10) | 11.88(0.09) | 10.73(0.05) | 9.99(0.08) | | 1407 | G009.028$-$00.310 | 14.70 | 0.137(0.062) | 0.329(0.108) | 0.83
25 | G009.1277$-$00.0047 | 11.95(0.11) | 10.28(0.07) | 9.19(0.03) | 8.60(0.02) | | 1409 | G009.125$-$00.002 | 14.91 | 0.117(0.059) | 0.275(0.096) | 0.71
26 | G009.2147$-$00.2021 | 12.04(0.04) | 9.62(0.06) | 8.64(0.04) | 8.80(0.09) | | 1412 | G009.212$-$00.202 | 64.29 | 1.018(0.088) | 4.599(0.368) | 6.13
27 | G009.8474$-$00.0322 | 10.89(0.05) | 8.77(0.05) | 7.66(0.03) | 6.63(0.02) | | 1425 | G009.850$-$00.032 | 35.30 | 0.210(0.063) | 0.787(0.154) | 1.27
28 | G010.2124$-$00.3238 | 12.93(0.08) | 10.25(0.08) | 8.44(0.04) | 6.81(0.04) | | 1466 | G010.214$-$00.324 | 84.60 | 1.556(0.134) | 8.300(0.634) | 9.38
29 | G010.2266$-$00.2091 | 12.47(0.07) | 10.23(0.05) | 8.80(0.04) | 7.68(0.04) | | 1467 | G010.226$-$00.208 | 98.76 | 0.997(0.083) | 7.897(0.576) | 6.01
30 | G010.2596$+$00.0755 | 13.60(0.14) | 11.57(0.15) | 9.96(0.22) | 8.59(0.19) | | 1472 | G010.262$+$00.074 | 64.92 | 0.240(0.067) | 1.312(0.227) | 1.45
31 | G010.3203$-$00.2589 | 11.41(0.12) | 8.85(0.11) | 7.23(0.09) | 6.00(0.30) | | 1479 | G010.320$-$00.258 | 62.32 | 0.857(0.074) | 3.361(0.277) | 5.16
32 | G010.4723$+$00.0272 | 12.48(0.13) | 9.98(0.09) | 7.83(0.08) | 6.04(0.08) | | 1497 | G010.472$+$00.026 | 37.59 | 9.398(0.580) | 20.789(1.314) | 56.64
33 | G010.6239$-$00.3842 | 11.40(0.11) | 9.32(0.24) | 7.78(0.12) | 6.28(0.12) | | 1508 | G010.625$-$00.384 | 31.39 | 9.722(0.597) | 20.020(1.256) | 58.59
34 | G010.6683$-$00.2001 | 14.73(0.20) | 12.29(0.10) | 10.81(0.17) | 9.99(0.28) | | 1516 | G010.670$-$00.198 | 49.89 | 0.275(0.047) | 1.295(0.152) | 1.66
35 | G010.8223$-$00.1031 | 9.67(0.06) | 7.59(0.05) | 6.45(0.03) | 5.81(0.03) | | 1543 | G010.825$-$00.102 | – | 0.119(0.043) | 0.182(0.056) | 0.72
36 | G010.9584$+$00.0219 | 11.15(0.10) | 8.71(0.14) | 7.05(0.07) | 5.53(0.08) | | 1559 | G010.959$+$00.020 | 49.81 | 0.783(0.073) | 2.702(0.239) | 4.72
37 | G011.0642$-$00.0993 | 13.66(0.10) | 11.20(0.07) | 9.98(0.05) | 9.52(0.05) | | 1580 | G011.063$-$00.096 | 77.96 | 0.343(0.047) | 2.633(0.234) | 2.07
38 | G011.0993$+$00.0702 | 13.95(0.14) | 11.60(0.13) | 10.21(0.09) | 9.12(0.07) | | 1587 | G011.101$+$00.072 | – | 0.130(0.043) | 0.327(0.077) | 0.78
39 | G011.1157$+$00.0512 | 9.71(0.06) | 6.83(0.06) | 5.25(0.05) | 4.35(0.03) | | 1591 | G011.115$+$00.052 | – | 0.282(0.046) | 0.548(0.085) | 1.70
40 | G011.1244$-$00.1297 | 14.02(0.14) | 11.59(0.09) | 10.4(0.06) | 9.80(0.05) | | 1592 | G011.121$-$00.128 | 68.78 | 0.440(0.048) | 2.674(0.222) | 2.65
41 | G011.9430$-$00.1563 | 12.26(0.09) | 10.09(0.08) | 8.96(0.04) | 8.74(0.06) | | 1657 | G011.941$-$00.154 | 39.1 | 0.530(0.062) | 1.696(0.178) | 3.19
42 | G012.0217$-$00.2073 | 12.96(0.10) | 10.79(0.20) | 9.25(0.05) | 8.80(0.05) | | 1668 | G012.023$-$00.206 | 37.95 | 0.352(0.054) | 0.969(0.138) | 2.12
43 | G012.1991$-$00.0334 | 10.50(0.18) | 7.67(0.11) | 6.30(0.04) | 5.96(0.10) | | 1682 | G012.201$-$00.034 | 52.77 | 0.694(0.076) | 2.064(0.237) | 4.18
44 | G012.3668$+$00.5116 | 13.32(0.07) | 11.54(0.07) | 10.59(0.08) | 9.67(0.07) | | 1699 | G012.367$+$00.510 | – | 0.078(0.049) | 0.103(0.052) | 0.47
45 | G012.4933$-$00.2231 | 11.55(0.07) | 9.03(0.06) | 7.85(0.04) | 7.85(0.05) | | 1720 | G012.497$-$00.222 | 42.00 | 0.438(0.051) | 1.341(0.147) | 2.64
46 | G012.5931$-$00.3788 | 13.55(0.14) | 11.55(0.10) | 10.48(0.08) | 9.95(0.11) | | 1734 | G012.593$-$00.382 | – | 0.072(0.039) | 0.192(0.064) | 0.43
47 | G012.6246$-$00.0167 | 11.42(0.06) | 7.87(0.07) | 6.54(0.04) | 6.26(0.04) | | 1742 | G012.627$-$00.016 | 78.87 | 1.146(0.085) | 5.708(0.413) | 6.91
48 | G012.7073$+$00.0612 | 11.26(0.19) | 9.21(0.08) | 7.74(0.03) | 6.95(0.06) | | 1756 | G012.709$+$00.064 | – | 0.082(0.046) | 0.185(0.070) | 0.49
49 | G012.8024$-$00.3192 | 12.19(0.13) | 9.66(0.15) | 7.88(0.05) | 6.42(0.08) | | 1778 | G012.805$-$00.318 | 72.81 | 0.664(0.070) | 3.174(0.301) | 4.00
50 | G012.8886$+$00.4890 | 12.08(0.10) | 8.44(0.19) | 6.57(0.04) | 6.15(0.05) | | 1803 | G012.889$+$00.490 | 53.31 | 2.534(0.173) | 7.895(0.564) | 15.27
51 | G012.9062$-$00.0310 | 15.31(0.32) | 12.65(0.16) | 10.03(0.05) | 8.21(0.05) | | 1809 | G012.905$-$00.030 | 65.04 | 1.032(0.077) | 3.558(0.271) | 6.22
52 | G013.0356$-$00.3207 | 12.49(0.06) | 10.19(0.06) | 8.98(0.04) | 8.39(0.05) | | 1841 | G013.037$-$00.318 | 50.18 | 0.177(0.046) | 0.652(0.116) | 1.07
53 | G013.0970$-$00.1447 | 10.16(0.06) | 7.53(0.05) | 6.10(0.03) | 5.30(0.03) | | 1853 | G013.097$-$00.146 | 59.34 | 0.445(0.056) | 1.887(0.199) | 2.68
54 | G013.1182$-$00.0966 | 11.02(0.06) | 8.62(0.05) | 7.67(0.04) | 7.04(0.04) | | 1857 | G013.121$-$00.094 | 64.68 | 0.386(0.048) | 2.208(0.206) | 2.33
55 | G013.1818$+$00.0610 | 11.23(0.08) | 8.90(0.06) | 7.63(0.03) | 6.78(0.03) | | 1865 | G013.179$+$00.060 | 58.37 | 1.423(0.108) | 5.123(0.378) | 8.58
56 | G013.2473$+$00.1578 | 14.72(0.16) | 12.02(0.08) | 10.07(0.05) | 8.70(0.06) | | 1877 | G013.245$+$00.158 | 13.18 | 0.123(0.044) | 0.276(0.076) | 0.74
57 | G014.1101$-$00.5626 | 14.33(0.11) | 11.92(0.06) | 10.62(0.09) | 9.54(0.10) | | 1998 | G014.107$-$00.563 | 15.68 | 0.593(0.000) | 1.247(0.178) | 3.57
58 | G014.1327$-$00.5222 | 13.47(0.10) | 10.91(0.10) | 10.26(0.06) | 8.86(0.05) | | 2002 | G014.133$-$00.521 | 39.36 | 0.241(0.093) | 1.071(0.225) | 1.45
59 | G014.1958$-$00.5070 | 14.60(0.11) | 11.97(0.09) | 10.85(0.06) | 9.99(0.08) | | 2012 | G014.195$-$00.509 | 50.13 | 0.618(0.088) | 2.854(0.268) | 3.72
60 | G014.4509$-$00.1024 | 11.50(0.14) | 9.42(0.10) | 7.86(0.05) | 6.89(0.05) | | 2045 | G014.450$-$00.101 | 96.53 | 1.030(0.091) | 8.996(0.636) | 6.21
61 | G014.6774$-$00.0410 | 13.66(0.21) | 11.43(0.07) | 9.80(0.07) | 8.77(0.10) | | 2091 | G014.678$-$00.044 | – | 0.115(0.059) | 0.171(0.070) | 0.69
62 | G014.7064$-$00.1568 | 11.41(0.07) | 9.33(0.08) | 8.26(0.06) | 7.25(0.04) | | 2096 | G014.708$-$00.154 | 61.14 | 0.402(0.063) | 2.021(0.220) | 2.42
63 | G014.8516$-$00.9890 | 13.22(0.15) | 10.72(0.12) | 10.17(0.08) | 9.43(0.05) | | 2124 | G014.849$-$00.992 | 56.03 | 0.883(0.101) | 2.881(0.332) | 5.32
64 | G015.0295$+$00.8533 | 13.18(0.10) | 10.44(0.06) | 9.12(0.04) | 8.27(0.03) | | 2154 | G015.029$+$00.852 | 28.59 | 0.243(0.084) | 0.595(0.155) | 1.46
65 | G015.2571$-$00.1560 | 12.15(0.07) | 9.23(0.05) | 7.86(0.03) | 6.96(0.04) | | 2199 | G015.258$-$00.156 | – | 0.112(0.057) | 0.235(0.089) | 0.67
66 | G016.1119$-$00.3036 | 10.50(0.13) | 8.06(0.06) | 6.53(0.03) | 5.67(0.03) | | 2246 | G016.114$-$00.301 | 38.80 | 0.116(0.048) | 0.383(0.103) | 0.70
67 | G016.3184$-$00.5313 | 11.89(0.06) | 9.57(0.05) | 8.36(0.04) | 7.56(0.03) | | 2264 | G016.317$-$00.533 | 30.48 | 0.319(0.073) | 0.845(0.157) | 1.92
68 | G016.5783$-$00.0814 | 11.24(0.05) | 8.95(0.05) | 7.60(0.04) | 6.89(0.03) | | 2291 | G016.580$-$00.081 | 77.59 | 0.396(0.075) | 2.554(0.309) | 2.39
69 | G016.5852$-$00.0507 | 12.72(0.13) | 9.33(0.15) | 7.69(0.05) | 7.43(0.10) | | 2292 | G016.586$-$00.051 | 31.69 | 1.451(0.115) | 3.424(0.292) | 8.74
70 | G016.6427$-$00.1194 | 12.22(0.11) | 9.88(0.12) | 8.80(0.05) | 8.31(0.04) | | 2297 | G016.641$-$00.119 | 25.67 | 0.271(0.068) | 0.809(0.155) | 1.63
71 | G017.8553$+$00.1190 | 10.77(0.07) | 8.74(0.05) | 7.62(0.03) | 6.98(0.03) | | 2354 | G017.856$+$00.120 | – | 0.118(0.069) | 0.182(0.083) | 0.71
72 | G018.2171$-$00.3426 | 13.55(0.09) | 11.06(0.06) | 9.85(0.05) | 8.92(0.03) | | 2381 | G018.218$-$00.342 | 72.95 | 0.622(0.095) | 3.497(0.372) | 3.75
73 | G018.8885$-$00.4746 | 12.96(0.08) | 10.41(0.20) | 9.33(0.07) | 9.40(0.12) | | 2467 | G018.888$-$00.475 | 100.92 | 1.356(0.106) | 9.848(0.692) | 8.17
74 | G019.0087$-$00.0293 | 11.38(0.12) | 7.89(0.07) | 6.44(0.03) | 6.20(0.03) | | 2499 | G019.010$-$00.029 | 44.10 | 0.761(0.070) | 2.401(0.211) | 4.59
75 | G019.8285$-$00.3302 | 8.97(0.21) | 6.71(0.10) | 5.33(0.03) | 4.63(0.02) | | 2630 | G019.827$-$00.329 | 81.51 | 0.446(0.047) | 2.734(0.240) | 2.69
76 | G019.8841$-$00.5351 | 9.25(0.18) | 6.75(0.05) | 5.45(0.03) | 4.87(0.02) | | 2636 | G019.884$-$00.535 | 40.73 | 2.106(0.147) | 5.221(0.394) | 12.69
77 | G019.9229$-$00.2581 | 9.22(0.09) | 6.93(0.07) | 5.25(0.02) | 4.10(0.03) | | 2641 | G019.926$-$00.257 | 55.03 | 0.964(0.073) | 3.427(0.263) | 5.81
78 | G020.0808$-$00.1356 | 11.66(0.10) | 9.26(0.17) | 7.44(0.05) | 5.95(0.06) | | 2659 | G020.082$-$00.135 | 59.03 | 1.977(0.127) | 5.934(0.407) | 11.91
79 | G020.2370$+$00.0653 | 11.66(0.10) | 8.65(0.08) | 7.41(0.03) | 7.72(0.03) | | 2665 | G020.238$+$00.065 | 37.39 | 0.427(0.044) | 1.153(0.112) | 2.57
80 | G020.5455$-$00.4390 | 14.30(0.14) | 11.92(0.13) | 9.97(0.06) | 8.99(0.07) | | 2696 | G020.545$-$00.443 | 55.14 | 0.104(0.035) | 0.483(0.095) | 0.63
81 | G020.7077$-$00.3136 | 14.33(0.14) | 11.96(0.09) | 10.01(0.06) | 8.97(0.06) | | 2711 | G020.708$-$00.311 | 57.17 | 0.338(0.047) | 1.357(0.157) | 2.04
82 | G020.7212$-$00.3580 | 12.93(0.11) | 10.57(0.06) | 9.24(0.04) | 8.34(0.03) | | 2713 | G020.718$-$00.359 | 85.71 | 0.354(0.043) | 2.644(0.230) | 2.13
83 | G022.0387$+$00.2222 | 14.05(0.14) | 11.05(0.17) | 9.46(0.07) | 9.60(0.12) | | 2837 | G022.041$+$00.221 | 87.90 | 0.953(0.074) | 4.676(0.361) | 5.74
84 | G022.3726$+$00.3758 | 12.78(0.10) | 10.29(0.07) | 9.24(0.04) | 8.76(0.04) | | 2858 | G022.371$+$00.379 | 65.89 | 0.409(0.063) | 2.409(0.256) | 2.46
85 | G022.4352$-$00.1694 | 13.12(0.11) | 10.90(0.14) | 9.61(0.05) | 8.78(0.04) | | 2865 | G022.436$-$00.171 | 53.64 | 0.444(0.054) | 1.596(0.174) | 2.68
86 | G022.5594$+$00.1692 | 13.16(0.17) | 11.82(0.22) | 10.38(0.11) | 9.61(0.15) | | 2890 | G022.559$+$00.169 | 51.32 | 0.270(0.054) | 0.928(0.154) | 1.63
87 | G022.7052$+$00.4046 | 13.35(0.07) | 11.21(0.08) | 9.95(0.05) | 9.15(0.04) | | 2904 | G022.705$+$00.404 | 74.62 | 0.279(0.044) | 1.755(0.187) | 1.68
88 | G023.2408$-$00.4814 | 11.17(0.08) | 9.22(0.10) | 7.76(0.04) | 6.87(0.03) | | 3022 | G023.242$-$00.482 | 19.07 | 0.189(0.048) | 0.388(0.087) | 1.14
89 | G023.2969$-$00.0720 | 12.24(0.11) | 10.65(0.10) | 9.54(0.05) | 9.44(0.04) | | 3034 | G023.300$-$00.074 | 75.90 | 0.298(0.058) | 2.073(0.242) | 1.80
90 | G023.4363$-$00.1842 | 14.12(0.14) | 10.23(0.15) | 8.34(0.04) | 8.00(0.08) | | 3071 | G023.437$-$00.184 | 118.24 | 2.146(0.140) | 13.906(0.905) | 12.93
91 | G023.4617$-$00.1567 | 13.53(0.10) | 11.46(0.08) | 10.32(0.08) | 9.78(0.10) | | 3081 | G023.462$-$00.156 | 58.81 | 0.349(0.052) | 2.034(0.191) | 2.10
92 | G023.7057$-$00.1999 | 13.64(0.11) | 12.30(0.14) | 10.53(0.14) | 9.81(0.14) | | 3153 | G023.708$-$00.198 | 58.05 | 0.614(0.058) | 2.196(0.209) | 3.70
93 | G023.9662$-$00.1093 | 14.05(0.23) | 10.76(0.10) | 9.14(0.05) | 9.97(0.08) | | 3202 | G023.968$-$00.110 | 57.34 | 1.029(0.080) | 3.349(0.272) | 6.20
94 | G023.9960$-$00.0997 | 14.36(0.24) | 10.81(0.08) | 9.58(0.09) | 9.39(0.10) | | 3208 | G023.996$-$00.100 | 35.53 | 0.681(0.068) | 1.870(0.187) | 4.10
95 | G024.0019$+$00.2511 | 14.68(0.17) | 12.90(0.18) | 11.09(0.14) | 9.58(0.09) | | 3209 | G024.001$+$00.250 | 34.41 | 0.105(0.042) | 0.305(0.086) | 0.63
96 | G024.2828$-$00.0094 | 9.59(0.19) | 7.58(0.04) | 6.55(0.04) | 5.87(0.02) | | 3274 | G024.282$-$00.008 | 20.57 | 0.127(0.057) | 0.497(0.129) | 0.77
97 | G024.3285$+$00.1440 | 12.79(0.10) | 9.11(0.34) | 7.41(0.06) | 6.95(0.07) | | 3284 | G024.329$+$00.142 | 42.32 | 1.564(0.112) | 4.381(0.330) | 9.43
98 | G024.6332$+$00.1531 | 11.13(0.06) | 8.28(0.07) | 7.33(0.03) | 6.76(0.03) | | 3383 | G024.632$+$00.155 | 50.03 | 0.618(0.060) | 1.980(0.189) | 3.72
99 | G024.6740$-$00.1538 | 11.75(0.25) | 9.40(0.24) | 7.95(0.12) | 7.02(0.17) | | 3394 | G024.676$-$00.151 | 68.17 | 1.658(0.126) | 7.278(0.531) | 9.99
100 | G024.7297$+$00.1530 | 9.43(0.11) | 7.18(0.07) | 5.64(0.02) | 4.79(0.02) | | 3402 | G024.728$+$00.153 | 28.35 | 0.418(0.061) | 1.051(0.142) | 2.52
101 | G024.7898$+$00.0836 | 11.55(0.12) | 7.78(0.06) | 6.19(0.02) | 6.34(0.04) | | 3413 | G024.791$+$00.083 | 73.40 | 4.790(0.301) | 17.786(1.141) | 28.87
102 | G024.9202$+$00.0878 | 13.57(0.10) | 11.29(0.11) | 10.13(0.05) | 9.80(0.07) | | 3437 | G024.920$+$00.085 | 51.67 | 0.975(0.080) | 3.685(0.296) | 5.88
103 | G025.1772$+$00.2111 | 10.92(0.08) | 8.63(0.06) | 7.28(0.03) | 6.60(0.04) | | 3466 | G025.179$+$00.213 | 64.15 | 0.215(0.056) | 1.189(0.185) | 1.30
104 | G025.3838$-$00.1477 | 11.43(0.12) | 9.14(0.07) | 7.85(0.05) | 7.46(0.14) | | 3503 | G025.388$-$00.147 | 33.50 | 1.390(0.115) | 3.173(0.266) | 8.38
105 | G025.3918$-$00.3640 | 12.55(0.10) | 10.44(0.05) | 9.24(0.05) | 8.61(0.08) | | 3504 | G025.394$-$00.363 | 38.93 | 0.361(0.054) | 1.103(0.145) | 2.18
106 | G025.3946$+$00.0341 | 9.52(0.16) | 7.29(0.10) | 5.50(0.05) | 4.53(0.07) | | 3505 | G025.395$+$00.033 | 63.10 | 0.645(0.070) | 2.889(0.266) | 3.89
107 | G025.5158$+$00.1411 | 14.32(0.26) | 11.57(0.08) | 10.30(0.07) | 9.85(0.10) | | 3528 | G025.515$+$00.141 | – | 0.177(0.051) | 0.288(0.075) | 1.07
108 | G025.5175$-$00.2060 | 11.05(0.12) | 8.27(0.14) | 6.51(0.02) | 5.44(0.03) | | 3530 | G025.516$-$00.205 | 32.07 | 0.164(0.044) | 0.454(0.093) | 0.99
109 | G026.5977$-$00.0236 | 10.11(0.19) | 7.45(0.08) | 5.72(0.05) | 5.04(0.12) | | 3699 | G026.597$-$00.025 | 46.31 | 0.509(0.065) | 1.682(0.191) | 3.07
110 | G026.8438$+$00.3729 | 8.77(0.11) | 6.83(0.06) | 5.59(0.03) | 4.87(0.03) | | 3721 | G026.843$+$00.375 | 18.23 | 0.142(0.045) | 0.354(0.083) | 0.86
111 | G027.0162$+$00.2001 | 11.53(0.14) | 8.75(0.09) | 7.49(0.04) | 7.34(0.03) | | 3741 | G027.019$+$00.201 | 78.78 | 0.332(0.055) | 1.698(0.212) | 2.00
112 | G027.2478$+$00.1079 | 11.25(0.06) | 8.91(0.09) | 7.66(0.03) | 7.22(0.04) | | 3771 | G027.249$+$00.109 | 38.85 | 0.208(0.048) | 0.662(0.114) | 1.25
113 | G027.7415$+$00.1710 | 14.53(0.14) | 11.68(0.10) | 10.44(0.06) | 9.82(0.06) | | 3822 | G027.743$+$00.170 | 28.96 | 0.182(0.041) | 0.502(0.090) | 1.10
114 | G027.7827$-$00.2585 | 12.05(0.09) | 10.01(0.09) | 8.99(0.04) | 8.98(0.04) | | 3833 | G027.783$-$00.258 | 40.48 | 0.428(0.046) | 1.244(0.132) | 2.58
115 | G027.9718$-$00.4222 | 14.07(0.18) | 10.89(0.06) | 9.86(0.06) | 9.37(0.07) | | 3863 | G027.972$-$00.422 | 67.50 | 0.367(0.040) | 1.623(0.158) | 2.21
116 | G028.0473$-$00.4562 | 12.92(0.12) | 10.49(0.11) | 9.29(0.10) | 8.88(0.08) | | 3876 | G028.047$-$00.460 | 23.30 | 0.237(0.041) | 0.546(0.085) | 1.43
117 | G028.1467$-$00.0043 | 11.37(0.28) | 8.76(0.07) | 7.06(0.03) | 5.58(0.04) | | 3897 | G028.147$-$00.006 | 91.07 | 0.645(0.052) | 3.656(0.282) | 3.89
118 | G028.2262$+$00.3589 | 13.46(0.23) | 11.43(0.15) | 10.18(0.14) | 9.08(0.10) | | 3917 | G028.222$+$00.358 | 58.12 | 0.135(0.041) | 0.664(0.126) | 0.81
119 | G028.3419$+$00.1421 | 12.32(0.07) | 9.66(0.06) | 7.66(0.03) | 6.51(0.04) | | 3938 | G028.341$+$00.140 | – | 0.118(0.040) | 0.204(0.061) | 0.71
120 | G028.3606$+$00.0520 | 12.92(0.06) | 10.61(0.06) | 9.63(0.06) | 9.06(0.05) | | 3946 | G028.361$+$00.054 | 48.70 | 0.467(0.051) | 1.961(0.176) | 2.81
121 | G028.4084$-$00.4387 | 13.46(0.10) | 10.93(0.09) | 9.88(0.05) | 9.02(0.05) | | 3959 | G028.407$-$00.436 | 43.32 | 0.157(0.034) | 0.644(0.098) | 0.95
122 | G028.5047$-$00.1399 | 10.52(0.07) | 8.24(0.06) | 7.20(0.03) | 6.34(0.03) | | 3985 | G028.504$-$00.142 | 74.73 | 0.214(0.033) | 1.164(0.142) | 1.29
123 | G028.5322$+$00.1288 | 9.53(0.05) | 7.45(0.04) | 5.88(0.02) | 4.85(0.02) | | 3994 | G028.533$+$00.128 | 30.30 | 0.098(0.035) | 0.288(0.074) | 0.59
124 | G028.5966$-$00.0208 | 9.89(0.06) | 7.88(0.04) | 6.62(0.03) | 5.99(0.04) | | 4003 | G028.597$-$00.022 | 79.18 | 0.263(0.039) | 1.693(0.165) | 1.58
125 | G028.7007$+$00.4033 | 12.28(0.06) | 9.87(0.06) | 8.56(0.04) | 7.96(0.08) | | 4020 | G028.701$+$00.406 | 26.25 | 0.259(0.042) | 0.593(0.093) | 1.56
126 | G028.9649$-$00.5952 | 10.51(0.04) | 8.26(0.04) | 6.61(0.03) | 5.57(0.04) | | 4082 | G028.963$-$00.597 | 24.03 | 0.176(0.065) | 0.383(0.113) | 1.06
127 | G029.1191$+$00.0288 | 12.20(0.12) | 9.52(0.07) | 8.15(0.04) | 7.18(0.03) | | 4106 | G029.117$+$00.025 | 65.07 | 0.289(0.041) | 1.286(0.149) | 1.74
128 | G029.2775$-$00.1283 | 9.12(0.10) | 7.02(0.08) | 5.89(0.03) | 5.32(0.03) | | 4133 | G029.277$-$00.131 | 71.35 | 0.218(0.038) | 1.452(0.163) | 1.31
129 | G029.3199$-$00.1615 | 11.16(0.12) | 8.54(0.07) | 7.11(0.03) | 6.37(0.03) | | 4139 | G029.318$-$00.165 | 21.24 | 0.156(0.040) | 0.338(0.076) | 0.94
130 | G029.7801$-$00.2594 | 12.46(0.08) | 10.43(0.08) | 9.38(0.08) | 8.66(0.09) | | 4219 | G029.781$-$00.262 | 45.50 | 0.224(0.036) | 0.910(0.115) | 1.35
131 | G030.0100$+$00.0356 | 12.04(0.06) | 9.99(0.06) | 8.15(0.03) | 7.03(0.03) | | 4284 | G030.010$+$00.034 | – | 0.127(0.044) | 0.304(0.084) | 0.77
132 | G030.2116$-$00.1885 | 14.20(0.22) | 11.35(0.10) | 9.50(0.10) | 8.69(0.24) | | 4321 | G030.215$-$00.188 | 83.95 | 0.917(0.070) | 5.916(0.407) | 5.53
133 | G030.3476$+$00.3917 | 10.75(0.10) | 8.27(0.09) | 7.05(0.04) | 6.69(0.04) | | 4366 | G030.347$+$00.390 | 77.98 | 0.387(0.039) | 1.973(0.178) | 2.33
134 | G030.4204$-$00.2283 | 12.66(0.15) | 10.36(0.30) | 9.01(0.06) | 8.29(0.08) | | 4398 | G030.419$-$00.232 | 87.20 | 1.500(0.102) | 7.704(0.522) | 9.04
135 | G030.6039$+$00.1760 | 12.43(0.06) | 9.07(0.18) | 7.14(0.03) | 6.92(0.03) | | 4472 | G030.603$+$00.175 | 128.72 | 1.462(0.098) | 12.965(0.830) | 8.81
136 | G030.6622$-$00.1393 | 13.47(0.13) | 11.22(0.09) | 9.74(0.11) | 9.12(0.17) | | 4492 | G030.666$-$00.139 | 53.89 | 0.298(0.034) | 1.316(0.123) | 1.80
137 | G030.6670$-$00.3318 | 9.72(0.18) | 7.42(0.08) | 6.12(0.02) | 4.09(0.03) | | 4497 | G030.667$-$00.331 | – | 0.116(0.027) | 0.192(0.044) | 0.70
138 | G030.8107$+$00.1895 | 14.58(0.13) | 12.43(0.08) | 10.90(0.08) | 9.94(0.08) | | 4556 | G030.812$+$00.191 | 47.60 | 0.225(0.035) | 0.841(0.093) | 1.36
139 | G030.8685$-$00.1188 | 14.42(0.33) | 12.13(0.24) | 10.57(0.15) | 9.52(0.26) | | 4581 | G030.868$-$00.121 | 73.45 | 0.420(0.040) | 3.184(0.226) | 2.53
140 | G030.9447$+$00.1574 | 11.54(0.07) | 9.34(0.05) | 8.19(0.03) | 7.37(0.03) | | 4621 | G030.948$+$00.159 | 28.30 | 0.125(0.032) | 0.351(0.066) | 0.75
141 | G030.9588$+$00.0863 | 9.05(0.19) | 6.89(0.07) | 5.26(0.04) | 4.58(0.13) | | 4627 | G030.960$+$00.085 | 54.60 | 0.600(0.046) | 2.191(0.166) | 3.62
142 | G030.9949$+$00.2339 | 14.15(0.13) | 11.55(0.10) | 10.43(0.06) | 9.89(0.07) | | 4642 | G030.998$+$00.235 | 63.53 | 0.378(0.037) | 2.153(0.163) | 2.28
143 | G031.0147$+$00.7783 | 14.52(0.12) | 11.59(0.10) | 8.62(0.05) | 7.06(0.10) | | 4649 | G031.013$+$00.781 | – | 0.140(0.068) | 0.184(0.076) | 0.84
144 | G031.0738$+$00.4596 | 13.79(0.08) | 11.44(0.08) | 10.22(0.05) | 9.93(0.05) | | 4673 | G031.077$+$00.459 | 66.76 | 0.417(0.040) | 2.136(0.173) | 2.51
145 | G031.1016$+$00.2644 | 13.93(0.12) | 11.16(0.08) | 9.72(0.07) | 9.21(0.12) | | 4678 | G031.103$+$00.265 | 64.32 | 0.184(0.029) | 1.178(0.121) | 1.11
146 | G031.1825$-$00.1479 | 13.04(0.09) | 10.75(0.07) | 9.43(0.05) | 8.38(0.05) | | 4701 | G031.182$-$00.145 | 73.20 | 0.289(0.034) | 1.824(0.154) | 1.74
147 | G031.3911$+$00.2037 | 12.47(0.10) | 10.41(0.13) | 8.68(0.05) | 7.60(0.07) | | 4759 | G031.394$+$00.207 | 87.18 | 0.207(0.032) | 1.407(0.146) | 1.25
148 | G031.5813$+$00.0788 | 13.31(0.14) | 11.12(0.09) | 9.15(0.05) | 7.40(0.04) | | 4812 | G031.582$+$00.077 | 73.18 | 1.053(0.072) | 3.928(0.278) | 6.35
149 | G031.9003$+$00.3410 | 13.03(0.09) | 10.85(0.08) | 10.39(0.06) | 9.90(0.07) | | 4892 | G031.900$+$00.343 | 53.48 | 0.164(0.028) | 0.679(0.088) | 0.99
150 | G032.6058$-$00.2557 | 14.51(0.22) | 11.69(0.11) | 9.66(0.06) | 8.77(0.07) | | 5008 | G032.605$-$00.253 | 88.71 | 0.207(0.027) | 1.963(0.163) | 1.25
151 | G032.7038$-$00.0560 | 12.25(0.23) | 9.48(0.14) | 8.25(0.06) | 7.59(0.12) | | 5032 | G032.704$-$00.059 | 46.07 | 0.333(0.034) | 1.151(0.107) | 2.01
152 | G032.8264$-$00.0824 | 12.66(0.20) | 10.22(0.06) | 9.26(0.06) | 8.87(0.06) | | 5061 | G032.829$-$00.081 | 71.84 | 0.227(0.029) | 1.147(0.119) | 1.37
153 | G032.9917$+$00.0339 | 10.59(0.06) | 8.55(0.13) | 7.40(0.04) | 6.93(0.04) | | 5100 | G032.991$+$00.037 | 75.21 | 0.759(0.055) | 3.401(0.249) | 4.57
154 | G033.3928$+$00.0097 | 10.69(0.08) | 7.62(0.04) | 6.55(0.03) | 5.95(0.03) | | 5167 | G033.390$+$00.008 | 98.64 | 0.813(0.063) | 6.914(0.471) | 4.90
155 | G033.4007$+$00.3713 | 13.50(0.13) | 11.38(0.07) | 9.86(0.06) | 8.69(0.06) | | 5170 | G033.404$+$00.370 | – | 0.068(0.026) | 0.194(0.048) | 0.41
156 | G033.6754$+$00.2031 | 14.49(0.14) | 12.34(0.11) | 10.83(0.07) | 9.92(0.04) | | 5230 | G033.672$+$00.201 | 65.65 | 0.144(0.033) | 0.799(0.116) | 0.87
157 | G033.7042$+$00.2821 | 14.10(0.12) | 11.12(0.09) | 10.40(0.05) | 9.74(0.07) | | 5240 | G033.704$+$00.285 | 40.95 | 0.269(0.036) | 0.963(0.105) | 1.62
158 | G033.7395$-$00.0198 | 13.87(0.19) | 11.48(0.12) | 9.71(0.04) | 8.06(0.02) | | 5252 | G033.740$-$00.017 | 90.78 | 0.708(0.062) | 5.136(0.375) | 4.27
159 | G033.8181$-$00.2121 | 11.26(0.11) | 9.24(0.06) | 8.65(0.03) | 8.24(0.03) | | 5265 | G033.817$-$00.215 | 29.71 | 0.152(0.037) | 0.430(0.077) | 0.92
160 | G033.8519$+$00.0180 | 13.54(0.10) | 10.70(0.12) | 9.38(0.04) | 9.25(0.04) | | 5270 | G033.850$+$00.017 | 64.54 | 0.260(0.032) | 1.111(0.121) | 1.57
161 | G034.4119$+$00.2343 | 14.15(0.20) | 11.36(0.17) | 10.55(0.12) | 9.99(0.12) | | 5373 | G034.410$+$00.232 | 96.10 | 3.337(0.210) | 20.777(1.303) | 20.11
162 | G034.9333$+$00.0194 | 13.99(0.19) | 11.76(0.08) | 9.25(0.04) | 7.79(0.04) | | 5501 | G034.932$+$00.022 | 71.21 | 0.254(0.037) | 1.292(0.147) | 1.53
163 | G034.9941$-$00.0446 | 12.11(0.10) | 9.62(0.10) | 8.46(0.04) | 7.77(0.04) | | 5516 | G034.991$-$00.046 | 46.34 | 0.163(0.037) | 0.689(0.104) | 0.98
164 | G035.2252$-$00.3596 | 11.47(0.06) | 9.33(0.05) | 8.04(0.04) | 7.10(0.03) | | 5572 | G035.228$-$00.358 | 22.89 | 0.364(0.043) | 0.784(0.095) | 2.19
165 | G035.2474$-$00.2368 | 11.77(0.19) | 9.37(0.10) | 8.74(0.04) | 7.78(0.03) | | 5577 | G035.247$-$00.238 | – | 0.087(0.038) | 0.123(0.047) | 0.52
166 | G035.3145$-$00.2254 | 12.16(0.12) | 10.10(0.09) | 9.15(0.04) | 8.12(0.03) | | 5594 | G035.316$-$00.222 | 58.23 | 0.149(0.036) | 0.673(0.111) | 0.90
167 | G035.7095$+$00.1631 | 12.84(0.13) | 10.49(0.07) | 9.39(0.04) | 8.82(0.04) | | 5691 | G035.707$+$00.164 | – | 0.065(0.032) | 0.084(0.036) | 0.39
168 | G036.0011$-$00.4644 | 13.50(0.13) | 11.43(0.08) | 10.00(0.05) | 8.99(0.03) | | 5720 | G035.997$-$00.466 | 79.56 | 0.163(0.034) | 1.266(0.146) | 0.98
169 | G036.0127$-$00.1974 | 13.03(0.19) | 9.70(0.14) | 9.66(0.07) | 9.57(0.04) | | 5722 | G036.012$-$00.198 | 36.14 | 0.242(0.037) | 0.674(0.094) | 1.46
170 | G036.7053$+$00.0962 | 10.75(0.08) | 8.70(0.07) | 8.09(0.04) | 7.70(0.03) | | 5782 | G036.704$+$00.094 | 70.92 | 0.156(0.037) | 1.088(0.147) | 0.94
171 | G037.3418$-$00.0591 | 9.93 (0.11) | 7.63(0.07) | 5.75(0.03) | 4.68(0.05) | | 5836 | G037.341$-$00.062 | 79.89 | 0.445(0.045) | 2.281(0.204) | 2.68
172 | G037.7632$-$00.2150 | 10.15(0.06) | 7.99(0.04) | 6.85(0.04) | 6.34(0.06) | | 5869 | G037.765$-$00.216 | 50.07 | 1.262(0.093) | 4.569(0.331) | 7.61
173 | G038.1616$-$00.0747 | 13.82(0.08) | 11.79(0.09) | 10.44(0.07) | 9.98(0.11) | | 5896 | G038.161$-$00.078 | 33.10 | 0.151(0.049) | 0.426(0.102) | 0.91
174 | G038.5548$+$00.1624 | 10.55(0.09) | 8.53(0.09) | 7.28(0.04) | 6.83(0.09) | | 5919 | G038.552$+$00.160 | – | 0.261(0.047) | 0.582(0.095) | 1.57
175 | G038.5977$-$00.2125 | 12.19(0.07) | 9.88(0.07) | 8.79(0.04) | 8.05(0.03) | | 5922 | G038.599$-$00.214 | 53.12 | 0.196(0.041) | 0.821(0.124) | 1.18
176 | G038.8471$-$00.4295 | 13.18(0.06) | 10.88(0.06) | 9.65(0.05) | 8.66(0.05) | | 5941 | G038.847$-$00.428 | 85.04 | 0.246(0.037) | 1.435(0.159) | 1.48
177 | G039.5875$-$00.2064 | 13.33(0.08) | 11.27(0.06) | 10.26(0.05) | 9.76(0.10) | | 5993 | G039.591$-$00.205 | 60.24 | 0.190(0.043) | 1.069(0.149) | 1.15
178 | G040.1579$+$00.1686 | 8.71(0.15) | 7.41(0.14) | 6.04(0.05) | 4.90(0.10) | | 6017 | G040.157$+$00.167 | – | 0.111(0.039) | 0.220(0.063) | 0.67
179 | G040.2782$-$00.2691 | 10.58(0.13) | 8.43(0.09) | 7.35(0.03) | 6.69(0.03) | | 6023 | G040.279$-$00.269 | 31.00 | 0.187(0.044) | 0.458(0.092) | 1.13
180 | G041.8828$+$00.4689 | 14.57(0.23) | 12.46(0.11) | 10.39(0.06) | 9.08(0.06) | | 6086 | G041.883$+$00.469 | – | 0.116(0.047) | 0.265(0.081) | 0.70
181 | G043.0386$-$00.4535 | 13.10(0.09) | 10.89(0.09) | 9.95(0.07) | 9.13(0.06) | | 6110 | G043.039$-$00.455 | 27.60 | 0.794(0.074) | 1.706(0.168) | 4.79
182 | G043.0757$-$00.0781 | 12.21(0.06) | 10.59(0.04) | 9.56(0.04) | 8.90(0.04) | | 6111 | G043.073$-$00.079 | 31.59 | 0.237(0.061) | 0.614(0.128) | 1.43
183 | G043.9293$-$00.3352 | 13.46(0.12) | 10.92(0.10) | 9.78(0.06) | 8.85(0.05) | | 6131 | G043.929$-$00.335 | 14.35 | 0.114(0.043) | 0.231(0.071) | 0.69
184 | G044.0967$+$00.1601 | 11.97(0.07) | 9.70(0.06) | 8.60(0.04) | 7.82(0.03) | | 6137 | G044.099$+$00.163 | 20.12 | 0.141(0.048) | 0.377(0.093) | 0.85
185 | G044.5215$+$00.3902 | 10.69(0.07) | 9.10(0.06) | 7.60(0.03) | 6.39(0.04) | | 6153 | G044.521$+$00.387 | 23.84 | 0.221(0.049) | 0.551(0.101) | 1.33
186 | G045.1669$+$00.0911 | 11.89(0.15) | 9.29(0.07) | 7.81(0.04) | 6.64(0.03) | | 6166 | G045.167$+$00.095 | 39.35 | 0.189(0.053) | 0.654(0.127) | 1.14
187 | G045.5683$-$00.1201 | 11.70(0.16) | 9.34(0.10) | 8.06(0.04) | 7.78(0.10) | | 6188 | G045.569$-$00.119 | 27.91 | 0.121(0.036) | 0.315(0.073) | 0.73
188 | G045.8818$-$00.5095 | 11.69(0.06) | 9.73(0.05) | 8.69(0.04) | 7.86(0.03) | | 6208 | G045.884$-$00.509 | – | 0.196(0.050) | 0.480(0.093) | 1.18
189 | G046.3163$-$00.2109 | 13.00(0.08) | 11.04(0.07) | 9.99(0.07) | 9.54(0.07) | | 6225 | G046.314$-$00.213 | 30.28 | 0.129(0.039) | 0.407(0.085) | 0.78
190 | G048.6113$+$00.2211 | 12.01(0.07) | 10.06(0.05) | 8.91(0.04) | 8.16(0.04) | | 6258 | G048.609$+$00.220 | – | 0.126(0.057) | 0.240(0.081) | 0.76
191 | G048.8398$-$00.4837 | 13.88(0.07) | 12.21(0.10) | 11.04(0.10) | 9.99(0.07) | | 6280 | G048.841$-$00.482 | 59.75 | 0.203(0.055) | 0.833(0.158) | 1.22
192 | G049.0721$-$00.3270 | 10.22(0.14) | 8.31(0.09) | 6.91(0.03) | 6.08(0.03) | | 6298 | G049.069$-$00.328 | 63.88 | 0.660(0.079) | 3.313(0.317) | 3.98
193 | G049.1073$-$00.2681 | 13.31(0.22) | 11.82(0.08) | 10.6(0.14) | 9.99(0.20) | | 6304 | G049.106$-$00.272 | 28.96 | 0.189(0.054) | 0.625(0.108) | 1.14
194 | G049.2634$-$00.3401 | 12.40(0.06) | 10.29(0.09) | 9.28(0.06) | 8.51(0.06) | | 6323 | G049.267$-$00.338 | 51.77 | 1.662(0.120) | 6.088(0.437) | 10.02
195 | G049.3811$-$00.1840 | 12.26(0.09) | 10.59(0.12) | 9.28(0.05) | 8.33(0.04) | | 6338 | G049.378$-$00.184 | 28.03 | 0.212(0.051) | 0.478(0.101) | 1.28
196 | G049.4065$-$00.3715 | 12.14(0.11) | 9.69(0.07) | 7.97(0.05) | 6.92(0.07) | | 6346 | G049.405$-$00.370 | 54.92 | 0.729(0.117) | 3.622(0.401) | 4.39
197 | G049.6006$-$00.2468 | 13.59(0.09) | 11.94(0.10) | 9.35(0.06) | 7.74(0.10) | | 6376 | G049.599$-$00.250 | 24.03 | 0.362(0.062) | 0.862(0.134) | 2.18
198 | G049.8149$+$00.4540 | 14.05(0.13) | 11.80(0.07) | 10.74(0.08) | 9.99(0.07) | | 6380 | G049.817$+$00.456 | – | 0.109(0.061) | 0.226(0.091) | 0.66
199 | G050.0644$+$00.0633 | 13.24(0.06) | 11.38(0.07) | 9.98(0.06) | 9.21(0.06) | | 6387 | G050.060$+$00.062 | 64.17 | 0.359(0.071) | 2.279(0.265) | 2.16
200 | G053.1398$+$00.0707 | 8.88(0.20) | 7.49(0.16) | 6.19(0.05) | 4.97(0.04) | | 6414 | G053.142$+$00.068 | 58.07 | 1.477(0.118) | 5.605(0.445) | 8.90
201 | G053.1632$-$00.2455 | 13.40(0.11) | 10.29(0.07) | 8.65(0.06) | 8.17(0.17) | | 6416 | G053.164$-$00.246 | 38.57 | 0.468(0.066) | 1.272(0.171) | 2.82
202 | G053.2480$-$00.0869 | 12.13(0.06) | 9.87(0.05) | 8.74(0.03) | 7.86(0.04) | | 6424 | G053.248$-$00.086 | – | 0.127(0.057) | 0.288(0.096) | 0.77
203 | G053.4552$+$00.0044 | 13.58(0.12) | 12.27(0.15) | 10.92(0.16) | 9.38(0.15) | | 6429 | G053.457$+$00.004 | 36.56 | 0.155(0.069) | 0.516(0.147) | 0.93
204 | G053.6180$+$00.0352 | 10.01(0.21) | 7.38(0.16) | 5.69(0.03) | 4.94(0.03) | | 6433 | G053.616$+$00.036 | 41.16 | 0.618(0.086) | 1.894(0.239) | 3.72
205 | G053.6316$+$00.0134 | 12.78(0.06) | 11.02(0.08) | 9.63(0.06) | 8.52(0.04) | | 6437 | G053.634$+$00.014 | 39.82 | 0.245(0.075) | 0.886(0.182) | 1.48
206 | G053.9436$-$00.0774 | 10.88(0.04) | 9.26(0.04) | 8.03(0.03) | 7.16(0.04) | | 6445 | G053.942$-$00.080 | 19.49 | 0.091(0.060) | 0.264(0.102) | 0.55
207 | G054.1098$-$00.0813 | 9.26(0.07) | 7.90(0.07) | 6.58(0.03) | 5.58(0.03) | | 6451 | G054.112$-$00.083 | 113.32 | 0.712(0.097) | 6.994(0.586) | 4.29
208 | G054.3890$-$00.0335 | 10.15(0.09) | 8.61(0.05) | 7.35(0.02) | 6.42(0.04) | | 6455 | G054.390$-$00.035 | 31.23 | 0.225(0.070) | 0.664(0.149) | 1.36
209 | G056.9631$-$00.2346 | 9.49(0.17) | 7.69(0.09) | 6.63(0.04) | 6.08(0.10) | | 6470 | G056.962$-$00.234 | 25.81 | 0.243(0.090) | 0.630(0.170) | 1.46
210 | G058.4719$+$00.4340 | 11.16(0.05) | 9.54(0.05) | 8.40(0.04) | 7.67(0.03) | | 6474 | G058.471$+$00.433 | 22.51 | 0.400(0.000) | 0.914(0.212) | 2.41
211 | G059.4978$-$00.2365 | 10.97(0.07) | 8.48(0.06) | 6.84(0.03) | 5.56(0.03) | | 6476 | G059.499$-$00.235 | 42.20 | 0.399(0.076) | 1.374(0.202) | 2.40
212 | G059.6366$-$00.1864 | 11.48(0.11) | 9.57(0.11) | 8.79(0.06) | 8.33(0.08) | | 6479 | G059.639$-$00.189 | 39.85 | 1.531(0.130) | 4.660(0.391) | 9.23
213 | G060.0162$+$00.1115 | 12.58(0.13) | 10.42(0.14) | 9.49(0.06) | 8.23(0.08) | | 6492 | G060.017$+$00.115 | 27.06 | 0.407(0.088) | 1.010(0.172) | 2.45
214 | G063.0768$+$00.1853 | 11.63(0.14) | 9.41(0.07) | 8.13(0.04) | 6.91(0.03) | | 6501 | G063.075$+$00.184 | – | 0.185(0.080) | 0.200(0.102) | 1.11
Note. — Column (1): source number which is organized by increasing galactic
longitude. Column (2): GLIMPSE point source name. Columns (3) – (6): the
magnitude of the GLIMPSE point source in the 3.6, 4.5, 5.8 and 8.0 $\mu$m
bands, respectively. Columns (7) and (8): the ID number and name of BGPS
source, respectively. Column (9): the radius of BGPS, sources which are
unresolved with the BGPS beam, are indicated with “–” in this column. Columns
(10) and (11): the aperture flux density within 40$\arcsec$ and the integrated
flux density of the BGPS sources. Note that a flux calibration correction
factor of 1.5 should be applied to the both the aperture and integrated flux
densities listed here to calculate BGPS gas mass and column/volume density
(see Section 4.2). In addition, an aperture correction of 1.46 is needed to
apply to aperture flux density within 40$\arcsec$ after applied a flux
calibration correction factor of 1.5 to calculate the beam-averaged column
density. Columns (12): the beam-averaged H2 column density (see section 4.2).
Table 2: Observed source positions and observing rms noise. Number | BGPS ID | R.A. (J2000) | Decl. (J2000) | $\sigma_{rms}$ (Jy) | Number | BGPS ID | R.A. (J2000) | Decl. (J2000) | $\sigma_{rms}$ (Jy)
---|---|---|---|---|---|---|---|---|---
1 | 1051 | 17 56 25.86 | -24 48 17.0 | 0.9 | 108 | 3530 | 18 38 35.11 | -06 41 27.1 | 1.2
2 | 1053 | 17 55 54.71 | -24 42 46.7 | 0.9 | 109 | 3699 | 18 39 56.20 | -05 38 48.3 | 0.9
3 | 1066 | 17 56 49.37 | -24 38 37.0 | 1.1 | 110 | 3721 | 18 38 57.67 | -05 14 41.2 | 1.0
4 | 1071 | 17 57 32.79 | -24 39 03.9 | 0.9 | 111 | 3741 | 18 39 54.38 | -05 10 05.3 | 0.4
5 | 1084 | 17 58 09.70 | -24 23 49.2 | 0.5 | 112 | 3771 | 18 40 39.47 | -05 00 21.0 | 0.9
6 | 1087 | 17 57 48.30 | -24 19 03.8 | 0.9 | 113 | 3822 | 18 41 20.84 | -04 32 20.6 | 1.0
7 | 1090 | 17 56 41.11 | -24 09 21.4 | 1.0 | 114 | 3833 | 18 42 56.88 | -04 41 57.6 | 1.0
8 | 1131 | 18 00 21.60 | -24 05 56.6 | 0.9 | 115 | 3863 | 18 43 52.92 | -04 36 19.3 | 1.1
9 | 1164 | 17 59 30.07 | -23 44 15.4 | 0.9 | 116 | 3876 | 18 44 09.20 | -04 33 24.9 | 1.0
10 | 1203 | 18 00 17.92 | -23 26 20.2 | 1.0 | 117 | 3897 | 18 42 43.01 | -04 15 37.5 | 0.8
11 | 1251 | 18 02 12.80 | -23 05 44.4 | 1.0 | 118 | 3917 | 18 41 33.50 | -04 01 34.5 | 0.9
12 | 1259 | 18 02 24.84 | -23 01 06.2 | 0.5 | 119 | 3938 | 18 42 33.13 | -04 01 16.0 | 1.0
13 | 1289 | 18 04 17.39 | -22 53 33.5 | 0.7 | 120 | 3946 | 18 42 53.73 | -04 02 33.6 | 0.9
14 | 1341 | 18 03 16.99 | -21 45 39.3 | 0.8 | 121 | 3959 | 18 44 43.65 | -04 13 32.8 | 0.8
15 | 1346 | 18 02 13.75 | -21 32 38.4 | 0.8 | 122 | 3985 | 18 43 51.51 | -04 00 15.3 | 0.9
16 | 1352 | 18 04 37.05 | -21 47 52.5 | 1.0 | 123 | 3994 | 18 42 56.83 | -03 51 21.2 | 1.0
17 | 1360 | 18 05 29.32 | -21 48 05.0 | 0.8 | 124 | 4003 | 18 43 35.95 | -03 52 03.3 | 1.0
18 | 1361 | 18 05 07.71 | -21 44 01.7 | 0.9 | 125 | 4020 | 18 42 15.86 | -03 34 45.5 | 0.4
19 | 1362 | 18 05 36.97 | -21 46 52.7 | 0.8 | 126 | 4082 | 18 46 18.99 | -03 48 15.7 | 0.8
20 | 1363 | 18 05 22.6 | -21 44 43.9 | 0.9 | 127 | 4106 | 18 44 22.86 | -03 22 59.5 | 1.1
21 | 1380 | 18 06 37.21 | -21 37 06.6 | 0.8 | 128 | 4133 | 18 45 13.80 | -03 18 43.9 | 0.8
22 | 1395 | 18 05 25.71 | -21 19 24.6 | 0.8 | 129 | 4139 | 18 45 25.69 | -03 17 25.4 | 0.8
23 | 1405 | 18 04 53.31 | -21 06 38.1 | 0.8 | 130 | 4219 | 18 46 37.29 | -02 55 23.8 | 0.8
24 | 1407 | 18 06 53.76 | -21 17 24.6 | 0.6 | 131 | 4284 | 18 45 59.25 | -02 35 00.7 | 0.9
25 | 1409 | 18 05 56.88 | -21 03 15.6 | 1.0 | 132 | 4321 | 18 47 09.08 | -02 30 12.3 | 0.9
26 | 1412 | 18 06 52.57 | -21 04 36.8 | 0.5 | 133 | 4366 | 18 45 20.04 | -02 07 19.3 | 0.8
27 | 1425 | 18 07 34.20 | -20 26 12.9 | 0.8 | 134 | 4398 | 18 47 40.85 | -02 20 31.2 | 0.9
28 | 1466 | 18 09 24.77 | -20 15 37.6 | 0.5 | 135 | 4472 | 18 46 33.95 | -01 59 31.8 | 0.8
29 | 1467 | 18 09 00.30 | -20 11 37.5 | 0.7 | 136 | 4492 | 18 47 47.84 | -02 04 48.9 | 0.9
30 | 1472 | 18 08 01.76 | -20 01 31.4 | 1.1 | 137 | 4497 | 18 48 29.10 | -02 09 57.8 | 0.5
31 | 1479 | 18 09 23.13 | -20 08 08.7 | 0.8 | 138 | 4556 | 18 46 53.32 | -01 47 59.2 | 0.7
32 | 1497 | 18 08 38.51 | -19 51 54.5 | 1.1 | 139 | 4581 | 18 48 06.12 | -01 53 32.3 | 1.0
33 | 1508 | 18 10 29.00 | -19 55 44.0 | 0.8 | 140 | 4621 | 18 47 15.06 | -01 41 36.1 | 0.9
34 | 1516 | 18 09 53.11 | -19 47 55.7 | 1.0 | 141 | 4627 | 18 47 32.19 | -01 42 59.2 | 1.0
35 | 1543 | 18 09 50.65 | -19 37 03.3 | 1.0 | 142 | 4642 | 18 47 04.30 | -01 36 51.1 | 0.9
36 | 1559 | 18 09 39.95 | -19 26 28.8 | 1.1 | 143 | 4649 | 18 45 09.42 | -01 21 02.7 | 0.8
37 | 1580 | 18 10 18.59 | -19 24 22.7 | 0.5 | 144 | 4673 | 18 46 25.22 | -01 26 26.8 | 0.5
38 | 1587 | 18 09 45.84 | -19 17 30.8 | 1.1 | 145 | 4678 | 18 47 09.51 | -01 30 22.3 | 0.5
39 | 1591 | 18 09 52.01 | -19 17 21.5 | 0.5 | 146 | 4701 | 18 48 45.64 | -01 37 25.9 | 1.1
40 | 1592 | 18 10 32.84 | -19 22 15.5 | 0.8 | 147 | 4759 | 18 47 53.66 | -01 16 28.6 | 1.0
41 | 1657 | 18 12 18.83 | -18 39 53.4 | 0.7 | 148 | 4812 | 18 48 42.02 | -01 09 59.9 | 0.9
42 | 1668 | 18 12 40.36 | -18 37 04.4 | 0.8 | 149 | 4892 | 18 48 20.04 | 00 45 44.3 | 0.8
43 | 1682 | 18 12 23.81 | -18 22 45.0 | 0.7 | 150 | 5008 | 18 51 44.64 | 00 24 21.7 | 1.0
44 | 1699 | 18 10 43.55 | -17 58 18.5 | 0.9 | 151 | 5032 | 18 51 14.16 | 00 13 42.8 | 0.8
45 | 1720 | 18 13 41.39 | -18 12 34.7 | 0.8 | 152 | 5061 | 18 51 32.43 | 00 07 41.7 | 0.5
46 | 1734 | 18 14 28.49 | -18 12 06.8 | 0.6 | 153 | 5100 | 18 51 24.97 | 00 04 11.1 | 0.5
47 | 1742 | 18 13 11.47 | -17 59 48.6 | 0.8 | 154 | 5167 | 18 52 14.76 | 00 24 46.3 | 0.5
48 | 1756 | 18 13 03.68 | -17 53 11.4 | 0.4 | 155 | 5170 | 18 50 58.77 | 00 35 19.5 | 0.9
49 | 1778 | 18 14 39.87 | -17 59 06.4 | 0.6 | 156 | 5230 | 18 52 04.30 | 00 45 02.4 | 0.9
50 | 1803 | 18 11 51.33 | -17 31 26.4 | 0.8 | 157 | 5240 | 18 51 49.85 | 00 49 02.7 | 0.8
51 | 1809 | 18 13 48.16 | -17 45 34.4 | 0.8 | 158 | 5252 | 18 52 58.32 | 00 42 42.8 | 0.9
52 | 1841 | 18 15 07.81 | -17 46 52.7 | 0.9 | 159 | 5265 | 18 53 49.15 | 00 41 28.0 | 0.7
53 | 1853 | 18 14 36.96 | -17 38 47.2 | 0.9 | 160 | 5270 | 18 53 03.08 | 00 49 31.1 | 0.5
54 | 1857 | 18 14 28.34 | -17 36 01.8 | 0.8 | 161 | 5373 | 18 53 18.61 | 01 25 16.6 | 0.7
55 | 1865 | 18 14 01.29 | -17 28 33.3 | 0.8 | 162 | 5501 | 18 55 00.61 | 01 47 24.6 | 0.8
56 | 1877 | 18 13 47.59 | -17 22 15.8 | 0.8 | 163 | 5516 | 18 55 21.71 | 01 48 45.2 | 0.7
57 | 1998 | 18 18 09.86 | -16 57 23.7 | 0.5 | 164 | 5572 | 18 56 54.23 | 01 52 48.9 | 0.5
58 | 2002 | 18 18 03.65 | -16 54 49.7 | 0.7 | 165 | 5577 | 18 56 30.78 | 01 57 10.0 | 1.0
59 | 2012 | 18 18 08.35 | -16 51 12.7 | 0.7 | 166 | 5594 | 18 56 34.81 | 02 01 14.1 | 1.0
60 | 2045 | 18 17 08.68 | -16 26 05.7 | 0.9 | 167 | 5691 | 18 55 55.25 | 02 32 43.8 | 0.6
61 | 2091 | 18 17 23.36 | -16 12 27.4 | 0.7 | 168 | 5720 | 18 58 41.71 | 02 30 57.5 | 0.9
62 | 2096 | 18 17 51.12 | -16 13 59.8 | 0.7 | 169 | 5722 | 18 57 45.97 | 02 39 02.8 | 0.8
63 | 2124 | 18 21 12.49 | -16 30 17.5 | 0.6 | 170 | 5782 | 18 57 59.47 | 03 23 59.0 | 0.9
64 | 2154 | 18 14 48.38 | -15 28 20.0 | 0.6 | 171 | 5836 | 18 59 42.95 | 03 53 45.3 | 0.5
65 | 2199 | 18 18 56.46 | -15 44 58.5 | 1.0 | 172 | 5869 | 19 01 02.49 | 04 12 06.8 | 0.5
66 | 2246 | 18 21 08.91 | -15 03 48.2 | 0.6 | 173 | 5896 | 19 01 16.50 | 04 37 01.8 | 0.9
67 | 2264 | 18 22 23.37 | -14 59 38.3 | 0.8 | 174 | 5919 | 19 01 08.68 | 05 04 28.5 | 0.7
68 | 2291 | 18 21 15.08 | -14 32 55.3 | 0.9 | 175 | 5922 | 19 02 33.87 | 04 56 40.0 | 0.6
69 | 2292 | 18 21 09.21 | -14 31 45.5 | 0.8 | 176 | 5941 | 19 03 47.06 | 05 04 00.9 | 0.9
70 | 2297 | 18 21 30.62 | -14 30 42.6 | 1.0 | 177 | 5993 | 19 04 21.68 | 05 49 47.5 | 0.9
71 | 2354 | 18 22 59.5 | -13 19 41.4 | 0.9 | 178 | 6017 | 19 04 04.51 | 06 30 12.3 | 1.2
72 | 2381 | 18 25 21.94 | -13 13 27.4 | 1.0 | 179 | 6023 | 19 05 51.60 | 06 24 42.4 | 0.8
73 | 2467 | 18 27 08.01 | -12 41 38.3 | 0.8 | 180 | 6086 | 19 06 11.27 | 08 10 31.7 | 0.6
74 | 2499 | 18 25 44.96 | -12 22 41.7 | 0.9 | 181 | 6110 | 19 11 39.57 | 08 46 30.4 | 0.6
75 | 2630 | 18 28 23.57 | -11 47 38.3 | 0.9 | 182 | 6111 | 19 10 22.40 | 08 58 44.8 | 1.1
76 | 2636 | 18 29 14.68 | -11 50 24.0 | 0.9 | 183 | 6131 | 19 12 53.93 | 09 37 11.3 | 1.0
77 | 2641 | 18 28 19.10 | -11 40 25.5 | 0.9 | 184 | 6137 | 19 11 25.56 | 10 00 03.4 | 0.5
78 | 2659 | 18 28 10.39 | -11 28 44.2 | 0.7 | 185 | 6153 | 19 11 24.68 | 10 28 43.3 | 1.1
79 | 2665 | 18 27 44.80 | -11 14 52.2 | 0.9 | 186 | 6166 | 19 13 40.95 | 10 54 57.8 | 0.8
80 | 2696 | 18 30 09.87 | -11 12 39.1 | 1.0 | 187 | 6188 | 19 15 12.95 | 11 10 21.5 | 1.0
81 | 2711 | 18 29 59.62 | -11 00 22.2 | 0.9 | 188 | 6208 | 19 17 13.41 | 11 16 14.0 | 1.1
82 | 2713 | 18 30 11.16 | -11 01 10.4 | 0.9 | 189 | 6225 | 19 16 58.42 | 11 47 20.2 | 1.0
83 | 2837 | 18 30 35.29 | -09 34 40.1 | 0.9 | 190 | 6258 | 19 19 48.70 | 14 01 11.2 | 0.8
84 | 2858 | 18 30 38.33 | -09 12 43.8 | 0.8 | 191 | 6280 | 19 22 48.67 | 13 53 34.3 | 0.9
85 | 2865 | 18 32 44.16 | -09 24 33.6 | 0.9 | 192 | 6298 | 19 22 41.65 | 14 09 59.4 | 1.0
86 | 2890 | 18 31 44.72 | -09 08 32.8 | 0.9 | 193 | 6304 | 19 22 33.86 | 14 13 35.1 | 0.9
87 | 2904 | 18 31 10.67 | -08 54 17.6 | 0.8 | 194 | 6323 | 19 23 06.95 | 14 20 11.1 | 0.5
88 | 3022 | 18 35 21.67 | -08 50 15.1 | 0.9 | 195 | 6338 | 19 22 46.39 | 14 30 28.0 | 0.7
89 | 3034 | 18 34 00.19 | -08 35 54.0 | 0.8 | 196 | 6346 | 19 23 30.08 | 14 26 34.6 | 0.8
90 | 3071 | 18 34 39.30 | -08 31 35.3 | 0.9 | 197 | 6376 | 19 23 26.56 | 14 40 14.1 | 0.8
91 | 3081 | 18 34 35.95 | -08 29 32.2 | 0.9 | 198 | 6380 | 19 21 17.62 | 15 11 50.1 | 1.0
92 | 3153 | 18 35 12.45 | -08 17 35.5 | 0.8 | 199 | 6387 | 19 23 12.41 | 15 13 30.5 | 1.1
93 | 3202 | 18 35 22.49 | -08 01 18.7 | 0.9 | 200 | 6414 | 19 29 18.22 | 17 56 19.0 | 0.9
94 | 3208 | 18 35 23.45 | -07 59 32.6 | 0.7 | 201 | 6416 | 19 30 30.35 | 17 48 26.1 | 0.8
95 | 3209 | 18 34 08.83 | -07 49 33.7 | 0.9 | 202 | 6424 | 19 30 05.13 | 17 57 28.0 | 1.0
96 | 3274 | 18 35 35.52 | -07 41 46.1 | 0.9 | 203 | 6429 | 19 30 10.65 | 18 11 06.7 | 1.0
97 | 3284 | 18 35 08.61 | -07 35 04.3 | 0.9 | 204 | 6433 | 19 30 22.74 | 18 20 20.9 | 0.6
98 | 3383 | 18 35 39.86 | -07 18 32.7 | 0.6 | 205 | 6437 | 19 30 29.80 | 18 20 39.7 | 0.9
99 | 3394 | 18 36 50.31 | -07 24 44.5 | 0.6 | 206 | 6445 | 19 31 28.15 | 18 34 08.9 | 1.0
100 | 3402 | 18 35 50.97 | -07 13 29.1 | 0.8 | 207 | 6451 | 19 31 49.34 | 18 43 01.6 | 1.1
101 | 3413 | 18 36 12.90 | -07 12 06.6 | 0.8 | 208 | 6455 | 19 32 12.69 | 18 59 01.5 | 1.1
102 | 3437 | 18 36 26.92 | -07 05 07.7 | 0.5 | 209 | 6470 | 19 38 16.80 | 21 08 02.2 | 0.6
103 | 3466 | 18 36 28.09 | -06 47 51.0 | 0.8 | 210 | 6474 | 19 38 58.22 | 22 46 34.6 | 0.9
104 | 3503 | 18 38 08.47 | -06 46 40.8 | 1.0 | 211 | 6476 | 19 43 42.32 | 23 20 20.3 | 0.9
105 | 3504 | 18 38 55.53 | -06 52 18.1 | 0.7 | 212 | 6479 | 19 43 50.15 | 23 28 59.7 | 0.9
106 | 3505 | 18 37 30.70 | -06 41 17.8 | 0.8 | 213 | 6492 | 19 43 30.46 | 23 57 44.9 | 1.0
107 | 3528 | 18 37 20.83 | -06 31 55.6 | 0.8 | 214 | 6501 | 19 50 03.19 | 26 38 23.3 | 0.8
Table 3: Observed properties of 95 GHz class I methanol maser sources detected
– The full table is available in the online journal.
| Gaussian Fit |
---|---|---
Number | VLSR | $\Delta V$ | $S$ | P | S${}_{int}^{m}$
| (km s-1) | (km s-1) | (Jy km s-1) | (Jy) | (Jy km s-1)
(1) | (2) | (3) | (4) | (5) | (6)
11 | 20.36(0.08) | 0.89(0.21) | 5.8(1.1) | 6.1 | 23.4
… | 21.35(0.07) | 0.51(0.16) | 3.2(1.0) | 5.8 |
… | 22.63(0.06) | 0.97(0.15) | 10.4(1.5) | 10.0 |
… | 24.02(0.18) | 1.08(0.43) | 4.1(1.4) | 3.5 |
13 | 20.08(0.89) | 11.54(2.26) | 15.0(2.5) | 1.2 | 17.3
… | 20.62(0.12) | 0.87(0.30) | 2.3(0.8) | 2.5 |
20 | 98.48(0.30) | 2.71(0.70) | 6.9(1.5) | 2.4 | 6.9
22 | -0.46(0.05) | 0.26(0.22) | 1.5(0.6) | 5.4 | 20.9
… | 0.39(0.03) | 0.73(0.10) | 8.2(1.1) | 10.7 |
… | 1.58(0.12) | 0.78(0.36) | 2.5(1.3) | 3.0 |
… | 1.80(0.84) | 5.71(1.88) | 8.7(2.8) | 1.4 |
29 | 11.95(0.20) | 2.93(0.47) | 8.4(1.2) | 2.7 | 8.4
31 | 32.70(0.01) | 0.33(0.01) | 15.2(1.0) | 43.4 | 45.3
… | 32.57(0.06) | 1.44(0.17) | 13.9(1.3) | 9.1 |
… | 34.93(0.04) | 1.23(0.09) | 16.1(1.0) | 12.3 |
32 | 66.91(0.19) | 9.38(0.46) | 78.3(3.4) | 7.8 | 78.3
33 | -3.42(0.21) | 8.90(0.40) | 80.9(3.7) | 8.5 | 113.4
… | -6.31(0.05) | 0.96(0.11) | 15.0(3.4) | 14.7 |
… | -7.59(0.18) | 1.75(0.36) | 17.5(4.2) | 9.4 |
39 | 21.81(0.05) | 1.16(0.12) | 6.5(0.6) | 5.2 | 6.5
41 | 43.00(0.13) | 2.59(0.31) | 10.3(1.1) | 3.7 | 10.3
43 | 49.47(0.18) | 1.64(0.43) | 4.2(1.0) | 2.4 | 8.7
… | 53.14(0.44) | 3.10(1.14) | 4.4(1.3) | 1.3 |
47 | 19.83(0.05) | 0.88(0.12) | 8.2(0.9) | 8.8 | 24.2
… | 21.17(0.13) | 0.90(0.39) | 3.6(2.2) | 3.8 |
… | 22.76(0.22) | 2.20(0.49) | 12.4(2.5) | 5.3 |
49 | 12.64(0.13) | 2.26(0.32) | 7.9(0.9) | 3.3 | 9.8
… | 15.02(0.08) | 0.63(0.18) | 1.9(0.5) | 2.9 |
50 | 31.66(0.03) | 0.47(0.08) | 4.4(0.9) | 8.7 | 27.6
… | 32.71(0.12) | 0.57(0.36) | 1.4(1.1) | 2.3 |
… | 32.94(0.19) | 3.87(0.40) | 21.8(2.3) | 5.3 |
51 | 55.27(0.14) | 1.20(0.30) | 5.3(1.6) | 4.2 | 27.1
… | 57.35(0.08) | 2.13(0.21) | 21.7(1.8) | 9.6 |
53 | 43.95(0.01) | 0.42(0.03) | 8.7(0.6) | 19.4 | 8.7
54 | 36.18(0.21) | 2.11(0.49) | 6.0(1.2) | 2.7 | 6.0
55 | 49.59(0.03) | 0.85(0.11) | 9.7(1.9) | 10.8 | 57.5
… | 49.62(0.05) | 3.32(0.18) | 47.8(2.2) | 13.5 |
57 | 81.38(0.05) | 0.40(0.13) | 1.3(0.4) | 3.0 | 4.3
… | 82.78(0.12) | 1.22(0.31) | 3.0(0.6) | 2.3 |
63 | 20.66(0.14) | 1.74(0.43) | 4.8(1.5) | 2.6 | 11.8
… | 24.87(1.38) | 7.76(2.91) | 7.0(2.5) | 0.9 |
69 | 61.37(0.03) | 0.84(0.07) | 17.1(1.3) | 19.0 | 29.9
… | 60.36(0.07) | 0.72(0.17) | 6.4(1.6) | 8.3 |
… | 58.68(0.20) | 1.84(0.54) | 6.4(1.5) | 3.3 |
72 | 46.08(0.02) | 0.30(0.06) | 6.2(1.5) | 19.4 | 21.7
… | 46.74(0.06) | 0.98(0.13) | 15.5(1.9) | 14.9 |
73 | 64.79(0.08) | 1.18(0.26) | 10.4(4.6) | 8.3 | 43.6
… | 66.17(0.02) | 0.51(0.07) | 7.9(1.5) | 14.4 |
… | 65.99(0.29) | 3.03(0.34) | 25.2(6.0) | 7.8 |
74 | 58.78(0.54) | 4.82(1.09) | 11.6(2.8) | 2.3 | 27.3
… | 59.73(0.04) | 1.19(0.12) | 15.7(2.0) | 12.3 |
76 | 41.38(0.06) | 0.67(0.14) | 4.4(0.8) | 6.1 | 72.3
… | 43.51(0.03) | 0.87(0.08) | 18.6(3.0) | 20.2 |
… | 44.06(0.07) | 2.28(0.10) | 49.2(3.4) | 20.3 |
77 | 63.14(0.04) | 0.66(0.10) | 6.8(1.0) | 9.7 | 25.5
… | 64.40(0.05) | 1.04(0.15) | 14.0(1.7) | 12.7 |
… | 65.92(0.18) | 1.20(0.44) | 4.8(1.5) | 3.8 |
78 | 42.82(0.24) | 7.94(0.58) | 13.0(0.8) | 1.5 | 13.0
83 | 50.67(0.04) | 0.85(0.12) | 10.6(1.5) | 11.7 | 54.8
… | 51.72(0.02) | 0.66(0.05) | 14.7(1.3) | 21.0 |
… | 52.19(0.23) | 5.11(0.47) | 29.5(2.8) | 5.4 |
90 | 96.91(0.02) | 1.24(0.07) | 24.0(1.5) | 18.2 | 136.3
… | 99.94(0.02) | 0.42(0.04) | 7.4(0.7) | 16.6 |
… | 101.38(0.13) | 5.81(0.31) | 73.7(3.0) | 11.9 |
… | 101.50(0.03) | 0.40(0.08) | 3.4(0.8) | 8.0 |
… | 102.63(0.02) | 1.05(0.05) | 27.7(1.6) | 24.8 |
93 | 72.98(0.47) | 6.30(1.18) | 14.7(2.3) | 2.2 | 17.6
… | 73.14(0.04) | 0.42(0.10) | 3.0(0.7) | 6.6 |
94 | 69.72(0.11) | 2.76(0.26) | 12.9(1.1) | 4.4 | 12.9
97 | 113.52(0.02) | 0.63(0.05) | 16.1(1.6) | 24.1 | 80.7
… | 114.33(0.05) | 2.34(0.09) | 58.9(2.4) | 23.7 |
… | 116.58(0.05) | 0.70(0.12) | 5.6(1.0) | 7.6 |
98 | 51.13(0.06) | 0.52(0.14) | 2.3(0.7) | 4.1 | 11.8
… | 52.19(0.14) | 0.98(0.40) | 2.5(1.0) | 2.4 |
… | 52.08(0.87) | 7.15(2.44) | 7.1(1.9) | 0.9 |
99 | 113.26(0.19) | 2.09(0.44) | 4.8(0.9) | 2.2 | 4.8
101 | 108.19(0.05) | 0.84(0.11) | 9.2(1.2) | 10.4 | 68.4
… | 109.45(0.13) | 1.02(0.14) | 5.7(1.7) | 5.3 |
… | 111.20(0.08) | 1.83(0.17) | 23.5(2.5) | 12.1 |
… | 112.66(0.11) | 0.77(0.21) | 6.7(2.0) | 8.1 |
… | 113.35(0.07) | 0.52(0.18) | 5.5(1.8) | 9.8 |
… | 114.13(0.05) | 0.73(0.17) | 8.7(1.8) | 11.2 |
… | 114.94(0.06) | 0.32(0.15) | 1.4(1.0) | 4.1 |
… | 115.69(0.55) | 2.30(0.21) | 8.0(1.8) | 3.3 |
102 | 46.69(0.24) | 2.59(0.58) | 4.2(0.8) | 1.5 | 4.2
104 | 93.88(0.06) | 0.61(0.15) | 6.8(1.4) | 10.6 | 23.2
… | 94.38(0.07) | 0.22(0.11) | 1.7(1.2) | 7.4 |
… | 96.19(0.11) | 2.03(0.27) | 14.6(1.6) | 6.7 |
114 | 104.52(0.33) | 4.89(0.77) | 13.0(2.3) | 2.3 | 14.3
… | 104.52(0.09) | 0.38(0.09) | 1.3(0.5) | 3.2 |
116 | 45.28(0.05) | 0.53(0.11) | 4.2(0.7) | 7.5 | 4.2
117 | 99.29(0.44) | 3.70(1.03) | 6.8(1.6) | 1.7 | 6.8
120 | 79.07(0.09) | 0.43(0.21) | 1.7(0.8) | 3.7 | 16.5
… | 79.71(0.07) | 0.48(0.17) | 2.6(0.9) | 5.0 |
… | 80.02(0.50) | 5.85(1.26) | 12.2(2.2) | 2.0 |
125 | 90.69(0.06) | 0.39(0.16) | 0.8(0.3) | 1.9 | 5.5
… | 90.26(0.77) | 6.87(1.90) | 4.7(1.1) | 0.6 |
126 | 76.43(0.08) | 1.56(0.18) | 10.1(1.0) | 6.1 | 10.1
128 | 60.28(0.01) | 0.51(0.02) | 14.8(0.6) | 27.3 | 14.8
133 | 93.10(0.09) | 1.05(0.20) | 4.8(0.8) | 4.3 | 4.8
134 | 101.16(0.09) | 0.61(0.21) | 2.6(0.8) | 3.9 | 45.1
… | 105.13(0.09) | 3.65(0.23) | 39.9(2.0) | 10.3 |
… | 109.35(0.43) | 1.75(1.02) | 2.7(1.4) | 1.4 |
135 | 103.69(0.07) | 0.53(0.17) | 2.0(0.6) | 3.6 | 10.2
… | 105.54(0.08) | 1.43(0.19) | 8.2(0.9) | 5.4 |
143 | 50.32(0.07) | 0.67(0.17) | 2.9(0.6) | 4.1 | 2.9
144 | 31.35(0.27) | 2.67(0.68) | 4.0(0.9) | 1.4 | 5.6
… | 35.75(0.68) | 2.72(1.71) | 1.6(0.9) | 0.6 |
148 | 96.30(0.08) | 3.62(0.20) | 36.4(1.7) | 9.4 | 36.4
153 | 83.22(0.07) | 1.55(0.17) | 6.3(0.6) | 3.8 | 6.3
154 | 104.09(0.49) | 7.29(1.17) | 9.5(1.3) | 1.2 | 9.5
158 | 106.46(0.25) | 3.63(0.70) | 12.6(1.9) | 3.2 | 14.9
… | 106.47(0.11) | 0.58(0.31) | 1.8(1.1) | 2.9 |
160 | 60.26(0.01) | 0.31(0.05) | 2.6(0.3) | 7.9 | 6.0
… | 61.77(0.04) | 0.76(0.11) | 3.4(0.4) | 4.2 |
161 | 58.09(0.09) | 5.01(0.18) | 61.4(2.3) | 11.5 | 64.9
… | 60.25(0.10) | 0.99(0.29) | 3.6(1.3) | 3.4 |
164 | 52.43(0.23) | 4.82(0.54) | 11.4(1.1) | 2.2 | 11.4
172 | 63.09(0.39) | 2.49(0.91) | 4.2(1.4) | 1.6 | 8.1
… | 65.96(0.32) | 2.05(0.70) | 3.8(1.3) | 1.7 |
181 | 58.10(0.19) | 3.49(0.49) | 10.0(1.2) | 2.7 | 11.1
… | 58.19(0.05) | 0.27(0.16) | 1.0(0.4) | 3.6 |
194 | 67.39(0.26) | 2.88(0.61) | 4.8(0.9) | 1.6 | 4.8
200 | 19.51(0.25) | 0.18(0.05) | 2.2(0.3) | 11.1 | 16.3
… | 20.35(0.55) | 2.51(0.38) | 5.7(1.3) | 2.1 |
… | 22.19(0.04) | 0.75(0.11) | 8.4(1.6) | 10.5 |
209 | 32.01(0.17) | 1.10(0.41) | 2.1(0.6) | 1.8 | 4.4
… | 33.49(0.05) | 0.53(0.12) | 2.3(0.5) | 4.1 |
210 | 36.28(0.11) | 1.37(0.27) | 6.7(1.1) | 4.6 | 10.0
… | 38.21(0.09) | 0.72(0.21) | 3.3(0.8) | 4.2 |
212 | 27.23(0.12) | 1.59(0.26) | 14.4(2.2) | 8.5 | 23.7
… | 28.88(0.18) | 1.20(0.39) | 5.8(2.2) | 4.6 |
… | 30.70(0.11) | 0.91(0.28) | 3.4(0.9) | 3.5 |
Note. — Column (1): source number. Columns. (2)-(5): the velocity at peak
VLSR, the line FWHM $\Delta$V, the integrated intensity $S$, and the peak flux
density $P$ of each maser feature estimated from Gaussian fits to the 95 GHz
class I methanol maser lines. The formal error from the Gaussian fit is given
in parenthesis. Col. (6): the total integrated flux density S${}_{int}^{m}$
(Jy) of the maser spectrum obtained from summing the integrated flux density
of all maser features in each source in column (4).
Table 4: Sources detected as class I methanol masers in previous surveys. Number | Source | Detections | Referee
---|---|---|---
| | 44 GHz | 95 GHz |
22 | 18024-2119 | Y | N | Fontani et al. (2010)
32 | G10.47+0.03 | Y | – | Kurtz et al. (2004)
33 | G10.62-0.38 | Y | – | Kurtz et al. (2004)
43 | EGO G12.20-0.03 | | Y | Chen et al. (2011)
51 | EGO G12.91-0.03 | | Y | Chen et al. (2011)
69 | EGO G16.59-0.05 | Y | Y | Slysh et al. (1994); Val’tts et al. (2000)
73 | EGO G18.89-0.47 | Y | Y | Chen et al. (2011)
74 | EGO G19.01-0.03 | Y | Y | Chen et al. (2011)
76 | EGO G19.88-0.53 | – | Y | Chen et al. (2011)
83 | EGO G22.04+0.22 | Y | Y | Chen et al. (2011)
90 | G23.43-0.19 | Y | – | Slysh et al. (1994)
93 | EGO G23.96-0.11 | Y | – | Chen et al. (2011)
94 | EGO G24.00-0.10 | – | Y | Chen et al. (2011)
97 | EGO G24.33+0.14 | – | Y | Chen et al. (2011)
98 | EGO G24.63+0.15 | – | Y | Chen et al. (2011)
101 | W42 | Y | – | Bachiller et al. (1990)
104 | EGO G25.38-0.15 | – | Y | Chen et al. (2011)
161 | EGO G34.41+0.24 | – | Y | Chen et al. (2011)
181 | EGO G43.04-0.45 | Y | – | Chen et al. (2011)
194 | EGO G49.27-0.34 | Y | – | Cyganowski et al. (2009)
Table 5: The related parameters of the detected class I methanol masers.
Number | Distance | Luminosity | BGPS source | IRDC | Class II maser
---|---|---|---|---|---
| | | ID | M | n(H2) | N(H2) | |
| (kpc) | ($10^{-6}L_{\odot}$) | | (M⊙) | (103 cm-3) | (1022 cm-2) | |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
11 | 13.0 | 124.3 | 1251 | 4900 | 1.6 | 1.5 | N | N
13 | 3.4 | 6.3 | 1289 | 310 | 5.3 | 1.4 | Y | N
20D | 4.3 | 4.0 | 1363 | 370 | 4.8 | 1.4 | Y | N
22G,D | 5.2 | 17.8 | 1395 | 2100 | 1.5 | 1.1 | Y | Y
29S | 1.9 | 0.9 | 1467 | 560 | 3.0 | 1.1 | Y | N
31 | 12.7 | 229.6 | 1479 | 11000 | 0.8 | 1.2 | N | Y
32 | 11.2 | 308.6 | 1497 | 52000 | 24.6 | 20.8 | N | Y
33 | 4.0∗ | 57.0 | 1508 | 6300 | 113.9 | 28.7 | Y | Y
39 | 13.7 | 38.2 | 1591 | 2000 | – | – | N | N
41G | 12.2 | 48.0 | 1657 | 5000 | 1.6 | 1.6 | N | Y
43G | 12.0 | 39.2 | 1682 | 5900 | 0.8 | 1.0 | N | Y
47G | 2.7 | 5.6 | 1742 | 820 | 3.0 | 1.3 | N | Y
49 | 14.7 | 66.6 | 1778 | 14000 | 0.4 | 0.8 | N | N
50G | 2.3 | 4.6 | 1803 | 830 | 15.9 | 3.9 | Y | Y
51S | 4.6 | 17.9 | 1809 | 1500 | 2.0 | 1.2 | Y | Y
53 | 12.4 | 42.0 | 1853 | 5700 | 0.5 | 0.8 | N | N
54S | 3.5 | 2.3 | 1857 | 540 | 1.6 | 0.7 | Y | N
55G,S | 4.1 | 30.4 | 1865 | 1700 | 4.4 | 2.1 | Y | Y
57 | 11.0 | 16.4 | 1998 | 3000 | 20.7 | 7.2 | N | N
63 | 13.7 | 69.5 | 2124 | 11000 | 0.8 | 1.3 | N | N
69G,S | 4.3 | 17.4 | 2292 | 1300 | 17.6 | 4.8 | Y | Y
72 | 12.5 | 106.4 | 2381 | 11000 | 0.5 | 0.9 | N | N
73G,S | 3.8 | 19.8 | 2467 | 2800 | 1.8 | 1.4 | Y | Y
74G | 12.0 | 123.6 | 2499 | 6800 | 1.6 | 1.7 | N | Y
76G,D | 3.3 | 24.7 | 2636 | 1100 | 16.5 | 4.4 | Y | Y
77D,S | 4.1 | 13.5 | 2641 | 1100 | 3.5 | 1.6 | Y | N
78D | 12.6 | 64.9 | 2659 | 19000 | 1.6 | 2.4 | N | –
83G | 3.3 | 18.8 | 2837 | 1000 | 1.5 | 0.9 | Y | Y
90G | 5.9 | 149.1 | 3071 | 9600 | 1.0 | 1.4 | Y | Y
93G | 11.4 | 72.0 | 3202 | 8600 | 1.1 | 1.4 | N | Y
94G | 11.4 | 52.7 | 3208 | 4800 | 2.6 | 2.1 | N | Y
97G,S | 9.5 | 228.9 | 3284 | 7800 | 4.3 | 3.5 | Y | Y
98S | 3.3 | 4.1 | 3383 | 430 | 3.4 | 1.1 | Y | N
99 | 5.8 | 5.1 | 3394 | 4800 | 2.8 | 2.2 | Y | –
101 | 9.6 | 199.1 | 3413 | 32000 | 3.3 | 4.7 | N | –
102 | 12.2 | 19.6 | 3437 | 11000 | 1.5 | 1.9 | N | –
104 | 5.1 | 18.9 | 3503 | 1600 | 11.6 | 4.0 | Y | –
114 | 5.5 | 13.0 | 3833 | 740 | 2.4 | 1.1 | Y | –
116S | 2.9 | 1.1 | 3876 | 91 | 10.5 | 1.4 | Y | –
117G,S | 5.3 | 6.0 | 3897 | 2000 | 0.6 | 0.6 | Y | Y
120 | 4.5 | 10.5 | 3946 | 790 | 2.7 | 1.2 | Y | –
125 | 4.9 | 4.2 | 4020 | 280 | 4.7 | 1.2 | Y | –
126 | 10.4 | 34.5 | 4082 | 820 | 1.9 | 0.9 | N | –
128 | 3.6 | 6.0 | 4133 | 370 | 0.8 | 0.4 | Y | –
133 | 5.1 | 4.0 | 4366 | 1000 | 0.6 | 0.5 | Y | –
134S | 5.7 | 46.1 | 4398 | 5000 | 1.4 | 1.4 | Y | –
135 | 8.7 | 24.4 | 4472 | 19000 | 0.5 | 1.1 | N | –
143 | 3.1 | 0.9 | 4649 | 35 | – | – | Y | –
144 | 12.3 | 26.6 | 4673 | 6400 | 0.4 | 0.7 | N | –
148D | 5.3 | 32.1 | 4812 | 2200 | 1.3 | 1.0 | Y | –
153D | 9.4 | 17.4 | 5100 | 5900 | 0.6 | 0.8 | N | –
154D | 5.7 | 9.7 | 5167 | 4400 | 0.9 | 1.0 | Y | –
158S | 6.5 | 19.1 | 5252 | 4300 | 0.7 | 0.9 | Y | –
160 | 10.3 | 19.9 | 5270 | 2300 | 0.3 | 0.4 | N | –
161 | 10.4 | 220.7 | 5373 | 44000 | 1.6 | 3.2 | N | –
164 | 10.6 | 40.2 | 5572 | 1700 | 4.3 | 2.1 | N | –
172S | 9.5 | 22.8 | 5869 | 8200 | 2.7 | 2.6 | N | –
181 | 8.6 | 25.8 | 6110 | 2500 | 6.6 | 3.2 | N | –
194S | 5.5∗ | 4.6 | 6323 | 3600 | 7.7 | 3.2 | N | –
200D | 1.6 | 1.3 | 6414 | 280 | 5.6 | 2.3 | Y | –
209 | 3.0 | 1.2 | 6470 | 110 | 8.6 | 1.3 | Y | –
210S | 4.4∗ | 6.1 | 6474 | 350 | 12.8 | 2.5 | N | –
212 | 5.9 | 25.9 | 6479 | 3200 | 8.8 | 4.1 | N | –
Note. — Column (1): source number. The sources which are marked by $G$, $D$ or
$S$ overlaid with that in Green & McClure-Griffiths (2011), Dunham et al.
(2011b) or Schlingman et al. (2011), respectively. Column (2): the kinematic
distance for the source, estimated from the Galactic rotation curve of Reid et
al. (2009). For sources overlapped with Green & McClure-Griffiths (2011), we
adpoted the distances estimated from their work. For the sources (marked by
$\ast$) of which distances cannot be derived from the Galactic rotation curve,
a distance of 4 kpc is adopted for source with an IRDC association (N33), and
that determined in Schlingman et al. (2011) for the other two sources (N194
and N210). Column (3): the integrated luminosity of 95 GHz methanol maser.
Columns (4) – (7): the ID number of BGPS source in the BGPS catalog, the
derived gas mass and averaged H2 volume and column densities of the BGPS
source, respectively. The gas volume and column densities can not be
determined due to absence of radius information for the sources which are
unresolved by the BGPS beam, we marked them with “–”. Column (8): association
with IRDC: Y = Yes, N = No. Column (9): association with a 6.7 GHz methanol
maser for which a precise position has been measured. The positions of the 6.7
GHz class II methanol masers were identified from published 6.7 GHz maser
catalogs (Caswell 2009; Caswell et al. 2010; Green et al. 2010; Caswell et al.
2011 ; Green et al. 2012): Y = Yes, N = No,“–” = no information.
Table 6: Trends with star formation activity for sources with and without methanol masers Property | Group | mean | standard deviation | minimum | median | maximum
---|---|---|---|---|---|---
With and without class I methanol masers
log($N_{H_{2}}^{beam}$) | class I | 22.7 | 0.4 | 21.9 | 22.7 | 23.8
$[cm^{-2}]$ | no class I | 21.9 | 0.3 | 21.4 | 21.9 | 22.7
log($S_{int}$) | class I | 0.7 | 0.4 | -0.6 | 0.7 | 1.5
$[Jy]$ | no class I | 0.0 | 0.4 | -0.9 | 0.0 | 1.1
Radius | class I | 57.8 | 25\. 1 | 15.7 | 56.0 | 128.7
$(^{\prime\prime})$ | no class I | 49.3 | 21.4 | 5.1 | 46.3 | 113.3
Class I methanol masers with and without class II maser associations
log($N_{H_{2}}^{beam}$) | only class I | 22.5 | 0.2 | 22.2 | 22.6 | 22.8
$[cm^{-2}]$ | class I+II | 22.9 | 0.3 | 22.5 | 22.8 | 23.8
log(Sint) | only class I | 0.48 | 0.29 | -0.09 | 0.47 | 1.07
$[Jy]$ | class I+II | 0.86 | 0.31 | 0.41 | 0.79 | 1.49
Table 7: The distributions of class I methanol maser numbers with BGPS source parameters Range | In our observing sample | | In the full BGPS catalog
---|---|---|---
| detections | total numbers | detection rate | | total numbers | expected detections
log($N_{H_{2}}^{beam}$) $[cm^{-2}]$ | | | | | |
21.0–21.2 | – | – | – | | 1 | 0
21.2–21.4 | – | – | – | | 88 | 0
21.4–21.6 | 0 | 1 | 0.00 | | 717 | 0
21.6–21.8 | 0 | 11 | 0.00 | | 2021 | 0
21.8–22.0 | 1 | 46 | 0.02 | | 2303 | 50
22.0–22.2 | 6 | 45 | 0.13 | | 1451 | 193
22.2–22.4 | 6 | 36 | 0.17 | | 804 | 134
22.4–22.6 | 11 | 29 | 0.38 | | 458 | 174
22.6–22.8 | 17 | 23 | 0.74 | | 240 | 177
22.8–23.0 | 13 | 14 | 0.93 | | 142 | 132
23.0–23.2 | 5 | 5 | 1.00 | | 69 | 69
23.2–23.4 | 1 | 1 | 1.00 | | 34 | 34
23.4–23.6 | 1 | 1 | 1.00 | | 8 | 8
23.6–23.8 | 2 | 2 | 1.00 | | 11 | 11
23.8–24.0 | – | – | – | | 7 | 7
24.0–24.2 | – | – | – | | 4 | 4
24.4–24.6 | – | – | – | | 1 | 1
sum | | | | | | 995
log($S_{int}$) $[Jy]$ | | | | | |
-1.4$-$-1.2 | – | – | – | | 10 | 0
-1.2$-$-1.0 | – | – | – | | 109 | 0
-1.0$-$-0.8 | 0 | 2 | 0.00 | | 410 | 0
-0.8$-$-0.6 | 0 | 2 | 0.00 | | 771 | 0
-0.6$-$-0.4 | 1 | 25 | 0.04 | | 1128 | 45
-0.4$-$-0.2 | 1 | 19 | 0.05 | | 1450 | 76
-0.2$-$0.0 | 4 | 28 | 0.14 | | 1296 | 185
0.0–0.2 | 3 | 26 | 0.12 | | 1103 | 127
0.2–0.4 | 6 | 30 | 0.20 | | 784 | 157
0.4–0.6 | 11 | 31 | 0.35 | | 531 | 188
0.6–0.8 | 14 | 23 | 0.61 | | 330 | 201
0.8–1.0 | 11 | 13 | 0.85 | | 223 | 189
1.0–1.2 | 6 | 9 | 0.67 | | 115 | 77
1.2–1.4 | 2 | 2 | 1.00 | | 47 | 47
1.4–1.6 | 4 | 4 | 1.00 | | 27 | 27
1.6–1.8 | – | – | – | | 12 | 12
1.8–2.0 | – | – | – | | 8 | 8
2.0–2.6 | – | – | – | | 6 | 6
2.6–2.8 | – | – | – | | 1 | 1
sum | | | | | | 1346
Figure 1: Number of sources as a function of the separations of the pair of
GLIMPSE point source and GBPS source in our observing sample.
Figure 2: Spectra of the 95 GHz methanol masers detected in the survey. The
dashed lines represent the Gaussian fitting of each maser feature, the bold-
solid line mark the sum of the Gaussian fitting of all maser feature.
Fig. 2.— Continued.
Fig. 2.— Continued.
Fig. 2.— Continued.
Figure 3: Comparison of the spectra of 95 GHz methanol maser emission in the
11 sources which have been detected in both the PMO 13.7-m survey (this work)
marked with black lines and the EGO-based Mopra survey by Chen et al. (2011)
marked with red lines. A color version of this figure is available in the
online journal.
Figure 4: Color-color diagrams of GLIMPSE point sources associated with and
without class I methanol maser detections in the survey. Filled and open
circles represent the sources with and without class I methanol maser
detections, respectively. The sources associated with EGOs (15 in total) are
enclosed by red triangles. The solid lines overlaid in [3.6]-[4.5] vs.
[5.8]-[8.0] diagram construct the regions occupied by various evolutionary-
stage (Stages I, II and III) YSOs according to the models of Robitaille et al.
(2006). The hatched region in the color-color plot is the region where models
of all evolutionary stages can be present. Note that the Stage II area in the
color-color plot is hatched to show that most models in this region are Stage
II models, however Stage I models can also be found within this area. The
reddening vectors in each panel show an extinction of Av=20, assuming the
Indebetouw et al. (2005) extinction law. A color version of this figure is
available in the online journal.
Figure 5: Left: logarithm of the 95 GHz class I methanol maser luminosity
versus GLIMPSE point source luminosity at 4.5 $\mu$m band; Right: color-color
diagram of [3.6]-log(Sm) versus [3.6]-[4.5] which combines the GLIMPSE point
sources and class I methanol maser emission. The line in each panel marks the
best fit to the corresponding distribution.
Figure 6: Logarithm of the 95 GHz class I methanol maser luminosity as a
function of the gas mass (left panel) and H2 volume density (right panel) of
the associated 1.1 mm BGPS sources. The line in each panel marks the best fit
from the linear regression analysis to the corresponding distribution.
Figure 7: Logarithm of the integrated flux density of the 95 GHz class I
methanol maser as a function of the beam-averaged H2 column density of the
BGPS source. The line marks the best fit from the linear regression analysis
to the distribution.
Figure 8: Number of sources as functions of the BGPS beam-averaged H2 column
density (left), integrated flux density of the BGPS source (middle) and BGPS
source radius (right) for the two groups with and without class I methanol
maser detections. For distributions in each BGPS property, the upper and lower
panels correspond to the BGPS sources with and without class I methanol maser
detections. The mean of each distribution is marked by the vertical dashed
line in the corresponding distribution.
Figure 9: Number of sources as functions of the BGPS beam-averaged H2 column
density (left) and BGPS integrated flux density (right) for the two subsamples
based on which class of methanol masers they are associated with. For
distributions in each BGPS property, the upper and lower panels correspond to
the BGPS sources associated with only class I methanol masers and associated
with both class I and II methanol masers, respectively. The mean of each
distribution is marked by the vertical dashed line in the corresponding
distribution.
Figure 10: Detection rates of class I methanol masers with 4.5 $\mu$m
magnitude (left panel) and [3.6]-[4.5] color (right panel) of the GLIMPSE
point sources. For each mid-IR property, the upper panel shows the histogram
distributions of number of total sample sources and detected class I methanol
maser sources marked with open bars and diagonal bars, respectively, and the
lower panel shows the corresponding detection rate of class I methanol maser
in each statistical bin.
Figure 11: As Figure 10, but for detection rates of class I methanol maser
with the BGPS properties of the beam-averaged H2 column density (left),
integrated flux density (right).
Figure 12: Left panel: Logarithm of the integrated flux densities versus beam-
averaged H2 column density of BGPS sources with and without class I methanol
maser detections (marked by red circles and blue triangles, respectively) in
our current survey sample, the class I methanol maser locating region is
enclosed by the red lines. Right panel: As Left panel, but for all cataloged
BGPS sources. (A color version of this figure is available in the online
journal.)
|
arxiv-papers
| 2012-02-29T08:24:42 |
2024-09-04T02:49:28.059416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xi Chen, Simon P. Ellingsen, Jin-Hua He, Ye Xu, Cong-Gui Gan,\n Zhi-Qiang Shen, Tao An, Yan Sun, Bing-Gang Ju",
"submitter": "Xi Chen",
"url": "https://arxiv.org/abs/1202.6478"
}
|
1202.6573
|
11institutetext: Dept. of Mathematics and Computer Science, University of the
Balearic Islands, E-07122 Palma. 11email:
{gabriel.cardona,arnau.mir,cesc.rossello}@uib.es
# Exact formulas for the variance of several balance indices under the Yule
model
Gabriel Cardona Arnau Mir Francesc Rosselló
###### Abstract
One of the main applications of balance indices is in tests of null models of
evolutionary processes. The knowledge of an exact formula for a statistic of a
balance index, holding for any number $n$ of leaves, is necessary in order to
use this statistic in tests of this kind involving trees of any size. In this
paper we obtain exact formulas for the variance under the Yule model of the
Sackin, the Colless and the total cophenetic indices of binary rooted
phylogenetic trees with $n$ leaves.
## 1 Introduction
One of the most thoroughly studied properties of the topology of phylogenetic
trees is their symmetry, that is, the degree to which both children of each
internal node tend to have the same number of descendant taxa. The symmetry of
a tree is usually quantified by means of _balance indices_. Many such indices
have been proposed so far in the literature [5, Chap. 33]. The most popular
are _Colless’ index_ $C$ [4], which is defined as the sum, over all internal
nodes $v$, of the absolute value of the difference between the number of
descendant leaves of $v$’s children, and _Sackin’s index_ $S$ [17], which is
defined as the sum of the depths of all leaves in the tree. We have recently
proposed an extension of Sackin’s index, the _total cophenetic index_ $\Phi$
[12]: the sum, over all pairs of different leaves of the tree, of the depth of
their least common ancestor. The main advantages of $\Phi$ over $S$ are that
it has a larger range of values and a smaller probability of ties. Moreover,
$\Phi$ retains other good properties of $S$: it makes sense for not
necessarily fully resolved phylogenetic trees (unlike Colless’ index), it can
be computed in linear time, and the statistical properties of its distribution
of values can be studied under different stochastic models of evolution, like
for instance the Yule [7, 23] and the uniform [3, 16, 20] models. This last
property is relevant because one of the main applications of balance indices
is their use as tools to test stochastic models of evolution [13, 18].
Exact formulas for the expected values under the Yule model of $C$, $S$, and
$\Phi$ on the space $\mathcal{T}_{n}$ of fully resolved rooted phylogenetic
trees with $n$ leaves have already been published. More specifically, if we
denote by $H_{n}$ the $n$-th _harmonic number_ ,
$H_{n}=\sum_{i=1}^{n}\frac{1}{i},$
these expected values are, respectively,
$\begin{array}[]{ll}E_{Y}(C_{n})=(n\mod
2)+n(H_{\lfloor\frac{n}{2}\rfloor}-1)&\mbox{
\cite[cite]{[\@@bibref{}{Heard92}{}{}]}}\\\ E_{Y}(S_{n})=2n(H_{n}-1)&\mbox{
\cite[cite]{[\@@bibref{}{KiSl:93}{}{}]}}\\\
E_{Y}(\Phi_{n})=n(n-1)-2n(H_{n}-1)&\mbox{
\cite[cite]{[\@@bibref{}{MRR}{}{}]}}\end{array}$
As we have already pointed out in [12], the last two formulas imply that the
expected value under the Yule model of the sum $\overline{\Phi}=S+\Phi$ on
$\mathcal{T}_{n}$ is
$E_{Y}(\overline{\Phi}_{n})=n(n-1),$
a quite simpler expression than those for $E_{Y}(S_{n})$ or $E_{Y}(\Phi_{n})$.
This index $\overline{\Phi}$ has the same good properties of $\Phi$, but the
formulas for its statistics under the Yule model tend to be simpler than the
corresponding formulas for other indices. We shall find here another example
of this fact: the variance.
The goal of this paper is to provide exact formulas for the variance of $S$,
$C$, $\Phi$ and $\overline{\Phi}$ on $\mathcal{T}_{n}$ under the Yule model.
As a byproduct of our computations, we shall also obtain the covariances of
$S$ with $\Phi$ and $\overline{\Phi}$. The variances of $S$ and $C$ on
$\mathcal{T}_{n}$ under this model were known so far only for their limit
distribution when $n\to\infty$ [1]:
$\begin{array}[]{l}\sigma_{Y}^{2}(C_{n})\sim\Big{(}3-\dfrac{\pi^{2}}{6}-\log(2)\Big{)}n^{2}\\\\[8.61108pt]
\sigma_{Y}^{2}(S_{n})\sim\Big{(}7-\dfrac{2\pi^{2}}{3}\Big{)}n^{2}\end{array}$
Also, Rogers [14, 15] found recursive formulas for the moment-generating
functions of $C$ and $S$ under this model, which allow one to compute
recursively as many values of $\sigma_{Y}^{2}(C_{n})$ and
$\sigma_{Y}^{2}(S_{n})$ as desired, but he did not produce explicit exact
formulas for them.
In this paper we obtain the following exact formulas for these variances:
$\begin{array}[]{l}\sigma_{Y}^{2}(C_{n}^{2})\displaystyle=\frac{5n^{2}+7n}{2}+(6n+1)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}^{2}+8\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}\\\
\qquad\displaystyle-8(n+1)\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}-6nH_{n}+\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-n(n-3)\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
\qquad\displaystyle-n^{2}H_{\lfloor\frac{n}{2}\rfloor}^{(2)}+\Big{(}n^{2}+3n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2nH_{\lfloor\frac{n}{4}\rfloor}\\\\[8.61108pt]
\displaystyle\sigma^{2}_{Y}(S_{n})=7n^{2}-4n^{2}H_{n}^{(2)}-2nH_{n}-n\\\\[4.30554pt]
\displaystyle\sigma^{2}_{Y}(\Phi_{n})=\frac{1}{12}(n^{4}-10n^{3}+131n^{2}-2n)-6nH_{n}-4nH_{n}^{2}-4n(n-1)H_{n}^{(2)}\\\\[8.61108pt]
\displaystyle\sigma^{2}_{Y}(\overline{\Phi}_{n})=2\binom{n}{4}\end{array}$
where $H_{n}^{(2)}=\sum\limits_{i=1}^{n}1/i^{2}$. We also obtain the following
exact formulas for the covariances, under the Yule model, of $S$ with $\Phi$
and $\overline{\Phi}$ on $\mathcal{T}_{n}$:
$\begin{array}[]{l}\displaystyle\textit{cov}_{Y}(S_{n},\Phi_{n})=4n(nH_{n}^{(2)}+H_{n})+\frac{1}{6}n(n^{2}-51n+2)\\\\[8.61108pt]
\displaystyle\textit{cov}_{Y}(S_{n},\overline{\Phi}_{n})=2nH_{n}+\frac{1}{6}n(n^{2}-9n-4)\end{array}$
All these formulas are valid for any number $n$ of leaves, and therefore they
can be used in a meaningful way in tests involving trees of any size. The
proofs consist mainly of elementary, although long and technically involved,
algebraic computations.
The rest of this paper is organized as follows. In a first section on
Preliminaries we gather some notations and conventions on phylogenetic trees
and some lemmas on probabilities of trees under the Yule model and on harmonic
numbers. In the next section, we establish a recursive formula for the
expected value under the Yule model of the square of a balance index
satisfying a certain kind or recursion (a _recursive shape index_ [11]) that
lies at the basis of all our computations. Then, we devote a series of
sections to compute the variances of $S$, $\Phi$, $\overline{\Phi}$, $C$ and
the covariance of $S$ with $\Phi$ and $\overline{\Phi}$, respectively. These
sections consist of long and tedious algebraic computations, without any
interest beyond the fact that they prove the formulas announced above. We
close the paper with a section on Conclusions and Discussion.
## 2 Preliminaries
### 2.1 Phylogenetic trees
In this paper, by a _phylogenetic tree_ on a set $S$ of taxa we mean a binary
rooted tree with its leaves bijectively labeled in the set $S$. We shall
always understand a phylogenetic tree as a directed graph, with its arcs
pointing away from the root. To simplify the language, we shall always
identify a leaf of a phylogenetic tree with its label. We shall also use the
term _phylogenetic tree with $n$ leaves_ to refer to a phylogenetic tree on
the set $\\{1,\ldots,n\\}$. We shall denote by $\mathcal{T}(S)$ the set of
isomorphism classes of phylogenetic trees on a set $S$ of taxa, and by
$\mathcal{T}_{n}$ the set $\mathcal{T}(\\{1,\ldots,n\\})$ of isomorphism
classes of phylogenetic trees with $n$ leaves. We shall denote by $V_{int}(T)$
the set of internal nodes of a phylogenetic tree $T$.
Whenever there exists a path from $u$ to $v$ in a phylogenetic tree $T$, we
shall say that $v$ is a _descendant_ of $u$ and that $u$ is an _ancestor_ of
$v$. The _lowest common ancestor_ $\textrm{LCA}_{T}(u,v)$ of a pair of nodes
$u,v$ in a phylogenetic tree $T$ is the unique common ancestor of them that is
a descendant of every other common ancestor of them.
The _depth_ $\delta_{T}(v)$ of a node $v$ in $T$ is the length (in number of
arcs) of the unique path from the root $r$ of $T$ to $v$. The _cophenetic
value_ $\varphi_{T}(i,j)$ of a pair of leaves $i,j$ is the depth of their
lowest common ancestor [19]:
$\varphi_{T}(i,j)=\delta_{T}(\textrm{LCA}_{T}(i,j)).$
To simplify the notations at some points, we shall also write
$\varphi_{T}(i,i)$ to denote the depth $\delta_{T}(i)$ of a leaf $i$.
Given two phylogenetic trees $T,T^{\prime}$ on disjoint sets of taxa
$S,S^{\prime}$, respectively, their _tree-sum_ is the tree $T\,\widehat{\
}\,T^{\prime}$on $S\cup S^{\prime}$ obtained by connecting the roots of $T$
and $T^{\prime}$ to a (new) common root. Every tree with $n$ leaves is
obtained as $T_{k}\widehat{\ }\,{}T^{\prime}_{n-k}$, for some $1\leqslant
k\leqslant n-1$, some subset $S_{k}\subseteq\\{1,\ldots,n\\}$ with $k$
elements, some tree $T_{k}$ on $S_{k}$ and some tree $T^{\prime}_{n-k}$ on
$S_{k}^{c}=\\{1,\ldots,n\\}\setminus S_{k}$; actually, every tree $T$ with $n$
leaves is obtained in this way _twice_.
The _Yule_ , or _Equal-Rate Markov_ , model of evolution [7, 23] is a
stochastic model of phylogenetic trees’ growth. It starts with a node, and at
every step a leaf is chosen randomly and uniformly and it is splitted into two
leaves. Finally, the labels are assigned randomly and uniformly to the leaves
once the desired number of leaves is reached. This corresponds to a model of
evolution where, at each step, each currently extant species can give rise,
with the same probability, to two new species. Under this model of evolution,
different trees with the same number of leaves may have different
probabilities. More specifically, if $T$ is a phylogenetic tree with $n$
leaves, and for every internal node $v$ we denote by $\kappa_{T}(v)$ the
number of its descendant leaves, then the probability of $T$ under the Yule
model is [2, 21]
$P_{Y}(T)=\frac{2^{n-1}}{n!}\prod_{v\in V_{int}(T)}\frac{1}{\kappa_{T}(v)-1}$
(1)
The following easy lemma on the probability of a tree-sum under the Yule model
will be used in our computations.
###### Lemma 1
Let $\emptyset\neq S_{k}\subsetneq\\{1,\ldots,n\\}$ with $|S_{k}|=k$, let
$T_{k}\in\mathcal{T}(S_{k})$ and
$T^{\prime}_{n-k}\in\mathcal{T}(\\{1,\ldots,n\\}\setminus S_{k})$. Then
$P_{Y}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})=\dfrac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})$
###### Proof
This equality is a direct consequence of applying equation (1) to compute
$P_{Y}(T_{k})$, $P_{Y}(T^{\prime}_{n-k})$ and $P_{Y}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})$, using the fact that $V_{int}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})$ is the disjoint union of $V_{int}(T_{k})$,
$V_{int}(T^{\prime}_{n-k})$ and the root $r$ of $T_{k}\widehat{\
}\,{}T^{\prime}_{n-k}$. ∎
### 2.2 Harmonic numbers
For every $n\geqslant 1$, let
$H_{n}=\sum_{i=1}^{n}\frac{1}{i},\quad
H_{n}^{(2)}=\sum_{i=1}^{n}\frac{1}{i^{2}}.$
Let, moreover, $H_{0}=H_{0}^{(2)}=0$. $H_{n}$ is called the $n$-th _harmonic
number_ , and $H_{n}^{(2)}$, the _generalized harmonic number of power $2$_.
It is known (see, for instance, [6, p. 264]) that
$\begin{array}[]{l}\displaystyle
H_{n}=\ln(n)+\gamma+\frac{1}{2n}-\frac{1}{12n^{2}}+O\Big{(}\frac{1}{n^{3}}\Big{)}\\\
\displaystyle
H_{n}^{(2)}=\frac{\pi^{2}}{6}-\frac{1}{n}+\frac{1}{2n^{2}}+O\Big{(}\frac{1}{n^{3}}\Big{)}\end{array}$
where $\gamma$ is Euler’s constant.
The following identities will be used in the proofs of our main results,
usually without any further notice.
###### Lemma 2
For every $n\geqslant 2$:
1. (1)
$\displaystyle\sum_{k=1}^{n-1}H_{k}=n(H_{n}-1)$
2. (2)
$\displaystyle\sum_{k=1}^{n-1}kH_{k}=\frac{1}{4}n(n-1)(2H_{n}-1)$
3. (3)
$\displaystyle\sum_{k=1}^{n-1}k^{2}H_{k}=\frac{1}{36}n(n-1)((12n-6)H_{n}-4n-1)$
4. (4)
$\displaystyle\sum_{k=1}^{n-1}\frac{H_{k}}{k+1}=\frac{1}{2}(H_{n}^{2}-H_{n}^{(2)})$
5. (5)
$\displaystyle\sum_{k=1}^{n-1}H_{k}^{2}=nH_{n}^{2}-(2n+1)H_{n}+2n$
6. (6)
$\displaystyle\sum_{k=1}^{n-1}H_{k}^{(2)}=nH_{n}^{(2)}-H_{n}$
7. (7)
$\displaystyle\sum_{k=1}^{n-1}H_{k}H_{n-k}=(n+1)(H_{n+1}^{2}-H_{n+1}^{(2)}-2H_{n+1}+2)$
8. (8)
$\displaystyle\sum_{k=1}^{n-1}kH_{k}H_{n-k}=\binom{n+1}{2}(H_{n+1}^{2}-H_{n+1}^{(2)}-2H_{n+1}+2)$
9. (9)
$\displaystyle\sum_{k=1}^{n-1}kH_{\lfloor
k/2\rfloor}=\frac{1}{2}n(n-1)H_{\lfloor
n/2\rfloor}-\Big{\lfloor}\frac{n}{2}\Big{\rfloor}^{2}$
10. (10)
$\displaystyle\sum_{k=1}^{n-1}\frac{H_{k}}{2k-1}=\sum_{k=1}^{n-1}\frac{H_{k}}{2k+1}-\left(\frac{4n-1}{2n-1}\right)H_{n}+2H_{2n}$
###### Proof
Identities (1)–(6) are well known and easily proved by induction on $n$: see,
for instance, the chapters on harmonic numbers in Knuth’s classical textbooks
[6, §6.3, 6.4] and [10, §1.2.7]. Identities (7) and (8) are proved in [22,
Thms. 1,2]. We shall prove (9) and (10) here.
As far as (9) goes, if $n$ is even,
$\begin{array}[]{l}\displaystyle\sum_{k=1}^{n-1}kH_{\lfloor
k/2\rfloor}=\sum_{j=1}^{(n-2)/2}2jH_{j}+\sum_{j=0}^{(n-2)/2}(2j+1)H_{j}\\\
\displaystyle\quad=4\sum_{j=1}^{n/2-1}jH_{j}+\sum_{j=1}^{n/2-1}H_{j}=\frac{1}{2}n(n-1)H_{\frac{n}{2}}-\Big{(}\frac{n}{2}\Big{)}^{2}\end{array}$
while if $n$ is odd,
$\begin{array}[]{l}\displaystyle\sum_{j=1}^{n-1}kH_{\lfloor
k/2\rfloor}=\sum_{j=1}^{(n-1)/2}2jH_{j}+\sum_{j=0}^{(n-1)/2-1}(2j+1)H_{j}\\\
\displaystyle\quad=4\sum_{j=1}^{(n-1)/2-1}jH_{j}+\sum_{j=1}^{(n-1)/2-1}H_{j}+(n-1)H_{\frac{n-1}{2}}=\frac{1}{2}n(n-1)H_{\frac{n-1}{2}}-\Big{(}\frac{n-1}{2}\Big{)}^{2}\end{array}$
Both equalities agree with identity (9).
As far as (10) goes,
$\begin{array}[]{l}\displaystyle\sum_{k=1}^{n-1}\frac{H_{k}}{2k+1}=\sum_{k=2}^{n}\frac{H_{k-1}}{2k-1}=\sum_{k=2}^{n}\frac{H_{k}-\frac{1}{k}}{2k-1}=\sum_{k=2}^{n}\frac{H_{k}}{2k-1}-\sum_{k=2}^{n}\frac{1}{k(2k-1)}\\\
\displaystyle\qquad=\sum_{k=1}^{n-1}\frac{H_{k}}{2k-1}-1+\frac{H_{n}}{2n-1}+\sum_{k=2}^{n}\left(\frac{1}{k}-\frac{2}{2k-1}\right)\\\
\displaystyle\qquad=\sum_{k=1}^{n-1}\frac{H_{k}}{2k-1}-1+\frac{H_{n}}{2n-1}+H_{n}-1-2\sum_{k=2}^{n}\frac{1}{2k-1}\\\
\displaystyle\qquad=\sum_{k=1}^{n-1}\frac{H_{k}}{2k-1}-1+\frac{H_{n}}{2n-1}+H_{n}-1-2(H_{2n}-\frac{1}{2}H_{n})+2\\\
\displaystyle\qquad=\sum_{k=1}^{n-1}\frac{H_{k}}{2k-1}+\left(\frac{4n-1}{2n-1}\right)H_{n}-2H_{2n}.\end{array}$
which is equivalent to (10).∎
## 3 Recursive shape indices
A _recursive shape index for phylogenetic trees_ [11] is a mapping $I$ that
associates to each phylogenetic tree $T$ a real number $I(T)\in\mathbb{R}$
satisfying the following two conditions:
1. (a)
It is invariant under tree isomorphisms and relabelings of leaves.
2. (b)
There exists a symmetrical mapping
$f_{I}:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$ such that, for every
phylogenetic trees $T,T^{\prime}$ on disjoint sets of taxa $S,S^{\prime}$,
respectively,
$I(T\,\widehat{\ }\,T^{\prime})=I(T)+I(T^{\prime})+f_{I}(|S|,|S^{\prime}|).$
As we shall see in later sections, the balance indices considered in this
paper are recursive shape indices in this sense. The following two results
extract a common part of the computation of their variances. In them, and
henceforth, $E_{Y}$ applied to a random variable will mean the expected value
of this random variable under the Yule model.
###### Lemma 3
Let $I$ be a recursive shape index for phylogenetic trees. For every
$n\geqslant 1$, let $I_{n}$ be the random variable that chooses a tree
$T\in\mathcal{T}_{n}$ and computes $I(T)$. Then,
$\begin{array}[]{rl}E_{Y}(I_{n}^{2})&\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{(}2E_{Y}(I^{2}_{k})+4f_{I}(k,n-k)E_{Y}(I_{k})+2E_{Y}(I_{k})E_{Y}(I_{n-k})\\\
&\displaystyle\qquad\qquad\qquad\quad+f_{I}(k,n-k)^{2}\Big{)}.\end{array}$
###### Proof
We compute $E_{Y}(I_{n}^{2})$ using its very definition and Lemma 1. Recall
that every tree in $\mathcal{T}_{n}$ is obtained _twice_ as $T_{k}\widehat{\
}\,{}T^{\prime}_{n-k}$, for some $1\leqslant k\leqslant n-1$, some subset
$S_{k}\subseteq\\{1,\ldots,n\\}$ with $k$ elements, some tree $T_{k}$ on
$S_{k}$ and some tree $T^{\prime}_{n-k}$ on $S_{k}^{c}$.
$\begin{array}[]{l}E_{Y}(I_{n}^{2})\displaystyle=\sum_{T\in\mathcal{T}_{n}}I(T)^{2}\cdot
P_{Y}(T)\\\
\quad\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\sum_{S_{k}\subsetneq\\{1,\ldots,n\\}\atop|S_{k}|=k}\sum_{T_{k}\in\mathcal{T}(S_{k})}\sum_{T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})}I(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})^{2}\cdot P_{Y}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\binom{n}{k}\sum_{T_{k}\in\mathcal{T}_{k}}\sum_{T^{\prime}_{n-k}\in\mathcal{T}_{n-k}}\big{(}I(T_{k})+I(T^{\prime}_{n-k})+f_{I}(k,n-k)\big{)}^{2}\\\
\quad\displaystyle\qquad\qquad\cdot\frac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}\big{[}I(T_{k})^{2}+I(T^{\prime}_{n-k})^{2}+f_{I}(k,n-k)^{2}+2I(T_{k})I(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad\qquad+2f_{I}(k,n-k)I(T_{k})+2f_{I}(k,n-k)I(T^{\prime}_{n-k})\big{]}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{[}\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}I(T_{k})^{2}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad+\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}I(T^{\prime}_{n-k})^{2}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad+\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}f_{I}(k,n-k)^{2}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}f_{I}(k,n-k)I(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}f_{I}(k,n-k)I(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\end{array}$
$\begin{array}[]{l}\quad\displaystyle\qquad+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}I(T_{k})I(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\Big{]}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{[}\sum_{T_{k}}I(T_{k})^{2}P_{Y}(T_{k})+\sum_{T^{\prime}_{n-k}}I(T^{\prime}_{n-k})^{2}P_{Y}(T^{\prime}_{n-k})+f_{I}(k,n-k)^{2}\\\
\quad\displaystyle\qquad+2\sum_{T_{k}}f_{I}(k,n-k)I(T_{k})P_{Y}(T_{k})+2\sum_{T^{\prime}_{n-k}}f_{I}(k,n-k)I(T^{\prime}_{n-k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle\qquad+2\Big{(}\sum_{T_{k}}I(T_{k})P_{Y}(T_{k})\Big{)}\Big{(}\sum_{T^{\prime}_{n-k}}I(T^{\prime}_{n-k})P_{Y}(T^{\prime}_{n-k})\Big{)}\Big{]}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{(}E_{Y}(I^{2}_{k})+E_{Y}(I^{2}_{n-k})+f_{I}(k,n-k)^{2}\\\
\quad\displaystyle\qquad+2f_{I}(k,n-k)(E_{Y}(I_{k})+E_{Y}(I_{n-k}))+2E_{Y}(I_{k})E_{Y}(I_{n-k})\Big{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{(}2E_{Y}(I_{k}^{2})+4f_{I}(k,n-k)E_{Y}(I_{k})+2E_{Y}(I_{k})E_{Y}(I_{n-k})\\\
\quad\displaystyle\qquad\qquad\qquad\qquad+f_{I}(k,n-k)^{2}\Big{)}\end{array}$
as we claimed. ∎
###### Corollary 1
Let $I$ be a recursive shape index for phylogenetic trees and, for every
$n\geqslant 1$, let $I_{n}$ be the random variable that chooses a tree
$T\in\mathcal{T}_{n}$ and computes $I(T)$. Set
$\begin{array}[]{l}\varepsilon_{I}(a,b-1)=f_{I}(a,b)-f_{I}(a,b-1)\mbox{ for
every $a\geqslant 1$ and $b\geqslant 2$}\\\
R_{I}(n-1)=E_{Y}(I_{n})-E_{Y}(I_{n-1})\mbox{ for every $n\geqslant
2$}\end{array}$
If $E_{Y}(I_{1})=0$, then
$\begin{array}[]{l}\displaystyle
E_{Y}(I_{n}^{2})=\frac{n}{n-1}E_{Y}(I_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{I}(k,n-1-k)E_{Y}(I_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}f_{I}(n-1,1)E_{Y}(I_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(I_{k})R_{I}(n-k-1)\\\
\quad\qquad\displaystyle+\frac{f_{I}(n-1,1)^{2}}{n-1}+\frac{1}{n-1}\sum_{k=1}^{n-2}(f_{I}(k,n-k)^{2}-f_{I}(k,n-k-1)^{2}).\end{array}$
###### Proof
By Lemma 3,
$\begin{array}[]{rl}E_{Y}(I_{n}^{2})&\displaystyle=\frac{2}{n-1}\sum_{k=1}^{n-1}E_{Y}(I^{2}_{k})+\frac{4}{n-1}\sum_{k=1}^{n-1}f_{I}(k,n-k)E_{Y}(I_{k})\\\
&\qquad\displaystyle+\frac{2}{n-1}\sum_{k=1}^{n-1}E_{Y}(I_{k})E_{Y}(I_{n-k})+\frac{1}{n-1}\sum_{k=1}^{n-1}f_{I}(k,n-k)^{2},\end{array}$
and in particular
$\begin{array}[]{rl}E_{Y}(I_{n-1}^{2})&\displaystyle=\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(I^{2}_{k})+\frac{4}{n-2}\sum_{k=1}^{n-2}f_{I}(k,n-1-k)E_{Y}(I_{k})\\\
&\quad\displaystyle+\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(I_{k})E_{Y}(I_{n-1-k})+\frac{1}{n-2}\sum_{k=1}^{n-2}f_{I}(k,n-k-1)^{2}.\end{array}$
Therefore
$\begin{array}[]{l}\displaystyle
E_{Y}(I_{n}^{2})=\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(I^{2}_{k})+\frac{2}{n-1}E_{Y}(I_{n-1}^{2})\\\
\quad\qquad\displaystyle+\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-2}E_{Y}(I_{k})(f_{I}(k,n-1-k)+\varepsilon_{I}(k,n-1-k))\\\
\quad\qquad\qquad\displaystyle+\frac{4}{n-1}f_{I}(n-1,1)E_{Y}(I_{n-1})\\\
\quad\qquad\displaystyle+\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(I_{k})(E_{Y}(I_{n-1-k})+R_{I}(n-k-1))\\\
\quad\qquad\qquad\displaystyle+\frac{2}{n-1}E_{Y}(I_{n-1})E_{Y}(I_{1})\\\
\quad\qquad\displaystyle+\frac{n-2}{n-1}\cdot\frac{1}{n-2}\sum_{k=1}^{n-2}f_{I}(k,n-k-1)^{2}+\frac{1}{n-1}\sum_{k=1}^{n-1}f_{I}(k,n-k)^{2}\\\
\quad\qquad\qquad\displaystyle-\frac{n-2}{n-1}\cdot\frac{1}{n-2}\sum_{k=1}^{n-2}f_{I}(k,n-k-1)^{2}\\\\[8.61108pt]
\quad\displaystyle=\frac{n-2}{n-1}E_{Y}(I_{n-1}^{2})+\frac{2}{n-1}E_{Y}(I_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{I}(k,n-1-k)E_{Y}(I_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}f_{I}(n-1,1)E_{Y}(I_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(I_{k})R_{I}(n-k-1)\\\
\quad\qquad\displaystyle+\frac{1}{n-1}\sum_{k=1}^{n-1}f_{I}(k,n-k)^{2}-\frac{1}{n-1}\sum_{k=1}^{n-2}f_{I}(k,n-k-1)^{2}\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(I_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{I}(k,n-1-k)E_{Y}(I_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}f_{I}(n-1,1)E_{Y}(I_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(I_{k})R_{I}(n-k-1)\\\
\quad\qquad\displaystyle+\frac{1}{n-1}\sum_{k=1}^{n-2}(f_{I}(k,n-k)^{2}-f_{I}(k,n-k-1)^{2})+\frac{1}{n-1}\cdot
f_{I}(n-1,1)^{2}\end{array}$
as we claimed. ∎
## 4 The variance of Sackin’s index
The _Sackin index_ of a phylogenetic tree $T\in\mathcal{T}_{n}$ is defined as
the sum of the depths of its leaves:
$S(T)=\sum_{i=1}^{n}\delta_{T}(i).$
It is well known (see, for instance, [15, Eq. (6)]) that if
$T_{k}\in\mathcal{T}(S_{k})$, for some $\emptyset\neq
S_{k}\subsetneq\\{1,\ldots,n\\}$, and
$T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})$, then
$S(T_{k}\widehat{\ }\,T^{\prime}_{n-k})=S(T_{k})+S(T^{\prime}_{n-k})+n.$
Let $S_{n}$ be the random variable that chooses a tree $T\in\mathcal{T}_{n}$
and computes $S(T)$. Its expected value under the Yule model is [8]
$E_{Y}(S_{n})=2n(H_{n}-1).$
In particular $E_{Y}(S_{1})=0$. Actually, the Sackin index of a tree with only
one node is 0. Notice moreover that $E_{Y}(S_{n})$ satisfies the recurrence
$E_{Y}(S_{n+1})=E_{Y}(S_{n})+2H_{n}.$
Indeed,
$\begin{array}[]{rl}E_{Y}(S_{n+1})-E_{Y}(S_{n})&=2(n+1)(H_{n+1}-1)-2n(H_{n}-1)\\\
&\displaystyle=2(n+1)(H_{n}+\frac{1}{n+1}-1)-2n(H_{n}-1)=2H_{n}.\end{array}$
In this section we prove that the variance of $S_{n}$ under this model is (see
Cor. 2)
$\sigma^{2}_{Y}(S_{n})=7n^{2}-4n^{2}H_{n}^{(2)}-2nH_{n}-n.$
###### Theorem 4.1
$\displaystyle
E_{Y}(S_{n}^{2})=4n^{2}(H_{n}^{2}-H_{n}^{(2)}-2H_{n})-2nH_{n}+11n^{2}-n$
###### Proof
As we have seen, Sackin’s index satisfies the hypotheses in Corollary 1, with
$f_{S}(k,n-k)=n$, and hence $\varepsilon_{S}(k,n-k-1)=1$, and
$R_{S}(k)=2H_{k}$. Therefore
$\begin{array}[]{l}\displaystyle
E_{Y}(S_{n}^{2})=\frac{n}{n-1}E_{Y}(S_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}nE_{Y}(S_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})2H_{n-k-1}\\\
\quad\qquad\displaystyle+\frac{n^{2}}{n-1}+\frac{1}{n-1}\sum_{k=1}^{n-2}(n^{2}-(n-1)^{2})\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(S_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})+8n(H_{n-1}-1)\\\
\quad\qquad\displaystyle+\frac{4}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})H_{n-k-1}+3n-2\end{array}$
Now, by Lemma 2,
$\begin{array}[]{l}\displaystyle\frac{4}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})=\frac{8}{n-1}\sum_{k=1}^{n-2}k(H_{k}-1)\\\
\qquad\displaystyle=\frac{8}{n-1}\Big{(}\frac{1}{4}(n-1)(n-2)(2H_{n-1}-1)-\frac{1}{2}(n-1)(n-2)\Big{)}\\\
\qquad\displaystyle=2(n-2)(2H_{n-1}-3)\end{array}$
and
$\begin{array}[]{l}\displaystyle\frac{4}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})H_{n-k-1}=\frac{8}{n-1}\sum_{k=1}^{n-2}k(H_{k}-1)H_{n-k-1}\\\
\qquad\displaystyle=\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}-\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{n-k-1}\\\
\qquad\displaystyle=\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}-\frac{8}{n-1}\sum_{k=1}^{n-2}(n-k-1)H_{k}\\\
\qquad\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)}-2H_{n}+2)-8\sum_{k=1}^{n-2}H_{k}+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}\\\
\qquad\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)}-2H_{n}+2)-8(n-1)(H_{n-1}-1)\\\
\qquad\qquad\displaystyle+2(n-2)(2H_{n-1}-1)\\\
\qquad\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)}-2H_{n-1}-2\cdot\frac{1}{n}+2)-4nH_{n-1}+6n-4\\\
\qquad\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)}-3H_{n-1})+14n-12\end{array}$
and thus
$\begin{array}[]{rl}\displaystyle
E_{Y}(S_{n}^{2})&\displaystyle=\frac{n}{n-1}E_{Y}(S_{n-1}^{2})+2(n-2)(2H_{n-1}-3)+8n(H_{n-1}-1)\\\
&\qquad\qquad\displaystyle+4n(H_{n}^{2}-H_{n}^{(2)}-3H_{n-1})+14n-12+3n-2\\\
&\displaystyle=\frac{n}{n-1}E_{Y}(S_{n-1}^{2})+4n(H_{n}^{2}-H_{n}^{(2)})-8H_{n-1}+3n-2\end{array}$
Setting $x_{n}=E_{Y}(S_{n}^{2})/n$, this equation becomes
$x_{n}=x_{n-1}+4(H_{n}^{2}-H_{n}^{(2)})-8\frac{H_{n-1}}{n}+3-\frac{2}{n}$
The solution of this recursive equation with $x_{1}=0$ is
$\begin{array}[]{rl}x_{n}&\displaystyle=\sum_{k=2}^{n}\Big{(}4(H_{k}^{2}-H_{k}^{(2)})-8\frac{H_{k-1}}{k}+3-\frac{2}{k}\Big{)}\\\
&\displaystyle=4\sum_{k=2}^{n}(H_{k}^{2}-H_{k}^{(2)})-8\sum_{k=1}^{n-1}\frac{H_{k}}{k+1}+3(n-1)-2\sum_{k=2}^{n}\frac{1}{k}\\\
&\displaystyle=4\sum_{k=2}^{n}(H_{k}^{2}-H_{k}^{(2)})-4(H_{n}^{2}-H_{n}^{(2)})+3(n-1)-2(H_{n}-1)\\\
&\displaystyle=4\sum_{k=2}^{n-1}(H_{k}^{2}-H_{k}^{(2)})-2H_{n}+3n-1\\\
&\displaystyle=4(nH_{n}^{2}-(2n+1)H_{n}+2n-nH_{n}^{(2)}+H_{n})-2H_{n}+3n-1\\\
&\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)}-2H_{n})-2H_{n}+11n-1\end{array}$
and therefore
$E_{Y}(S_{n}^{2})=nx_{n}=4n^{2}(H_{n}^{2}-H_{n}^{(2)}-2H_{n})-2nH_{n}+11n^{2}-n$
as we claimed. ∎
###### Corollary 2
The variance of $S_{n}$ under the Yule model is
$\sigma^{2}_{Y}(S_{n})=7n^{2}-4n^{2}H_{n}^{(2)}-2nH_{n}-n.$
###### Proof
This formula is obtained by replacing
$\begin{array}[]{rl}E_{Y}(S_{n}^{2})&=4n^{2}(H_{n}^{2}-H_{n}^{(2)}-2H_{n})-2nH_{n}+11n^{2}-n\\\
E_{Y}(S_{n})&=2n(H_{n}-1)\end{array}$
in the identity $\sigma^{2}_{Y}(S_{n})=E_{Y}(S_{n}^{2})-E_{Y}(S_{n})^{2}$. ∎
From this exact formula we can obtain an $O(1/n)$ approximation of
$\sigma^{2}_{Y}(S_{n})$, which refines the limit formula obtained in [1].
###### Corollary 3
$\displaystyle\sigma^{2}_{Y}(S_{n})=\Big{(}7-\frac{2\pi^{2}}{3}\Big{)}n^{2}+n(3-2\ln(n)-2\gamma)-3+O\Big{(}\frac{1}{n}\Big{)}.$
## 5 The variance of the total cophenetic index $\Phi$
The _total cophenetic index_ of a phylogenetic tree $T\in\mathcal{T}_{n}$ is
defined as the sum of the cophenetic values of its pairs of leaves:
$\Phi(T)=\sum_{1\leqslant i<j\leqslant n}\varphi_{T}(i,j).$
By [12, Lem. 4], if $T_{k}\in\mathcal{T}(S_{k})$, for some $\emptyset\neq
S_{k}\subsetneq\\{1,\ldots,n\\}$ with $k$ elements, and
$T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})$, then
$\Phi(T_{k}\widehat{\
}\,{}T_{n-k})=\Phi(T_{k})+\Phi(T_{n-k})+\binom{k}{2}+\binom{n-k}{2}.$
Therefore, $\Phi$ is a recursive shape index with
$f_{\Phi}(k,n-k)=\binom{k}{2}+\binom{n-k}{2}$, and in particular
$\varepsilon_{\Phi}(k,n-k-1)=n-k-1$.
Let $\Phi_{n}$ be the random variable that chooses a tree
$T\in\mathcal{T}_{n}$ and computes its total cophenetic index $\Phi(T)$. The
expected value under the Yule model of $\Phi_{n}$ is [12]
$E_{Y}(\Phi_{n})=n(n-1)-2n(H_{n}-1)=n(n+1-2H_{n}).$
In particular, $E_{Y}(\Phi_{1})=0$. Actually, the total cophenetic index of a
tree with only one node is 0. Moreover, we have that
$E_{Y}(\Phi_{n})=E_{Y}(\Phi_{n-1})+2(n-1-H_{n-1}),$
and therefore $R(k)=2(k-H_{k})$.
In this section we prove that the variance of $\Phi_{n}$ under this model is
(see Cor. 4)
$\sigma^{2}_{Y}(\Phi_{n})=\frac{1}{12}(n^{4}-10n^{3}+131n^{2}-2n)-4n^{2}H_{n}^{(2)}-6nH_{n}$
###### Theorem 5.1
$\begin{array}[]{rl}E_{Y}(\Phi_{n}^{2})&=\displaystyle\frac{1}{12}(13n^{4}+14n^{3}+143n^{2}-2n)+4n^{2}(H_{n}^{2}-H_{n}^{(2)})\\\\[8.61108pt]
&\quad-2(2n^{3}+2n^{2}+3n)H_{n}\end{array}$
###### Proof
As we have seen, $\Phi$ satisfies the hypotheses of Corollary 1, with
$\varepsilon_{\Phi}(k,n-k-1)=n-k-1,\quad R(k)=2(k-H_{k}).$
Therefore, by the aforementioned result,
$\begin{array}[]{l}\displaystyle
E_{Y}(\Phi_{n}^{2})=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}(n-k-1)E_{Y}(\Phi_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}\binom{n-1}{2}E_{Y}(\Phi_{n-1})+\frac{1}{n-1}\binom{n-1}{2}^{2}\\\
\quad\qquad\displaystyle+\frac{2}{n-1}\sum_{k=1}^{n-2}2E_{Y}(\Phi_{k})((n-k-1)-H_{n-k-1})\\\
\quad\qquad\displaystyle+\frac{1}{n-1}\sum_{k=1}^{n-2}\Big{(}\Big{(}\binom{k}{2}+\binom{n-k}{2}\Big{)}^{2}-\Big{(}\binom{k}{2}+\binom{n-k-1}{2}\Big{)}^{2}\Big{)}\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})+\frac{8}{n-1}\sum_{k=1}^{n-2}(n-k-1)(k^{2}+k-2kH_{k})\\\
\quad\qquad\displaystyle+2(n-2)(n-1)(n-2H_{n-1})\\\
\qquad\quad\displaystyle-\frac{4}{n-1}\sum_{k=1}^{n-2}H_{n-k-1}(k^{2}+k-2kH_{k})+\frac{1}{12}(n-2)(7n^{2}-21n+12)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})-4(n-2)(n-1)H_{n-1}\\\
\qquad\quad\displaystyle-16\sum_{k=1}^{n-2}kH_{k}+\frac{16}{n-1}\sum_{k=1}^{n-2}k^{2}H_{k}\\\
\qquad\quad\displaystyle-\frac{4}{n-1}\sum_{k=1}^{n-2}k^{2}H_{n-k-1}-\frac{4}{n-1}\sum_{k=1}^{n-2}kH_{n-k-1}\\\
\qquad\quad\displaystyle+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}+\frac{1}{12}(n-2)(39n^{2}-37n+12)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})-4(n-2)(n-1)H_{n-1}\\\
\qquad\quad\displaystyle-16\sum_{k=1}^{n-2}kH_{k}+\frac{16}{n-1}\sum_{k=1}^{n-2}k^{2}H_{k}\\\
\qquad\quad\displaystyle-\frac{4}{n-1}\sum_{k=1}^{n-2}(n-k-1)^{2}H_{k}-\frac{4}{n-1}\sum_{k=1}^{n-2}(n-k-1)H_{k}\\\
\qquad\quad\displaystyle+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}+\frac{1}{12}(n-2)(39n^{2}-37n+12)\\\\[8.61108pt]
\end{array}$
$\begin{array}[]{l}\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})-4(n-2)(n-1)H_{n-1}\\\
\qquad\quad\displaystyle-16\sum_{k=1}^{n-2}kH_{k}+\frac{16}{n-1}\sum_{k=1}^{n-2}k^{2}H_{k}+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}\\\
\qquad\quad\displaystyle-4(n-1)\sum_{k=1}^{n-2}H_{k}+8\sum_{k=1}^{n-2}kH_{k}-\frac{4}{n-1}\sum_{k=1}^{n-2}k^{2}H_{k}-4\sum_{k=1}^{n-2}H_{k}\\\
\qquad\quad\displaystyle+\frac{4}{n-1}\sum_{k=1}^{n-2}kH_{k}+\frac{1}{12}(n-2)(39n^{2}-37n+12)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})-4(n-2)(n-1)H_{n-1}\\\
\qquad\quad\displaystyle+\frac{12}{n-1}\sum_{k=1}^{n-2}k^{2}H_{k}+\frac{12-8n}{n-1}\sum_{k=1}^{n-2}kH_{k}-4n\sum_{k=1}^{n-2}H_{k}\\\
\qquad\quad\displaystyle+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}+\frac{1}{12}(n-2)(39n^{2}-37n+12)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})-4(n-2)(n-1)H_{n-1}\\\
\qquad\quad\displaystyle+\frac{12}{n-1}\cdot\frac{1}{36}(n-1)(n-2)\big{(}(12n-18)H_{n-1}-4n+3\big{)}\\\
\qquad\quad\displaystyle+\frac{12-8n}{n-1}\cdot\frac{1}{4}(n-1)(n-2)(2H_{n-1}-1)-4n(n-1)(H_{n-1}-1)\\\
\qquad\quad\displaystyle+\frac{8}{n-1}\binom{n}{2}(H_{n}^{2}-H_{n}^{(2)}-2H_{n}+2)+\frac{1}{12}(n-2)(39n^{2}-37n+12)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})+4n(H_{n}^{2}-H_{n}^{(2)})-8nH_{n}\\\
\qquad\quad\displaystyle-8(n-1)^{2}H_{n-1}+\frac{1}{12}(39n^{3}-59n^{2}+94n+24)\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\Phi_{n-1}^{2})+4n(H_{n}^{2}-H_{n}^{(2)})-8(n^{2}-n+1)H_{n-1}\\\
\qquad\quad\displaystyle+\frac{1}{12}(39n^{3}-59n^{2}+94n-72)\end{array}$
Setting $x_{n}=E_{Y}(\Phi_{n}^{2})/n$, this equation becomes
$x_{n}=x_{n-1}+4(H_{n}^{2}-H_{n}^{(2)})-8\Big{(}n-1+\frac{1}{n}\Big{)}H_{n-1}+\frac{1}{12}\Big{(}39n^{2}-59n+94-\frac{72}{n}\Big{)}$
The solution of this recursive equation with $x_{1}=0$ is
$\begin{array}[]{rl}x_{n}&\displaystyle=\sum_{k=2}^{n}\Big{(}4(H_{k}^{2}-H_{k}^{(2)})-8\Big{(}k-1+\frac{1}{k}\Big{)}H_{k-1}+\frac{1}{12}\Big{(}39k^{2}-59k+94-\frac{72}{k}\Big{)}\Big{)}\\\
&\displaystyle=4\sum_{k=2}^{n}H_{k}^{2}-4\sum_{k=2}^{n}H_{k}^{(2)}-8\sum_{k=1}^{n-1}kH_{k}-8\sum_{k=1}^{n-1}\frac{H_{k}}{k+1}\\\
&\qquad\displaystyle+\frac{1}{12}\sum_{k=2}^{n}\Big{(}39k^{2}-59k+94-\frac{72}{k}\Big{)}\\\
&\displaystyle=4\sum_{k=2}^{n-1}H_{k}^{2}-4\sum_{k=2}^{n-1}H_{k}^{(2)}-8\sum_{k=1}^{n-1}kH_{k}+\frac{1}{12}\sum_{k=2}^{n}\Big{(}39k^{2}-59k+94-\frac{72}{k}\Big{)}\end{array}$
$\begin{array}[]{rl}\hphantom{x_{n}}&\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)})-8nH_{n}+8n-2n(n-1)(2H_{n}-1)-6H_{n}\\\
&\qquad\displaystyle+\frac{1}{12}(13n^{3}-10n^{2}+71n-2)\\\
&\displaystyle=4n(H_{n}^{2}-H_{n}^{(2)})-2(2n^{2}+2n+3)H_{n}+\frac{1}{12}(13n^{3}+14n^{2}+143n-2)\\\
\end{array}$
Therefore
$\begin{array}[]{rl}E_{Y}(\Phi_{n}^{2})&\displaystyle=nx_{n}=4n^{2}(H_{n}^{2}-H_{n}^{(2)})-2(2n^{3}+2n^{2}+3n)H_{n}\\\
&\qquad\displaystyle+\frac{1}{12}(13n^{4}+14n^{3}+143n^{2}-2n)\end{array}$
as we claimed. ∎
###### Corollary 4
The covariance of $\Phi_{n}$ under the Yule model is
$\sigma^{2}_{Y}(\Phi_{n})=\frac{1}{12}(n^{4}-10n^{3}+131n^{2}-2n)-4n^{2}H_{n}^{(2)}-6nH_{n}$
###### Proof
Simply replace in the formula
$\sigma^{2}_{Y}(\Phi_{n})=E_{Y}(\Phi_{n}^{2})-E_{Y}(\Phi_{n})^{2}$ the value
of $E_{Y}(\Phi_{n}^{2})$ obtained in the last theorem and the value of
$E_{Y}(\Phi_{n})$ recalled above. ∎
###### Corollary 5
$\begin{array}[]{rl}\sigma^{2}_{Y}(\Phi_{n})&=\displaystyle\frac{1}{12}n^{4}-\frac{5}{6}n^{3}+\Big{(}\frac{131}{12}-\frac{2\pi^{2}}{3}\Big{)}n^{2}-6n\ln(n)+\Big{(}\frac{23}{6}-6\gamma)n-5\\\
&\quad\displaystyle+O\Big{(}\frac{1}{n}\Big{)}\end{array}$
## 6 The variance of $\overline{\Phi}$
For every $T\in\mathcal{T}_{n}$, let
$\overline{\Phi}(T)=S(T)+\Phi(T)=\sum\limits_{1\leqslant i\leqslant j\leqslant
n}\varphi_{T}(i,j)$
###### Lemma 4
If $T_{k}\in\mathcal{T}(S_{k})$, with $\emptyset\neq
S_{k}\subsetneq\\{1,\ldots,n\\}$ and $|S_{k}|=k$, and
$T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})$, then
$\overline{\Phi}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})=\overline{\Phi}(T_{k})+\overline{\Phi}(T^{\prime}_{n-k})+\binom{k+1}{2}+\binom{n-k+1}{2}.$
###### Proof
Since $S(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})=S(T_{k})+S(T^{\prime}_{n-k})+n$ and
$\Phi(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})=\Phi(T_{k})+\Phi(T^{\prime}_{n-k})+\binom{k}{2}+\binom{n-k}{2},$
we have that
$\begin{array}[]{rl}\overline{\Phi}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})&\displaystyle=\overline{\Phi}(T_{k})+\overline{\Phi}(T^{\prime}_{n-k})+\binom{k}{2}+\binom{n-k}{2}+n\\\
&\displaystyle=\overline{\Phi}(T_{k})+\overline{\Phi}(T^{\prime}_{n-k})+\binom{k+1}{2}+\binom{n-k+1}{2}.\end{array}$
∎
So, $\overline{\Phi}$ is a recursive shape index for phylogenetic trees with
$f_{\overline{\Phi}}(a,b)=\binom{a+1}{2}+\binom{b+1}{2}$, and hence
$\varepsilon_{\overline{\Phi}}(a,b)=b+1$.
Let $\overline{\Phi}_{n}$ be the random variable that chooses a tree
$T\in\mathcal{T}_{n}$ and computes $\overline{\Phi}(T)$. Its expected value
under the Yule model is [12]
$E_{Y}(\overline{\Phi}_{n})=n(n-1).$
In particular, $E_{Y}(\overline{\Phi}_{1})=0$ (actually,
$\overline{\Phi}_{1}=0$) and $R_{\overline{\Phi}}(k)=2k$. In this section we
compute the variance of $\overline{\Phi}_{n}$.
In this section we prove that the variance of $\overline{\Phi}_{n}$ under this
model is (see Cor. 6)
$\sigma^{2}_{Y}(\overline{\Phi}_{n})=2\binom{n}{4}$
###### Theorem 6.1
$E_{Y}(\overline{\Phi}_{n}^{2})=\frac{1}{12}(13n^{4}-30n^{3}+23n^{2}-6n).$
###### Proof
$\overline{\Phi}$ is a recursive shape index for phylogenetic trees with
$\varepsilon_{\overline{\Phi}}(k,n-k-1)=n-k,\quad R(k)=2k.$
Then, by Corollary 1,
$\begin{array}[]{l}\displaystyle
E_{Y}(\overline{\Phi}_{n}^{2})=\frac{n}{n-1}E_{Y}(\overline{\Phi}_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}(n-k)E_{Y}(\overline{\Phi}_{k})+\frac{1}{n-1}\Big{(}\binom{n}{2}+1\Big{)}^{2}\\\
\qquad\displaystyle+\frac{4}{n-1}\Big{(}\binom{n}{2}+1\Big{)}E_{Y}(\overline{\Phi}_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}2E_{Y}(\overline{\Phi}_{k})(n-k-1)\\\
\qquad\displaystyle+\frac{1}{n-1}\sum_{k=1}^{n-2}\Big{(}\Big{(}\binom{k+1}{2}+\binom{n-k+1}{2}\Big{)}^{2}-\Big{(}\binom{k+1}{2}+\binom{n-k}{2}\Big{)}^{2}\Big{)}\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\overline{\Phi}_{n-1}^{2})\\\
\qquad\displaystyle+\frac{4}{n-1}\sum_{k=1}^{n-2}(n-k)k(k-1)+\frac{4}{n-1}\Big{(}\binom{n}{2}+1\Big{)}(n-1)(n-2)\\\
\qquad\displaystyle+\frac{4}{n-1}\sum_{k=1}^{n-2}k(k-1)(n-k-1)+\frac{1}{n-1}\Big{(}\binom{n}{2}+1\Big{)}^{2}\\\
\qquad\displaystyle+\frac{1}{n-1}\sum_{k=1}^{n-2}(2(n-k)\Big{(}\binom{k+1}{2}+\binom{n-k}{2}\Big{)}+(n-k)^{2})\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(\overline{\Phi}_{n-1}^{2})+\frac{1}{4}n(13n^{2}-33n+22)\end{array}$
Setting $x_{n}=E_{Y}(\overline{\Phi}_{n}^{2})/n$, this recurrence becomes
$x_{n}=x_{n-1}+\frac{1}{4}(13n^{2}-33n+22)$
and the solution of this recursive equation with $x_{1}=0$ is
$x_{n}=\frac{1}{12}(13n^{3}-30n^{2}+23n-6)$
from where we deduce that
$E_{Y}(\overline{\Phi}_{n}^{2})=nx_{n}=\frac{1}{12}(13n^{4}-30n^{3}+23n^{2}-6n)$
as we claimed. ∎
###### Corollary 6
The variance of $\overline{\Phi}_{n}$ under the Yule model is
$\sigma^{2}_{Y}(\overline{\Phi}_{n})=2\binom{n}{4}$
###### Proof
Simply apply that
$\sigma^{2}_{Y}(\overline{\Phi}_{n})=E_{Y}(\overline{\Phi}_{n}^{2})-E_{Y}(\overline{\Phi}_{n})^{2}$.
∎
## 7 The variance of Colless’ index
The _Colless index_ $C(T)$ of a phylogenetic tree $T\in\mathcal{T}_{n}$ is
defined as the sum, over all its internal nodes $v$, of the absolute value of
the difference between the number of descendant leaves of $v$’s children. In
other words, if for every internal node $v$ we let $v_{1},v_{2}$ denote its
children, then
$C(T)=\sum_{v\in V_{int}(T)}|\kappa_{T}(v_{1})-\kappa_{T}(v_{2})|$
It is well known (see, for instance, [15, p. 100]) that if
$T_{k}\in\mathcal{T}(S_{k})$, for some $\emptyset\neq
S_{k}\subsetneq\\{1,\ldots,n\\}$ with $k$ elements, and
$T^{\prime}_{n-k}\in\mathcal{T}(\\{1,\ldots,n\\}\setminus S_{k})$, then
$C(T_{k}\widehat{\ }\,T^{\prime}_{n-k})=C(T_{k})+C(T^{\prime}_{n-k})+|n-2k|.$
In particular, it is a recursive shape index with
$f_{C}(a,b)=|b-a|.$
We have, then,
$\varepsilon_{C}(a,b-1)=f_{C}(a,b)-f_{C}(a,b-1)=|b-a|-|b-1-a|=\left\\{\begin{array}[]{ll}1&\mbox{if
$b\geqslant a+1$}\\\ -1&\mbox{if $b\leqslant a$}\end{array}\right.$
Let $C_{n}$ be the random variable that chooses a tree $T\in\mathcal{T}_{n}$
and computes $C(T)$. Its expected value under the Yule model is [8]
$E_{Y}(C_{n})=n(H_{\lfloor\frac{n}{2}\rfloor}-1)+(n\mod
2)=nH_{\lfloor\frac{n}{2}\rfloor}-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}.$
In particular, $E_{Y}(C_{1})=0$ and $E_{Y}(C_{n})$ satisfies the recurrence
$E_{Y}(C_{n+1})=E_{Y}(C_{n})+H_{\lfloor\frac{n}{2}\rfloor}.$
Indeed
$\begin{array}[]{rl}E_{Y}(C_{n+1})-E_{Y}(C_{n})&=(n+1)(H_{\lfloor\frac{n+1}{2}\rfloor}-1)+((n+1)\mod
2)\\\ &\qquad-n(H_{\lfloor\frac{n}{2}\rfloor}-1)-(n\mod 2)=(*)\end{array}$
Now we distinguish two cases, depending on the parity of $n$
* •
If $n$ is even
$(*)=(n+1)(H_{\frac{n}{2}}-1)+1-n(H_{\frac{n}{2}}-1)=H_{\frac{n}{2}}=H_{\lfloor\frac{n}{2}\rfloor}$
* •
If $n$ is odd
$\begin{array}[]{rl}(*)&=(n+1)(H_{\frac{n+1}{2}}-1)-n(H_{\frac{n-1}{2}}-1)-1\\\
&\displaystyle=(n+1)\Big{(}H_{\frac{n-1}{2}}+\frac{2}{n+1}\Big{)}-nH_{\frac{n-1}{2}}-2=H_{\frac{n-1}{2}}=H_{\lfloor\frac{n}{2}\rfloor}\end{array}$
In this section we prove that the variance of $C_{n}$ under this model is (see
Cor. 7)
$\begin{array}[]{l}\sigma_{Y}^{2}(C_{n}^{2})\displaystyle=\frac{5n^{2}+7n}{2}+(6n+1)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}^{2}+8\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}\\\
\qquad\displaystyle-8(n+1)\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}-6nH_{n}+\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-n(n-3)\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
\qquad\displaystyle-n^{2}H_{\lfloor\frac{n}{2}\rfloor}^{(2)}+\Big{(}n^{2}+3n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2nH_{\lfloor\frac{n}{4}\rfloor}\end{array}$
###### Theorem 7.1
$\displaystyle\begin{array}[]{l}E_{Y}(C_{n}^{2})\displaystyle=\frac{5n^{2}+7n}{2}+(6n+1)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}^{2}+8\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}\\\
\qquad\displaystyle+n^{2}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})-6nH_{n}+\Big{(}3n-n^{2}-(4n-2)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
\qquad\displaystyle+\Big{(}n^{2}+3n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2nH_{\lfloor\frac{n}{4}\rfloor}\end{array}$
###### Proof
As we have mentioned, Colless’ index satisfies the hypotheses in Corollary 1
with
$\begin{array}[]{c}R_{C}(n)=H_{\lfloor\frac{n}{2}\rfloor},\
f_{C}(k,n-k)=|n-2k|,\\\\[8.61108pt]
\varepsilon_{C}(k,n-k-1)=\left\\{\begin{array}[]{ll}1&\mbox{ if $2k\leqslant
n-1$}\\\ -1&\mbox{ if $2k>n-1$}\end{array}\right.\end{array}$
Therefore, by the aforementioned lemma, for $n\geqslant 2$
$\begin{array}[]{l}\displaystyle
E_{Y}(C_{n}^{2})=\frac{n}{n-1}E_{Y}(C_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{C}(k,n-1-k)E_{Y}(C_{k})\\\
\quad\qquad\displaystyle+\frac{4}{n-1}(n-2)E_{Y}(C_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(C_{k})H_{\lfloor\frac{n-k-1}{2}\rfloor}\\\
\quad\qquad\displaystyle+\frac{(n-2)^{2}}{n-1}+\frac{1}{n-1}\sum_{k=1}^{n-2}\Big{(}(n-2k)^{2}-(n-2k-1)^{2}\Big{)}\\\\[8.61108pt]
\quad\displaystyle=\frac{n}{n-1}E_{Y}(C_{n-1}^{2})+\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{C}(k,n-1-k)E_{Y}(C_{k})\\\
\quad\qquad\displaystyle+4(n-2)H_{\lfloor\frac{n-1}{2}\rfloor}+\frac{4(n-2)}{n-1}((n-1)\mod
2)\\\
\quad\qquad\displaystyle+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(C_{k})H_{\lfloor\frac{n-k-1}{2}\rfloor}-3(n-2)\end{array}$
We need to compute now
$X_{n}=\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{C}(k,n-1-k)E_{Y}(C_{k}),\
Y_{n}=\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(C_{k})H_{\lfloor\frac{n-k-1}{2}\rfloor}$
In both of them, we shall have to distinguish several cases, depending on the
parity of $n$.
1. (1)
Set
$X_{n}=\displaystyle\frac{4}{n-1}\sum_{k=1}^{n-2}\varepsilon_{C}(k,n-1-k)E_{Y}(C_{k})$.
Notice that
$\begin{array}[]{l}\varepsilon_{C}(k,n-k-1)=\left\\{\begin{array}[]{ll}1&\mbox{
if $k\leqslant(n-1)/2$}\\\ -1&\mbox{ if
$k>(n-1)/2$}\end{array}\right.\\\\[8.61108pt]
\varepsilon_{C}(k,n-k-2)=\left\\{\begin{array}[]{ll}1&\mbox{ if
$k\leqslant(n-2)/2$}\\\ -1&\mbox{ if
$k>(n-2)/2$}\end{array}\right.\end{array}$
Now, on the one hand, if $n-1$ is even,
$\varepsilon_{C}(k,n-k-1)-\varepsilon_{C}(k,n-k-2)=\left\\{\begin{array}[]{ll}0&\mbox{
if $k<(n-1)/2$}\\\ 2&\mbox{ if $k=(n-1)/2$}\\\ 0&\mbox{ if
$k>(n-1)/2$}\end{array}\right.$
Then,
$\begin{array}[]{rl}X_{n}&\displaystyle=\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-3}\varepsilon_{C}(k,n-k-1)E_{Y}(C_{k})\\\
&\displaystyle\qquad\qquad+\frac{4}{n-1}\varepsilon_{C}(n-2,1)E_{Y}(C_{n-2})\\\
&\displaystyle=\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-3}\varepsilon_{C}(k,n-k-2)E_{Y}(C_{k})+\frac{8}{n-1}E_{Y}(C_{\frac{n-1}{2}})\\\
&\displaystyle\qquad-\frac{4}{n-1}E_{Y}(C_{n-2})\\\
&\displaystyle=\frac{n-2}{n-1}X_{n-1}+\frac{8}{n-1}\Big{(}\frac{n-1}{2}(H_{\lfloor\frac{n-1}{4}\rfloor}-1)+\Big{(}\frac{n-1}{2}\mod
2\Big{)}\Big{)}\\\
&\qquad\qquad\displaystyle-\frac{4}{n-1}\big{(}(n-2)(H_{\lfloor\frac{n-2}{2}\rfloor}-1)+((n-2)\mod
2)\big{)}\\\
&\displaystyle=\frac{n-2}{n-1}X_{n-1}+4H_{\lfloor\frac{n-1}{4}\rfloor}-\frac{4(n-2)}{n-1}H_{\frac{n-3}{2}}\\\
&\qquad\qquad\displaystyle+\frac{8}{n-1}\Big{(}\Big{(}\frac{n-1}{2}\mod
2\Big{)}-1\Big{)}\end{array}$
On the other hand, if $n-1$ is odd, then
$\varepsilon_{C}(k,n-k-1)=\varepsilon_{C}(k,n-k-2)\quad\mbox{for every
$k=1,\ldots,n-3$}$
and therefore
$\begin{array}[]{rl}X_{n}&\displaystyle=\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-3}\varepsilon_{C}(k,n-k-1)\cdot
E_{Y}(C_{k})\\\
&\qquad\qquad\displaystyle+\frac{4}{n-1}\varepsilon_{C}(n-2,1)E_{Y}(C_{n-2})\\\
&\displaystyle=\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-3}\varepsilon_{C}(k,n-k-2)\cdot
E_{Y}(C_{k})\\\
&\qquad\qquad\displaystyle-\frac{4}{n-1}\big{(}(n-2)(H_{\lfloor\frac{n-2}{2}\rfloor}-1)+((n-2)\mod
2)\big{)}\\\
&\displaystyle=\frac{n-2}{n-1}X_{n-1}-\frac{4(n-2)}{n-1}(H_{\frac{n-2}{2}}-1)\end{array}$
Setting $x_{n}=(n-1)X_{n}$, we have
$x_{n}=\left\\{\begin{array}[]{ll}x_{n-1}-4(n-2)(H_{\frac{n-2}{2}}-1)&\ \mbox{
if $n$ is even}\\\
x_{n-1}+4(n-1)H_{\lfloor\frac{n-1}{4}\rfloor}-4(n-2)H_{\frac{n-3}{2}}&\\\
\qquad\qquad+8\big{(}(\frac{n-1}{2}\mod 2)-1\big{)}&\ \mbox{ if $n$ is odd}\\\
\end{array}\right.$
Iterating these recurrences we obtain that:
* •
If $n$ is even
$\begin{array}[]{rl}x_{n}&=x_{n-2}+4(n-2)H_{\lfloor\frac{n-2}{4}\rfloor}-4(n-3)H_{\frac{n-4}{2}}\\\
&\qquad\qquad+8\big{(}(\frac{n-2}{2}\mod
2)-1\big{)}-4(n-2)(H_{\frac{n-2}{2}}-1)\\\
&=x_{n-2}+4(n-2)H_{\lfloor\frac{n-2}{4}\rfloor}-4(n-3)H_{\frac{n-4}{2}}\\\
&\qquad\qquad+8\big{(}(\frac{n-2}{2}\mod
2)-1\big{)}-4(n-2)(H_{\frac{n-4}{2}}+\frac{2}{n-2}-1)\\\
&=x_{n-2}+4(n-2)H_{\lfloor\frac{n-2}{4}\rfloor}-4(2n-5)H_{\frac{n-4}{2}}+4(n-6)\\\
&\qquad\qquad+8(\frac{n-2}{2}\mod 2)\end{array}$
* •
If $n$ is odd
$\begin{array}[]{rl}x_{n}&=x_{n-2}-4(n-3)(H_{\frac{n-3}{2}}-1)+4(n-1)H_{\lfloor\frac{n-1}{4}\rfloor}\\\
&\qquad\qquad-4(n-2)H_{\frac{n-3}{2}}+8\big{(}(\frac{n-1}{2}\mod
2)-1\big{)}\\\
&=x_{n-2}+4(n-1)H_{\lfloor\frac{n-1}{4}\rfloor}-4(2n-5)H_{\frac{n-3}{2}}+4(n-5)\\\
&\qquad\qquad+8(\frac{n-1}{2}\mod 2)\end{array}$
From these recurrences, and noticing that $x_{1}=x_{2}=0$, we obtain that, for
every $m\geqslant 1$
$\begin{array}[]{l}x_{2m}\displaystyle=\sum_{k=1}^{m-1}\Big{(}8kH_{\lfloor\frac{k}{2}\rfloor}-4(4k-1)H_{k-1}+8(k-2)+8(k\mod
2)\Big{)}\\\
\quad\displaystyle=8\sum_{k=1}^{m-1}kH_{\lfloor\frac{k}{2}\rfloor}-16\sum_{k=1}^{m-1}(k-1)H_{k-1}-12\sum_{k=1}^{m-1}H_{k-1}+8\sum_{k=1}^{m-1}(k-2)\\\
\qquad\quad\displaystyle+8\sum_{k=1}^{m-1}(k\mod 2)\end{array}$
$\begin{array}[]{l}\quad\displaystyle=8\sum_{k=1}^{m-1}kH_{\lfloor\frac{k}{2}\rfloor}-16\sum_{k=1}^{m-2}kH_{k}-12\sum_{k=1}^{m-2}H_{k}+4(m-1)(m-4)+8\Big{\lfloor}\frac{m}{2}\Big{\rfloor}\\\
\quad\displaystyle=4m(m-1)H_{\lfloor\frac{m}{2}\rfloor}-8\Big{\lfloor}\frac{m}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{m}{2}\Big{\rfloor}-1\Big{)}-4(m-1)(2m-1)H_{m-1}\\\
\displaystyle\quad\qquad+4(m-1)(2m-3)\\\\[8.61108pt]
x_{2m+1}\displaystyle=\sum_{k=1}^{m}\Big{(}8kH_{\lfloor\frac{k}{2}\rfloor}-4(4k-3)H_{k-1}+8(k-2)+8(k\mod
2)\Big{)}\\\
\quad\displaystyle=8\sum_{k=1}^{m}kH_{\lfloor\frac{k}{2}\rfloor}-4\sum_{k=1}^{m}(4k-3)H_{k-1}+8\sum_{k=1}^{m}(k-2)+8\sum_{k=1}^{m}(k\mod
2)\\\
\quad\displaystyle=8\sum_{k=1}^{m}kH_{\lfloor\frac{k}{2}\rfloor}-16\sum_{k=1}^{m-1}kH_{k}-4\sum_{k=1}^{m-1}H_{k}+4m(m-3)+8\Big{\lfloor}\frac{m+1}{2}\Big{\rfloor}\\\
\quad\displaystyle=4m(m+1)H_{\lfloor\frac{m+1}{2}\rfloor}-8\Big{\lfloor}\frac{m+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{m+1}{2}\Big{\rfloor}-1\Big{)}-4m(2m-1)H_{m}\\\
\quad\qquad\displaystyle+4m(2m-3)\end{array}$
Both formulas correspond to:
$\begin{array}[]{rl}x_{n}&\displaystyle=4\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}-1)\Big{)}H_{\lfloor\frac{n+1}{4}\rfloor}-2(n-1)(n-2)H_{\lfloor\frac{n-1}{2}\rfloor}\\\\[8.61108pt]
&\qquad\qquad\displaystyle-8\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}-1\Big{)}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-3\Big{)}\end{array}$
Finally,
$\begin{array}[]{rl}X_{n}&=\displaystyle\frac{1}{n-1}x_{n}\\\\[8.61108pt]
&\displaystyle=\frac{1}{n-1}\Big{\\{}4\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}-1)\Big{)}H_{\lfloor\frac{n+1}{4}\rfloor}-2(n-1)(n-2)H_{\lfloor\frac{n-1}{2}\rfloor}\\\\[6.45831pt]
&\qquad\qquad\displaystyle-8\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}-1\Big{)}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-3\Big{)}\Big{\\}}\end{array}$
2. (2)
Set
$y_{n}=\sum_{k=1}^{n-2}E_{Y}(C_{k})H_{\lfloor\frac{n-k-1}{2}\rfloor}=\sum_{k=1}^{n-2}(k(H_{\lfloor\frac{k}{2}\rfloor}-1)+(k\mod
2))H_{\lfloor\frac{n-k-1}{2}\rfloor}$
so that $Y_{n}=2y_{n}/(n-1)$. If $n=2m$, then
$\begin{array}[]{l}y_{2m}\displaystyle=\sum_{j=1}^{m-1}2j(H_{j}-1)H_{m-j-1}+\sum_{j=0}^{m-2}((2j+1)(H_{j}-1)+1)H_{m-j-1}\\\
\quad=\displaystyle\sum_{j=1}^{m-2}2j(H_{j}-1)H_{m-j-1}+\sum_{j=1}^{m-2}(2j(H_{j}-1)+H_{j})H_{m-j-1}\\\
\quad=\displaystyle
4\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}-4\sum_{j=1}^{m-2}jH_{m-j-1}\\\
\quad=\displaystyle
4\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}-4\sum_{j=1}^{m-2}(m-1-j)H_{j}\end{array}$
$\begin{array}[]{l}\quad=\displaystyle
4\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}-4(m-1)\sum_{j=1}^{m-2}H_{j}+4\sum_{j=1}^{m-2}jH_{j}\\\
\quad\displaystyle=4\binom{m}{2}(H_{m}^{2}-H_{m}^{(2)}-2H_{m}+2)+m(H_{m}^{2}-H_{m}^{(2)}-2H_{m}+2)\\\
\quad\qquad\displaystyle-4(m-1)^{2}(H_{m-1}-1)+(m-1)(m-2)(2H_{m-1}-1)\\\
\quad\displaystyle=m(2m-1)(H_{m}^{2}-H_{m}^{(2)})-2m(3m-2)H_{m}+m(7m-5)\end{array}$
If $n=2m+1$, then
$\begin{array}[]{l}y_{2m+1}\displaystyle=\sum_{j=1}^{m-1}2j(H_{j}-1)H_{m-j}+\sum_{j=0}^{m-1}((2j+1)(H_{j}-1)+1)H_{m-j-1}\\\
\quad=\displaystyle
2\sum_{j=1}^{m-1}jH_{j}H_{m-j}-2\sum_{j=1}^{m-1}jH_{m-j}+2\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}\\\
\quad\displaystyle\qquad-2\sum_{j=1}^{m-2}jH_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}\\\
\quad=\displaystyle
2\sum_{j=1}^{m-1}jH_{j}H_{m-j}+2\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}\\\
\quad\displaystyle\qquad-2\sum_{j=1}^{m-1}(m-j)H_{j}-2\sum_{j=1}^{m-2}(m-1-j)H_{j}\\\
\quad=\displaystyle
2\sum_{j=1}^{m-1}jH_{j}H_{m-j}+2\sum_{j=1}^{m-2}jH_{j}H_{m-j-1}+\sum_{j=1}^{m-2}H_{j}H_{m-j-1}\\\
\quad\displaystyle\qquad-2(2m-1)\sum_{j=1}^{m-2}H_{j}+4\sum_{j=1}^{m-2}jH_{j}-2H_{m-1}\\\
\end{array}$ $\begin{array}[]{l}\quad=\displaystyle
2\binom{m+1}{2}(H_{m+1}^{2}-H_{m+1}^{(2)}-2H_{m+1}+2)\\\
\quad\displaystyle\qquad+2\binom{m}{2}(H_{m}^{2}-H_{m}^{(2)}-2H_{m}+2)\\\
\quad\displaystyle\qquad+m(H_{m}^{2}-H_{m}^{(2)}-2H_{m}+2)-2(2m-1)(m-1)(H_{m-1}-1)\\\
\quad\displaystyle\qquad+(m-1)(m-2)(2H_{m-1}-1)-2H_{m-1}\\\
\quad=\displaystyle
m(2m+1)(H_{m}^{2}-H_{m}^{(2)})-6m^{2}H_{m}+m(7m-1)\end{array}$
Both formulas can be summarized into
$\begin{array}[]{l}\displaystyle
y_{n}=\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+1\Big{)}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})\\\
\displaystyle\qquad-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}3\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}-1\Big{)}\end{array}$
and therefore
$\begin{array}[]{l}\displaystyle
Y_{n}=\frac{2}{n-1}\Big{\\{}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+1\Big{)}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})\\\
\displaystyle\qquad-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}3\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}-1\Big{)}\Big{\\}}\end{array}$
We can return now to our recursive formula for $E_{Y}(C_{n}^{2})$, which now
becomes
$\begin{array}[]{l}\displaystyle
E_{Y}(C_{n}^{2})=\frac{n}{n-1}E_{Y}(C_{n-1}^{2})+4(n-2)H_{\lfloor\frac{n-1}{2}\rfloor}-3(n-2)\\\
\qquad\displaystyle+\frac{4(n-2)}{n-1}((n-1)\mod 2)\\\
\qquad\displaystyle+\frac{1}{n-1}\Big{\\{}4\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}-1)\Big{)}H_{\lfloor\frac{n+1}{4}\rfloor}-2(n-1)(n-2)H_{\lfloor\frac{n-1}{2}\rfloor}\\\
\qquad\qquad\displaystyle-8\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}-1\Big{)}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-3\Big{)}\Big{\\}}\\\
\qquad\displaystyle+\frac{2}{n-1}\Big{\\{}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+1\Big{)}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})\\\
\displaystyle\qquad\qquad-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}3\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+4\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}-1\Big{)}\Big{\\}}\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(C_{n-1}^{2})+(n-2)(2H_{\lfloor\frac{n-1}{2}\rfloor}-3)\\\
\qquad\displaystyle+\frac{1}{n-1}\Big{\\{}4\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}-1)\Big{)}H_{\lfloor\frac{n+1}{4}\rfloor}-8\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}-1\Big{)}\\\
\qquad\qquad\displaystyle-12\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+1\Big{)}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})\\\
\qquad\qquad\displaystyle-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}3\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+8\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}-1\Big{)}\\\
\displaystyle\qquad\qquad+4(n-2)\big{(}(n-1)\mod 2\big{)}\Big{\\}}\end{array}$
Setting $z_{n}=E_{Y}(C_{n}^{2})/n$, this identity becomes
$\begin{array}[]{l}\displaystyle
z_{n}=z_{n-1}+\frac{n-2}{n}(2H_{\lfloor\frac{n-1}{2}\rfloor}-3)\\\
\qquad\displaystyle+\frac{1}{n(n-1)}\Big{\\{}4\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{2}\Big{\rfloor}-1)\Big{)}H_{\lfloor\frac{n+1}{4}\rfloor}-8\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n+1}{4}\Big{\rfloor}-1\Big{)}\\\
\qquad\qquad\displaystyle-12\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}+1\Big{)}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})\\\
\qquad\qquad\displaystyle-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{(}3\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+8\Big{\lfloor}\frac{n-1}{2}\Big{\rfloor}-1\Big{)}\\\
\displaystyle\qquad\qquad+4(n-2)\big{(}(n-1)\mod 2\big{)}\Big{\\}}\end{array}$
Writing this equation as $z_{n}=z_{n-1}+f(n)$, its solution with $z_{1}=0$ is
$z_{n}=\sum_{k=2}^{n}f(k)$
and it remains to compute this sum. To do it, we split it into eight terms,
$\begin{array}[]{l}\displaystyle
S_{1}(n)=\sum_{k=2}^{n}\frac{k-2}{k}\left(2H_{\lfloor\frac{k-1}{2}\rfloor}-3\right)\\\
\displaystyle
S_{2}(n)=4\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k+1}{2}\Big{\rfloor}\left(\Big{\lfloor}\frac{k+1}{2}\Big{\rfloor}-1\right)H_{\lfloor\frac{k+1}{4}\rfloor}\\\
\displaystyle
S_{3}(n)=8\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k+1}{4}\Big{\rfloor}\left(\Big{\lfloor}\frac{k+1}{4}\Big{\rfloor}-1\right)\end{array}$
$\begin{array}[]{l}\displaystyle
S_{4}(n)=12\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k-1}{2}\Big{\rfloor}\\\
\displaystyle
S_{5}(n)=2\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k}{2}\Big{\rfloor}\left(2\Big{\lfloor}\frac{k-1}{2}\Big{\rfloor}+1\right)\left(H_{\lfloor\frac{k}{2}\rfloor}^{2}-H_{\lfloor\frac{k}{2}\rfloor}^{(2)}\right)\\\
\displaystyle
S_{6}(n)=4\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k}{2}\Big{\rfloor}\left(\Big{\lfloor}\frac{k}{2}\Big{\rfloor}+2\Big{\lfloor}\frac{k-1}{2}\Big{\rfloor}\right)H_{\lfloor\frac{k}{2}\rfloor}\\\
\displaystyle
S_{7}(n)=2\sum_{k=2}^{n}\frac{1}{k(k-1)}\Big{\lfloor}\frac{k}{2}\Big{\rfloor}\left(3\Big{\lfloor}\frac{k}{2}\Big{\rfloor}+8\Big{\lfloor}\frac{k-1}{2}\Big{\rfloor}-1\right)\\\
\displaystyle S_{8}(n)=4\sum_{k=2}^{n}\frac{1}{k(k-1)}(k-2)((k-1)\mod
2)\end{array}$
in such a way that
$z_{n}=S_{1}(n)+S_{2}(n)-S_{3}(n)-S_{4}(n)+S_{5}(n)-S_{6}(n)+S_{7}(n)+S_{8}(n).$
Now, we compute each one of these sums. To simplify the results, set
${\cal S}_{m}=\sum_{l=1}^{m-1}\frac{H_{l}}{2l+1}.$
_Sum $S_{1}$_. We consider two cases, depending on the parity of $n$. If $n$
is even, say $n=2m$, then
$\begin{array}[]{l}\displaystyle
S_{1}(2m)=\sum_{l=1}^{m}\frac{l-1}{l}\left(2H_{l-1}-3\right)+\sum_{l=1}^{m-1}\frac{2l-1}{2l+1}\left(2H_{l}-3\right)\\\
\displaystyle\quad=\sum_{l=1}^{m-1}\left(2-\frac{1}{l+1}-\frac{2}{2l+1}\right)\left(2H_{l}-3\right)\\\
\displaystyle\quad=-10m-3+4mH_{m}+6H_{2m}-(H_{m}^{2}-H_{m}^{(2)})-4{\cal
S}_{m}\end{array}$
If $n$ is odd, say $n=2m+1$, then
$\begin{array}[]{l}S_{1}(2m+1)\displaystyle=S_{1}(2m)+\frac{2m-1}{2m+1}(2H_{m}-3)\\\
\quad\displaystyle=-10m-3+4mH_{m}+6H_{2m+1}-\frac{6}{2m+1}-(H_{m}^{2}-H_{m}^{(2)})-4{\cal
S}_{m}\\\
\qquad\qquad\displaystyle+\frac{4m-2}{2m+1}H_{m}-\frac{6m-3}{2m+1}\\\
\quad\displaystyle=-10m-6+\Big{(}4m+2-\frac{4}{2m+1}\Big{)}H_{m}+6H_{2m+1}-(H_{m}^{2}-H_{m}^{(2)})-4{\cal
S}_{m}\end{array}$
Both formulas agree with
$\begin{array}[]{rl}S_{1}(n)&\displaystyle=-3n-3-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+\Big{(}2n-4+\frac{8}{n}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}+6H_{n}\\\
&\qquad\displaystyle-(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})-4{\cal
S}_{\lfloor\frac{n}{2}\rfloor}\end{array}$
_Sum $S_{2}$_. We consider four cases, depending of the class of $n$ mod $4$.
If $n=4m-2$, then
$\begin{array}[]{l}\displaystyle
S_{2}(4m-2)\displaystyle=4\Big{(}\sum_{l=1}^{m}\frac{l-1}{4l-3}H_{l-1}+\sum_{l=1}^{m-1}\frac{l}{4l-1}H_{l}+\sum_{l=1}^{m-1}\frac{2l-1}{2(4l-1)}H_{l}\\\
\quad\qquad\quad\qquad\qquad\displaystyle+\sum_{l=1}^{m-1}\frac{2l+1}{2(4l+1)}H_{l}\Big{)}\\\
\quad\displaystyle=4\Big{(}\sum_{l=1}^{m-1}\frac{l}{4l+1}H_{l}+\sum_{l=1}^{m-1}\frac{l}{4l-1}H_{l}+\sum_{l=1}^{m-1}\frac{2l-1}{2(4l-1)}H_{l}+\sum_{l=1}^{m-1}\frac{2l+1}{2(4l+1)}H_{l}\Big{)}\\\
\quad\displaystyle=4\sum_{l=1}^{m-1}\Big{(}\frac{l}{4l+1}+\frac{l}{4l-1}+\frac{2l-1}{2(4l-1)}+\frac{2l+1}{2(4l+1)}\Big{)}H_{l}\\\
\quad\displaystyle=4\sum_{l=1}^{m-1}H_{l}=4m(H_{m}-1)\end{array}$
Now, if $n=4m-1$, then
$S_{2}(4m-1)=S_{2}(4m-2)+4\frac{2m(2m-1)}{(4m-1)(4m-2)}H_{m}=\frac{16m^{2}}{4m-1}H_{m}-4m$
If $n=4m$, then
$S_{2}(4m)=S_{2}(4m-1)+4\frac{2m(2m-1)}{(4m-1)4m}H_{m}=(4m+2)H_{m}-4m$
And finally, if $n=4m+1$, then
$S_{2}(4m+1)=S_{2}(4m)+4\frac{2m(2m+1)}{4m(4m+1)}H_{m}=\frac{(4m+2)^{2}}{4m+1}H_{m}-4m$
These four formulas agree with
$S_{2}(n)=\Big{(}n+3-\frac{2}{n}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-4\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}.$
_Sum $S_{3}$_. We consider the same four cases as in the previous sum. If
$n=4m-2$, then
$\begin{array}[]{l}\displaystyle
S_{3}(4m-2)=8\left(\sum_{l=1}^{m}\frac{(l-1)(l-2)}{(4l-2)(4l-3)}+\sum_{l=1}^{m-1}\frac{l(l-1)}{(4l-1)(4l-2)}\right.\\\
\displaystyle\qquad\qquad\left.+\sum_{l=1}^{m-1}\frac{l(l-1)}{4l(4l-1)}+\sum_{l=1}^{m-1}\frac{l(l-1)}{4l(4l+1)}\right)\\\
\displaystyle\qquad=8\left(\sum_{l=1}^{m-1}\frac{l(l-1)}{(4l+2)(4l+1)}+\sum_{l=1}^{m-1}\frac{l(l-1)}{(4l-1)(4l-2)}+\sum_{l=1}^{m-1}\frac{l(l-1)}{4l(4l-1)}\right.\\\
\displaystyle\qquad\qquad\left.+\sum_{l=1}^{m-1}\frac{l(l-1)}{4l(4l+1)}\right)\end{array}$
$\begin{array}[]{l}\displaystyle\qquad=8\sum_{l=1}^{m-1}\frac{l(l-1)}{4l^{2}-1}=\sum_{l=1}^{m-1}\Big{(}2-\frac{1}{2l-1}-\frac{3}{2l+1}\Big{)}\\\
\displaystyle\qquad=2H_{m-1}-4H_{2m-2}+\frac{4m^{2}-4}{2m-1}=2H_{m-1}-4H_{2m-1}+\frac{4m^{2}}{2m-1}\end{array}$
If $n=4m-1$, then
$\begin{array}[]{l}\displaystyle
S_{3}(4m-1)=S_{3}(4m-2)+\frac{8m(m-1)}{(4m-1)(4m-2)}\\\
\displaystyle\qquad=2H_{m-1}-4H_{2m-1}+\frac{4m^{2}}{2m-1}+\frac{8m(m-1)}{(4m-1)(4m-2)}\\\
\displaystyle\qquad=2H_{m-1}-4H_{2m-1}+\frac{4m(2m+1)}{4m-1}\end{array}$
If $n=4m$, then
$\begin{array}[]{l}\displaystyle
S_{3}(4m)=S_{3}(4m-1)+\frac{8m(m-1)}{4m(4m-1)}\\\
\displaystyle\qquad=2H_{m-1}-4H_{2m-1}+\frac{4m(2m+1)}{4m-1}+\frac{2(m-1)}{4m-1}\\\
\displaystyle\qquad=2H_{m}-4H_{2m}+2(m+1)\end{array}$
And if $n=4m+1$, then
$\begin{array}[]{l}\displaystyle
S_{3}(4m+1)=S_{3}(4m)+\frac{8m(m-1)}{4m(4m+1)}\\\
\displaystyle\qquad=2H_{m}-4H_{2m}+2(m+1)+\frac{2(m-1)}{4m+1}\\\
\displaystyle\qquad=2H_{m}-4H_{2m}+\frac{4m(2m+3)}{4m+1}\end{array}$
These four formulas correspond to
$S_{3}(n)=2H_{\lfloor\frac{n}{4}\rfloor}-4H_{\lfloor\frac{n}{2}\rfloor}+\frac{4}{n}\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}\Big{(}n+2-2\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}\Big{)}$
_Sum $S_{4}$_. If $n=2m$, then
$\begin{array}[]{l}\displaystyle
S_{4}(2m)=6\left(\sum_{l=1}^{m}\frac{l-1}{l(2l-1)}+\sum_{l=1}^{m-1}\frac{1}{2l+1}\right)\\\
\displaystyle\qquad=6\sum_{l=1}^{m-1}\left(\frac{l}{(l+1)(2l+1)}+\frac{1}{2l+1}\right)=6\sum_{l=1}^{m-1}\frac{1}{l+1}=6H_{m}-6\end{array}$
If $n=2m+1$, then
$\begin{array}[]{l}\displaystyle
S_{4}(2m+1)=S_{4}(2m)+12\cdot\frac{m}{2m(2m+1)}=6(H_{m}-1)+\frac{6}{2m+1}\\\
\displaystyle\qquad=6H_{m}-\frac{12m}{2m+1}\end{array}$
Both formulas agree with
$S_{4}(n)=6H_{\lfloor\frac{n}{2}\rfloor}-\frac{12}{n}\cdot\Big{\lfloor}\frac{n}{2}\Big{\rfloor}$
_Sum $S_{5}$_. If $n=2m$, then
$\begin{array}[]{l}\displaystyle
S_{5}(2m)=2\sum_{l=1}^{m}\frac{j(2j-1)}{2j(2j-1)}(H_{l}^{2}-H_{l}^{(2)})+2\sum_{l=1}^{m-1}\frac{j(2j+1)}{2j(2j+1)}(H_{l}^{2}-H_{l}^{(2)})\\\
\displaystyle\qquad=\sum_{l=1}^{m}(H_{l}^{2}-H_{l}^{(2)})+\sum_{l=1}^{m-1}(H_{l}^{2}-H_{l}^{(2)})\\\
\displaystyle\qquad=2\sum_{l=1}^{m-1}(H_{l}^{2}-H_{l}^{(2)})+H_{m}^{2}-H_{m}^{(2)}\\\
\displaystyle\qquad=(2m+1)(H_{m}^{2}-H_{m}^{(2)})-4mH_{m}+4m\end{array}$
If $n=2m+1$, then
$\begin{array}[]{l}\displaystyle
S_{5}(2m+1)=S_{5}(2m)+2\cdot\frac{m(2m+1)}{2m(2m+1)}(H_{m}^{2}-H_{m}^{(2)})\\\
\displaystyle\qquad=(2m+2)(H_{m}^{2}-H_{m}^{(2)})-4mH_{m}+4m\end{array}$
This shows that
$S_{5}(n)=(n+1)(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}(H_{\lfloor\frac{n}{2}\rfloor}-1)$
_Sum $S_{6}$_. If $n=2m$, then
$\begin{array}[]{l}\displaystyle
S_{6}(2m)=2\left(\sum_{l=1}^{m}\frac{3l-2}{2l-1}H_{l}+\sum_{l=1}^{m-1}\frac{3l}{2l+1}H_{l}\right)\\\
\displaystyle\qquad=\sum_{l=1}^{m-1}\left(6-\frac{1}{2l-1}-\frac{3}{2l+1}\right)H_{l}+\frac{2(3m-2)}{2m-1}H_{m}\\\
\displaystyle\qquad=6m(H_{m}-1)-\Big{(}{\cal
S}_{m}-\frac{4m-1}{2m-1}H_{m}+2H_{2m}\Big{)}-3{\cal
S}_{m}+\frac{6m-4}{2m-1}H_{m}\\\ \displaystyle\qquad=(6m+5)H_{m}-4{\cal
S}_{m}-2H_{2m}-6m\end{array}$
If $n=2m+1$, then
$\begin{array}[]{l}\displaystyle S_{6}(2m+1)=S_{6}(2m)+\frac{6m}{2m+1}H_{m}\\\
\displaystyle\qquad=(6m+5)H_{m}-4{\cal
S}_{m}-2H_{2m}-6m+\frac{6m}{2m+1}H_{m}\\\
\displaystyle\qquad=\left(6m+5+\frac{6m}{2m+1}\right)H_{m}-4{\cal
S}_{m}-2H_{2m+1}+\frac{2}{2m+1}-6m\end{array}$
Both formulas correspond to
$S_{6}(n)=\Big{(}3n+2+\frac{6}{n}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}-4{\cal
S}_{\lfloor\frac{n}{2}\rfloor}-2H_{n}-6\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+\frac{2}{n}\Big{(}n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}$
_Sum $S_{7}$_. If $n=2m$,
$\begin{array}[]{l}\displaystyle
S_{7}(2m)=\sum_{l=1}^{m}\frac{11l-9}{2l-1}+\sum_{l=1}^{m-1}\frac{11l-1}{2l+1}\\\
\displaystyle\qquad=\sum_{l=1}^{m-1}\left(11-\frac{1}{2}\left(\frac{7}{2l-1}+\frac{13}{2l+1}\right)\right)+\frac{11m-9}{2m-1}\\\
\displaystyle\qquad=11(m-1)-10\sum_{l=1}^{m}\frac{1}{2l-1}+\frac{7}{2(2m-1)}+\frac{13}{2}+\frac{11m-9}{2m-1}\\\
\displaystyle\qquad=11m+1+5H_{m}-10H_{2m}\end{array}$
If $n=2m+1$,
$\begin{array}[]{l}\displaystyle
S_{7}(2m+1)=S_{7}(2m)+2\cdot\frac{m(11m-1)}{2m(2m+1)}\\\
\displaystyle\qquad=11m+1+5H_{m}-10H_{2m}+\frac{11m-1}{2m+1}\\\
\displaystyle\qquad=5H_{m}-10H_{2m+1}+11m+1+\frac{11m+9}{2m+1}\end{array}$
Both formulas agree with
$S_{7}(n)=5H_{\lfloor\frac{n}{2}\rfloor}-10H_{n}+5n+\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+1+\frac{1}{2}\Big{(}1+\frac{7}{n}\Big{)}\Big{(}n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}$
_Sum $S_{8}$_. If $n=2m$,
$S_{8}(2m)=4\sum_{l=1}^{m}\frac{l-1}{l(2l-1)}=4\sum_{l=1}^{m}\left(\frac{1}{l}-\frac{1}{2l-1}\right)=6H_{m}-4H_{2m},$
and if $n=2m+1$,
$S_{8}(2m+1)=S(2m)+\frac{(2m-1)\cdot
0}{2m(2m+1)}=6H_{m}-4H_{2m}=6H_{m}-4H_{2m+1}+\frac{4}{2m+1}$
Therefore,
$S_{8}(n)=6H_{\lfloor\frac{n}{2}\rfloor}-4H_{n}+\frac{4}{n}\Big{(}n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}$
Now, once we have computed $S_{1},\ldots,S_{8}$, we can compute $z_{n}$:
$\begin{array}[]{rl}z_{n}&=S_{1}(n)+S_{2}(n)-S_{3}(n)-S_{4}(n)+S_{5}(n)-S_{6}(n)+S_{7}(n)+S_{8}(n)\\\
&\displaystyle=\frac{5n+7}{2}+\Big{(}6+\frac{1}{n}\Big{)}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+\frac{8}{n}\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}-\frac{8(n+1)}{n}\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}\\\
&\quad\displaystyle+n(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})-6H_{n}+\Big{(}3-n-\Big{(}4-\frac{2}{n}\Big{)}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
&\quad\displaystyle+\Big{(}n+3-\frac{2}{n}\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2H_{\lfloor\frac{n}{4}\rfloor}\end{array}$
And finally
$\displaystyle\begin{array}[]{l}E_{Y}(C_{n}^{2})=nz_{n}\\\
\quad\displaystyle=\frac{5n^{2}+7n}{2}+(6n+1)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}+8\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}-8(n+1)\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}\\\
\qquad\displaystyle+n^{2}(H_{\lfloor\frac{n}{2}\rfloor}^{2}-H_{\lfloor\frac{n}{2}\rfloor}^{(2)})-6nH_{n}+\Big{(}3n-n^{2}-(4n-2)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
\qquad\displaystyle+\Big{(}n^{2}+3n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2nH_{\lfloor\frac{n}{4}\rfloor}\end{array}$
as we claimed. ∎
###### Corollary 7
The variance of $C_{n}$ under the Yule model is
$\begin{array}[]{l}\sigma_{Y}^{2}(C_{n}^{2})\displaystyle=\frac{5n^{2}+7n}{2}+(6n+1)\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-4\Big{\lfloor}\frac{n}{2}\Big{\rfloor}^{2}+8\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}^{2}\\\
\qquad\displaystyle-8(n+1)\Big{\lfloor}\frac{n+2}{4}\Big{\rfloor}-6nH_{n}+\Big{(}2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}-n(n-3)\Big{)}H_{\lfloor\frac{n}{2}\rfloor}\\\
\qquad\displaystyle-n^{2}H_{\lfloor\frac{n}{2}\rfloor}^{(2)}+\Big{(}n^{2}+3n-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{)}H_{\lfloor\frac{n+2}{4}\rfloor}-2nH_{\lfloor\frac{n}{4}\rfloor}\end{array}$
###### Proof
Simply replace in the formula
$\sigma^{2}_{Y}(C_{n})=E_{Y}(C_{n}^{2})-E_{Y}(C_{n})^{2}$ the value of
$E_{Y}(C_{n}^{2})$ obtained in the last theorem and the value of
$E_{Y}(C_{n})=n(H_{\lfloor\frac{n}{2}\rfloor}-1)+(n\mod
2)=nH_{\lfloor\frac{n}{2}\rfloor}-2\Big{\lfloor}\frac{n}{2}\Big{\rfloor}$
recalled above. ∎
###### Corollary 8
$\begin{array}[]{l}\displaystyle\sigma^{2}_{Y}(C_{n})=-\frac{8}{3}\Big{(}-18+\pi^{2}+\log(64)\Big{)}\Big{\lfloor}\frac{n}{4}\Big{\rfloor}^{2}-8\Big{\lfloor}\frac{n}{4}\Big{\rfloor}\log\Big{(}\Big{\lfloor}\frac{n}{4}\Big{\rfloor}\Big{)}\\\
\qquad\displaystyle+\Big{(}20-8\gamma-32\log(2)+\Big{(}24-\frac{4}{3}\pi^{2}-8\log(2)\Big{)}(n\mod
4)\Big{)}\Big{\lfloor}\frac{n}{4}\Big{\rfloor}+O(1).\end{array}$
###### Proof
We expand the expression for $\sigma^{2}_{Y}(C_{n})$ given in the previous
corollary, taking into account the value of $n$ module $4$.
If there exists some $m$ such that $n=4m$, then
$\begin{array}[]{l}\sigma^{2}_{Y}(C_{n})=8m\left(2mH_{m}-2(m-1)H_{2m}-3H_{4m}-2mH_{2m}^{(2)}+6m+1\right)\\\
\quad\displaystyle=-\frac{8}{3}m^{2}\left(-18+\pi^{2}+\log(64)\right)+m\left(-8\log(m)-8\gamma+20-32\log(2)\right)\\\
\qquad\displaystyle-2+O\left(\frac{1}{m}\right)\end{array}$
If there exists some $m$ such that $n=4m+1$, then
$\begin{array}[]{l}\sigma^{2}_{Y}(C_{n})=2\left(8m^{2}+4m+1\right)H_{m}+\left(-16m^{2}+8m+2\right)H_{2m}-24mH_{4m+1}\\\
\qquad-6H_{4m+1}-16m^{2}H_{2m}^{(2)}-8mH_{2m}^{(2)}-H_{2m}^{(2)}+48m^{2}+32m+6\\\
\quad\displaystyle=-\frac{8}{3}m^{2}\left(-18+\pi^{2}+\log(64)\right)-\frac{4}{3}m(6\log(m)+\pi^{2}+6\gamma-33+30\log(2))\\\
\qquad\displaystyle+\left(-2\log(m)-\frac{\pi^{2}}{6}-2\gamma+4-10\log(2)\right)+O\left(\frac{1}{m}\right)\end{array}$
If there exists some $m$ such that $n=4m+2$, then
$\begin{array}[]{l}\sigma^{2}_{Y}(C_{n})=16mH_{2m+2}-24mH_{4m+2}+4(2m+1)^{2}H_{m+1}-4(2m+1)^{2}H_{2m+1}\\\
\qquad\displaystyle+8H_{2m+2}-12H_{4m+2}-16m^{2}H_{2m+1}^{(2)}-16mH_{2m+1}^{(2)}-4H_{2m+1}^{(2)}\\\
\qquad+48m^{2}+48m+10\\\
\quad\displaystyle=-\frac{8}{3}m^{2}\left(-18+\pi^{2}+\log(64)\right)+m\left(-8\log(m)-\frac{8\pi^{2}}{3}-8\gamma+68-48\log(2)\right)\\\
\qquad\displaystyle-\frac{2}{3}(6\log(m)+\pi^{2}+6\gamma-24+30\log(2))+O\left(\frac{1}{m}\right)\end{array}$
Finally, if there exists some $m$ such that $n=4m+3$, then
$\begin{array}[]{l}\displaystyle\sigma^{2}_{Y}(C_{n})=(4m+3)\big{[}4(m+1)H_{m+1}-4(m+1)H_{2m+1}+4H_{2m+2}-6H_{4m+3}\\\
\qquad\displaystyle-(4m+3)H_{2m+1}^{(2)}+10m+11\big{]}+(2m+1)\left(-2H_{m+1}+2H_{2m+1}+24m+19\right)\\\
\qquad\displaystyle-24(m+1)^{2}-4(2m+1)^{2}\\\
\quad\displaystyle=-\frac{8}{3}m^{2}\left(-18+\pi^{2}+\log(64)\right)-4m\left(2\log(m)+\pi^{2}+2\gamma-23+7\log(4)\right)\\\
\qquad\displaystyle+\left(-6\log(m)-\frac{3\pi^{2}}{2}-6\gamma+34-17\log(4)\right)+O\left(\frac{1}{m}\right)\end{array}$
Now, using that $m=\lfloor n/4\rfloor$ and $n\mod 4=n-4\lfloor n/4\rfloor$, it
can be easily seen that these formulas are consistent with the development of
$\sigma^{2}_{Y}(C_{n})$ until $O(1)$ given in the statement. ∎
## 8 Some covariances
In this section we obtain the covariance under the Yule model of $S_{n}$ and
$\Phi_{n}$ from the formulas obtained in the previous sections for
$E_{Y}(\overline{\Phi}^{2}_{n})$, $E_{Y}(S_{n}^{2})$ and
$E_{Y}(\Phi_{n}^{2})$.
###### Corollary 9
$\textit{cov}_{Y}(S_{n},\Phi_{n})=4n(nH_{n}^{(2)}+H_{n})+\frac{1}{6}n(n^{2}-51n+2)$.
###### Proof
Notice that
$\begin{array}[]{l}\textit{cov}_{Y}(S_{n},\Phi_{n})=E_{Y}(S_{n}\cdot\Phi_{n})-E_{Y}(S_{n})\cdot
E_{Y}(\Phi_{n})\\\
\qquad\displaystyle=\frac{1}{2}\big{(}E_{Y}((\Phi_{n}+S_{n})^{2})-E_{Y}(S_{n}^{2})-E_{Y}(\Phi_{n}^{2})\big{)}-E_{Y}(S_{n})\cdot
E_{Y}(\Phi_{n})\\\\[8.61108pt]
\qquad\displaystyle=\frac{1}{2}\big{(}E_{Y}(\overline{\Phi}^{2}_{n})-E_{Y}(S_{n}^{2})-E_{Y}(\Phi_{n}^{2})\big{)}-E_{Y}(S_{n})\cdot
E_{Y}(\Phi_{n})\end{array}$
The formula in the statement is obtained by replacing in this identity
$E_{Y}(\overline{\Phi}^{2}_{n})$, $E_{Y}(S_{n}^{2})$, $E_{Y}(\Phi_{n}^{2})$,
$E_{Y}(S_{n})$, and $E_{Y}(\Phi_{n})$ by their values. ∎
###### Corollary 10
$\displaystyle\textit{cov}_{Y}(S_{n},\overline{\Phi}_{n})=2nH_{n}+\frac{1}{6}n(n^{2}-9n-4)$
###### Proof
By the bilinearity of covariances,
$\textit{cov}_{Y}(S_{n},\overline{\Phi}_{n})=\textit{cov}_{Y}(S_{n},S_{n}+\Phi_{n})=\sigma_{Y}^{2}(S_{n})+\textit{cov}_{Y}(S_{n},\Phi_{n})$.
∎
###### Corollary 11
$\begin{array}[]{rl}\displaystyle\textit{cov}_{Y}(S_{n},\Phi_{n})&=\displaystyle\frac{1}{6}n^{3}+\Big{(}\frac{2\pi^{2}}{3}-\frac{17}{2}\Big{)}n^{2}+4n\ln(n)+\frac{1}{3}(12\gamma-11)n+4+O\Big{(}\frac{1}{n}\Big{)}\\\\[8.61108pt]
\textit{cov}_{Y}(S_{n},\overline{\Phi}_{n})&=\displaystyle\frac{1}{6}n^{3}-\frac{3}{2}n^{2}+2n\ln(n)+\frac{1}{3}(6\gamma-2)n+1+O\Big{(}\frac{1}{n}\Big{)}\end{array}$
From the formulas for $\sigma_{Y}^{2}(\Phi_{n})$,
$\sigma_{Y}^{2}(\overline{\Phi}_{n})$, and $\textit{cov}_{Y}(S_{n},\Phi_{n})$,
we can compute Pearson’s correlation coefficient between $S_{n}$ and
$\Phi_{n}$,
$cor_{Y}(S_{n},\Phi_{n})=\frac{\textit{cov}_{Y}(S_{n},\Phi_{n})}{\sqrt{\sigma_{Y}^{2}(\Phi_{n})\cdot\sigma_{Y}^{2}(\overline{\Phi}_{n})}}.$
The exact formula for this coefficient is
$cor_{Y}(S_{n},\Phi_{n})\textstyle=\frac{4n(nH_{n}^{(2)}+H_{n})+\frac{1}{6}n(n^{2}-51n+2)}{\sqrt{\big{(}7n^{2}-4n^{2}H_{n}^{(2)}-2nH_{n}-n\big{)}\big{(}\frac{1}{12}(n^{4}-10n^{3}+131n^{2}-2n)-4n^{2}H_{n}^{(2)}-6nH_{n}\big{)}}}$
and in the limit it is equal to
$cor_{Y}(S_{n},\Phi_{n})\sim\frac{1}{6\sqrt{\big{(}(7-\frac{2\pi^{2}}{3})\cdot\frac{1}{12}\big{)}}}=0.89059$
## 9 Conclusions
In this paper we have obtained exact formulas for the variance under the Yule
model of the Colless index $C$, the Sackin index $S$, the total cophenetic
index $\Phi$, and the sum $\overline{\Phi}=S+\Phi$, as well as for the
covariances of $S$ and $\Phi,\overline{\Phi}$. Our formulas are explicit and
hold on spaces $\mathcal{T}_{n}$ of binary phylogenetic trees with any number
$n$ of leaves, unlike other expressions published so far in the literature,
which were either recursive or asymptotic.
The proofs consist of elementary, although long and involved, algebraic
computations. Since it is not difficult to slip some mistake in such long
algebraic computations, to double-check the results we have directly computed
these variances and covariances on $\mathcal{T}_{n}$ for $n=3,\ldots,9$ and
confirmed that our formulas give the right results. The values obtained are
given in the next table. The Python scripts used to compute them are available
at the Supplementary Material web page
http:/bioinfo.uib.es/~recerca/phylotrees/Yulevariances/.
| 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---
$\sigma_{Y}^{2}(C_{n})$ | 0 | 2 | 3.5 | 6.8 | 10.072222
$\sigma_{Y}^{2}(S_{n})$ | 0 | 0.222222 | 0.805556 | 1.84 | 3.877778
$\sigma_{Y}^{2}(\Phi_{n})$ | 0 | 0.888889 | 5.138889 | 17.04 | 42.787778
$\sigma_{Y}^{2}(\overline{\Phi}_{n})$ | 0 | 2 | 10 | 30 | 70
$\textit{cov}_{Y}(S_{n},\Phi_{n})$ | 0 | 0.444444 | 2.0277778 | 5.56 | 11.912222
$cor_{Y}(S_{n},\Phi_{n})$ | - | 1 | 0.996639 | 0.992958 | 0.989408
| 8 | 9 | | |
$\sigma_{Y}^{2}(C_{n})$ | 15.765079 | 21.089881 | | |
$\sigma_{Y}^{2}(S_{n})$ | 5.49424 | 8.193827 | | |
$\sigma_{Y}^{2}(\Phi_{n})$ | 90.522812 | 170.350969 | | |
$\sigma_{Y}^{2}(\overline{\Phi}_{n})$ | 140 | 252 | | |
$\textit{cov}_{Y}(S_{n},\Phi_{n})$ | 21.991474 | 36.727602 | | |
$cor_{Y}(S_{n},\Phi_{n})$ | 0.986101 | 0.983053 | | |
Table 1: Values of $\sigma_{Y}^{2}(C_{n})$, $\sigma_{Y}^{2}(S_{n})$,
$\sigma_{Y}^{2}(\Phi_{n})$, $\sigma_{Y}^{2}(\overline{\Phi}_{n})$,
$\textit{cov}_{Y}(S_{n},\Phi_{n})$, and $cor_{Y}(S_{n},\Phi_{n})$ for
$n=3,\ldots,9$. They agree with those given by our formulas.
It can be seen in this table that the values of the variances of $S_{n}$ are
smaller than those of the variance of $\Phi$ or $\overline{\Phi}$. Actually,
as we have recalled in the Introduction, $\sigma_{Y}^{2}(S_{n})$ has order
$O(n^{2})$, while $\sigma_{Y}^{2}(\Phi_{n})$ and
$\sigma_{Y}^{2}(\overline{\Phi}_{n})$ are $O(n^{4})$. This is consistent with
the fact that $\Phi$ and $\overline{\Phi}$ have larger spans of values than
$S$, $O(n^{3})$ instead of $O(n^{2})$, and much less ties. It is also deduced
from the formulas obtained in this paper, and from this table for small values
of $n$, that there is a strong direct linear correlation between $S_{n}$ and
$\Phi_{n}$, although in the limit Pearson’s coefficient between them decreases
to 0.89.
It remains to compute exact formulas for covariances of $C$ with $S$ and
$\Phi$. These formulas can surely be obtained using a recurrence for the
expected value of the product of two recursive shape indices similar in spirit
to Corollary 1, but the computations seem to be even longer than those leading
to the computation of $\sigma_{Y}^{2}(C_{n})$.
## Acknowledgements
This research has been partially supported by the Spanish government and the
UE FEDER program, through projects MTM2009-07165 and TIN2008-04487-E/TIN. We
thank J. Miró and M. Lewis for several comments on a previous version of this
work. Most computations in this paper have been carried out or checked with
the aid of Mathematica.
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|
arxiv-papers
| 2012-02-29T15:28:47 |
2024-09-04T02:49:28.097671
|
{
"license": "Public Domain",
"authors": "Gabriel Cardona, Arnau Mir, Francesc Rossello",
"submitter": "Francesc Rossell\\'o",
"url": "https://arxiv.org/abs/1202.6573"
}
|
1202.6579
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-051 LHCb-PAPER-2011-036
Measurement of $\varUpsilon$ production in $pp$ collisions at
$\sqrt{s}=7~{}\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration 111Authors are listed on the following pages.
The production of $\varUpsilon(1S)$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$
mesons in proton-proton collisions at the centre-of-mass energy of
${\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}}$ is studied with the LHCb
detector. The analysis is based on a data sample of $25~{}\mbox{\,pb}^{-1}$
collected at the Large Hadron Collider. The $\varUpsilon$ mesons are
reconstructed in the decay mode
$\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$ and the signal yields are
extracted from a fit to the $\mu^{+}\mu^{-}$ invariant mass distributions. The
differential production cross-sections times dimuon branching fractions are
measured as a function of the $\varUpsilon$ transverse momentum $p_{\rm T}$
and rapidity $y$, over the range $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$. The cross-
sections times branching fractions, integrated over these kinematic ranges,
are measured to be
$\displaystyle\sigma(pp\rightarrow\varUpsilon(1S)\,X)\times\mathcal{B}(\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-})=2.29\phantom{0}\pm
0.01\phantom{0}\pm 0.10\phantom{0}\,\,_{-0.37}^{+0.19}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(2S)\,X)\times\mathcal{B}(\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-})=0.562\pm
0.007\pm 0.023\,_{-0.092}^{+0.048}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(3S)\,X)\times\mathcal{B}(\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-})=0.283\pm
0.005\pm 0.012\,_{-0.048}^{+0.025}~{}{\rm nb},$
where the first uncertainty is statistical, the second systematic and the
third is due to the unknown polarisation of the three $\varUpsilon$ states.
Published in Eur. Phys. J. C volume 72,6 (June 2012)
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S.
Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
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Falabella16,e,14, E. Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38,
S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30,
C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2,
M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
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Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
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Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
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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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The measurement of heavy quark production in hadron collisions probes the
dynamics of the colliding partons. The study of heavy quark-antiquark
resonances, such as the $b\overline{b}$ bound states $\varUpsilon(1S)$,
$\varUpsilon(2S)$ and $\varUpsilon(3S)$ (indicated generically as
$\varUpsilon$ in the following) is of interest as these mesons have large
production cross-sections and can be produced in different spin
configurations. In addition, the thorough understanding of these states is the
first step towards the study of recently discovered new states in the
$b\bar{b}$ system [1, 2, 3, 4]. Although $\varUpsilon$ production was studied
by several experiments in the past, the underlying production mechanism is
still not well understood. Several models exist but fail to reproduce both the
cross-section and the polarisation measurements at the Tevatron [5, 6, 7].
Among these are the Colour Singlet Model (CSM) [8, 9, 10], recently improved
by adding higher order contributions (NLO CSM), the standard truncation of the
nonrelativistic QCD expansion (NRQCD) [11], which includes contributions from
the Colour Octet Mechanism [12, 13], and the Colour Evaporation Model (CEM)
[14]. Although the disagreement of the theory with the data is less pronounced
for bottomonium than for charmonium, the measurement of $\varUpsilon$
production is important as the theoretical calculations are more robust due to
the heavier bottom quark.
There are two major sources of $\varUpsilon$ production in $pp$ collisions:
direct production and feed-down from the decay of heavier prompt bottomonium
states, like $\chi_{b}$, or higher-mass $\varUpsilon$ states. This study
presents measurements of the individual inclusive production cross-sections of
the three $\varUpsilon$ mesons decaying into a pair of muons. The measurements
are performed in $7~{}\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass $pp$
collisions as a function of the $\varUpsilon$ transverse momentum
($\mbox{$p_{\rm T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) and rapidity
($2<y<4.5$), in 15 bins of $p_{\rm T}$ and five bins of $y$. This analysis is
complementary to those recently presented by the ATLAS collaboration, who
measured the $\varUpsilon(1S)$ cross section for $|y|<2.4$ [15], and the CMS
collaboration, who measured the $\varUpsilon(1S),\varUpsilon(2S)$ and
$\varUpsilon(3S)$ cross sections in the rapidity region $|y|<2.0$ [16].
## 2 The LHCb detector and data
The results presented here are based on a dataset of $25.0\pm
0.9~{}\mbox{\,pb}^{-1}$ collected at the Large Hadron Collider (LHC) in 2010
with the LHCb detector at a centre-of-mass energy of 7
$\mathrm{\,Te\kern-1.00006ptV}$.
The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers. The trigger
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage which applies a full event
reconstruction. This analysis uses events triggered by one or two muons. At
the hardware level one or two muon candidates are required with $p_{\rm T}$
larger than 1.4 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for one muon, and 0.56
and 0.48 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for two muons. At the software
level, the combined dimuon mass is required to be greater than 2.9
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and both the tracks and the vertex
have to be of good quality. To avoid the possibility that a few events with a
high occupancy dominate the trigger processing time, a set of global event
selection requirements based on hit multiplicities is applied.
The Monte Carlo samples used are based on the Pythia 6.4 generator [18], with
a choice of parameters specifically configured for LHCb [19]. The EvtGen
package [20] describes the decay of the $\varUpsilon$ resonances, and the
Geant4 package [21] simulates the detector response. The prompt bottomonium
production processes activated in Pythia are those from the leading-order
colour-singlet and colour-octet mechanisms for the $\varUpsilon(1S)$, and
colour-singlet only for the $\varUpsilon(2S)$ and the $\varUpsilon(3S)$. QED
radiative corrections to the decay
$\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$ are generated with the Photos
package [22].
## 3 Cross-section determination
The double differential cross-section for the inclusive $\varUpsilon$
production of the three different states is computed as
$\frac{{\rm d}^{2}\sigma^{iS}}{{\rm d}\mbox{$p_{\rm T}$}{\rm
d}y}\,\times\,\mathcal{B}^{iS}=\frac{N^{iS}}{\mathcal{L}\times\varepsilon^{iS}\times\Delta
y\times\Delta\mbox{$p_{\rm T}$}},\quad i=1,2,3;$ (1)
where $\sigma^{iS}$ is the inclusive cross section
$\sigma(pp\rightarrow\varUpsilon(iS)X)$, $\mathcal{B}^{iS}$ is the dimuon
branching fraction $\mathcal{B}(\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-})$,
$N^{iS}$ is the number of observed $\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$
decays in a given bin of $p_{\rm T}$ and $y$, $\varepsilon^{iS}$ is the
$\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$ total detection efficiency
including acceptance effects, $\mathcal{L}$ is the integrated luminosity and
$\Delta y=0.5$ and $\Delta\mbox{$p_{\rm T}$}=1~{}{\rm GeV}/c$ are the rapidity
and $p_{\rm T}$ bin sizes, respectively. In order to estimate $N^{iS}$, a fit
to the reconstructed invariant mass distribution is performed in each of the
15 $\mbox{$p_{\rm T}$}\times 5~{}y$ bins. $\varUpsilon$ candidates are formed
from pairs of oppositely charged muon tracks which traverse the full
spectrometer and satisfy the trigger requirements. Each track must have
$\mbox{$p_{\rm T}$}>1~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, be identified
as a muon and have a good quality of the track fit. The two muons are required
to originate from a common vertex with a good $\chi^{2}$ probability. The
three $\varUpsilon$ signal yields are determined from a fit to the
reconstructed invariant mass $m$ of the selected $\varUpsilon$ candidates in
the interval 8.9–10.9 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The mass
distribution is described by a sum of three Crystal Ball functions [23] for
the $\varUpsilon(1S)$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$ signals and an
exponential function for the combinatorial background. The Crystal Ball
function is defined as
$f_{\mathrm{CB}}=\begin{dcases}\frac{\Big{(}\frac{n}{|a|}\Big{)}^{n}e^{-\frac{1}{2}a^{2}}}{\Big{(}\frac{n}{|a|}-|a|-\frac{m-M}{\sigma}\Big{)}^{n}}&{\mathrm{if}}\,\,\,\frac{m-M}{\sigma}<-|a|\\\
\exp\Bigg{(}-\frac{1}{2}\Big{(}\frac{m-M}{\sigma}\Big{)}^{2}\Bigg{)}&{\mathrm{otherwise}},\end{dcases}$
(2)
with $f_{\mathrm{CB}}=f_{\mathrm{CB}}(m;M,\sigma,a,n)$, where $M$ and $\sigma$
are the mean and width of the gaussian. The parameters $a$ and $n$ describing
the radiative tail of the $\varUpsilon$ mass distribution are fixed to
describe a tail dominated by QED photon emission, as confirmed by simulation.
The distribution in Fig. 1 shows the results of the fit performed in the full
range of $p_{\rm T}$ and $y$.
Figure 1: Invariant mass distribution of the selected
$\varUpsilon\rightarrow\mu^{+}\mu^{-}$ candidates in the range $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and $2.0<y<4.5$. The three
peaks correspond to the $\varUpsilon(1S)$, $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ signals (from left to right). The superimposed curves are
the result of the fit as described in the text.
The signal yields obtained from the fit are $\varUpsilon(1S)=26\,410\pm 212$,
$\varUpsilon(2S)=6726\pm 142$ and $\varUpsilon(3S)=3260\pm 112$ events. The
mass resolution of the $\varUpsilon(1S)$ peak is $\sigma=53.9\pm
0.5~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The resolutions of the
$\varUpsilon(2S)$ and $\varUpsilon(3S)$ peaks are fixed to the resolution of
the $\varUpsilon(1S)$, scaled by the ratio of the masses, as expected from
resolution effects. The masses are allowed to vary in the fit and are measured
to be $M(\varUpsilon(1S))=9448.3\pm 0.5$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $M(\varUpsilon(2S))=10\,010.4\pm
1.4$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$M(\varUpsilon(3S))=10\,338.7\pm 2.6$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the quoted uncertainties are
statistical only. The fit is repeated independently for each of the bins in
$p_{\rm T}$ and $y$. When fitting the individual bins, due to the reduced
dataset, the masses and widths of the three $\varUpsilon$ states in the fit
are fixed to the values obtained when fitting the full range. Bins with fewer
than 36 entries are excluded from the analysis. The total efficiency
$\varepsilon$ entering the cross-section expression of Eq. (1) is the product
of the geometric acceptance, the reconstruction and selection efficiency and
the trigger efficiency. All efficiency terms have been evaluated using Monte
Carlo simulations in each ($\mbox{$p_{\rm T}$},y$) bin separately, with the
exception of that related to the global event selection which has been
determined from data. In the simulation the $\varUpsilon$ meson is produced in
an unpolarised state. The absolute luminosity scale was measured at specific
periods during the 2010 data taking using both van der Meer scans and a beam-
gas imaging method [24, 25]. The uncertainty on the integrated luminosity for
the analysed sample due to this method is estimated to be 3.5% [25]. The
knowledge of the absolute luminosity scale is used to calibrate the number of
tracks in the vertex detector, which is found to be stable throughout the
data-taking period and can therefore be used to monitor the instantaneous
luminosity of the entire data sample. The integrated luminosity of the data
sample used in this analysis is determined to be $25.0~{}\mbox{\,pb}^{-1}$.
## 4 Systematic uncertainties
Extensive studies on dimuon decays [26, 16, 15] have shown that the total
efficiency depends strongly on the initial polarisation state of the vector
meson. In this analysis, the influence of the unknown polarisation is studied
in the helicity frame [27] using Monte Carlo simulation. The angular
distribution of the muons from the $\varUpsilon$, ignoring the azimuthal part,
is
$\displaystyle\frac{{\rm d}N}{{\rm
d}\cos\theta}\,=\,\frac{1+\alpha\cos^{2}\theta}{2+2\alpha/3},$ (3)
where $\theta$ is the angle between the direction of the $\mu^{+}$ momentum in
the $\varUpsilon$ centre-of-mass frame and the direction of the $\varUpsilon$
momentum in the colliding proton centre-of-mass frame. The values
$\alpha=+1,-1,0$ correspond to fully transverse, fully longitudinal, and no
polarisation respectively. Figure 2 shows the $\varUpsilon(1S)$ total
efficiency for these three scenarios, and indicates that the polarisation
significantly affects the efficiencies and that the effect depends on $p_{\rm
T}$ and $y$. A similar behaviour is observed for the $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ efficiencies.
Figure 2: Total efficiency $\varepsilon$ of the $\varUpsilon(1S)$ as a
function of (a) the $\varUpsilon(1S)$ transverse momentum and (b) rapidity,
estimated using the Monte Carlo simulation, for three different
$\varUpsilon(1S)$ polarisation scenarios, indicated by the parameter $\alpha$
described in the text.
Following this observation, in each $(\mbox{$p_{\rm T}$},y)$ bin the maximal
difference between the polarised scenarios ($\alpha=\pm 1$) and the
unpolarised scenario ($\alpha=0$) is taken as a systematic uncertainty on the
efficiency. This results in an uncertainty of up to $17\%$ on the integrated
cross-sections and of up to 40% in the individual bins. Several other sources
of possible systematic effects were studied. They are summarised in Table 1.
Table 1: Summary of the relative systematic uncertainties on the cross-section measurements. Ranges indicate variations depending on the ($\mbox{$p_{\rm T}$},y$) bin and the $\varUpsilon$ state. All uncertainties are fully correlated among the bins. Source | Uncertainty (%)
---|---
Unknown $\varUpsilon$ polarisation | 0.3–41.0
Trigger | 3.0
Track reconstruction | 2.4
Track quality requirement | 0.5
Vertexing requirement | 1.0
Muon identification | 1.1
Global event selection requirements | 0.6
$p_{\rm T}$ binning effect | 1.0
Fit function | 1.1–2.1
Luminosity | 3.5
The trigger efficiency is determined on data using an unbiased sample of
events that would trigger if the $\varUpsilon$ candidate were removed. The
efficiency obtained with this method is compared with the efficiency
determined in the simulation. The difference of 3.0% is assigned as a
systematic uncertainty.
The uncertainty on the muon track reconstruction efficiency has been estimated
using a data driven tag-and-probe approach based on partially reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
[28], and found to be 2.4% per muon pair. Additional uncertainties are
assigned, which account for the different behaviour in data and simulation of
the track and vertex quality requirements. The muon identification efficiency
is measured using a tag-and-probe approach, which gives an uncertainty on the
efficiency of 1.1% [26].
The measurement of the global event selection efficiency is taken as an
additional uncertainty associated with the trigger. An uncertainty of 1.0% is
considered to account for the difference in the $p_{\rm T}$ spectra in data
and Monte Carlo simulation for the three $\varUpsilon$ states, which might
have an effect on the correct bin assignment (“binning effect”).
The influence of the choice of the fit function describing the shape of the
invariant mass distribution includes two components. The uncertainty on the
shape of the background distribution is estimated using a different fit model
(1.0–1.5%). The systematic associated with fixing the parameters of the
Crystal Ball function is estimated by varying the central values within the
parameters uncertainties, obtained when leaving them free to vary in the fit
(0.5–1.4%).
## 5 Results
The double differential cross-sections as a function of $p_{\rm T}$ and $y$
are shown in Fig. 3 and Tables 2-4. The integrated cross-sections times
branching fractions in the range $\mbox{$p_{\rm
T}$}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0~{}<~{}y~{}<~{}4.5$ are
measured to be
$\displaystyle\sigma(pp\rightarrow\varUpsilon(1S)\,X)\times\mathcal{B}^{1S}=2.29\phantom{0}\pm
0.01\phantom{0}\pm 0.10\phantom{0}\,\,_{-0.37}^{+0.19}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(2S)\,X)\times\mathcal{B}^{2S}=0.562\pm
0.007\pm 0.023\,_{-0.092}^{+0.048}~{}{\rm nb},$
$\displaystyle\sigma(pp\rightarrow\varUpsilon(3S)\,X)\times\mathcal{B}^{3S}=0.283\pm
0.005\pm 0.012\,_{-0.048}^{+0.025}~{}{\rm nb},$
where the first uncertainties are statistical, the second systematic and the
third are due to the unknown polarisation of the three $\varUpsilon$ states.
The integrated $\varUpsilon(1S)$ cross-section is about a factor one hundred
smaller than the integrated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
cross-section in the identical region of $p_{\rm T}$ and $y$ [26], and a
factor three smaller than the integrated $\varUpsilon(1S)$ cross-section in
the central region, as measured by CMS [16] and ATLAS [15].
Figure 3: Double differential $\nobreak{\varUpsilon\rightarrow\mu^{+}\mu^{-}}$
cross-sections times dimuon branching fractions as a function of $p_{\rm T}$
in bins of rapidity for (a) the $\varUpsilon(1S)$, (b) the $\varUpsilon(2S)$
and (c) the $\varUpsilon(3S)$. The error bars correspond to the total
uncertainty for each bin.
Figure 4 compares the LHCb measurement of the differential
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ production cross-section with
several theory predictions in the LHCb acceptance region. In Fig. 4(a) the
data are compared to direct production as calculated from a NNLO* colour-
singlet model [29, 30], where the notation NNLO* denotes an evaluation that is
not a complete next-to-next leading order computation and that can be affected
by logarithmic corrections, which are not easily quantifiable. Direct
production as calculated from NLO CSM is also represented. In Fig. 4(b) the
data are compared to two model predictions for the $\varUpsilon(1S)$
production: the calculation from NRQCD at NLO, including contributions from
$\chi_{b}$ and higher $\varUpsilon$ states decays, summing the colour-singlet
and colour-octet contributions [31], and the calculation from the NLO CEM,
including contributions from $\chi_{b}$ and higher $\varUpsilon$ states decays
[14]. Note that the NNLO* theoretical model computes the direct
$\varUpsilon(1S)$ production, whereas the LHCb measurement includes
$\varUpsilon(1S)$ from $\chi_{b}$, $\varUpsilon(2S)$ and $\varUpsilon(3S)$
decays. However, taking into account the feed-down contribution, which has
been measured to be of the order of 50% [32], a satisfactory agreement is
found with the theoretical predictions.
Figure 4: Differential $\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ production
cross-section times dimuon branching fraction as a function of $p_{\rm T}$
integrated over $y$ in the range 2.0–4.5, compared with the predictions from
(a) the NNLO* CSM [29] for direct production, and (b) the NLO NRQCD [31] and
CEM [14]. The error bars on the data correspond to the total uncertainties for
each bin, while the bands indicate the uncertainty on the theory prediction.
Figure 5 compares the LHCb measurement of the differential $\varUpsilon(2S)$
and $\varUpsilon(3S)$ production cross-sections times branching fraction with
the NNLO* theory predictions of direct production. It can be seen that the
agreement with the theory is better for the $\varUpsilon(3S)$, which is
expected to be less affected by feed-down. At present there is no measurement
of the contribution of feed-down to the $\varUpsilon(2S)$ and
$\varUpsilon(3S)$ inclusive rate.
Figure 5: Differential (a) $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ and (b)
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ production cross-sections times
dimuon branching fractions as a function of $p_{\rm T}$ integrated over $y$ in
the range 2.0–4.5, compared with the predictions from the NNLO* CSM for direct
production [29]. The error bars on the data correspond to the total
uncertainties for each bin, while the bands indicate the uncertainty on the
theory prediction.
The cross-sections times the dimuon branching fractions for the three
$\varUpsilon$ states are compared in Fig. 6 as a function of rapidity and
transverse momentum.
Figure 6: Differential cross-sections of $\varUpsilon(1S),\varUpsilon(2S)$
and $\varUpsilon(3S)$ times dimuon branching fractions as a function of (a)
$p_{\rm T}$ integrated over $y$ and (b) $y$ integrated over $p_{\rm T}$. The
error bars on the data correspond to the total uncertainties for each bin.
The cross-section results are used to evaluate the ratios $R^{iS/1S}$ of the
$\varUpsilon(2S)$ to $\varUpsilon(1S)$ and $\varUpsilon(3S)$ to
$\varUpsilon(1S)$ cross-sections times the dimuon branching fractions. Most of
the systematic uncertainties on the cross-sections cancel in the ratio, except
those due to the size of the data sample, the choice of fit function and the
unknown polarisation of the different states. The polarisation uncertainty has
been evaluated for the scenarios in which one of the two $\varUpsilon$ states
is completely polarised (either transversely or longitudinally) and the other
is not polarised. The maximum difference of these two cases ranges between 15%
and 26%. The ratios $R^{iS/1S},i=2,3,$ are given in Table 5 and shown in Fig.
7. The polarisation uncertainty is not included in these figures. The results
agree well with the corresponding ratio measurements from CMS [16] in the
$p_{\rm T}$ range common to both experiments.
Figure 7: Ratios of $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ and
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ with respect to
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of $p_{\rm T}$ of the
$\varUpsilon$ in the range $2.0<y<4.5$, assuming no polarisation. The error
bars on the data correspond to the total uncertainties for each bin except for
that due to the unknown polarisation, which ranges between 15% and 26% as
listed in Table 5.
## 6 Conclusions
The differential cross-sections $\varUpsilon(iS)\rightarrow\mu^{+}\mu^{-}$,
for $i=1,2,3$, are measured as a function of the $\varUpsilon$ transverse
momentum and rapidity in the region $\mbox{$p_{\rm
T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, 2.0 $<y<$ 4.5 in the LHCb
experiment. The analysis is based on a data sample corresponding to an
integrated luminosity of 25 $\mbox{\,pb}^{-1}$ collected at the Large Hadron
Collider at a centre-of-mass energy of
$\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$. The results obtained are
compatible with previous measurements in $pp$ collisions at the same centre-
of-mass energy, performed by ATLAS and CMS in a different region of rapidity
[16, 15]. This is the first measurement of $\varUpsilon$ production in the
forward region at $\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$. A comparison
with theoretical models shows good agreement with the measured $\varUpsilon$
cross-sections. The measurement of the differential cross-sections is not
sufficient to discriminate amongst the various models, and studies of other
observables such as the $\varUpsilon$ polarisations will be necessary.
## 7 Acknowledgements
We thank P. Artoisenet, M. Butenschön, K.-T. Chao, B. Kniehl, J.-P. Lansberg
and R. Vogt for providing theoretical predictions of $\varUpsilon$ cross-
sections in the LHCb acceptance range. 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
(The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom
(Russia); MICINN, XuntaGal 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 Region Auvergne.
Table 2: Double differential cross-section
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(1S)$.
$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | | | | |
0–1 | 53.1 $\pm$ 4.0 $\pm$ 2.5 ${}_{-17.3}^{+8.9}$ | 62.6 $\pm$ 3.0 $\pm$ 2.9 ${}_{-11.5}^{+6.1}$ | 48.0 $\pm$ 2.4 $\pm$ 2.2 ${}_{-5.8}^{+3.1}$ | 40.1 $\pm$ 2.4 $\pm$ 1.9 ${}_{-7.0}^{+3.9}$ | 22.9 $\pm$ 2.7 $\pm$ 1.1 ${}_{-5.9}^{+3.4}$
1–2 | 152.5 $\pm$ 6.8 $\pm$ 7.2 ${}_{-50.4}^{+25.7}$ | 148.8 $\pm$ 4.7 $\pm$ 7.0 ${}_{-27.5}^{+14.6}$ | 120.5 $\pm$ 3.8 $\pm$ 5.6 ${}_{-14.0}^{+7.5}$ | 93.3 $\pm$ 3.7 $\pm$ 4.3 ${}_{-14.8}^{+8.1}$ | 64.5 $\pm$ 4.5 $\pm$ 3.0 ${}_{-15.0}^{+8.7}$
2–3 | 211.0 $\pm$ 8.0 $\pm$ 10.0 ${}_{-67.2}^{+34.3}$ | 185.3 $\pm$ 5.2 $\pm$ 8.7 ${}_{-34.4}^{+18.1}$ | 150.0 $\pm$ 4.3 $\pm$ 7.0 ${}_{-17.4}^{+9.2}$ | 116.1 $\pm$ 4.1 $\pm$ 5.4 ${}_{-15.5}^{+8.4}$ | 69.8 $\pm$ 4.6 $\pm$ 3.3 ${}_{-14.6}^{+8.3}$
3–4 | 184.3 $\pm$ 7.3 $\pm$ 8.8 ${}_{-56.3}^{+28.8}$ | 167.7 $\pm$ 4.9 $\pm$ 7.9 ${}_{-29.3}^{+15.6}$ | 141.9 $\pm$ 4.2 $\pm$ 6.6 ${}_{-15.0}^{+8.0}$ | 109.7 $\pm$ 4.0 $\pm$ 5.1 ${}_{-11.9}^{+6.3}$ | 70.6 $\pm$ 4.6 $\pm$ 3.3 ${}_{-12.2}^{+6.7}$
4–5 | 187.3 $\pm$ 7.3 $\pm$ 8.9 ${}_{-54.8}^{+27.9}$ | 158.4 $\pm$ 4.8 $\pm$ 7.4 ${}_{-26.4}^{+14.0}$ | 120.9 $\pm$ 3.9 $\pm$ 5.7 ${}_{-11.3}^{+6.0}$ | 84.6 $\pm$ 3.5 $\pm$ 4.0 ${}_{-7.0}^{+3.7}$ | 50.4 $\pm$ 3.8 $\pm$ 2.4 ${}_{-7.0}^{+3.7}$
5–6 | 138.0 $\pm$ 6.2 $\pm$ 6.6 ${}_{-38.3}^{+19.4}$ | 134.5 $\pm$ 4.4 $\pm$ 6.3 ${}_{-20.8}^{+11.0}$ | 94.2 $\pm$ 3.5 $\pm$ 4.4 ${}_{-7.3}^{+3.8}$ | 70.6 $\pm$ 3.2 $\pm$ 3.3 ${}_{-4.0}^{+2.1}$ | 45.3 $\pm$ 3.6 $\pm$ 2.1 ${}_{-4.9}^{+2.5}$
6–7 | 105.3 $\pm$ 5.3 $\pm$ 5.0 ${}_{-27.6}^{+14.0}$ | 95.2 $\pm$ 3.7 $\pm$ 4.5 ${}_{-13.7}^{+7.2}$ | 73.5 $\pm$ 3.0 $\pm$ 3.5 ${}_{-4.6}^{+2.4}$ | 57.0 $\pm$ 2.9 $\pm$ 2.7 ${}_{-1.9}^{+1.0}$ | 29.5 $\pm$ 2.8 $\pm$ 1.4 ${}_{-2.5}^{+1.2}$
7–8 | 78.3 $\pm$ 4.5 $\pm$ 3.7 ${}_{-19.4}^{+9.8}$ | 72.9 $\pm$ 3.2 $\pm$ 3.4 ${}_{-9.6}^{+5.0}$ | 60.2 $\pm$ 2.7 $\pm$ 2.8 ${}_{-3.0}^{+1.6}$ | 38.3 $\pm$ 2.3 $\pm$ 1.8 ${}_{-0.8}^{+0.4}$ | 21.6 $\pm$ 2.4 $\pm$ 1.0 ${}_{-1.5}^{+0.7}$
8–9 | 63.5 $\pm$ 4.0 $\pm$ 3.0 ${}_{-14.8}^{+7.5}$ | 57.0 $\pm$ 2.8 $\pm$ 2.7 ${}_{-6.8}^{+3.6}$ | 43.3 $\pm$ 2.3 $\pm$ 2.0 ${}_{-1.9}^{+1.0}$ | 24.7 $\pm$ 1.9 $\pm$ 1.2 ${}_{-0.6}^{+0.3}$ | 13.6 $\pm$ 1.9 $\pm$ 0.6 ${}_{-0.8}^{+0.4}$
9–10 | 50.1 $\pm$ 3.5 $\pm$ 2.4 ${}_{-10.8}^{+5.5}$ | 43.2 $\pm$ 2.4 $\pm$ 2.0 ${}_{-5.0}^{+2.6}$ | 29.8 $\pm$ 1.9 $\pm$ 1.4 ${}_{-1.0}^{+0.5}$ | 19.4 $\pm$ 1.6 $\pm$ 0.9 ${}_{-0.6}^{+0.3}$ | 6.1 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.3}^{+0.1}$
10–11 | 35.4 $\pm$ 2.9 $\pm$ 1.7 ${}_{-7.3}^{+3.7}$ | 28.2 $\pm$ 1.9 $\pm$ 1.3 ${}_{-3.0}^{+1.6}$ | 23.9 $\pm$ 1.7 $\pm$ 1.1 ${}_{-0.8}^{+0.4}$ | 12.3 $\pm$ 1.3 $\pm$ 0.6 ${}_{-0.5}^{+0.2}$ | 6.8 $\pm$ 1.3 $\pm$ 0.3 ${}_{-0.4}^{+0.2}$
11–12 | 29.3 $\pm$ 2.6 $\pm$ 1.4 ${}_{-5.8}^{+2.9}$ | 19.4 $\pm$ 1.6 $\pm$ 0.9 ${}_{-1.9}^{+1.0}$ | 14.7 $\pm$ 1.3 $\pm$ 0.7 ${}_{-0.6}^{+0.3}$ | 6.7 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 4.3 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
12–13 | 20.3 $\pm$ 2.1 $\pm$ 1.0 ${}_{-3.7}^{+1.9}$ | 13.7 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.3}^{+0.7}$ | 10.3 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.3}^{+0.2}$ | 6.7 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.8 $\pm$ 0.8 $\pm$ 0.1 ${}_{-0.2}^{+0.1}$
13–14 | 10.4 $\pm$ 1.5 $\pm$ 0.5 ${}_{-1.9}^{+0.9}$ | 11.6 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.1}^{+0.6}$ | 8.6 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.2}^{+0.1}$ | 5.0 $\pm$ 0.8 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 0.8 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.1}^{+0.0}$
14–15 | 11.2 $\pm$ 1.5 $\pm$ 0.5 ${}_{-2.0}^{+1.0}$ | 8.9 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.8}^{+0.4}$ | 5.7 $\pm$ 0.8 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.2 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 1.8 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$
Table 3: Double differential cross-section
$\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(2S)$. Regions where the number of
events was not sufficient to perform a measurement are indicated with a dash.
$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | | | | |
0–1 | 8.2 $\pm$ 1.7 $\pm$ 0.4 ${}_{-3.1}^{+1.5}$ | 15.8 $\pm$ 1.6 $\pm$ 0.7 ${}_{-2.8}^{+1.5}$ | 7.8 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.8}^{+0.4}$ | 8.6 $\pm$ 1.2 $\pm$ 0.4 ${}_{-1.5}^{+0.8}$ | -
1–2 | 25.8 $\pm$ 2.9 $\pm$ 1.2 ${}_{-9.2}^{+4.6}$ | 31.2 $\pm$ 2.2 $\pm$ 1.5 ${}_{-5.6}^{+3.1}$ | 23.0 $\pm$ 1.7 $\pm$ 1.1 ${}_{-2.9}^{+1.6}$ | 18.3 $\pm$ 1.6 $\pm$ 0.9 ${}_{-2.8}^{+1.6}$ | 10.4 $\pm$ 1.8 $\pm$ 0.5 ${}_{-2.3}^{+1.4}$
2–3 | 39.3 $\pm$ 3.6 $\pm$ 1.9 ${}_{-12.9}^{+6.4}$ | 45.7 $\pm$ 2.6 $\pm$ 2.1 ${}_{-8.2}^{+4.5}$ | 24.4 $\pm$ 1.8 $\pm$ 1.1 ${}_{-2.9}^{+1.5}$ | 26.3 $\pm$ 2.0 $\pm$ 1.2 ${}_{-3.4}^{+1.9}$ | 14.9 $\pm$ 2.2 $\pm$ 0.7 ${}_{-3.2}^{+1.8}$
3–4 | 55.8 $\pm$ 4.2 $\pm$ 2.6 ${}_{-17.4}^{+8.9}$ | 42.1 $\pm$ 2.5 $\pm$ 2.0 ${}_{-7.3}^{+3.8}$ | 37.8 $\pm$ 2.2 $\pm$ 1.8 ${}_{-4.3}^{+2.2}$ | 20.8 $\pm$ 1.8 $\pm$ 1.0 ${}_{-2.4}^{+1.3}$ | 11.9 $\pm$ 1.9 $\pm$ 0.6 ${}_{-2.1}^{+1.2}$
4–5 | 54.5 $\pm$ 4.1 $\pm$ 2.6 ${}_{-15.9}^{+8.2}$ | 39.2 $\pm$ 2.4 $\pm$ 1.8 ${}_{-6.7}^{+3.6}$ | 22.6 $\pm$ 1.7 $\pm$ 1.1 ${}_{-2.0}^{+1.1}$ | 18.3 $\pm$ 1.6 $\pm$ 0.9 ${}_{-1.6}^{+0.8}$ | 12.2 $\pm$ 1.9 $\pm$ 0.6 ${}_{-1.8}^{+1.0}$
5–6 | 39.1 $\pm$ 3.4 $\pm$ 1.9 ${}_{-10.3}^{+5.4}$ | 44.8 $\pm$ 2.6 $\pm$ 2.1 ${}_{-7.6}^{+3.9}$ | 32.8 $\pm$ 2.1 $\pm$ 1.5 ${}_{-2.8}^{+1.5}$ | 18.1 $\pm$ 1.6 $\pm$ 0.8 ${}_{-1.2}^{+0.6}$ | 7.8 $\pm$ 1.5 $\pm$ 0.4 ${}_{-0.9}^{+0.4}$
6–7 | 28.8 $\pm$ 2.9 $\pm$ 1.4 ${}_{-8.3}^{+4.1}$ | 25.1 $\pm$ 1.9 $\pm$ 1.2 ${}_{-3.9}^{+2.0}$ | 22.3 $\pm$ 1.7 $\pm$ 1.0 ${}_{-1.4}^{+0.7}$ | 11.6 $\pm$ 1.3 $\pm$ 0.5 ${}_{-0.5}^{+0.3}$ | 5.2 $\pm$ 1.2 $\pm$ 0.2 ${}_{-0.5}^{+0.2}$
7–8 | 21.9 $\pm$ 2.4 $\pm$ 1.0 ${}_{-5.4}^{+2.7}$ | 23.4 $\pm$ 1.9 $\pm$ 1.1 ${}_{-3.5}^{+1.8}$ | 16.3 $\pm$ 1.4 $\pm$ 0.8 ${}_{-0.9}^{+0.4}$ | 5.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.1}^{+0.1}$ | 5.4 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.4}^{+0.2}$
8–9 | 22.9 $\pm$ 2.4 $\pm$ 1.1 ${}_{-4.8}^{+2.6}$ | 17.1 $\pm$ 1.5 $\pm$ 0.8 ${}_{-2.0}^{+1.0}$ | 12.4 $\pm$ 1.2 $\pm$ 0.6 ${}_{-0.6}^{+0.3}$ | 7.6 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.2}^{+0.1}$ | 4.3 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
9–10 | 12.8 $\pm$ 1.8 $\pm$ 0.6 ${}_{-2.9}^{+1.5}$ | 12.9 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.2}^{+0.6}$ | 9.8 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.5}^{+0.2}$ | 7.0 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 1.2 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$
10–11 | 10.3 $\pm$ 1.6 $\pm$ 0.5 ${}_{-2.1}^{+1.1}$ | 9.5 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.9}^{+0.5}$ | 4.3 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 6.4 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.2}^{+0.1}$ | 2.6 $\pm$ 0.8 $\pm$ 0.1 ${}_{-0.2}^{+0.1}$
11–12 | 8.6 $\pm$ 1.5 $\pm$ 0.4 ${}_{-2.4}^{+1.2}$ | 10.0 $\pm$ 1.1 $\pm$ 0.5 ${}_{-0.9}^{+0.5}$ | 4.4 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.0}$ | 1.2 $\pm$ 0.4 $\pm$ 0.1 ${}_{-0.0}^{+0.0}$ | -
12–13 | 5.8 $\pm$ 1.2 $\pm$ 0.3 ${}_{-0.9}^{+0.5}$ | 5.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.5}^{+0.3}$ | 4.1 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.0}$ | - | -
13–14 | 4.4 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.7}^{+0.4}$ | 1.7 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | 2.6 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | - | -
14–15 | 1.9 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.3}^{+0.2}$ | 4.9 $\pm$ 0.8 $\pm$ 0.2 ${}_{-0.5}^{+0.3}$ | 3.9 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$ | - | -
Table 4: Double differential cross-section
$\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ as a function of rapidity and
transverse momentum, in pb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$). The first
uncertainty is statistical, the second is systematic, and the third is due to
the unknown polarisation of the $\varUpsilon(3S)$. Regions where the number of
events was not sufficient to perform a measurement are indicated with a dash.
$p_{\rm T}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | | | | |
0–1 | 7.0 $\pm$ 1.5 $\pm$ 0.3 ${}_{-2.6}^{+1.3}$ | 6.3 $\pm$ 1.0 $\pm$ 0.3 ${}_{-1.0}^{+0.6}$ | 3.1 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.4}^{+0.2}$ | 5.0 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.9}^{+0.5}$ | -
1–2 | 14.1 $\pm$ 2.2 $\pm$ 0.7 ${}_{-5.3}^{+2.6}$ | 5.6 $\pm$ 0.9 $\pm$ 0.3 ${}_{-1.1}^{+0.6}$ | 11.6 $\pm$ 1.2 $\pm$ 0.6 ${}_{-1.3}^{+0.7}$ | 12.7 $\pm$ 1.4 $\pm$ 0.6 ${}_{-2.1}^{+1.2}$ | 10.2 $\pm$ 1.9 $\pm$ 0.5 ${}_{-2.6}^{+1.4}$
2–3 | 17.6 $\pm$ 2.3 $\pm$ 0.9 ${}_{-5.3}^{+2.7}$ | 22.3 $\pm$ 1.8 $\pm$ 1.1 ${}_{-4.1}^{+2.1}$ | 15.2 $\pm$ 1.4 $\pm$ 0.7 ${}_{-1.6}^{+0.8}$ | 6.7 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.9}^{+0.5}$ | 9.9 $\pm$ 1.7 $\pm$ 0.5 ${}_{-2.1}^{+1.2}$
3–4 | 24.9 $\pm$ 2.7 $\pm$ 1.2 ${}_{-7.7}^{+4.0}$ | 17.6 $\pm$ 1.6 $\pm$ 0.8 ${}_{-3.1}^{+1.6}$ | 13.5 $\pm$ 1.3 $\pm$ 0.6 ${}_{-1.6}^{+0.8}$ | 6.8 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.8}^{+0.4}$ | 7.5 $\pm$ 1.5 $\pm$ 0.4 ${}_{-1.3}^{+0.7}$
4–5 | 16.7 $\pm$ 2.2 $\pm$ 0.8 ${}_{-5.1}^{+2.6}$ | 17.5 $\pm$ 1.6 $\pm$ 0.8 ${}_{-3.0}^{+1.6}$ | 6.9 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.6}^{+0.3}$ | 6.1 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.5}^{+0.3}$ | 7.6 $\pm$ 1.5 $\pm$ 0.4 ${}_{-1.2}^{+0.6}$
5–6 | 16.6 $\pm$ 2.1 $\pm$ 0.8 ${}_{-4.6}^{+2.4}$ | 21.3 $\pm$ 1.8 $\pm$ 1.0 ${}_{-3.5}^{+1.8}$ | 12.1 $\pm$ 1.2 $\pm$ 0.6 ${}_{-1.1}^{+0.6}$ | 7.8 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.5}^{+0.3}$ | 7.6 $\pm$ 1.4 $\pm$ 0.4 ${}_{-0.9}^{+0.5}$
6–7 | 22.2 $\pm$ 2.5 $\pm$ 1.1 ${}_{-5.6}^{+3.0}$ | 19.1 $\pm$ 1.7 $\pm$ 0.9 ${}_{-3.0}^{+1.5}$ | 8.4 $\pm$ 1.0 $\pm$ 0.4 ${}_{-0.6}^{+0.3}$ | 7.1 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.3}^{+0.2}$ | 3.1 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$
7–8 | 20.6 $\pm$ 2.4 $\pm$ 1.0 ${}_{-5.4}^{+2.7}$ | 10.5 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.6}^{+0.8}$ | 9.2 $\pm$ 1.1 $\pm$ 0.4 ${}_{-0.6}^{+0.3}$ | 5.2 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.1}^{+0.1}$ | 1.4 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$
8–9 | 13.7 $\pm$ 1.9 $\pm$ 0.7 ${}_{-3.3}^{+1.7}$ | 10.7 $\pm$ 1.2 $\pm$ 0.5 ${}_{-1.6}^{+0.8}$ | 6.8 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.3}^{+0.1}$ | 2.4 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 0.6 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.0}^{+0.0}$
9–10 | 11.3 $\pm$ 1.7 $\pm$ 0.5 ${}_{-2.5}^{+1.3}$ | 6.9 $\pm$ 1.0 $\pm$ 0.3 ${}_{-0.8}^{+0.4}$ | 5.7 $\pm$ 0.8 $\pm$ 0.3 ${}_{-0.3}^{+0.2}$ | 2.5 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | 3.2 $\pm$ 0.9 $\pm$ 0.2 ${}_{-0.1}^{+0.1}$
10–11 | 8.4 $\pm$ 1.5 $\pm$ 0.4 ${}_{-2.0}^{+1.0}$ | 5.5 $\pm$ 0.9 $\pm$ 0.3 ${}_{-0.6}^{+0.3}$ | 4.3 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 2.6 $\pm$ 0.6 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | -
11–12 | 8.7 $\pm$ 1.4 $\pm$ 0.4 ${}_{-1.7}^{+0.9}$ | 4.4 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.3}^{+0.2}$ | 3.2 $\pm$ 0.6 $\pm$ 0.2 ${}_{-0.2}^{+0.1}$ | 1.8 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | -
12–13 | 4.5 $\pm$ 1.0 $\pm$ 0.2 ${}_{-0.9}^{+0.4}$ | 3.2 $\pm$ 0.6 $\pm$ 0.2 ${}_{-0.3}^{+0.1}$ | 3.5 $\pm$ 0.7 $\pm$ 0.2 ${}_{-0.1}^{+0.1}$ | - | -
13–14 | 2.4 $\pm$ 0.7 $\pm$ 0.1 ${}_{-0.4}^{+0.2}$ | 0.7 $\pm$ 0.3 $\pm$ 0.0 ${}_{-0.1}^{+0.0}$ | 2.1 $\pm$ 0.5 $\pm$ 0.1 ${}_{-0.1}^{+0.0}$ | - | -
14–15 | 0.7 $\pm$ 0.4 $\pm$ 0.0 ${}_{-0.1}^{+0.1}$ | 1.5 $\pm$ 0.4 $\pm$ 0.1 ${}_{-0.1}^{+0.1}$ | 0.9 $\pm$ 0.3 $\pm$ 0.0 ${}_{-0.0}^{+0.0}$ | - | -
Table 5: Ratios of cross-sections $\varUpsilon(2S)\rightarrow\mu^{+}\mu^{-}$
and $\varUpsilon(3S)\rightarrow\mu^{+}\mu^{-}$ with respect to
$\varUpsilon(1S)\rightarrow\mu^{+}\mu^{-}$ as a function of $p_{\rm T}$ in the
range $2.0<y<4.5$, assuming no polarisation. The first uncertainty is
statistical, the second is systematic and the third is due to the unknown
polarisation of the three states.
$p_{\rm T}$ | $R^{2S/1S}$ | $R^{3S/1S}$
---|---|---
(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | |
0–1 | 0.202 $\pm$ 0.015 $\pm$ 0.006 $\pm$ 0.052 | 0.099 $\pm$ 0.010 $\pm$ 0.003 $\pm$ 0.025
1–2 | 0.192 $\pm$ 0.009 $\pm$ 0.005 $\pm$ 0.051 | 0.089 $\pm$ 0.006 $\pm$ 0.003 $\pm$ 0.024
2–3 | 0.207 $\pm$ 0.008 $\pm$ 0.006 $\pm$ 0.052 | 0.098 $\pm$ 0.005 $\pm$ 0.003 $\pm$ 0.025
3–4 | 0.247 $\pm$ 0.010 $\pm$ 0.007 $\pm$ 0.056 | 0.099 $\pm$ 0.006 $\pm$ 0.003 $\pm$ 0.023
4–5 | 0.234 $\pm$ 0.010 $\pm$ 0.007 $\pm$ 0.047 | 0.087 $\pm$ 0.005 $\pm$ 0.003 $\pm$ 0.017
5–6 | 0.305 $\pm$ 0.013 $\pm$ 0.009 $\pm$ 0.058 | 0.136 $\pm$ 0.007 $\pm$ 0.005 $\pm$ 0.023
6–7 | 0.260 $\pm$ 0.013 $\pm$ 0.007 $\pm$ 0.048 | 0.160 $\pm$ 0.009 $\pm$ 0.006 $\pm$ 0.027
7–8 | 0.268 $\pm$ 0.015 $\pm$ 0.008 $\pm$ 0.048 | 0.162 $\pm$ 0.011 $\pm$ 0.006 $\pm$ 0.027
8–9 | 0.309 $\pm$ 0.019 $\pm$ 0.009 $\pm$ 0.046 | 0.166 $\pm$ 0.013 $\pm$ 0.006 $\pm$ 0.028
9–10 | 0.303 $\pm$ 0.022 $\pm$ 0.009 $\pm$ 0.045 | 0.187 $\pm$ 0.016 $\pm$ 0.007 $\pm$ 0.032
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arxiv-papers
| 2012-02-29T15:43:27 |
2024-09-04T02:49:28.109184
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y.\n Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L.\n Arrabito, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, D. S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook,\n H. Brown, K. de Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L.\n Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P.\n Collins, A. Comerma-Montells, F. Constantin, A. Contu, A. Cook, M. Coombes,\n G. Corti, B. Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P.\n N. Y. David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De\n Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D.\n Esperante Pereira, A. Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C.\n Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S.\n Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L.\n Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon,\n Ph. 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, E. Greening, S. Gregson, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.\n C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, K. Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V. N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu,\n J. Luisier, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O.\n Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe,\n C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D. A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Nedos, M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M.\n Nicol, V. Niess, N. Nikitin, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, K. Pal, J. Palacios,\n A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.\n 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, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler,\n A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin,\n F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V. K. Subbiah, 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, N. Torr,\n E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B.\n Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang,\n W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Giulia Manca",
"url": "https://arxiv.org/abs/1202.6579"
}
|
1202.6668
|
# Complexity of complexity and maximal plain versus prefix-free Kolmogorov
complexity
Bruno Bauwens 111 Instituto de Telecomunicações Faculdade de Ciência da
Universidade do Porto. Supported by the Portuguese science foundation FCT
(SFRH/BPD/75129/2010), and is also partially supported by the project
$CSI^{2}$ (PTDC/EIAC/099951/2008). The author is grateful to Elena Kalinina
and (Nikolay) Kolia Vereshchagin for giving the text [3]. The author is also
grateful to (Alexander) Sasha Shen for his very generous help: for reading
earlier texts on these results, for discussion, for providing a clear
exposition of section 1 and some parts of section 2, and for his permission to
publish it (with small modifications).
###### Abstract
Peter Gacs showed [1] that for every $n$ there exists a bit string $x$ of
length $n$ whose plain complexity $\operatorname{\mathit{C}\,}(x)$ has almost
maximal conditional complexity relative to $x$, i.e.,
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x)|x)\geq\log
n-\log^{(2)}n-O(1)$. Here $\log^{2}(i)=\log\log i$ etc. Following Elena
Kalinina [3], we provide a game-theoretic proof of this result; modifying her
argument, we get a better (and tight) bound $\log n-O(1)$. We also show the
same bound for prefix-free complexity.
Robert Solovay’s showed [10] that infinitely many strings $x$ have maximal
plain complexity but not maximal prefix-free complexity (among the strings of
the same length); i.e. for some $c$: $|x|-\operatorname{\mathit{C}\,}(x)\leq
c$ and
$|x|+\operatorname{\mathit{K}\,}(|x|)-\operatorname{\mathit{K}\,}(x)\geq\log^{(2)}|x|-c\log^{(3)}|x|$.
Using the result above, we provide a short proof of Solovay’s result. We also
generalize it by showing that for some $c$ and for all $n$ there are strings
$x$ of length $n$ with $n-\operatorname{\mathit{C}\,}(x)\leq c$, and
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)\geq\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)-3\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)-c$.
This is very close to the upperbound
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(1)$ proved by
Solovay.
## Introduction
Plain Kolmogorov complexity $\operatorname{\mathit{C}\,}(x)$ of a binary
string $x$ was defined in [4] as the minimal length of a program that computes
$x$. (See the preliminaries or [2, 5, 9] for the details.) It was clear from
the beginning (see, e.g., [12]) that complexity function is not computable: no
algorithm can compute $\operatorname{\mathit{C}\,}(x)$ given $x$. In [1] a
stronger non-uniform version of this result was proven: for every $n$ there
exists a string $x$ of length $n$ such that conditional complexity
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x)|x)$, i.e., the
minimal length of a program that maps $x$ to $\operatorname{\mathit{C}\,}(x)$,
is at least $\log n-O(\log^{(2)}n)$. (If complexity function were computable,
this conditional complexity would be bounded.)
In Section 1 we revisit this classical result and improve it a bit by removing
the $\log^{(2)}n$ term. No further improvement is possible because
$\operatorname{\mathit{C}\,}(n)\leq n+O(1)$, therefore
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(n)|x)\leq\log n+O(1)$
for all $x$. We use a game technique that was developed by Andrej Muchnik (see
[8, 7, 11]) and turned out to be useful in many cases. Recently Elena Kalinina
(in her master thesis [3]) used it to provide a proof of Gacs’ result. We use
a more detailed analysis of essentially the same game to get a better bound.
For some $c$, a bit string $x$ is $\operatorname{\mathit{C}\,}$-random if
$n-\operatorname{\mathit{C}\,}(x)\leq c$. Note that $n+O(1)$ is the smallest
upper bound for $\operatorname{\mathit{C}\,}(x)$. A variant of plain
complexity is prefix-free or self-delimiting complexity, which is defined as
the shortest program that produces $x$ on a Turing machine with binary input
tape, i.e. without blanc or terminating symbol. (See the preliminaries or [2,
5, 9] for the details.) The smallest upper bound for
$\operatorname{\mathit{K}\,}(x)$ for strings of length $n$ is
$n+\operatorname{\mathit{K}\,}(n)+O(1)$. For some $c$, the string $x$ is
defined to be $\operatorname{\mathit{K}\,}$-random if
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)\leq c$.
Robert Solovay [10] observed that $\operatorname{\mathit{K}\,}$-random strings
are also $\operatorname{\mathit{C}\,}$-random strings (for some
$c^{\prime}\leq O(c)$), but not vice versa. Moreover, he showed that some $c$
and infinitely many $x$ satisfy $|x|-\operatorname{\mathit{C}\,}(x)\leq c$ and
$|x|+\operatorname{\mathit{K}\,}(|x|)-\operatorname{\mathit{K}\,}(x)\geq\log^{(2)}|x|-c\log^{(3)}|x|\,.$
He also showed that for $\operatorname{\mathit{C}\,}$-random $x$ the left-hand
side of the equation is upper-bounded by
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(1)$, which is
bounded by $\log^{(2)}n+O(1)$. Later Joseph Miller [6] and Alexander Shen [8]
generalized this, by showing that every co-enumerable set (i.e., the
complement is enumerable) containing strings of every length, also contains
infinitely many $x$ such that the above equation holds. (Note that the set of
$\operatorname{\mathit{C}\,}$-random strings is co-enumerable but the set of
$\operatorname{\mathit{K}\,}$-random strings not.)
In Section 2 we provide a short proof for Solovay’s result using the improved
version of Gacs’ theorem. Then we generalize it by showing that for some $c$
and every $n$ there are strings $x$ of length $n$ with
$n-\operatorname{\mathit{C}\,}(x)\leq c$ and
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)\geq\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)-3\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)-c\,.$
This is very close to the upperbound
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)-O(1)$, which
was shown by Solovay [10]. By the improved version of Gacs’ result, we can
choose $n$ such that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)=\log^{(2)}n+O(1)$.
For such $n$ we obtain Solovay’s theorem with the $c\log^{(3)}|x|$ term
replaced by a $O(1)$ constant.
Preliminaries: Let $U$ be a Turing machine. The plain (Kolmogorov) complexity
relative to $U$ is defined by
$\operatorname{\mathit{C}\,}_{U}(x|y)=\min\left\\{|p|:U(p,y)=x\right\\}\,.$
If the machine $U$ is prefix-free (i.e., for every $p,y$ such that $U(p,y)$
halts, there is no prefix $q$ of $p$ such that $U(q,y)$ halts) then we write
$\operatorname{\mathit{K}\,}_{U}(x|y)$ rather than
$\operatorname{\mathit{C}\,}_{U}(x|y)$, and refer to it as prefix-free
(Kolmogorov) complexity relative to $U$. There exist plain and prefix-free
Turing machines $U$ and $V$ for which $\operatorname{\mathit{C}\,}_{U}(x|y)$
and $\operatorname{\mathit{K}\,}_{V}(x|y)$ are minimal within an $O(1)$
constant. We fix such machines and omit the indexes $U$,$V$. If $y$ is the
empty string we use the notation $\operatorname{\mathit{C}\,}(x)$ and
$\operatorname{\mathit{K}\,}(x)$.
## 1 Complexity of complexity can be high
###### Theorem 1.
There exist some constant $c$ such that for every $n$ there exists a string
$x$ of length $n$ such that
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x)|x)\geq\log n-c$.
To prove this theorem, we first define some game and show a winning strategy
for the game. (The connection between the game and the statement that we want
to prove will be explained later.)
### 1.1 The game
Game $G_{n}$ has parameter $n$ and is played on a rectangular board divided
into cells. The board has $2^{n}$ columns and $n$ rows numbered
$0,1,\ldots,n-1$ (the bottom row has number $0$, the next one has number $1$
and so on, the top row has number $n-1$), see Fig. 1.
Initially the board is empty. Two players: White and Black, alternate their
moves. At each move, a player can pass or place a pawn (of his color) on the
board. The pawn can not be moved or removed afterwards. Also Black may blacken
some cell instead. Let us agree that White starts the game (though it does not
matter).
The position of the game should satisfy some restrictions; the player who
violates these restrictions, loses the game immediately. Formally the game is
infinite, but since the number of (non-trivial) moves is a priori bounded, it
can be considered as finite, and the winner is determined by the last (limit)
position on the board.
_Restrictions_ : (1) each player may put at most $2^{i}$ pawns in row $i$
(thus the total number of black and white pawns in a row can be at most
$2^{i}+2^{i}$); (2) in each column Black may blacken at most half of the
cells.
We say that a white pawn is _dead_ if either it is on a blackened cell or has
a black pawn in the same column strictly below it.
_Winning rule_ : Black wins if he killed all white pawns, i.e., if each white
pawn is dead in the final position.
Figure 1: Game board
For example, if the game ends in the position shown at Fig. 1, the
restrictions are not violated (there are $3\leq 2^{2}$ white pawns in row $2$
and $1\leq 2^{1}$ white pawn in row $1$, as well as $1\leq 2^{2}$ black pawn
in row $2$ and $1\leq 2^{0}$ black pawn in row $0$). Black loses because the
white pawn in the third column is not dead: it has no black pawn below and the
cell is not blackened. (There is also one living pawn in the fourth column.)
### 1.2 How White can win
The strategy is quite simple. White starts by placing a white pawn in an upper
row of some column and waits until Black kills it, i.e., blackens the cell or
places a black pawn below. In the first case White puts her pawn one row down
and waits again. Since Black has no right to make all cells in a column black
(at most half may be blackened), at some point he will be forced to place a
black pawn below the white pawn in this column. After that White switches to
some other column. (The ordering of columns is not important; we may assume
that White moves from left to right.)
Note that when White switches to a next column, it may happen that there is a
black pawn in this column or some cells are already blackened. If there is
already a black pawn, White switches again to the next column; if some cell is
blackened, White puts her pawn in the topmost white (non-blackened) cell.
This strategy allows White to win. Indeed, Black cannot place his pawns in all
the columns due to the restrictions (the total number of his pawns is
$\sum_{i=0}^{n-1}2^{i}=2^{n}-1$, which is less than the number of columns).
White also cannot violate the restriction for the number of her pawns on some
row $i$: all dead pawns have a black pawns strictly below them, so the number
of them on row $i$ is $\sum_{j=0}^{i-1}2^{j}=2^{i}-1$, hence White can put an
additional pawn.
In fact we may even allow Black to blacken all the cells except one in each
column, and White will still win, but this is not needed (and the $n/2$
restriction will be convenient later).
### 1.3 Proof of Gacs’ theorem
Let us show that for each $n$ there exists a string $x$ of length $n$ such
that $\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x|n)|x)\geq\log
n-O(1)$. Note that here $\operatorname{\mathit{C}\,}(x|n)$ is used instead of
$\operatorname{\mathit{C}\,}(x)$; the difference between these two numbers is
$O(\log n)$ since $n$ can be described by $\log n$ bits, so the difference
between the complexities of these two numbers is $O(\log\log n)$.
Consider the following strategy for Black (assuming that the columns of the
table are indexed by strings of length $n$):
* •
Black blackens the cell in column $x$ and row $i$ as soon as he discovers that
$\operatorname{\mathit{C}\,}(i|x)<\log n-1$. (The constant $1$ guarantees that
less than half of the cells will be blackened.) Note that Kolmogorov
complexity is an upper semicomputable function, and Black approximates it from
above, so more and more cells are blackened.
* •
Black puts a black pawn in a cell $(x,i)$ when he finds a program of length
$i$ that produces $x$ with input $n$ (this implies that
$\operatorname{\mathit{C}\,}(x|n)\leq i$). Note that there are at most $2^{i}$
programs of length $i$, so Black does not violate the restriction for the
number of pawns on any row $i$.
Let White play against this strategy (using the strategy described above).
Since the strategy is computable, the behavior of White is also computable.
One can construct a decompressor $V$ for the strings of length $n$ as follows:
each time White puts a pawn in a cell $(x,i)$, a program of length $i$ is
assigned to $x$. By White’s restriction, no more than $2^{i}$ programs need to
be assigned. By universality, a white pawn on cell $(x,i)$ implies that
$\operatorname{\mathit{C}\,}(x|n)\leq i+O(1)$. If White’s pawn is alive in
column $x$, there is no black pawn below, so
$\operatorname{\mathit{C}\,}(x|n)\geq i$, and therefore
$\operatorname{\mathit{C}\,}(x|n)=i+O(1)$. Moreover, for a winning pawn, the
cell $(x,i)$ is not blackened, so $\operatorname{\mathit{C}\,}(i|x)\geq\log
n-1$. Therefore,
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x|n)|x)\geq\log
n-O(1)$.
Remark: the construction also guarantees that
$\operatorname{\mathit{C}\,}(x|n)\geq n/2-O(1)$ for that $x$. (Here the factor
$1/2$ can be replaced by any $\alpha<1$ if we change the rules of the game
accordingly.) Indeed, according to white’s strategy, he always plays in the
highest non-black cell of some column, and at most half of the cells in a
column can be blackened, therefore no white pawns appear in the lower half of
the board.
### 1.4 Modified game and the proof of Theorem 1
Now we need to get rid of the condition $n$ and show that for every $n$ there
is some $x$ such that
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{C}\,}(x)|x)\geq\log
n-O(1)$. Imagine that White and Black play simultaneously all the games
$G_{n}$. Black blackens the cell $(x,i)$ in game $G_{|x|}$ when he discovers
that $\operatorname{\mathit{C}\,}(i|x)<\log n-1$, as he did before, and puts a
black pawn in a cell $(x,i)$ when he discovers an _unconditional_ program of
length $i$ for $x$. If Black uses this strategy, he satisfies the stronger
restriction: the total number of pawns in row $i$ _on all boards_ is bounded
by $2^{i}$.
Assume that White uses the described strategy on each board. What can be said
about the total number of white pawns in row $i$? The dead pawns have black
pawns strictly below them and hence the total number of them does not exceed
$2^{i}-1$. On the other hand, there is at most one live white pawn on each
board. We know also that in $G_{n}$ white pawns never appear below row
$n/2-1$, so the number of live white pawns does not exceed $2i+O(1)$.
Therefore we have $O(2^{i})$ white pawns on the $i$-th row in total.
For each $n$ there is a cell $(x,i)$ in $G_{n}$ where White wins in $G_{n}$.
Hence, $\operatorname{\mathit{C}\,}(x)<i+O(1)$ (because of property just
mentioned and the computability of White’s behavior),
$\operatorname{\mathit{C}\,}(x)\geq i$ and
$\operatorname{\mathit{C}\,}(i|x)\geq\log n-1$ (by construction of Black’s
strategies and the winning condition). Theorem 1 is proven.
### 1.5 Version for prefix complexity
###### Theorem 2.
There exist some constant $c$ such that for every $n$ there exists a string
$x$ of length $n$ such that
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(x)|x)\geq\log n-c$
and $\operatorname{\mathit{K}\,}(x)\geq n/2$. This also implies that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(x)|x)\geq\log n-c$.
The proof of
$\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(x)|x)\geq\log n-c$
goes in the same way. Black places a pawn in cell $(i,x)$ if some program of
length $i$ for a prefix-free (unconditional) machine computes $x$ (and hence
$\operatorname{\mathit{K}\,}(x)\leq i$); White uses the same strategy as
described above. The sum of $2^{-i}$ for all black pawns is less than $1$
(Kraft-inequality); some white pawns are dead, i.e., strictly above black
ones, and for each column the sum of $2^{-j}$ where $j$ is the row number,
does not exceed $\sum_{j>i}^{n}2^{-j}<2^{-i}$. Hence the corresponding sum for
all dead white pawns is less than $1$; for the rest the sum is bounded by
$\sum_{n}2^{-n/2+1}$, so the total sum is bounded by a constant, and we
conclude that for $x$ in the winning column the row number is
$\operatorname{\mathit{K}\,}(x)+O(1)$, and this cell is not blackened.
## 2 Strings with maximal plain and
non-maximal prefix-free complexity
In this section we compare two measures of non-randomness. Let $x$ be a string
of length $n$; we know that $\operatorname{\mathit{C}\,}(x)\leq n+O(1)$, and
the difference $n-\operatorname{\mathit{C}\,}(n)$ measures how “nonrandom” $x$
is. Let us call it $\operatorname{\mathit{C}\,}$-deficiency of $x$. On the
other hand, $\operatorname{\mathit{K}\,}(x)\leq
n+\operatorname{\mathit{K}\,}(n)+O(1)$, so
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)$ also
measures “nonrandomness” in some other way; we call this quantity
$\operatorname{\mathit{K}\,}$-deficiency of $x$.
The following proposition means that $\operatorname{\mathit{K}\,}$-random
strings (for which $\operatorname{\mathit{K}\,}$-deficiency is small; they are
also called “Chaitin random”) are always $\operatorname{\mathit{C}\,}$-random
($\operatorname{\mathit{C}\,}$-deficiency is small; such strings are also
called “Kolmogorov random”).
###### Proposition 3 (Solovay [10]).
$|x|+K(|x|)-K(x)\leq c$ implies $|x|-C(x)\leq O(c)$.
###### Proof.
We use a result of Levin: for every string $u$
$\operatorname{\mathit{K}\,}(u|\operatorname{\mathit{C}\,}(u))=\operatorname{\mathit{C}\,}(u)+O(1),$
and, on the other hand, for any positive or negative integer number $c$:
$\operatorname{\mathit{K}\,}(u|i)=i+c,$
implies $\operatorname{\mathit{C}\,}(u)=i+O(c)$222Textbooks like [5, Lemma
3.1.1] mention only the first statement. To show the second, note that the
function $i\mapsto\operatorname{\mathit{K}\,}(x|i)$ maps numbers at distance
$c$ to numbers at distance $O(\log c)$, hence, the fixed point
$\operatorname{\mathit{C}\,}(x)$ must be unique within an $O(1)$ constant.
Furthermore, for any $i$, the fixed point must be within distance
$O(\log|i-\operatorname{\mathit{K}\,}(u|i)|)$ from $i$, hence
$|\operatorname{\mathit{C}\,}(u)-i|\leq
O(\log|i-\operatorname{\mathit{K}\,}(u|i)|)=O(\log c)$. .
Let $n=|x|$. Notice that
$n+K(n)\leq K(x)-c=K(x,n)-O(c)\leq K(x|n)+K(n)-O(c)\,.$
Hence, $K(x|n)\geq n-O(c)$, thus $K(x|n)=n+O(c)$ and thus: $C(x)=n+O(c)$. ∎
R. Solovay showed that the reverse statement is not always true: a
$\operatorname{\mathit{C}\,}$-random string may be not
$\operatorname{\mathit{K}\,}$-random. However, as the following result shows,
the $\operatorname{\mathit{K}\,}$-deficiency still can be bounded for
$\operatorname{\mathit{C}\,}$-random strings:
###### Proposition 4 (Solovay [10]).
For any $x$ of length $n$ the inequality $\operatorname{\mathit{C}\,}(x)\geq
n-c$ implies:
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)\leq\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(c)\,.$
Note that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\leq\log^{(2)}n+O(1).$
###### Proof.
The proof uses another result of Levin [1, 2, 5]: for all $u,v$ we have the
additivity property
$\operatorname{\mathit{K}\,}(u,v)=\operatorname{\mathit{K}\,}(u)+\operatorname{\mathit{K}\,}(v|u,\operatorname{\mathit{K}\,}(u))+O(1)\,.$
To prove Proposition 4, notice that
$n=\operatorname{\mathit{C}\,}(x)=\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{C}\,}(x))=\operatorname{\mathit{K}\,}(x|n)$
with $O(c)$-precision. By additivity we have:
$\operatorname{\mathit{K}\,}(x)=\operatorname{\mathit{K}\,}(n,x)=\operatorname{\mathit{K}\,}(n)+\operatorname{\mathit{K}\,}(x|n,\operatorname{\mathit{K}\,}(n))$.
Putting these observations together, we get (with $O(c)$-precision)
$\displaystyle
n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)$
$\displaystyle=\operatorname{\mathit{K}\,}(x|n)+\operatorname{\mathit{K}\,}(n)-(\operatorname{\mathit{K}\,}(n)+\operatorname{\mathit{K}\,}(x|n,K(n)))=$
$\displaystyle=\operatorname{\mathit{K}\,}(x|n)-\operatorname{\mathit{K}\,}(x|n,\operatorname{\mathit{K}\,}(n))\,.$
(1)
Observe that
$\operatorname{\mathit{K}\,}(x|n)\leq\operatorname{\mathit{K}\,}(x|n,\operatorname{\mathit{K}\,}(n))+\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(1)$,
hence the $\operatorname{\mathit{K}\,}$-deficiency is bounded by
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(c)$. ∎
The following theorem shows that for all $n$ the bound
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)$ for
$\operatorname{\mathit{K}\,}$-deficiency for
$\operatorname{\mathit{C}\,}$-random strings can almost be achieved. The error
is at most
$O(\log\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n))$.
###### Theorem 5.
For some $c$ and all $n$ there are strings $x$ of length $n$ such that
$n-\operatorname{\mathit{C}\,}(x)\leq c$, and
$n+\operatorname{\mathit{K}\,}(n)-\operatorname{\mathit{K}\,}(x)\geq\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)-3\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)-c\,.$
By corollary, infinitely many $\operatorname{\mathit{C}\,}$-random strings
have $\operatorname{\mathit{K}\,}$-deficiency $\log^{(2)}|x|+O(1)$. Indeed,
for $n$ such that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)=\log^{(2)}n+O(1)$,
we have
$\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)\leq
O(1)$, and hence, a slightly stronger statement than proved by Solovay [10] is
obtained.
###### Corollary 6.
There exists a constant $c$ and infinitely many $x$ such that
$|x|-\operatorname{\mathit{C}\,}(x)\leq c$ and
$|x|+\operatorname{\mathit{K}\,}(|x|)-\operatorname{\mathit{K}\,}(x)\geq\log^{(2)}|x|-c$.
Before proving Theorem 5, we prove the corollary directly.
###### Proof.
First we choose $n$, the length of string $x$. It is chosen in such a way that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)=\log^{(2)}n+O(1)$
and $\operatorname{\mathit{K}\,}(n)\geq(\log n)/2$ (Theorem 2). (So the bound
of Proposition 4 is not an obstacle.) We know already (see equation 1) that
for a string $x$ with $\operatorname{\mathit{C}\,}$-deficiency $c$ the value
of $\operatorname{\mathit{K}\,}$-deficiency is $O(c)$-close to
$\operatorname{\mathit{K}\,}(x|n)-\operatorname{\mathit{K}\,}(x|n,\operatorname{\mathit{K}\,}(n))$.
This means that adding $\operatorname{\mathit{K}\,}(n)$ in the condition
should decrease the complexity, so let us include
$\operatorname{\mathit{K}\,}(n)$ in $x$ somehow. We also have to guarantee
maximal $\operatorname{\mathit{C}\,}$-complexity of $x$. This motivates the
following choice:
* •
choose $r$ of length $n-\log^{(2)}n$ such that
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n))\geq|r|$. Note
that this implies
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n))=|r|+O(1)$,
since the length of $r$ is determined by the condition;
* •
let $x=\langle K(n)\rangle r$, the concatenation of $K(n)$ (in binary) with
$r$. Note that $\langle K(n)\rangle$ has at most $\log^{(2)}n+O(1)$ bits for
every $n$, and by choice of $n$ has at least $\log^{(2)}n-1$ bits, hence
$|x|=n+O(1)$.
As we have seen (looking at equation (1)), it is enough to show that
$\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{K}\,}(n),n)\leq
n-\log^{(2)}n$ and $\operatorname{\mathit{K}\,}(x|n)\geq n$ (the latter
equality implies $\operatorname{\mathit{C}\,}(x)=n$); all the equalities here
and below are up to $O(1)$ additive term.
* •
Knowing $n$, we can split $x$ in two parts
$\langle\operatorname{\mathit{K}\,}(n)\rangle$ and $r$. Hence,
$\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{K}\,}(n),n)=\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n),r|n,\operatorname{\mathit{K}\,}(n))$,
and this equals
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n))$, i.e.,
$n-\log^{(2)}n$ by choice of $r$.
* •
To compute $\operatorname{\mathit{K}\,}(x|n)$, we use additivity:
$\operatorname{\mathit{K}\,}(x|n)=\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n),r|n)=\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+\operatorname{\mathit{K}\,}(r|\operatorname{\mathit{K}\,}(n),\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n),n)\,.$
By choice of $n$, we have
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)=\log^{(2)}n$,
and the last term simplifies to
$\operatorname{\mathit{K}\,}(r|\operatorname{\mathit{K}\,}(n),\log^{(2)}n,n)$,
and this equals
$\operatorname{\mathit{K}\,}(r|\operatorname{\mathit{K}\,}(n),n)=n-\log^{(2)}n$
by choice of $r$. Hence
$\operatorname{\mathit{K}\,}(x|n)=\log^{(2)}n+(n-\log^{(2)}n)=n$.
∎
Remark 1: One can also ask how many strings exist that satisfy the conditions
of Corollary 6. By Proposition 4, the length $n$ of such a string must satisfy
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\geq\log^{(2)}n-O(1)$.
By Theorem 2, there is at least one such an $n$ for every length of $n$ in
binary. Hence such $n$, can be found within exponential intervals.
Remark 2: One can ask for these $n$, how many strings $x$ of length $n$
satisfy the conditions of Corollary 6. By a theorem of Chaitin [5], there are
at least $O(2^{n-k})$ strings with $\operatorname{\mathit{K}\,}$-deficiency
$k$, hence we can have at most $O(2^{n-\log^{(2)}n})$ such strings. It turns
out that indeed at least a fraction $1/O(1)$ of them satisfy the conditions of
Corollary 6. To show this, note that in the proof Theorem 5, every different
$r$ of length $n-|q|=|n|-\log^{(2)}n+O(1)$ leads to the construction of a
different $x$. For such $r$ we essentially need
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n),q)\geq|r|-O(1)$,
and hence there are $O(2^{n-\log^{(2)}n})$ of them.
Proof of Theorem 5. In the proof above, in order to obtain a large value
$\operatorname{\mathit{K}\,}(x|n)-\operatorname{\mathit{K}\,}(x|n,\operatorname{\mathit{K}\,}(n))$,
we incorporated $\operatorname{\mathit{K}\,}(n)$ in a direct way (as
$\langle\operatorname{\mathit{K}\,}(n)\rangle$) in $x$. To show that
$C(x)=K(x|n)+O(1)$ is large we essentially used that the length of
$\langle\operatorname{\mathit{K}\,}(n)\rangle$ equals
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(1)$. For
general $n$, this trick does not work anymore, but we can use a shortest
program for $\operatorname{\mathit{K}\,}(n)$ given $n$ (on a plain machine).
For every $n$ we can construct $x$ as follows:
* •
let $q$ be a shortest program that computes $\operatorname{\mathit{K}\,}(n)$
from $n$ on a plain machine (if there are several shortest programs, we choose
the one with shortest running time). Note that
$|q|=\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(n)|n)+O(1)=\operatorname{\mathit{C}\,}(q|n)+O(1)$
(remind that by adding some fixed instructions, a program can print itself,
and that a shortest program is always incompressible, thus up to $O(1)$
constants:
$|q|\geq\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(n)|n)\geq\operatorname{\mathit{C}\,}(q|n)\geq|q|$),
by Levin’s result (conditional version), the last term also equals
$\operatorname{\mathit{K}\,}(q|n,|q|)+O(1)$;
* •
let $r$ have length $n-|q|$, such that
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n),q)\geq|r|$.
Note that this implies
$\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n),q)=|r|+O(1)$,
(since the length of $r$ is determined by the condition).
* •
We define $x$ as the concatenation $qr$.
We show that $\operatorname{\mathit{C}\,}(x)=n+O(1)$ and that the
$\operatorname{\mathit{K}\,}$-deficiency is at least
$|q|-\operatorname{\mathit{K}\,}(|q|\,|n)+O(1)$. To show that this implies the
theorem, we need that
$\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)-3\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)\leq\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(n)|n)-\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{C}\,}(\operatorname{\mathit{K}\,}(n)|n)\;|n)+O(1)\,,$
which is for $a=\operatorname{\mathit{K}\,}(n)$ the conditioned version of
Lemma 7:
$\operatorname{\mathit{K}\,}(a|n)-3\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{K}\,}(a|n)\;|n)\leq\operatorname{\mathit{C}\,}(a|n)-\operatorname{\mathit{K}\,}(\;\operatorname{\mathit{C}\,}(a|n)\;|n)+O(1)\,.$
Following the same structure as the proof above, it remains to show that
$\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{K}\,}(n),n)\leq
n-|q|+\operatorname{\mathit{K}\,}(|q|\,|n)$ and
$\operatorname{\mathit{K}\,}(x|n)\geq n$ (the latter equality implies
$\operatorname{\mathit{C}\,}(x)=n$); all the equalities here and below are up
to $O(1)$ additive term.
* •
Knowing $|q|$, we can split $x$ in two parts $q$ and $r$. Hence,
$\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{K}\,}(n),n,|q|)=\operatorname{\mathit{K}\,}(q,r|n,\operatorname{\mathit{K}\,}(n),|q|)$.
Given $n,\operatorname{\mathit{K}\,}(n),|q|$ we can search for a program of
length $|q|$ that on input $n$ outputs $\operatorname{\mathit{K}\,}(n)$; the
one with shortest computation time is $q$. Hence,
$\operatorname{\mathit{K}\,}(q,r|n,\operatorname{\mathit{K}\,}(n),|q|)=\operatorname{\mathit{K}\,}(r|n,\operatorname{\mathit{K}\,}(n),|q|)$,
i.e., $n-|q|$ by choice of $r$, and therefore
$\operatorname{\mathit{K}\,}(x|\operatorname{\mathit{K}\,}(n),n)\leq
n-|q|+\operatorname{\mathit{K}\,}(|q|\,|n)$.
* •
To compute $\operatorname{\mathit{K}\,}(x|n)$, we use additivity:
$\operatorname{\mathit{K}\,}(x|n)\geq\operatorname{\mathit{K}\,}(x|n,|q|)=\operatorname{\mathit{K}\,}(q,r|n,|q|)=\operatorname{\mathit{K}\,}(q|n,|q|)+\operatorname{\mathit{K}\,}(r|q,\operatorname{\mathit{K}\,}(q|n,|q|),n)\,.$
By choice of $q$ we have $\operatorname{\mathit{C}\,}(q|n)=|q|$, and hence by
Levin’s result $\operatorname{\mathit{K}\,}(q|n,|q|)=|q|$. The last term is
$\operatorname{\mathit{K}\,}(r|q,|q|,n)$ which equals
$\operatorname{\mathit{K}\,}(r|q,n)=n-|q|$ by choice of $r$. Hence,
$\operatorname{\mathit{K}\,}(x|n)\geq|q|+(n-|q|)=n$. ∎
###### Lemma 7.
$\operatorname{\mathit{K}\,}(a)-3\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a))\leq\operatorname{\mathit{C}\,}(a)-\operatorname{\mathit{K}\,}(\operatorname{\mathit{C}\,}(a))+O(1)$
###### Proof.
Note that
$\operatorname{\mathit{K}\,}(a)-\operatorname{\mathit{C}\,}(a)\leq\operatorname{\mathit{K}\,}(\operatorname{\mathit{C}\,}(a))$.
Indeed, any program for a plain machine can be considered as a program for a
prefix-free machine conditional to it’s length. Hence, we can transform a
plain program $p$ to a prefix-free program by adding a description of $|p|$ of
length $\operatorname{\mathit{K}\,}(|p|)$ to $p$. Hence it remains to show
$2\operatorname{\mathit{K}\,}(\operatorname{\mathit{C}\,}(a))\leq
3\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a))+O(1)$. Solovay
[10] showed that
$\operatorname{\mathit{K}\,}(a)-\operatorname{\mathit{C}\,}(a)=\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a))+O(\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a))))\,,$
hence,
$|\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a))-\operatorname{\mathit{K}\,}(\operatorname{\mathit{C}\,}(a))|\leq
O(\log\operatorname{\mathit{K}\,}(\operatorname{\mathit{K}\,}(a)))\,.$
∎
## References
* [1] P. Gács. On the symmetry of algorithmic information. Soviet Math. Dokl., 15(5):1477–1480 (1974).
* [2] P. Gács. Lecture notes on descriptional complexity and randomness. http://www.cs.bu.edu/faculty/gacs/papers/ait-notes.pdf, 1988-2011.
* [3] E. Kalinina. Some applications of the method of games in Kolmogorov complexity. Master thesis. Moscow State University, 2011.
* [4] A.N. Kolmogorov, Three approaches to the quantitative definition of information, _Problemy peredachi Informatsii_ , vol. 1, no. 1, pp. 3–11 (1965)
* [5] M. Li and P.M.B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, New York, 2008.
* [6] J.S. Miller, Contrasting plain and prefix-free complexities. Preprint available at http://www.math.wisc.edu/~jmiller/downloads.html.
* [7] A. Muchnik, On the basic structures of the descriptive theory of algorithms, _Soviet Math. Dokl._ , 32, p. 671–674 (1985).
* [8] An.A. Muchnik, I. Mezhirov, A. Shen, N. Vereshchagin, _Game interpretation of Kolmogorov complexity_ (2010), arxiv:1003.4712v1
* [9] A. Shen, _Algorithmic Information theory and Kolmogorov complexity_. Technical report TR2000-034, Uppsala University (2000).
* [10] R.Solovay, Draft of a paper (or series of papers) on Chaitin’s work. Unpublished notes, 215 pages, (1975).
* [11] N. Vereshchagin, Kolmogorov complexity and Games, _Bulletin of the European Association for Theoretical Computer Science_ , 94, Feb. 2008, p. 51–83.
* [12] A.K. Zvonkin, L.A. Levin, The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, _Russian Math. Surveys_ , 25, issue 6(156):83–124, 1970.
|
arxiv-papers
| 2012-02-29T20:09:17 |
2024-09-04T02:49:28.123827
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bruno Bauwens, Alexander Shen",
"submitter": "Bruno Bauwens",
"url": "https://arxiv.org/abs/1202.6668"
}
|
1203.0018
|
# The Rational Number $\mathbf{\frac{n}{p}}$ as a sum of two unit fractions
Konstantine Zelator
Department of Mathematics, Statistics, and Computer Science
212 Ben Franklin Hall
Bloomsburg University of Pennsylvania
400 East 2nd Street
Bloomsburg, PA 17815
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
e-mails: kzelator@bloomu.edu
konstantine zelator@yahoo.com
## 1 Introduction
In a 2011 paper in the journal Asian Journal of Algebra (see [1]), the authors
consider, among other equations, the diophantine equations
$2xy=n(x+y)\ \ {\rm and}\ \ 3xy=n(x+y).$
For the first equation, with $n$ being an odd positive integer, they give the
solution (in positive integers $x$ and $y$)
$\dfrac{n+1}{2}=x=\dfrac{(n-1)}{2}+1$,
$y=n\left(\dfrac{(n-1)}{2}+1\right)=n\left(\dfrac{n+1}{2}\right)$.
For the second equation, with $n\equiv 2({\rm mod}3)$, they present the
particular solution,
$\dfrac{n+1}{3}=x=\dfrac{(n-2)}{3}+1,\ \
y=n\left(\dfrac{(n-2)}{3}+1\right)=n\left(\dfrac{n+1}{3}\right).$
If in the above equations we assume $n$ to be prime, then these two equations
become special cases of the diophantine equation, $nxy=p(x+y)$, with $p$ being
a prime and $n$ a positive integer with $n\geq 2$.
This two-variable symmetric diophantine equation is the subject matter of this
article; with the added condition that the integer $n$ is not divisible by the
prime $p$. Observe that this equation can be written equivalently in fraction
form:
$\dfrac{n}{p}=\dfrac{1}{x}+\dfrac{1}{y}.$
This problem then can be approached from the point of view of decomposing a
positive rational number into a sum of two unit fractions (i.e., two rational
numbers whose numerators are equal to $1$). The ancient Egyptians left behind
an entire body of work involving the decomposition of a given fraction into a
sum of two or more unit fractions. They did so by creating tables containing
the decomposition of specific fractions into sums of unit fractions. An
excellent source on the subject of the work of the ancient Egyptians on unit
fractions is the book by David M. Burton, “The History of Mathematics, An
Introduction” (see [2]). Note that thanks to the identity
$\dfrac{1}{k}=\dfrac{1}{k+1}+\dfrac{1}{k(k+1)}$, a unit fraction can always be
written as a sum of two unit fractions.
We state our theorem.
###### Theorem 1.
Let $p$ be a prime, $n$ a positive integer, $n\geq 2$. Also, assume that
gcd$(p,n)=1$ (equivalently, $n$ is not divisible by $p$). Consider the two-
variable symmetric diophantine equation,
$nxy=p(x+y)$ (1)
with the two variables $x$ and $y$ taking values from the set
${\mathbb{Z}}^{+}$ of positive integers. Then,
1. (i)
If $n=2$ and $p\geq 3$, equation (1) has exactly three distinct solutions, the
following positive integer pairs:
$(x,y)=(p,p),\ (x,y)=\left(p\left(\dfrac{p+1}{2}\right),\
\dfrac{p+1}{2}\right),$
and its symmetric counterpart
$(x,y)=\left(\dfrac{p+1}{2},\ p\left(\dfrac{p+1}{2}\right)\right).$
2. (ii)
If $n\geq 3$, and $n$ is a divisor of $p+1$. Then equation (1) has exactly two
distinct solutions:
$(x,y)=\left(p\left(\dfrac{p+1}{n}\right),\ \dfrac{p+1}{n}\right)\ {\rm and}\
(x,y)=\left(\dfrac{p+1}{n},\ p\left(\dfrac{p+1}{n}\right)\right).$
3. (iii)
If $n$ is not a divisor of $p+1$, Equation (1) has no solution.
## 2 A lemma from number theory
The following lemma, commonly referred to as Euclid’s lemma, is of great
significance in number theory.
###### Lemma 1.
(Euclid’s lemma): Suppose that $a,b,c$ are positive integers such that $a$ is
a divisor of the product $bc$; and gcd$(a,b)=1$ (i.e., $a$ and $b$ are
relatively prime), then $a$ must be a divisor of $c$.
Typically, this lemma and its proof can be found in an introductory number
theory book. For example, see reference [3].
## 3 Proof of Theorem 1
First we show that the positive integer pairs listed in Theorem 1 are indeed
solutions to Equation (1).
If $n=2$ and $p\geq 3$, then for $(x,y)=(p,p)$, a straightforward calculation
shows both sides of (1) are equal to $2p^{2}$; and for
$(x,y)=\left(\dfrac{p(p+1)}{2},\dfrac{p+1}{2}\right)$, a calculation shows
that both sides of (1) are equal to $\dfrac{p(p+1)^{2}}{2}$.
If $n\geq 3$ and $n$ is a divisor of $p+1$, then for
$(x,y)=\left(p\left(\dfrac{p+1}{n}\right),\dfrac{p+1}{n}\right)$, a
calculation shows that both sides of equation (1) are equal to
$\dfrac{p(p+1)^{2}}{n}$.
In the second part of this proof, we show that there are no other solutions to
equation (1). To do so, we will demonstrate that if $(t_{1},t_{2})$ is a
solution to (1), then it must be one of the solutions listed in Theorem 1\.
So, let $(t_{1},t_{2})$ be a positive integer solution to equation (1).
We have,
$\left\\{\begin{array}[]{c}p(t_{1}+t_{2})=nt_{1}t_{2}\\\ \\\
t_{1},t_{2}\in{\mathbb{Z}}^{+}\end{array}\right\\}$ (2)
Let $d$ be the greatest common divisor of $t_{1}$ and $t_{2}$. Then
$\left\\{\begin{array}[]{l}t_{1}=du_{1},\ t_{2}=du_{2};\\\ {\rm for\
relatively\ prime\ positive\ integers}\ u_{1}\ {\rm and}\ u_{2};\\\ {\rm
gcd}(u_{1},u_{2})=1\end{array}\right\\}$ (3)
From (2) and (3) we obtain,
$p(u_{1}+u_{2})=nd\,u_{1}u_{2}$ (4)
Since the prime $p$ is relatively prime to $n$. By (4) and Lemma 1, it follows
that $p$ must divide the product $du_{1}u_{2}$. Since $p$ is a prime number,
it must divide at least one of $d$ and $u_{1}u_{2}$. We distinguish between
two cases: The case wherein $p$ divides the product $u_{1}u_{2}$; and the case
in which $p$ is a divisor of $d$.
Case 1: $p$ is a divisor of $u_{1}u_{2}$.
Since $p$ is a prime, and the integers $u_{1}$ and $u_{2}$ are relatively
prime by (3), and also in view of the fact that $p$ divides the product
$u_{1}u_{2}$, it follows that $p$ must divide exactly one of $u_{1},u_{2}$. It
must divide one but not the other. Thus, there are two subcases in Case 1.
Subcase 1a being the one with $p|u_{1}$ (i.e., $p$ divides $u_{1}$);
Subcase 1b: $p$ divides $u_{2}$.
But these two subcases are symmetric since equation (4) is symmetric in
$u_{1}$ and $u_{2}$. Thus, without loss of generality, we need only consider
the subcase $p|u_{1}$. So we set
$\left(u_{1}=pv_{1},\ v_{1}\ {\rm a\ positive\ integer}\right)$ (5)
Combining (5) with (4) we get,
$\left\\{\begin{array}[]{c}pv_{1}+u_{2}=nd\,v_{1}u_{2}\\\ \\\ {\rm or\
equivalently},\ u_{2}=v_{1}\cdot(ndu_{2}-p)\end{array}\right\\}$ (6)
According to (6), the positive integer $v_{1}$ is a divisor of $u_{2}$. But,
by (5) $v_{1}$ is also a divisor of $u_{1}$. Since $u_{1}$ and $u_{2}$ are
relatively prime by (3), it follows that
$v_{1}=1$ (7)
Hence, by (7) and (6), we further obtain,
$p=u_{2}(nd-1)$ (8)
According to (8), $u_{2}$ is a divisor of $p$, and since $p$ is a prime it
follows that either $u_{2}=1$ or $u_{2}=p$. If $u_{2}=1$, then (8) yields
$p+1=nd$ which implies that $n$ is a divisor of $p+1$. Using
$d=\dfrac{p+1}{n},\ v_{1}=1,u_{2}=1$, we also get $u_{1}=p$ (by (5)). So, by
(3) we obtain the solution
$t_{1}=p\left(\dfrac{p+1}{n}\right),t_{2}=\dfrac{p+1}{n}$ (already a verified
solution in the first part of the proof). Now, if $u_{2}=p$ in (8), then
$2=nd$ which implies either $n=2$ and $d=1$, or $n=1$ and $d=2$. But $n\geq
2$, so the latter possibility is ruled out. Thus, $u_{2}=p,\ n=2$, and $d=1$.
Also, by (7) we have $v_{1}=1$ and so $u_{1}=p$ by (5).
Hence, (3) yields $t_{1}=p=t_{2}$; $(p,p)$ with $n=2$ being a solution
verified in the first part of the proof.
Case 2: $p$ is a divisor of $d$
We set
$(d=p\delta,\ \delta\ {\rm is\ a\ positive\ integer})$ (9)
by (9) and (4) we have,
$u_{1}+u_{2}=n\delta u_{1}u_{2}$ (10)
Clearly, by inspection, we see that equation (10) implies that the positive
integers $u_{1}$ and $u_{2}$ must divide each other. Since they are relatively
prime, it follows that
$u_{1}=u_{2}=1$ (11)
Equations (10) and (11) yield
$2=n\delta$ (12)
Due to the fact $n\geq 2$, (12) implies that $n=2$ and $\delta=1$. So, by
(11), (9), and (3), it is clear that (since $d=p$) $u_{1}=u_{2}=p$. This
produces $(u_{1},u_{2})=(p,p)$, with $n=2$. An already verified solution. The
proof is complete. $\Box$
## References
* [1] Kishan, Hari, Rani, Megha and Agarwal, Smiti, The Diophantine Equations of Second and Higher Degree of the Form $3xy=n(x+y)$ and $3xyz=n(xy+yz+zx)$, etc., Asian Journal of Algebra 4(1), (2011), pp. 31-37.
* [2] Burton, David M., “The History of Mathematics, An Introduction”, Sixth Edition, McGraw Hill, (2007), p. 40.
* [3] Rose, Kenneth H., “Elementary Number Theory and Its Applications”, 5th Edition, Pearson, Addison Wesley, (2005), p. 109.
|
arxiv-papers
| 2012-02-29T21:11:08 |
2024-09-04T02:49:28.134672
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1203.0018"
}
|
1203.0028
|
# Epitaxial Ferromagnetic Nanoislands of Cubic GdN in Hexagonal GaN
T. F. Kent Department of Materials Science and Engineering, The Ohio State
University, Columbus, Ohio 43210, USA J. Yang Department of Materials
Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA
L. Yang Department of Materials Science and Engineering, The Ohio State
University, Columbus, Ohio 43210, USA M. J. Mills Department of Materials
Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA
R. C. Myers Department of Materials Science and Engineering, The Ohio State
University, Columbus, Ohio 43210, USA Deparment of Electrical and Computer
Engineering, The Ohio State University, Columbus, Ohio 43210, USA
###### Abstract
Periodic structures of GdN particles encapsulated in a single crystalline GaN
matrix were prepared by plasma assisted molecular beam epitaxy. High
resolution X-ray diffractometery shows that GdN islands, with rock salt
structure are epitaxially oriented to the wurtzite GaN matrix. Scanning
transmission electron microscopy combined with in-situ reflection high energy
electron diffraction allows for the study of island formation dynamics, which
occurs after 1.2 monolayers of GdN coverage. Magnetometry reveals two
ferromagnetic phases, one due to GdN particles with Curie temperature of 70K
and a second, anomalous room temperature phase.
In this work, the epitaxial integration of discrete cubic GdN nanoparticles in
a continuous, high crystalline quality GaN matrix is reported. Although the
growth of coalesced epitaxial GdN films on III-nitrides by molecular beam
epitaxy (MBE)Scarpulla ; Natali and by reactive ion sputtering on
AlNYoshitomi has been previously reported, formation of discrete GdN islands
within a continuous, epitaxial III-nitride matrix has until now, yet to be
explored. Epitaxial growth of dissimilar crystal structures is in general met
with many challengesSands , but the rare earth pnictides (RE-Pn:EuO, ErAs.)
have been shown to grow well on zincblende III-V semiconductor
compoundsPalmstrom . Most widely studied has been the epitaxial integration of
semi-metallic ErAs in InGaAsKadow , which has resulted in high speed
photodetector and THz source applicationsGriebel . The mechanism of ErAs
embedded nanoparticle growth in GaAs has been proposedCrook to proceed first
by incomplete layer formation of ErAs islands on the surface followed by
epitaxial lateral overgrowth of the uncovered GaAs.
---
Figure 1: (a) Cross sectional z-contrast HAADF STEM image of calibration
sample. (b) Atomic resolution STEM images for selected layers in the
calibration structure. The top image shows a GdN nanoparticle of clearly cubic
structure surrounded by a continuous GaN matrix. The bottom atomic resolution
data shows the normal wurtzite structure prior to island formation. ( c) RHEED
patterns taken during growth of GdN layer for different thicknesses showing
evolution of the surface reconstruction from 1$\times$2 to 2$\times$4
reconstruction and back to the 1$\times$2 with increasing layer thickness.
Figure 2: (a) Structural diagram of 50 period 10nm GaN/2.4ML GdN superlattice.
(b) Cross sectional z-contrast HAADF STEM image showing all periods of
superlattice heterostructure. (c) High resolution X-ray diffraction
$\omega$-2$\theta$ scan showing epitaxial orientation of the GdN (111) peak to
the wurtzite (0002) of GaN as well as superlattice fringes, indicating precise
layer thickness control.
The epitaxial integration of GdN with GaN is attractive for a number of
reasons. First, the dilute doping of III-Nitrides with Gd has attracted a
large amount of attention in recent years initially for its promise of
utilization of the intra-f-shell UV optical transitions of Gd in AlNGruber
and subsequently for the search of a room temperature dilute magnetic
semiconductor following the report of room temperature ferromagnetism in
Gd:GaNDhar ; Bedoya ; Davies . Gd is attractive for its magnetic properties,
possessing the most strongly correlated electronic structure of the
lanthanides, with 4f ground state of spin 7/2 . Devices for semiconductor
spintronics require efficient ferromagnetic spin injection and detection
layers, which currently are composed of either dilute magnetic
semicondutor(DMS) (GaMnAs) or metallic layers. No epitaxial spin injector is
currently available in the III-nitride materials system. For the preservation
of spin coherence through the device, interface and crystalline quality are
key considerations. Dilute Gd:GaN, though offering the promise of room
temperature ferromagnetism and realization of a nitride based DMS has proven
to be a difficult material to controlRoever due to its poorly understood
defect mediated mechanism of ferromagnetism. GdN, in contrast is a well
understood classical ferromagnetGambino with Tc around 70K. Furthermore,
unlike most other RE-Pn, which are well established to be semimetals, thin GdN
layers has been predictedMitra ; Duan ; Lambrecht to be indirect gap
semiconductors, a claim which is consistent with recently reported absorption
features for thin GdN filmsYoshitomi . This leads to the possibility of a
controllable ferromagnetic semiconductor which can be epitaxially integrated
with GaN. In addition to intriguing magnetic properties, embedded GdN
nanoparticles in GaN could potentially function as carrier recombination
centers, giving rise to ultrafast photoconductivity in the same fashion as RE-
As particles in III-arsenides.
For the epitaxial structure of the matrix to remain single crystalline, the
layer coverage of the rock salt GdN must remain incomplete, allowing for
epitaxial laterall overgrowth of the surrounding matrix. This is due to the
lower symmetry of the rock salt structure (Fm$\overline{3}$m) than the host
(P63mc in the case of wurtzite). Complete films of GdN on wurtzite III-
Nitrides have been shown to epitaxial with the relationship
GdN[111]$||$GaN[0001] but containing two rotational variants due to crystal
symmetry considerations, which can be observed by an off-axis $\phi$ scan in
x-ray diffractometry and resulting in a polycrystalline overlayer of
GaNScarpulla .
Samples were prepared using the technique of plasma assisted molecular beam
epitaxy (PAMBE). In a Veeco GEN930 PAMBE system equipped with a Ga, Gd
effusion cell, and nitrogen plasma source. To study the GdN island formation
threshold, a calibration stack consisting of increasing effective thicknesses
of GdN are deposited from 0.2ML to 2.4ML in between 10nm GaN spacers on a GaN
buffer layer grown on an AlN on sapphire (KYMA) template at a substrate
temperature of 730C, beam equivalent pressure of 2$\times 10^{-5}$Torr and
III/V ratio of 2. During the period of GdN growth, the Ga shutter is closed,
meaning only Gd and N are being deposited. There is, however, a residual
amount of Ga present on the surface, due to growth of the GaN spacer under
metal rich conditions. To analyze the onset of GdN island formation, the
samples were characterized by cross-sectional, atomic resolution TEM using an
FEI Titan3 80-300 Probe-Corrected Monochromated (S)TEM, as can be seen Fig 1a.
Up to 1.2ML GdN, no change in the structure of the heavily Gd doped region is
observed, however at 1.2ML, discrete clusters of highly concentrated Gd atoms
appear. From the atomic resolution data shown in Fig.1b, cubic particles of
GdN are observed in the 1.4ML layer. From image analysis of the STEM
dataImageJ , the lattice parameter of cubic GdN in GaN is measured to be
4.8$\pm 0.1\text{\AA}$ and the nanoparticle size is roughly 2.6nm x 3.6nm.
After 1.2ML and up to 2.4ML, GdN particles with clearly cubic structure
surrounded by a hexagonal GaN matrix can be seen with the major change with
additional Gd deposition being increased lateral growth, suggesting that the
height of the nanoparticle is self limited and further growth will proceed by
lateral expansion of the GdN islands, which is similar to what has been
observed for Er-Pn in III-As nanoparticle structuresHanson .
Figure 3: Diamagnetic background corrected magnetization hysteresis loops for
low (a) and high (b) temperatures from SQUID magnetometry for in-plane and
out-of-plane film orientations relative to the applied field. Inserts show the
low field data and open nature of the loops. The low temperature scan clearly
shows the highly symmetric GdN magnetic phase with the expected saturation
magnetization of 7$\mu$B/Gd3+. (c). Diamagnetic background corrected, constant
field magnetization behavior with temperature after field cooling from 300K to
5K at 5T for in plane and out of plane orientation of the film with the
applied field.
During growth, the surface reconstruction was monitored using reflection high
energy electron diffraction (RHEED) operating at 10kV and cathode current of
1.4A, results are shown in Fig. 1c. For the first 0.4ML of GdN coverage, the
pattern is representative of the wurtzite Ga-face 1$\times$2 reconstruction.
After 0.5ML and until 1.2ML of GdN the pattern changes to a 2$\times$4
reconstruction. Past 1.2ML of coverage, the wurtzite 1$\times$2 pattern again
is visible, indicating a temporary change in the surface structure during
growth of the GdN layer.
After calibration of the GdN precipitation threshold, the heterostructure show
in Fig. 2a. consisting of a GaN buffer on an AlN template on sapphire (KYMA)
and a 50 period superlattice of alternating 10nm uid-GaN and 2.4ML GdN layers
was prepared under identical growth conditions as the calibration structure.
Cross sectional STEM images, shown in Fig.2b, using a Technai F20 operating in
HAADF imaging mode, which provides atomic number contrast, shows expected
discrete GdN particles in a GaN matrix with GaN spacer thickness of
11.8$\pm$0.4nm and GdN layer thickness of 5.6$\pm$0.3nm obtained from image
analysis.The structure of the sample was further characterized by high
resolution x-ray diffractometry using a Bruker D8 triple axis system.
Diffraction data shown in Fig. 2c. exhibits clear epitaxial orientation of the
GdN [111] to the wurtzite [0001]. Also visible are superlattice fringes,
indicating precise layer thickness control of the GaN spacing layers between
the GdN regions. From analysis of the superlattice fringes, the GaN spacer
thickness can be determined to be 10.98nm which is close to the value
determined from STEM. Furthermore, from the diffraction angle, we can
determine the lattice parameter of GdN to be 4.97Å, which is in very good
agreement with the value obtained from STEM of the nanoparticles and with
values for bulk GdNNatali .
The magnetic properties of the sample were analyzed by superconducting quantum
interference device (SQUID) magnetometry using a Quantum Design MPMS XL.
Results, depicted in Fig. 3 clearly show evidence of two distinct
ferromagnetic phases in the sample. The dominant phase at low temperature
(Fig. 3a) can be identified as rocksalt GdN due to a saturation magnetization
of nearly exactly 7$\mu_{B}$/Gd3+ (158.2 emu/cm3), the expected configuration
for GdN. The low remanent magnetization but correct saturation of
7$\mu_{\text{B}}$ is indicative that a large fraction of Gd is paramagnetic,
which is further supported by a temperature dependence containing both a mean
field like behavior with a T${}_{\text{c}}$ 70K but an additional 1/T
contribution. Samples were characterized in both the in-plane
($\vec{B}$$||$GaN [0001]) and out-of-plane ($\vec{B}$$\perp$GaN [0001])
configuration. The low temperature phase shows very little anisotropy which is
consistent with small particles of a cubic structure, which should be free
from the shape anisotropy of a fully coalesced film. The coercive field,
H${}_{\text{c}}$ is measured to be 363Oe for the in-plane configuration and
170Oe in the out-of-plane configuration, respectively.
Past the curie point of GdN, a second, weaker and anisotropic ferromagnetic
phase persists to room temperature. This anomalous phase is hypothesized to be
the result of interaction of the Gd with local point defects in the GaN matrix
and is of the same type as observed by Dhar, et. al.Dhar . It was previously
reportedDavies that ferromagnetic films of dilutely doped Gd:GaN exhibit
anisotropy in their saturation magnetization between the in-plane and out of
plane orientations of the film. As observed in Fig 3b, the room temperature
phase exhibits anisotropy with M${}_{\text{s}}$=6.84emu/cm3 and
H${}_{\text{c}}$ = 100Oe for the out-of-plane configuration and
M${}_{\text{s}}$ =2.8emu/cm3, H${}_{\text{c}}$ = 30.6Oe for the in-plane
orientation of the film.
Measurement of the magnetization behavior with temperature, after cooling in a
5T field, is shown in Fig 3c. These data reveal a sharp decrease in the
magnetization with temperature up to 70K, the reported Curie point of
GdNScarpulla . After 70K and up to the highest temperature measured, 350K, a
residual amount of magnetization persists, again pointing to the possibility
of an anomalous room temperature phase. In the M vs. T data, anisotropy is
observed at low temperatures, which is consistent with the low field
anisotropy present in the 5K hysteresis scan. For the out-of-plane
configuration, a distinct knee is visible in the M vs. T scan which could be
due to error in the orientation of the film as mounted in the magnetometer,
causing signal from the in-plane configuration to contribute slightly to the
measured magnetization. Due to the sample mounting technique employed, the
out-of-plane configuration has a larger uncertainty in the absolute
orientation of the film.
In summary, we have extended the growth of embedded rare earth pnictide
nanoparticles in III-V semiconductors to the family of the III-nitrides.
Samples show clear rocksalt structure in cross sectional TEM above a threshold
value of 1.2ML GdN and x-ray diffractometry indicates epitaxial orientation of
the [111] direction of GdN to the wurtzite c-axis. Magnetic characterization
shows evidence of two magnetic phases, one due to the rocksalt ferromagnet GdN
with Curie temperature of 70K and a second, anisotropic phase whose
magnetization persists past room temperature. The room temperature phase is
hypothesized to be the same type of defect mediated ferromagnetism reported in
Gd:GaN and shows a prominent out-of plane easy axisDhar ; Bedoya ; Davies .
Funding provided by the Center for Emergent Materials under NSF Award Number
DMR-0820414 and the Institute for Materials Research at OSU under the
Interdisciplinary Materials Research Grants (IMRG) program. Jim O’Brien of
Quantum Design is thanked for valuable discussions about advanced SQUID
measurement techniques.
## References
* (1) M.A. Scarpulla, C. S. Gallinat, S. Mack, J. S. Speck, A. C. Gossard, J. Crys. Grow. 311, 1239 (2009)
* (2) F. Natali, N. O. V. Plank, J. Galipaud, B. J. Ruck, J. J. Trodahl, F. Semond, S. Sorieul, L. Hirsch, J. Crys. Growth 312 3583
* (3) H. Yoshitomi, S. Kitayama, T. Kita, O. Wada, M. Fujisawa, H. Ohta, T. Sakurai, Phys. Rev. B 83, 155202 (2011)
* (4) T. Sands, C. J. Palmstr$\o$m, J. P. Harbison, V. G. Keramidas, N. Tabatabaie, T. L. Cheeks, R. Ramesh, Y. Silberberg, Mat. Sci. Rep. v5, 3, 99-170 (1990)
* (5) C. J. Palmstr$\o$m, Annu. Rev. Mater. Sci. 25: 389-415 (1995)
* (6) C. Kadow, S. B. Fleischer, J. P. Ibbetson, J. E. Bowers, A. C. Gossard, J. W. Dong, C. J. Palmstr$\o$m, Appl. Phys. lett. 75, 3548 (1999)
* (7) M. Griebel, J. H. Smet, D. C. Driscoll, J. Kuhl, C. A. Diez, N. Freytag, C. Kadow, A. C. Gossard, K. von Kiltzing, Nature Materials 2, 122 - 126 (2003)
* (8) A. M. Crook, H. P. Nair. D. A. Ferrer. S. R. Bank, Appl. Phys. Lett. 99, 072120 (2011)
* (9) J. B. Gruber, U. Vetter, H. Hofsäss, B. Zandi, M. F. Reid, Phys. Rev. B 69, 195202 (2004)
* (10) C. Mitra and W. R. L. Lambrecht, Phys. Rev. B 78, 195203 (2008)
* (11) W. R. Lambrecht, Phys. Rev. B. 62, 13538 (2000)
* (12) R. P. Davis, B. P. Gila, C. R. Abernathy, S. J. Pearton, C. J. Stanton, Appl. Phys. Lett. 96, 212502 (2010)
* (13) C. Duan, R. F. Sabiryanov, J. Liu, W. N. Mei, Pa. A. Dowben, J. R. Hardy, Phys. Rev. Lett. 94 237201 (2005)
* (14) S. Dhar, O. Brandt, M. Ramsteiner, V. F. Sapega, K. H. Ploog, Phys. Rev. Lett. 94 037205 (2005)
* (15) A. Bedoya-Pinto, J. Malindretos, M. Roever, D. D. Mai, A. Rizzi, Phys. Rev. B. 80, 195208 (2009)
* (16) M. Roever, J. Malindretos, A. Bedoya-Pinto, A. Rizzi, C. Rauch, F. Tuomisto, Phys. Rev. B. 84, 081201(R) (2011)
* (17) M. Hanson, Ph.D. thesis, University of California at Santa Barbara (2007).
* (18) R. J. Gambino, T. R. McGuire, H. A. Alperin, S. J. Pickart, J. Appl. Phys. 41, 933 (1970)
* (19) M. D. Abramoff, Biophotonics International 11 36, (2004)
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|
arxiv-papers
| 2012-02-29T21:53:21 |
2024-09-04T02:49:28.140264
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. F. Kent, J. Yang, L. Yang, M. J. Mills, and R. C. Myers",
"submitter": "Roberto Myers",
"url": "https://arxiv.org/abs/1203.0028"
}
|
1203.0157
|
# Berry phase, semiclassical quantization and Landau levels
A.Yu. Ozerin Verechagin Institute of the High Pressure Physics, Troitsk
142190, Russia L.A. Falkovsky Verechagin Institute of the High Pressure
Physics, Troitsk 142190, Russia Landau Institute for Theoretical Physics,
Moscow 119334, Russia
###### Abstract
We propose the semiclassical quantization for complicated electron systems
governed by a many-band Hamiltonian. An explicit analytical expression of the
corresponding Berry phase is derived. This impact allows us to evaluate the
Landau magnetic levels when the rigorous quantization fails, for instance, for
bilayer graphene and graphite with the trigonal warping. We find that the
magnetic breakdown can be observed for the certain type of classical electron
orbits.
###### pacs:
81.05.ue
The most accurate investigation of the band structure of metals and
semiconductors is studying the Landau levels in magneto-transport and magneto-
optical experiments. However, the theoretical solution of the band problem in
magnetic fields cannot often be exactly found. A typical example is presented
by graphene layers. For bilayer graphene and graphite, the effective
Hamiltonian is a $4\times 4$ matrix giving four energy bands. Fig. 1 shows
nearest two bands of the level structure together with semiclassical orbits.
The trigonal warping described by the effective Hamiltonian with a relatively
small parameter $\gamma_{3}$ provides an evident effect (see right panel).
Another important parameter is the gate-tunable bandgap $U$ in bilayer
graphene. In this situation, the quantization problem cannot be solved within
a rigorous method. To overcome this difficulty one can use a perturbation
theory, however this theory becomes quite complicated for the many-band
Hamiltonian.
Alternatively, the semiclassical quantization can be applied. Thus, we can use
the Bohr-Zommerfeld condition as
$\frac{c}{e\hbar
B}S(\varepsilon)=2\pi\left[n+\frac{\mathcal{T}}{4}+\delta(\varepsilon)\right]\,.$
(1)
Here $S(\varepsilon)$ is the cross-section area of the electron orbit in the
${\bf k}$ space for the energy $\varepsilon$ in absence of the magnetic field
B and for the constant momentum projection $k_{z}$ on the magnetic field, $n$
is an integer supposed to be large. $\mathcal{T}$ is the number of the smooth
turning points on the electron orbit. There are two smooth turning points for
the Landau levels and only one for skipping electrons reflected by the hard
edge.
The goal of this letter is an explicit analytical expression for the
$\delta(\varepsilon)-$phase within the band scheme of the matrix Hamiltonian.
The semiclassical approach is used for the magnetic field normal to the
layered system when the quantization of in-layer momentum components is only
essential and the size of the Fermi surface is small compared with the
Brillouin zone size. We illustrate our results for bilayer graphene. Notice,
that the $\delta(\varepsilon)-$phase depends on the energy and can be taken in
the interval $0\leq|\delta|\leq 1/2$. If the spin is neglected, $\delta=0$ and
$\mathcal{T}=2$ for the Landau levels, and $\delta=1/2$ and $\mathcal{T}=2$
for monolayer graphene. In these two cases, the semiclassical result coincides
with the rigorous quantization and it is closely connected with the
topological Berry phase Be . This $\delta-$phase was evaluated for bismuth in
Ref. Fal , preceding Berry’s work by almost two decades, and it was considered
again for bismuth in Ref. MS . For graphite, the semiclassical quantization
was applied in Ref. Dr . However, in the general case, the evaluation of the
$\delta-$phase is still attracted a widespread interest TA ; CU ; KEM ; PM ;
PS ; LBM ; ZFA .
Figure 1: (Color online) (a) The energy dispersion $\varepsilon(k,\alpha)$ of
two nearest bands (the electron band shown in solid line and the hole band in
dashed line) in bilayer graphene for two polar angles $\alpha$ with the local
extrema at $k\neq 0$ (”mexican hat”) represented. The band parameters are
given in the figure, others are $\gamma_{0}=3.05$ eV, $\gamma_{1}=360$ meV,
$\gamma_{4}=-150$ meV PP ; GAW . (b) Cross-sections $k(\alpha,\varepsilon)$ of
the electron band for energies of 80 meV (dashed-dotted line) and 40 meV
(solid line).
The problem under consideration is described by the Hamiltonian in the band
representation
${(\bf V\cdot\tilde{k}}+\Gamma-\varepsilon)\Psi=0\,,$ (2)
where the column $\Psi$ consists of functions corresponding with a number of
bands included and is labelled by the band subscript which we omit together
with the matrix subscripts on $\Gamma$ and ${\bf V}$; a summation over them is
implied in Eq. (2). Matrices $\Gamma$ and ${\bf V}$ are the first two terms in
a series expansion of the Hamiltonian in the power of quasi-momentum $\bf{k}$.
In the magnetic field, the momentum operator ${\bf\tilde{k}}$ depends on the
vector-potential ${\bf A}$ by means the Peierls substitution,
${\bf\tilde{k}}=-i\hbar\nabla-e{\bf A}/c,$
providing the gauge invariance of the theory. The magnetic field can also
enter explicitly describing the magnetic interaction with the spin of a
particle. However, for the graphene family, the magnetic interaction is weak
and omitted here.
A simple example of Eq. (2) is given by the graphene monolayer. There are two
sublattices in it, and Eq. (2) is represented by a $2\times 2$ matrix if the
spin of carriers is neglected. Another example considered below is bilayer
graphene with the $4\times 4$ matrix Hamiltonian. For the monolayer and
bilayer graphene, both ${\bf V}$ and ${\bf{\tilde{k}}}$ are two-dimensional
vectors, e.g., with $x$ and $y$ components.
We seek for $\Psi$ in the form
$\Psi=\Phi\exp{(is/\hbar)}\,,$
where the function $s$ is assumed to be common for all components of the
column $\Psi$. The equation for $\Phi$ is reduced to
$\displaystyle[{\bf V\cdot(k}-i\hbar\nabla)+\Gamma-\varepsilon]\Phi=0\,,$ (3)
$\displaystyle\text{with}\quad{\bf k}=\nabla s-e{\bf A}/c\,.$ (4)
The function $\Phi$ is expanded in series of $\hbar/i$:
$\Phi=\sum_{m=0}^{\infty}\left(\frac{\hbar}{i}\right)^{m}\varphi_{m}\,.$
Comparing the terms involving the same powers of $\hbar$ in Eq. (3) we have
$({\bf V\cdot k}+\Gamma-\varepsilon)\varphi_{m}=-{\bf
V\nabla}\varphi_{m-1}\,.$ (5)
For $m=0$, we get a homogeneous system of algebraic equations
$({\bf V\cdot k}+\Gamma-\varepsilon)\varphi_{0}=0\,$ (6)
which has a solution under the condition
$\text{Det}({\bf V\cdot k}+\Gamma-\varepsilon)=0\,.$ (7)
This equation determines the classical electron orbit,
$\varepsilon(k_{x},k_{y})=\varepsilon$, in presence of the magnetic field
while the electron energy $\varepsilon$ is constant. At the same time, the
equation yields the electron dispersion equation with ${\bf k}$ as the
momentum without any magnetic field. In 3d case, the electron dispersion
depends as well on the momentum projection $k_{z}$ on the magnetic field and
our scheme can be implied in this case without the expansion in $k_{z}$.
It is convenient to choose the vector-potential in the Landau gauge
$A_{x}=-By,A_{y}=A_{z}=0$ in such a way that the Hamiltonian does not depend
on the $x-$coordinate. Then, the $x-$momentum component $K_{x}$ becomes a
conserved quantum number and the function $s$ in Eq. (4) can be written as
$s=xK_{x}+\sigma(y)\,.$ (8)
The equations (4) are reduced to
$k_{x}=K_{x}+\frac{e}{c}By\,,\quad k_{y}=\frac{d\sigma}{dy}.$
These equations enable us to use the variable $k_{x}$ instead of $y$ and to
obtain
$\displaystyle\sigma(k_{x})=\frac{c}{eB}\int\limits^{k_{x}}k_{y}(k_{x}^{\prime})dk_{x}^{\prime}\,,$
(9)
where $k_{y}$ as a function of $k_{x}$ is determined by the dispersion
equation (7).
The eigenfunction column obeying Eq. (6) can be multiplied by the scalar
function $C$ common for all elements of the column
$\varphi_{0}\rightarrow C\varphi_{0}\,$
where $\varphi_{0}$ is any eigen-column of Eq. (6). The function $C$ is
determined by Eq. (5) with $m=1$. Left-to-right multiplying both sides of this
equation by $\varphi^{*}_{0}$ and using the Hamiltonian hermiticity, i.e. the
complex conjugations of Eq. (6), we get the consistency condition
$\varphi^{*}_{0}{\bf V\cdot\nabla}(C\varphi_{0})=0\,,$ (10)
where the derivative with respect to $y$ (i.e. to $k_{x}$) is only to be
taken. The left hand-side of this equation can be written as
$\frac{1}{C}\frac{dC}{dk_{x}}+\frac{1}{2\varphi^{*}_{0}V_{y}\varphi_{0}}\frac{d\varphi^{*}V_{y}\varphi}{dk_{x}}+\frac{i}{\varphi^{*}_{0}V_{y}\varphi_{0}}\text{Im}\,\varphi^{*}_{0}V_{y}\frac{d\varphi_{0}}{dk_{x}}$
Using the identity $\varphi^{*}_{0}{\bf
V}\varphi_{0}=\varphi^{*}_{0}\varphi_{0}{\bf v}$ with the electron velocity
${\bf v}=\partial\varepsilon/\partial{\bf k}$, one can write the solution of
Eq. (10) as
$C=c_{0}(\varphi^{*}_{0}\varphi_{0}v_{y})^{-1/2}\exp(-i\theta)\,,$ (11)
where
$\theta=\text{Im}\int\frac{dk_{x}}{\varphi^{*}_{0}\varphi_{0}v_{y}}\varphi^{*}_{0}V_{y}\frac{d\varphi_{0}}{dk_{x}}$
(12)
and $c_{0}$ is the normalization factor.
The quantization condition can be written as usual from the requirement that
the wave function has to be single-valued. Continuing Eqs. (9), (11), and (12)
along the orbit and making the bypass in the complex plane around the turning
points where $v_{y}=0$ to obtain the decreasing solutions in the classically
unaccessible region, one obtains $\mathcal{T}=2$ and $\delta-$phase as a
contour integral along the classical orbit
$\delta(\varepsilon)=\frac{1}{2\pi}\text{Im}\oint\frac{dk_{x}}{\varphi^{*}_{0}\varphi_{0}v_{y}}\varphi^{*}_{0}V_{y}\frac{d\varphi_{0}}{dk_{x}}\,.$
(13)
Using the Hamiltonian hermiticity, after the simple algebra (see Ref. Fal ),
Eq. (13) can be rewrite as
$\delta(\varepsilon)=\frac{1}{4\pi}\text{Im}\oint\frac{dk}{\varphi_{0}^{*}\varphi_{0}v}\varphi^{*}_{0}\left[{\bf
V}\times\frac{d}{d{\bf k}}\right]_{z}\varphi_{0}$
called usually the Berry phase.
Now let us calculate the $\delta-$phase for bilayer graphene. In simplest
case, the effective Hamiltonian can be written (see, for instance Refs. PP ;
GAW ) as
$H(\mathbf{k})=\left(\begin{array}[]{cccc}U&q_{+}&\gamma_{1}&0\\\
q_{-}&U&0&0\\\ \gamma_{1}&0&-U&q_{-}\\\ 0&0&q_{+}&-U\end{array}\right),$ (14)
where the parameter $U$ describes the tunable gap, $\gamma_{1}$ is the
nearest-neighbor hopping integral energy, the matrix elements are expanded in
the momentum $k_{\pm}=\mp ik_{x}-k_{y}$ near the $K$ points of the Brillouin
zone, and the constant velocity parameter $v$ is incorporated in the notation
$q_{\pm}=vk_{\pm}$.
Here, the orbit is the circle defined by Eq. (7), written in the following
form
$[(U+\varepsilon)^{2}-q^{2}][(U-\varepsilon)^{2}-q^{2}]-\gamma_{1}^{2}(\varepsilon^{2}-U^{2})=0\,.$
(15)
The eigenfunction ${\mathbf{\varphi}_{0}}$ of the Hamiltonian (14) can be
taken as
${\mathbf{\varphi}_{0}}=\left(\begin{array}[]{c}(U-\varepsilon)[(\varepsilon+U)^{2}-q^{2}]\\\
q_{-}[q^{2}-(\varepsilon+U)^{2}]\\\ \gamma_{1}(U^{2}-\varepsilon^{2})\\\
\gamma_{1}q_{+}(U-\varepsilon)\end{array}\right),$ (16)
with the norm squared
$\displaystyle\varphi_{0}^{*}\varphi_{0}=[(\varepsilon+U)^{2}-q^{2}]^{2}[(\varepsilon-U)^{2}+q^{2}]$
$\displaystyle+\gamma_{1}^{2}(\varepsilon-U)^{2}[(\varepsilon+U)^{2}+q^{2}]\,.$
(17)
The derivatives for Eq. (13) are calculated along the trajectory where the
energy $\varepsilon$ and consequently the trajectory radius $q$ are constant.
The equation (15 ) has only one solution for $q^{2}$ if
$|U|<|\varepsilon|<\sqrt{U^{2}+\gamma_{1}^{2}}.$ First, let us consider this
case.
Figure 2: (Color online) Semiclassical phase vs energy in the conduction band
of bilayer graphene without trigonal warping (solid line) and with warping
(dashed line).
(i) there is only one orbit at given energy $\varepsilon$ with the radius
squared
$q^{2}=U^{2}+\varepsilon^{2}+\sqrt{4U^{2}\varepsilon^{2}+(\varepsilon^{2}-U^{2})\gamma_{1}^{2}}\,.$
The matrix $V_{y}=\partial H/\partial k_{y}$ in Eq. (13) has only four nonzero
elements $V_{y12}=V_{y21}=V_{y34}=V_{y43}=-1$. Using Eqs. (15) and (16), we
find
$\text{Im}\,\varphi_{0}^{*}V_{y}\frac{d\varphi_{0}}{dk_{x}}=4U\varepsilon(U-\varepsilon)[(\varepsilon+U)^{2}-q^{2}]\,.$
(18)
This expression is constant on the trajectory as well as
$\varphi_{0}^{*}\varphi_{0}$, Eq. (17). Therefore, in order to find $\delta$,
Eq. (13), we have to integrate along the trajectory
$\oint\frac{dk_{x}}{v_{y}}\,.$
This integral equals $-dS(\varepsilon)/d\varepsilon$, where
$S(\varepsilon)=\pi q^{2}$ is the cross-section area, Eq. (1), with
$\frac{dS(\varepsilon)}{d\varepsilon}=\pi\varepsilon\frac{2(q^{2}+U^{2}-\varepsilon^{2})+\gamma_{1}^{2}}{q^{2}-U^{2}-\varepsilon^{2}}\,.$
Now we have to substitute Eqs. (17), (18), and (Berry phase, semiclassical
quantization and Landau levels) into Eq. (13). Thus, we find the Berry phase
$\delta(\varepsilon)=\frac{-\varepsilon
U}{q^{2}-\varepsilon^{2}-U^{2}}=\frac{-\varepsilon
U}{\sqrt{4U^{2}\varepsilon^{2}+(\varepsilon^{2}-U^{2})\gamma_{1}^{2}}}$ (19)
shown in Fig. 2, where $\delta-$phase of bilayer graphene with trigonal
warping is also shown, the detailed calculations will be elsewhere published.
Figure 3: (Color online) Energy levels $\varepsilon_{sn_{L}}$ for the $K$
valley in magnetic fields for bilayer graphene within rigorous quantization
(solid lines) and in the semiclassical approximation (dashed-dotted lines); in
the notation $|sn_{L}\rangle$, $n_{L}$ is the Landau number and $s=1,2,3,4$ is
the band number, only two nearest bands ($s=2,3$) are shown at given $n_{L}$
from 2 to 7. There is only one level, $|10\rangle$, with $n_{L}=0$ and three
levels ($s=1,2,3$) with $n_{L}=1$. The levels for the $K^{\prime}$ valley are
obtained by mirror reflection with respect to the $\varepsilon=0$ axis.
For the ungaped bilayer, $U=0$, the Berry phase $\delta(\varepsilon)=0$. The
Berry phase depends on the energy and $\delta=\mp 1/2$ at $\varepsilon=\pm U$.
At the larger energy, $\varepsilon\gg U$, the Berry phase
$\delta\rightarrow\mp U/\gamma_{1}$.
Substituting Eq. (19) in the semiclassical quantization condition, Eq. (1),
and solving the equation obtained for $\varepsilon$, we get energy levels as a
function of the magnetic field. We have to notice that the Landau numbers
$n_{L}$ listed in Fig. 3 do not coincide with the numbers $n$ in the
semiclassical condition (1). The rigorous quantization shows that there are
only one Landau level with $n_{L}=0$ and three Landau levels with $n_{L}=1$ Fa
. These levels are not correctly described within the semiclassical approach.
However, for $n_{L}\geq 2$, there are levels in all four bands $s$ (two
nearest bands with $s=2,3$ are shown in Fig. 3). They correspond with the
quantum number $n=n_{L}-1$, and the semiclassical levels become in excellent
agreement with the rigorous solution for the larger $n$.
(ii) for $|U|/\sqrt{1+(2U/\gamma_{1})^{2}}<|\varepsilon|<|U|\,,$ at the given
energy, there are two orbits with the radius squared
$q_{1,2}^{2}=U^{2}+\varepsilon^{2}\pm r\,,\text{where}\quad
r=\sqrt{4U^{2}\varepsilon^{2}+(\varepsilon^{2}-U^{2})\gamma_{1}^{2}}\,.$
This is an effect of ”the mexican hat”. Then we seek for the general solution
as a sum of two solutions $\varphi_{0}^{j}(1)$ and $\varphi_{0}^{j}(2)$
corresponding to these two contours,
$\varphi_{0}^{j}=C_{1}\varphi_{0}^{j}(1)+C_{2}\varphi_{0}^{j}(2)$
with two scalars $C_{1}$ and $C_{2}.$ Instead of Eq. (10) we have a system of
two equations written in the $2\times 2$ matrix form as follows
$a\frac{dC}{dq_{x}}+bC=0$ (20)
where the notations of the matrix elements are introduced
$a_{ik}=\varphi_{0}^{*}(i)V_{y}\varphi_{0}(k)\,,\quad{\displaystyle
b_{ik}=\varphi_{0}^{*}(i)V_{y}\frac{d\varphi_{0}(k)}{dq_{x}}}\,.$
The off-diagonal matrix elements $a_{ik}$ vanish, $a_{12}=a_{21}=0$. Thus, the
first equation of the system (20) becomes
$2q_{1y}r\frac{dC_{1}}{dq_{x}}+(2i\varepsilon
U-rq_{x}/q_{1y})C_{1}+i(2\varepsilon U+r)C_{2}=0\,,$
and the second equation can be obtained with the index replacement
$1\leftrightarrow 2$ and $r\rightarrow-r$ .
These equations can be simplified with the substitution
$C_{i}=\tilde{C}_{i}(q_{i}^{2}-q_{x}^{2})^{-1/4}\,.$ (21)
For the new functions $\tilde{C}_{i}$, we get the equation system
$\begin{array}[]{c}{\displaystyle
q_{1y}\frac{d\tilde{C}_{1}}{dq_{x}}+iE\tilde{C}_{1}+i\sqrt{\frac{q_{1y}}{q_{2y}}}(E+\frac{1}{2})\tilde{C}_{2}=0\,,}\\\
{\displaystyle
q_{2y}\frac{d\tilde{C}_{2}}{dq_{x}}-iE\tilde{C}_{2}-i\sqrt{\frac{q_{2y}}{q_{1y}}}(E-\frac{1}{2})\tilde{C}_{1}=0\,},\end{array}$
where the parameter $q_{iy}=\sqrt{q_{i}^{2}-q_{x}^{2}},\quad i=1,2$ and
$E=\varepsilon U/r$ .
For the minimum of conduction band (maximum of valence band), where
$r\rightarrow 0$, there is a simple limit,
$q_{1y}\frac{d\tilde{C}_{1}}{dq_{x}}-\frac{i}{2}\tilde{C}_{1}=0\quad\text{with}\quad\tilde{C}_{2}=-\tilde{C}_{1}\,.$
Solving this equation, one gets
$\tilde{C}_{1}=c_{0}\exp\left(\frac{i}{2}\arcsin{\frac{q_{x}}{q_{1}}}\right)\,.$
(22)
Going with $q_{x}$ along the trajectories and making the bypass in the complex
plane around the turning points $q_{x}=\pm q_{1}$ and $q_{x}=\pm q_{2}$, we
see that both $C_{1}$ and $C_{2}$ acquire from two turning points in Eq. (21)
the additional phase $-\pi$ with $\mathcal{T}=2$. At the same time, we have
$-1/2$ from Eq. (22) for $\delta-$phase. Thus, at the boundaries of the narrow
interval considered, the $\delta-$phase obtains the same value, $\delta=-1/2.$
Taking into account the phases of the functions $\varphi_{0}^{j}(i)$, we see,
that the area $S(\varepsilon)$ in Eq. (1) can play the different role. In weak
magnetic fields, slower oscillations with the smaller $S(\varepsilon)$
corresponding to $q_{2}$ should be observed in oscillating phenomena. However,
when the magnetic field becomes larger and the semiclassical condition is
fulfilled only for the larger cross-section $S(\varepsilon)$, calculated with
$q_{1}$, the larger frequency oscillations should be observed. This is nothing
but the magnetic breakdown CF which should be utilized if the chemical
potential belongs to the interval where the effect of ”the mexican hat”
appears.
In conclusion, our study shows that the semiclassical approach gives a
powerful tool for probing the electron magnetic properties in metals. The
Berry phase depending on the energy can be calculated and observed even for
complicated band scheme. The method presented here should be useful for many
electron systems.
We thank I. Luk’yanchuk for helpful discussions. This work was supported by
the SCOPES grant IZ73Z0$\\_$128026 of Swiss NSF, by the grant SIMTECH No.
246937, and by the Russian Foundation for Basic Research (grant No.
10-02-00193-a).
## References
* (1) M.V. Berry, Proc. Roy. Soc. London, Ser. A 392, 45 (1984)
* (2) L.A. Falkovsky, Zh. Eksp. Teor. Fiz. 49, 609 (1965) [Sov. Phys. JETP 22, 423 (1966)].
* (3) G.P. Mikitik, Yu.V. Sharlai, Zh. Eksp. Teor. Fiz. 114, 1357 (1998)[Sov. Phys. JETP 87, 747 (1998)]; Phys. Rev. B 67, 115114 (2003).
* (4) G. Dresselhaus, Phys. Rev. B 10, 3602 (1974).
* (5) P. Carmier, D. Ullmo, Phys. Rev. B 77, 245413 (2008).
* (6) A.A. Taskin, Y. Ando, Phys. Rev. B 84, 035301 (20011).
* (7) E.V. Kurganova, H.J. van Eleferen, A. McCollam, L.A. Ponomarenko, K.S. Novoselov, A. Veligura, B.J. van Wees, J.C. Maan, U. Zeitler, Phys. Rev. B 84, 121407 (20011).
* (8) Cheol-Hwan Park, N. Marzari, Phys. Rev. B 84, 205440 (2011).
* (9) Singhun Park, H.-S. Sim, Phys. Rev. B 84, 235432 (2011).
* (10) Y. Liu, G. Bian, T. Miller, T.-C. Chiang, Phys. Rev. Lett. 107, 166803 (2011).
* (11) L.M. Zhang, M.M. Fogel, D.P. Arovas, Phys. Rev. B 84, 075451 (2011).
* (12) B. Partoens, F.M. Peeters, Phys. Rev. B 74, 075404 (2006).
* (13) A. Grüneis, C. Attaccalite, L. Wirtz, H. Shiozawa, R. Saito, T. Pichler, A. Rubio, Phys. Rev. B 78, 205425 (2008).
* (14) L.A. Falkovsky Phys. Rev. B 84, 115414 (2011).
* (15) M.N. Cohen, L.M. Falikov, Phys. Rev. Lett. 7, 231 (1961).
|
arxiv-papers
| 2012-03-01T11:40:41 |
2024-09-04T02:49:28.149806
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.Yu. Ozerin, L.A. Falkovsky",
"submitter": "L. A. Falkovsky",
"url": "https://arxiv.org/abs/1203.0157"
}
|
1203.0180
|
# Flexible and robust networks
S. Vakulenko1 and O. Radulescu2
1 Saint Petersburg State University of Technology and Design, St.Petersburg,
Russia,
2 DIMNP UMR CNRS 5235, University of Montpellier 2, Montpellier, France.
Abstract We consider networks with two types of nodes. The $v$-nodes, called
centers, are hyperconnected and interact one to another via many $u$-nodes,
called satellites. This centralized architecture, widespread in gene networks,
possesses two fundamental properties. Namely, this organization creates
feedback loops that are capable to generate practically any prescribed
patterning dynamics, chaotic or periodic, or having a number of equilibrium
states. Moreover, this organization is robust with respect to random
perturbations of the system.
## 1 Introduction
Flexibility and robustness are important properties of biological systems.
Flexibility means the capacity to adapt with respect to changes of environment
whereas robustness is the capacity to support homeostasis in spite of
environmental changes. Intriguingly, it seems that biological systems could be
in the same time robust and flexible. Development of an organism is robust to
variations of initial conditions and environment, species can diversify in
order to better satisfy constraints imposed by a varying environment.
We discuss here flexibility and robustness problems for genetical networks of
a special topological structure as a model for flexible and robust systems. In
these networks, highly connected hubs play the role of organizing centers
(centralized networks). The hubs receive and dispatch interactions. Each
center interacts with many weakly connected nodes (satellites). Similar ideas,
that such a ”bow-tie” connectivity can play a role in robustness, have been
proposed by (Zhao et al. 2006). In the field of random boolean networks
(Kauffman 1969), the phase transitions from chaotic to frozen (robust) phases
were related to scale-freeness and heterogeneity of the network by (Aldana
2003). We show that centralized network are capable to produce a number of
patterns, while being protected against environment fluctuations.
Network models usually involve interactions between transcription factors
(TFs) (Reinitz et al. 1991). In the last years, a great attention has been
focused on microRNAs (He and Hannon 2004, Bartel 2009, Hendrikscon et al.
2009, Ihui et al. 2010). MicroRNAs (miRNAs) are short ribonucleic acid (RNA)
molecules, on average only 22 nucleotides long and are found in all eukaryotic
cells. miRNAs are post-transcriptional regulators that bind to complementary
sequences on target messenger RNA transcripts (mRNAs) and repress translation
or trigger mRNA cleavage and degradation. Thus, miRNAs have an impact on gene
expression and it was shown recently that they contribute to canalization of
development (Li et al. 2009). (Shalgi et al. 2007) shows the existence of many
genes submitted to extensive miRNA regulation with many TF among these ”target
hubs”. Without excluding other applications, we consider regulation of TF by
miRNAs as a possible example of a centralized network. For this particular
situation we generalize the TF network models (Reinitz et al. 1991) to take
into account miRNA satellites and the centralized architecture. The
interaction between network nodes is defined by sigmoidal functions that can
be defined by two parameters: the maximum rates of production $r_{i}$, and
sharpness constants $K_{i}$. Other important parameters, play a key role,
namely, the degradation constants $\lambda_{i}>0$ of centers and satellites.
We obtain a fundamental relation between the main network parameters. This
relation ensures maximal robustness of the network with respect to random
internal and external fluctuations, given a certain amount of flexibility
defined as the number of attractors that are accessible to the network
dynamics. Our mathematical results have a transparent biological
interpretation: centralized motifs can be simultaneously flexible and robust.
One can expect that miRNA molecules, being smaller with respect to TF, are
more mobile and react faster to perturbations. This property plays a key role
in the flexible and robust functioning of centralized motifs.
The paper is organized as follows. Centralized networks are introduced in
Section 2. We also formulate here an important assertion on the flexibility of
general centralized networks. We show that these networks are capable to
generate practically all dynamics, chaotic or periodic, with any number of
equilibrium states. To study robustness with respect to random fluctuations,
in Section 3 we consider a toy model of simple centralized TF - miRNA networks
with a single center. We show here, by an elementary way, that centralized
networks with mutually repressive hub-satellite interaction can produce many
different robust patterns.
## 2 Centralized networks
Centralized networks have been empirically identified in molecular biology,
where the centers can be, for example, transcription factors, while the
satellite regulators can be small regulatory molecules such as microRNAs (Li
et al. 2010). Notice that, in the last decades, the theory of so-called scale-
free networks has become very popular. Scale-free networks (Barabasi and
Albert 2002, Lesne 2006) occur in many areas, in economics, biology and
sociology. In the scale-free networks the probability $P(k)$ that a node is
connected with $k$ neighbors, has the asymptotics $Ck^{-\gamma}$, with
$\gamma\in(2,3)$. Such networks typically contain a few strongly connected
nodes and a number of satellite nodes. Hence, scale-free networks are, in a
sense, centralized.
In order to model dynamics of centralized networks we adapt a gene circuit
model proposed to describe early stages of Drosophila (fruit-fly)
morphogenesis (Mjolness et al. 1991, Reinitz and Sharp 1995). To take into
account the two types of the nodes, we use distinct variables $v_{j}$, $u_{i}$
for the centers and the satellites. The real matrix entry $A_{ij}$ defines the
intensity of the action of a center node $j$ on a satellite node $i$. This
action can be either a repression $A_{ij}<0$ or an activation $A_{ij}>0$.
Similarly, the matrices ${\bf B}$ and ${\bf C}$ define the action of the
centers on the satellites and the satellites on the centers, respectively. Let
us assume that a satellite can not act directly on another satellite (the
principle of divide et impera). We also assume that satellites respond more
rapidly to perturbations and are more diffusive/mobile than the centers. Both
these assumptions are natural if we identify satellites as microRNAs.
Let $M,N$ be positive integers, and let ${\bf A},{\bf B}$ and ${\bf C}$ be
matrices of the sizes $N\times M,M\times M$ and $M\times N$ respectively. We
denote by ${\bf A}_{i},{\bf B}_{j}$ and ${\bf C}_{j}$ the rows of these
matrices. To simplify formulas, we use the notation
$\sum_{j=1}^{M}A_{ij}v_{j}={\bf A}_{i}v,\quad\sum_{l=1}^{M}B_{jl}v_{l}={\bf
B}_{j}v,\quad\sum_{k=1}^{N}C_{jk}u_{k}={\bf C}_{j}u.$
Then, the network model reads (we exclude diffusion effects):
$\frac{du_{i}}{dt}=\tilde{r}_{i}\sigma\left({\bf
A}_{i}v-\tilde{h}_{i}\right)-\tilde{\lambda}_{i}u_{i},$ (1)
$\frac{dv_{j}}{dt}=r_{j}\sigma\left({\bf B}_{j}v+{\bf
C}_{j}u-h_{j}\right)-\lambda_{j}v_{j}.$ (2)
We assume that the rate coefficients $r_{j},\tilde{r}_{i}$ are non-negative:
$r_{i},\tilde{r}_{i}\geq 0$. Here $i=1,...,N$, $j=1,...,M$ and $\sigma$ is a
monotone and smooth (at least twice differentiable) sigmoidal function such
that
$\sigma(-\infty)=0,\quad\sigma(+\infty)=1.$ (3)
Typical examples can be given by the Fermi and Hill functions:
$\sigma(x)=\frac{1}{1+\exp(-x)},\quad\sigma_{H}(x)=\frac{x^{p}}{K_{a}^{p}+x^{p}},$
(4)
where $K_{a}$, $p>0$ are parameters and in the second case $x>0$. For $x<0$ we
set $\sigma_{H}(x)=0$. Analytical and computer simulation results are similar
for both variants $\sigma$ and $\sigma_{H}$.
The parameters $\lambda_{i},\tilde{\lambda}_{i}$ are degradation coefficients,
and $h_{i},\tilde{h}_{i}$ are thresholds for activation.
Let us prove that the gene network dynamics defines a dissipative dynamics. In
fact, there exists an absorbing set ${\cal B}$ defined by
${\cal B}=\\{w=(u,v):0\leq v_{j}\leq r_{j}\lambda_{j}^{-1},\ 0\leq
u_{i}\leq\tilde{r}_{i}\tilde{\lambda}_{i}^{-1},\ j=1,...,M,\ i=1,...,N\\}.$
One can show, by comparison principles for ordinary differential equations,
that
$\begin{split}0\leq
u_{i}(x,t)\leq\tilde{\phi}_{i}(x)\exp(-\tilde{\lambda}_{i}t)+\tilde{r}_{i}\tilde{\lambda}_{i}^{-1}(1-\exp(-\tilde{\lambda}_{i}t)),\\\
0\leq
v_{i}(x,t)\leq\phi_{i}(x)\exp(-\lambda_{i}t)+r_{i}\lambda_{i}^{-1}(1-\exp(-\lambda_{i}t)).\end{split}$
(5)
Therefore, solutions of (1), (2) exist for all times $t$ and they enters for
the set ${\cal B}$ at a time moment $t_{0}$ and then stays in this set for all
$t>t_{0}$. So, our system defines a dissipative dynamics and all
concentrations are positive if they are positive at the initial moment. In
mathematical terms, the Cauchy problem (initial value problem) for our system
is well posed.
## 3 Complex dynamics of centralized networks
Let us show that the centralized networks have a formidable power in dynamics
generation. First, we will find an asymptotic simplification of the dynamics,
then show that any dynamics, periodic, chaotic, or with a number of stable
steady states can be approximated by centralized networks.
### 3.1 Simplified dynamics when satellites are fast
We suppose here that the $u$-variables are fast and the $v$-ones are slow.
Then the fast $u$ variables are slaved, for large times, by the slow $v$
modes: one has $u=U(v)+\tilde{u}$, where $\tilde{u}$ is a small correction.
This means that, for large times, the satellite dynamics is defined almost
completely by the center dynamics.
To realize this approach, let us assume that the parameters of the system
satisfy the following conditions:
$\ |A_{jl}|,|B_{il}|,|C_{ij}|,|\tilde{h}_{i}|,|h_{j}|<C_{0},$ (6)
where $i=1,2,...,N,\ \ i,l=1,...,M,\ j=1,...,N$,
$0<C_{1}<\tilde{\lambda}_{j},$ (7)
and
$r_{i}=\kappa R_{i},\quad\tilde{r}_{i}=\kappa\tilde{R}_{i},$ (8)
where
$|R_{i}|,|\tilde{R}_{i}|<C_{5},\quad\lambda_{i}=\kappa\bar{\lambda}_{i},\
|\bar{\lambda}|<C_{6},$ (9)
where $\kappa$ is a small parameter, and where all positive constants $C_{k}$
are independent of $\kappa$.
Assertion 2.1. Under assumptions (6), (7), (8) for sufficiently small
$\kappa<\kappa_{0}$ solutions $(u,v)$ of (1), (2) satisfy
$u=U(v(t))+\tilde{u}(t),$ (10)
where the $j$-th component $U_{j}$ of $U$ is defined by
$-\tilde{\lambda}_{j}U_{j}=\kappa G_{j}(v),$ (11)
where
$G_{j}=\tilde{R}_{j}\sigma\left({\bf A}_{j}v(t)-\tilde{h}_{j}\right)$
The function $\tilde{u}$ satisfies estimates
$|\tilde{u}|<c\kappa^{2}+R\exp(-\beta t),\quad\beta>0.$ (12)
The $v$ dynamics for large times $t>C_{1}|\log\kappa|$ takes the form
$\frac{dv_{i}}{dt}=\kappa F_{i}(u,v)+w_{i},$ (13)
where $w_{i}$ satisfy
$|w_{i}|<c\kappa^{2}$
and
$F_{i}(u,v)=R_{i}\sigma\left({\bf B}_{i}v+{\bf
C}_{i}U(v)-h_{i}\right)-\bar{\lambda}_{i}v_{i}.$
This assertion, known in computational biology as the quasi-steady state
assumption, can be proved by well known methods from the theory of
differential equations (Henry 1981).
### 3.2 Realization of prescribed dynamics by networks
Our next goal is to show that dynamics (13) can realize, in a sense, arbitrary
dynamics of the centers. To precise this, let us describe the method of
realization of the vector fields for dissipative systems (proposed by Poláčik
1991, for applications see, for example, Dancer - Poláčik 1999, Rybakowski
1994, Vakulenko 2000). This method is based on the well developed theory of
invariant and inertial manifolds, see Marion 1989, Mane 1977, Constantin,
Foias, Nicolaenko and Temam, 1989, Chow-Lu 1988, Babin-Vishik 1988). One can
show that there are systems enjoying the following properties:
A These systems generate global semiflows $S_{\cal P}^{t}$ in an ambient phase
space $H$. These semiflows depend on some parameters $\cal P$ (which could be
elements of another parameter space $\cal B$). They have global attractors and
finite dimensional local attracting invariant $C^{1}$ (continuously
differentiable) - manifolds $\cal M$, at least for some $\cal P$.
B Dynamics of $S^{t}_{\cal P}$ reduced on these invariant manifolds is, in a
sense, ”almost completely controllable”. It can be described as follows.
Assume the differential equations
$\frac{dp}{dt}=F(p),\quad F\in C^{1}(B^{n})$ (14)
define a dynamical system in the unit ball ${B}^{n}\subset{\bf R}^{n}$.
For any prescribed dynamics (14) and any $\delta>0$, we can choose suitable
parameters ${\cal P}={\cal P}(n,F,\delta)$ such that
B1 The semiflow $S_{\cal P}^{t}$ has a $C^{1}$\- smooth locally attracting
invariant manifold ${\cal M}_{\cal P}$ diffeomorphic to the ball ${B}^{n}$;
B2 The reduced dynamics $S_{\cal P}^{t}|_{{\cal M}_{\cal P}}$ is defined by
equations
$\frac{dp}{dt}=\tilde{F}(p,{\cal P}),\quad\tilde{F}\in C^{1}(B^{n})$ (15)
where the estimate
$|F-\tilde{F}|_{C^{1}({B}^{n})}<\delta$ (16)
holds. In other words, one can say that, by $\cal P$, the dynamics can be
specified to within an arbitrarily small error.
Thus, all dynamics can occur as inertial forms of these systems. Such systems
can be named maximally dynamically flexible, or, for brevity, MDF systems.
Such dynamics can be chaotic. There is a rather wide broad in different
definitions of ”chaos”. In principle, one can use here any concept of chaos.
If this chaos is stable under small $C^{1}$ -perturbations this kind of chaos
occurs in the dynamics of MDF systems. To fix ideas, we use here, following
Ruelle and Takens 1971, Newhouse, Ruelle and Takens 1971 Smale 1980, Anosov
1995), such a definition. We say that a finite dimensional dynamics is chaotic
if this generates a non-quasiperiodic hyperbolic invariant set $\Gamma$. If,
moreover, this set $\Gamma$ is attracting we say that $\Gamma$ is a chaotic
(strange) attractor. (For definition of hyperbolic sets, see Ruelle 1989,
Anosov 1995). In this paper, we use only the following basic property of
hyperbolic sets, so-called Persistence (Ruelle 1989, Anosov 1995). This means
that the hyperbolic sets are, in a sense, stable(robust): if (14) generates
the hyperbolic set $\Gamma$ and $\delta$ is sufficiently small, then dynamics
(14) also generates another hyperbolic set $\tilde{\Gamma}$. Dynamics (14) and
(15) restricted to $\Gamma$ and $\tilde{\Gamma}$ respectively, are
topologically orbitally equivalent (on definition of this equivalence, see
Ruelle 1989, Anosov 1995). It is important to mention that a chaos in
dissipative systems may be stable, in the sense of structural stability, and
although not yet observed in gene networks, structurally stable chaotic
itineracy is thought to play a functional role in neuroscience (Rabinovitch
1998).
Therefore, any possible chaotic robust dynamics can be generated by the MDF
systems, for example, the Smale horseshoes, Anosov flows, the Ruelle-Takens-
Newhouse chaos, see Newhouse, Ruelle, and Takens, 1971, Smale 1980, Ruelle
1989. Some examples of the MDF systems were given in Dancer- Poláčik 1999,
Rybakowski 1994, Vakulenko 2000.
Assertion 2.1 allows us to apply this approach to centralized network
dynamics. To this end, assume that (8) and (9) hold. Moreover, let us assume
$\lambda_{i}=\kappa^{2}\bar{\lambda}_{i},\quad h_{i}=\kappa\bar{h}_{i}$ (17)
where all coefficients $\bar{h}_{i}$ are uniform in $\kappa$ as $\kappa\to 0$.
We also assume that all direct interactions between centers are absent, ${\bf
B}={\bf 0}$. This constraint is not essential.
Since $U_{j}=O(\kappa)$ for small $\kappa$, we can use the Taylor expansion
for $\sigma$ in (13). Then these equations reduce to
$\frac{dv_{i}(\tau)}{d\tau}=\rho_{i}({\bf
C}_{i}V(v)-\bar{h}_{i})-\bar{\lambda}_{i}v_{i}+\tilde{w}_{i}(t),$ (18)
where $\rho_{i}=\bar{r}_{i}\sigma^{\prime}(0)$, $i=1,2,...,M$ and $\tau$ is a
slow rescaling time: $\tau=\kappa^{2}t$. Due to conditions (17), the
corrections $\tilde{w}_{i}$ satisfy
$|\tilde{w}_{i}|<c\kappa.$
Let us focus now our attention to non-perturbed equation (18) with
$\tilde{w}_{i}=0$. Let us fix the number of centers $M$. The number of
satellites $N$ will be considered as a parameter.
The next important assertion immediately follows from well known approximation
theorems of the multilayered network theory, see, for example, Barron 1993,
Funahashi and Nakamura 1993.
Assertion 2.2. Given a number $\delta>0$, an integer $M$ and a vector field
$F=(F_{1},...,F_{M})$ defined on the ball $B^{M}=\\{|v|\leq 1\\}$, $F_{i}\in
C^{1}(B^{M})$, there are a number $N$, an $N\times M$ matrix ${\bf A}$, an
$M\times N$ matrix ${\bf C}$ and coefficients $h_{i}$, where $i=1,2,...,N$,
such that
$|F_{j}(\cdot)-{\bf C}_{j}W(\cdot)|_{C^{1}(B^{M})}<\delta,$ (19)
where
$W_{i}(v)=\sigma\left({\bf A}_{i}v-h_{i}\right),$ (20)
where $v=(v_{1},...,v_{M})\in{\bf R}^{M}$.
This assertion gives us a tool to control network dynamics. Assume
$\bar{h}_{i}=0$. Then equations (18) with $\tilde{w}_{i}=0$ reduce to the
Hopfield-like equations for variables $v_{i}\equiv v_{i}(\tau)$ that depend
only on $\tau$:
$\frac{dv_{l}}{d\tau}={\bf K}_{l}W(v)-\bar{\lambda}_{l}v_{l},$ (21)
where $l=1,...,M$, the matrix $\bf K$ is defined by
$K_{lj}=\rho_{l}C_{lj}R_{j}\tilde{\lambda}_{j}^{-1}$. The parameters $\cal P$
of (21) are $\bf K$, $M$, $h_{j}$ and $\bar{\lambda}_{j}$.
In this case one can formulate the following result.
Assertion 2.3. Let us consider a $C^{1}$-smooth vector field $Q(p)$ defined on
a ball $B_{R}\subset{\bf R}^{M}$ and directed strictly inside this ball at the
boundary $\partial B^{M}$:
$F(p)\cdot p<0,\quad p\in\partial B^{M}.$ (22)
Then, for each $\delta>0$, there is a choice of parameters $\cal P$ such that
(21) $\delta$ -realizes system (14). This means that (21) is a MDF system.
This follows from Assertions 2.1 and 2.2.
## 4 A toy model of centralized network
In this section we consider a simple centralized network that, nonetheless,
can produce a number of point attractors (stable steady states). Due to its
simple structure, we can investigate here the robustness of this system.
Let us consider a central node interacting with many satellites. This motif
can appear as a subnetwork in a larger scale-free network. In order to study
robustness, we add noise to the model. We consider two types of stochastic
perturbations. The first type of perturbations is a Langevin type additive
noise that can simulate intrinsic stochastic fluctuations of gene expression
dynamics. The choice of additive noise is for the sake of simplicity, however
more general multiplicative noise can be used with no change of the results.
The second type of noise is a shot-like perturbation that can simulate the
external contributions to noise, caused by the environment. Furthermore, we
replace the sigmoid in (2) by a linear function. This is justified in TF -
miRNAs networks, where the action of satellites (miRNA’s) on centers (TF’s) is
post-transcriptional and produces a modulation of the production rate of the
center protein. This modulation can be modeled by a soft sigmoid or even by a
linear function. Moreover, to simplify our model, we assume that all
satellites are, in a sense, equivalent.
The network dynamics can be described then by the following equations:
$\frac{du_{i}}{dt}=-\lambda u_{i}+f_{i}(v)+\xi_{i}(t),$ (23)
$\frac{dv}{dt}=-\nu v+Q(u)+\xi_{0}(t),$ (24)
where $f_{i}$, $Q$ are defined by
$Q(u)=a_{0}+a\sum_{i=1}^{n}u_{i},\quad f_{i}(u)=r\sigma(b(v-h_{i})),$
Here $\xi_{i}$ are noises, the coefficient $\lambda>0$ is a satellite mobility
(degradation rate), $r>0$ is the satellite maximum production rate, $b$
defines a sharpness of center action on the satellites, $\nu>0$ is a center
mobility (degradation rate), $a$ is the strength of the satellites feedback
action on the center.
We consider the following type of noises: non-correlated white noise
$\langle\xi_{i}(t),\xi_{j}(t^{\prime})\rangle=\beta_{i}\delta_{ij}\delta(t-t^{\prime})$
(25)
where $\beta_{i}>0$ are intensities, and shot-like noise
$\xi_{i}(t)=\beta_{i}\eta_{i}\delta(t-\tau_{j})$ (26)
where $\tau_{j}$ are random shot times following a Poisson process,
$\beta_{i}$ are noise amplitude coefficients, and $\eta_{i}$ are random
variables distributed uniformly on $[0,1]$. In numerical simulations we set
$\delta(t-\tau_{j})=1$ with a probability $p_{0}<<1$ and
$\delta(t-\tau_{j})=0$ with the probability $1-p_{0}$, where $\tau_{j}=j\delta
t$, $\delta t$ is a time step. Such noises $\xi_{i}$ can summarize the effect
of a strong environment fluctuations on the satellite and center expression.
We study the problem under the following
Assumption. Let the derivatives of $f_{i}$ and $Q$ satisfy
$f_{i}^{{}^{\prime}}(v)Q^{\prime}(u)>0\ for\ all\ i,u,v.$
Then one can show, following (Hirsch, 1988) that the dynamics is monotone,
and, therefore, all trajectories converge to equilibria. The numerical
simulations confirm this fact. Notice that the above assumption is not needed
when satellites are fast, because in this case the asymptotic dynamics is one
dimensional and in dimension one all the attractors are stable steady states
(point attractors). Although this simple system can not generate chaos or
periodic behavior, the number of point attractors can be arbitrarily large,
and thus this system is nonetheless flexible.
### 4.1 Multistationarity of the toy model
Let us fix the signs of the satellite actions on the center assuming that
$a<0$. This restriction is fulfilled in gene networks, where the centers are
transcription factors (TF) and the satellites are microRNAs (indeed, usually
microRNA can only repress transcription factors). Let us show that the toy
model admits coexistence of any number of point attractors.
Let us make a transformation reducing (23) and (24) to a system of two
equations introducing a new variable $Z$ by
$Z=\sum_{j=1}^{N}u_{j},\quad G(v)=\sum_{j=1}^{N}r\sigma(b(v-h_{j})).$
Then, by summarizing eqs. (23), one obtains
$\frac{dZ}{dt}=-\lambda Z+G(v),$ (27) $\frac{dv}{dt}=-\nu v+a_{0}+aZ.$ (28)
This system is relatively simple and it can be studied analytically and
numerically. Since all trajectories are convergent we obtain that the
attractor consists of equilibria defined by
$\lambda Z+F(Z)=0,\quad F(Z)=G(\nu^{-1}(a_{0}+aZ)).$ (29)
Let ${\cal P}=\\{b,h_{i},N,a_{0},a\\}$ be free parameters that can be
adjusted. Like to the previous section, we can ”control” the nonlinearity $F$
by ${\cal P}$ and use the fact that $F(Z)$ can approximate arbitrary smooth
functions. The following assertion shows that the system is multi-stationarity
with an arbitrary number of point attractors:
Assertion 3.1. Let $N$ be a positive integer. Then there are coefficients
$b,\lambda>0,r>0$, where $i=1,...,N$, $\nu>0$ and $h_{i},a_{0},a$ in such a
way that equation (29) has at least $n+1$ stable roots that can be placed in
any given positions in the $Z$-space.
The main idea of the proof can be illustrated by Fig. 1 and holds in both
cases of the Fermi and the Hill sigmoids. Let us make a variable change
$w=\lambda Z/r$. The steady states are solutions of the equation $rw=F(w)$,
where the function $F(w)$ is close to a step function with $N$ steps; each
step is given by the function $\sigma(\gamma(w-\bar{h}_{i}))$ that is close to
Heaviside step function for large $\gamma$. Here $\gamma$ is a parameter that
defines the sigmoid sharpness:
$\gamma=abr(\nu\lambda)^{-1}.$ (30)
The steady states of the system are given by the intersections between the
graph of $F(w)$ and the straight line of slope $r$. An elementary argument
shows that the intersections lying on horizontal segments of the graph of
$F(w)$ are stable attractors, whereas the intersections on ascending vertical
segments correspond to repellers.
The position of the $i$-th step in $w$-space is $\bar{h}_{i}$ and its height
is $r$. Under an appropriate choice of $\bar{h}_{i}$ this entails our
assertion (see Fig. 1). In the neural network theory, $\gamma$ is known as
gain parameter. This quantity, defined as the product of rates on sharpness
divided on the product of degradation coefficients, gives the maximal possible
density of the equilibrium states in $w$-space.
It is useful to note that one gets $n+1$ attractors on the horizontal segments
of the step function provided that $\bar{h}_{i}$ decrease with $i$.
Notice that the main condition to obtain flexibility (multistationarity) is
the sharpness of the sigmoidal function, meaning that the gain parameter
$\gamma$ should be large. The construction is robust: we can vary
$w_{i},b,\bar{h}_{i}$ but the number of equilibria is conserved.
### 4.2 Robustness and stability of attractors
The roots of Eq.(29) are point attractors and then they are dynamically
stable, otherwise, they are repellers and unstable. In Fig. 1, attractors
correspond to intersections of the straight line $y=rw$ with the curve
$y=F(w)$, lying on horizontal segments of the graph of $F$. A simple argument
suggests that the positions of these attractors are robust with respect to
variations of the thresholds $\bar{h}_{i}$. Indeed, a perturbation of
$\bar{h}_{i}$ induces a horizontal shift of the step
$\sigma(\gamma(w-\bar{h}_{i}))$, and the positions of the attractors are only
slightly affected.
Figure 1: Intersections of the curve $y=F(w)$ and the straight line $y=rw$
correspond to steady states; intersections on horizontal segments of the graph
of $F$ correspond to stable steady states.
More insight into robustness of the centralized toy model can be obtained by
considering the noisy case $\xi_{i}\neq 0$.
First, let us consider the case of the Langevin noise (25). We are interested
in the robustness of the number and positions of the attractors with respect
to noises $\xi_{i}(t)$. Near a point attractor, the equations (27), (28) can
be linearized. The linearized dynamics is defined by the following matrix
${\bf H}$:
$\left(\begin{array}[]{ccc}-\lambda&\mu\\\ a&-\nu\\\ \end{array}\right)$
where $\mu=G^{\prime}(v_{eq})$. For large $\gamma$ and for stable stationary
states $\mu$ is small, $\mu=G^{\prime}(v_{eq})\to 0$ as $\gamma\to\infty$. Let
us assume that the noises $\xi_{i}$ are independent white noises. Using
standard results from the theory of linear stochastic differential equations,
see, for example, Keizer 1987, it follows that small deviations $\delta
Z,\delta v$ from the equilibrium are normally distributed with the density
$\rho(\delta Z,\delta v)=const\exp(-X\cdot{\bf M}^{-1}\cdot X^{tr}),\quad
X=(\delta Z,\delta v),$ (31)
where ${\bf M}$ is a symmetric, positively defined, $2\times 2$ covariation
matrix with entries $m_{11},m_{22},m_{12}=m_{21}$. This matrix can be defined
by the well known relation (the fluctuation-dissipation theorem):
${\bf HM+MH}^{tr}=-{\bf B},$ (32)
where ${\bf B}=diag(B_{1}^{2},B_{2}^{2})$ and, since the noises are non-
correlated, $B_{1}=\sqrt{\sum_{i=1}^{N}\beta_{i}^{2}}$, $B_{2}=\beta_{0}.$
As a result, a characteristic fluctuation amplitude $F_{A}$ is proportional to
the maximum $\max\\{\theta_{1}^{1/2},\theta_{2}^{1/2}\\}$ where $\theta_{i}$
are eigenvalues of ${\bf M}$. Eq. (32) can be resolved explicitly and
$\theta_{i}$ can be found.
Now we can investigate the following problem: how to tune the parameters
$\lambda,\nu$ and $a$ to obtain the minimal fluctuation amplitude $F_{A}$ with
respect to the noise under a given multistationarity level (this means
$\gamma=\gamma_{0}>>1$ is fixed but we can vary the degradation rates
$\nu,\lambda$). This optimization problem can be resolved numerically. The
results, which describe the optimal $\lambda_{opt},\nu_{opt}$ as functions of
$\gamma$, are as follows.
The case A), $B_{2}>>B_{1}$, the center is under a stronger noise than the
satellites. Then the center degradation rate $\nu_{opt}$ should be large, and
$\lambda_{opt}$ is a small, decreasing in $\gamma$ function.
The case B), $B_{2}\leq B_{1}$, the center is under smaller noise than the
satellites. Then the center degradation rate $\nu_{opt}$ should be smaller,
$\lambda_{opt}>\nu_{opt}$, and the both parameters are decreasing in $\gamma$.
This situation is illustrated by Fig. 2.
Figure 2: Optimal degradation parameters $\nu_{opt}$, $\lambda_{opt}$
(minimizing the eigenvalues of the matrix ${\bf H}$) as functions of the gain
parameter $\gamma$ in the case $B_{2}\leq B_{1}$, when the center is under
smaller noise than the satellites.
The classical ideas of the invariant manifold theory, discussed in the
preceding section, allow us to systematize these results. The centralized
network can function under two main and quite opposite regimes. The first one
arises when $\lambda>>\nu$. Then the satellite dynamics is slaved by the
center motion. The center dominates and such a regime can be named power of
the center. This regime is stable if $\beta_{0}$ is small, but $\beta_{i}$ are
large (the noises act on satellites mainly, case B). Considering that the
noise intensity is larger for those components that are expressed in larger
copy numbers, the case should be representative for miRNA-TF networks, when
miRNA are in smaller copy numbers than the transcription factors. In this case
the noise perturb satellite states ($u_{i}$) but, since the satellites are
controlled by the center state $v$, satellites return to the normal states and
dynamics is robust, the noise does not damage the attractor. The opposite
regime is when $\lambda<<\nu$. Then, opposite to the previous situation, the
center dynamics is slaved by the satellites motion. Such a regime can be named
satellite democracy. This regime is stable when $\beta_{0}$ is large, but
$\beta_{i}$ are small (the noise acts stronger on the center, case A). Here
the noise can perturb the center state ($v$) but this state can be restored by
satellites. The large time dynamics is robust, again the noise does not damage
the attractor.
Similar results are illustrated in the case of a shot noise in Fig. 3.
So, we obtain an interesting connection between robustness, multistationarity
and network rate: to support robustness and multistationarity in a noisy
situation, we should decrease the degradation constants. Multistationarity of
molecular switches is important in decision making processes in
differentiation, development, and immune response of the organisms. Our
finding means that noise protected switches are necessarily slow.
Figure 3: Numerical simulations of the system’s trajectories under shot noise.
The parameters were as follows: $N=6$, $\sigma(z)=\sigma_{H}(z)$ with $p=4$,
$b=20$, $K_{a}=1$, $h_{i}=i$, $r=1$, $\lambda=5$, $a_{0}=0.3\nu$, $x\in[0,10]$
and $t\in[0,120]$. The parameter $\beta_{i}=0$ for all $i$ beside $i=3$, where
$\beta_{3}=50$. The two functioning regimes correspond to different values of
$\nu,a$, namely, $a=50,\ \nu=5$ (satellites democracy (SD), a)), and $a=5,\
\nu=0.5$ (power of the center (PC), b)). In the both cases the system shows
multistationarity. For the chosen initial data trajectories converge to an
attractor $v\approx 5.3$ as $t>>1$ in the PC regime and also in the SD regime
$v\to 3.3$. This means that the fast center loses the attractor control, while
the slow center controls dynamics even under large deviations.
## 5 Conclusion
We have considered networks with two types of nodes. The $v$-nodes, called
centers, are hyperconnected and interact one to another via many $u$-nodes,
called satellites. We show, by recently advanced mathematical methods, that
this centralized network architecture, allows us to control network dynamics
to create complicated dynamical regimes. This network organization creates
feedback loops that are capable to generate practically all kinds of dynamics,
chaotic or periodic, or having a number of equilibrium states. This strong
flexibility could be crucial for adaptive biological functions of these
networks.
Using the simple example of a motif with a single center, we also argued that
centralized networks can perform trade-offs between flexibility and
robustness. To support both flexibility and robustness in a noisy situation,
the network should function in a slow manner, i.e, we propose slow-down as a
way to increase stability.
Which of the nodes should be slowed-down depends on the fluctuations. Basic
ideas from the invariant manifold theory show that if the noises act on the
satellites, then, in order to conserve dynamics and the attractor structure,
the center should be slow and controls the satellites (we called this regime
power of the center). In the opposite case, when the noise acts on the center,
the satellites should be slow in order to control the center and the global
dynamics (we called this regime satellites democracy).
We did not consider here extrinsic noise or parametric variability of the
system, that we plan to study in the future. We also think that the slow-down
effect could be observed in all systems where there is a separation into slow
and fast variables, independently of architecture.
Acknowledgements. The authors are grateful to Maria Samsonova and Vitaly
Gursky for useful discussions. We are thankful to M. S. Gelfand and his
colleagues for interesting discussions in Moscow.
The first author was supported by the Russian Foundation for Basic Research
(Grant Nos. 10-01- 00627 s and 10-01-00814 a) and the CDRF NIH (Grant No.
RR07801). We are grateful to the anonymous referees for important remarks.
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|
arxiv-papers
| 2012-03-01T13:33:39 |
2024-09-04T02:49:28.158685
|
{
"license": "Public Domain",
"authors": "S.A. Vakulenko, O. Radulescu",
"submitter": "Ovidiu Radulescu",
"url": "https://arxiv.org/abs/1203.0180"
}
|
1203.0207
|
#
Relativistic correction to color Octet $J/\psi$ production at hadron colliders
Guang-Zhi Xu (a,b) still200@gmail.com Yi-Jie Li (a,b) yijiegood@gmail.com Kui-
Yong Liu (b) liukuiyong@lnu.edu.cn Yu-Jie Zhang (a) nophy0@gmail.com (a)
School of Physics, Beihang University, Beijing 100191, China
(b) Department of Physics, Liaoning University, Shenyang 110036 , China
###### Abstract
The relativistic corrections to the color-octet $J/\psi$ hadroproduction at
the Tevatron and LHC are calculated up to $\mathcal{O}(v^{2})$ in
nonrelativistic QCD factorization frame. The short distance coefficients are
obtained by matching full QCD with NRQCD results for the partonic subprocess
$g+g\to J/\psi({}^{1}S_{0}^{[8]},{}^{3}S_{1}^{[8]},{}^{3}P_{J}^{[8]})+g$,
$q+\bar{q}\to J/\psi({}^{1}S_{0}^{[8]},{}^{3}S_{1}^{[8]},{}^{3}P_{J}^{[8]})+g$
and $g+q({\bar{q}})\to
J/\psi({}^{1}S_{0}^{[8]},{}^{3}S_{1}^{[8]},{}^{3}P_{J}^{[8]})+q({\bar{q}})$.
The short distance coefficient ratios of relativistic correction to leading
order for color-octet states ${{}^{1}}S_{0}^{[8]}$, ${{}^{3}}S_{1}^{[8]}$, and
${{}^{3}}P_{J}^{[8]}$ at large $p_{T}$ are approximately -5/6, -11/6, and
-31/30, respectively, for each subprocess, and it is 1/6 for color-singlet
state ${{}^{3}}S_{1}^{[1]}$. If the higher order long distance matrix elements
are estimated through velocity scaling rule with adopting $v^{2}=0.23$ and the
lower order long distance matrix elements are fixed, the leading order cross
sections of color-octet states are reduced by about a factor of $20\sim 40\%$
at large $p_{T}$ at both the Tevatron and the LHC. Comparing with QCD
radiative corrections to color-octet states, relativistic correction is
ignored along with $p_{T}$ increasing. Using long distance matrix elements
extracted from the fit to $J/\psi$ production at the Tevatron, we can find the
unpolarization cross sections of $J/\psi$ production at the LHC taking into
account both QCD and relativistic corrections are changed by about $20\sim
50\%$ of that considering only QCD corrections. These results indicate that
relativistic corrections may play an important role in $J/\psi$ production at
the Tevatron and the LHC.
###### pacs:
12.38.Bx,12.39.St,13.85.Ni
## I Introduction
Heavy quarkonium is an excellent candidate to probe quantum chromodynamics
(QCD) from the high energy to the low energy regimes. Nonrelativistic QCD
(NRQCD) factorization formalism was establishedBodwin:1994jh to describe the
production and decay of heavy quarkonium. In the NRQCD approach, the
production and decay of heavy quarkonium is factored into short distance
coefficients and long distance matrix elements(LDMEs). The short distance
coefficients indicate the creation or annihilation of a heavy quark pair can
be calculated perturbatively with the expansions by the strong coupling
constant $\alpha_{s}$. However, the LDMEs, which represent the evolution of a
free heavy quark pair into a bound state, can be scaled by the relative
velocity $v$ between the quark and antiquark and obtained by lattice QCD or
extracted from the experiment. $v^{2}$ is about $0.2\sim 0.3$ for charmonium
and about $0.08\sim 0.1$ for bottomonium. The color-octet mechanism (COM) was
introduced here. The heavy quark pair should be a color-singlet (CS) bound
state at long distances, but it may be in a color-octet (CO) state at short
distances. NRQCD had achieved great success since it was proposed. The COM was
applied to cancel the infrared divergences in the decay widths of $P$-wave
Huang ; Petrelli:1997 and $D$-waveHe:2008xb ; Fan:2009cj heavy quarkonium.
However, difficulties were still encountered. The large discrepancy between
the experimental data and the theoretical calculation of $J/\psi$ and
$\psi^{\prime}$ unpolarization and polarization production at Tevatron is an
interesting phenomenon that can verify NRQCD when solvedCDF:1992 ;
arXiv:0704.0638 . Theoretical prediction with COM contributions was introduced
and was found to fit with the experimental data on $J/\psi$ production at
TevatronBraaten:1994 . However, the CO contributions from gluon fragmentation
indicated that the $J/\psi$ was transversely polarized at large $p_{T}$, which
is inconsistent with the experimental dataCDF:1992 .
The next-to-leading order (NLO) QCD corrections and other possible solutions
for $J/\psi$ hadroproduction were calculated to resolve the $J/\psi$ hadronic
production and polarization puzzleCampbell:2007 ; Gong:2008 . The calculation
enhanced the CS cross sections at large $p_{T}$ by approximately an order of
magnitude. However, the large discrepancy between the CS predictions and
experimental data remains unsolved. The relativistic correction to CS $J/\psi$
hadroproduction was insignificantFan:2009zq . The NLO QCD corrections of COM
$J/\psi$ hadroproduction were also calculated to formulate a possible solution
to the long-standing $J/\psi$ polarization puzzle Chao:2012iv ; Ma:2010jj ;
Butenschoen:2011yh . The spin-flip interactions in the spin density matrix of
the hardronization of a color-octet charm quark pair had been examined in
Ref.Liu:2006hc . A similar large discrepancy was found in double-charmonium
production at $B$ factoriesAbe:2002rb ; BaBar:2005 ; Braaten:2002fi . A great
deal of work had been performed on this area, and these discrepancies can
apparently be resolved by including NLO QCD correctionsZhang:2005cha ;
Zhang:2006ay ; Gong:2007db ; zhangma08 and relativistic
correctionsBodwin:2006dm ; He:2007te ; Jia:2009np ; He:2009uf . The data from
$B$ factories highlight that the COM LDMEs of $J/\psi$ production may be
smaller than previously expectedMa:2008gq ; Gong:2009kp ; Jia:2009np ;
He:2009uf ; Zhang:2009ym . Relativistic corrections have also been studied in
Ref.Huang:1996bk for heavy quarkonium decay, in Ref.Paranavitane:2000if for
$J/\psi$ photoproduction, in Ref.Ma:2000qn for $J/\psi$ production in $b$
decay, and in Ref.Bodwin:2003wh for gluon fragmentation into spin triplet $S$
wave quarkonium. More information about heavy quarkonium physics can be found
in Refs.Kramer:2001 ; Lansberg:2006dh ; Brambilla:2010cs .
In this paper, the effects of relativistic corrections to the COM $J/\psi$
hadroproduction at Tevatron and LHC were estimated based on NRQCD. The short
distance coefficients were calculated up to $\mathcal{O}(v^{2})$. Many free
LDMEs were realized at $\mathcal{O}(v^{2})$, which were estimated according to
the velocity scaling rules of NRQCD with $v^{2}=0.23$velocity .
The paper is organized as follows. In Sec. II, the frame of calculation is
introduced for the relativistic correction of both the $S$\- and $P$-wave
states in NRQCD frame. Section III provides the numerical result. Finally, a
brief summary of this work is presented.
## II Relativistic Corrections of Cross Section in NRQCD
We only consider $J/\psi$ direct production at high energy hadron colliders,
which contributes $70\%$ to the prompt cross section. The differential cross
section of direct production can be obtained by integrating the cross sections
of parton level as the following expression:
$\displaystyle
d\sigma\big{(}p+p(\bar{p}){\rightarrow}J/\psi+X\big{)}=\sum_{a,b,d}{\int}dx_{1}dx_{2}{f_{a/p}(x_{1})}{f_{b/p(\bar{p})}(x_{2})}d\hat{\sigma}(a+b{\rightarrow}J/\psi+d).$
(1)
where $f_{a(b)/p(\bar{p})}(x_{i})$ is the parton distribution function(PDF),
and $x_{i}$ is the parton momentum fraction denoted the fraction parton
carried from proton or antiproton. The sum is over all the partonic
subprocesses including
$\displaystyle g+g{\rightarrow}J/\psi+g$ $\displaystyle
g+q(\bar{q}){\rightarrow}J/\psi+q(\bar{q})$ $\displaystyle
q+\bar{q}{\rightarrow}J/\psi+g.$
As shown at the beginning of this paper, under the NRQCD frame, the
computation to cross section of each subprocess can be divided into two parts:
short distance coefficients and LDMEs:
$d\hat{\sigma}(a(k_{1})+b(k_{2}){\rightarrow}J/\psi(P)+d(k_{3}))=\sum_{n}\frac{F_{n}(ab)}{m_{c}^{d_{n}-4}}\langle
0|\mathcal{O}_{n}^{J/\psi}|0\rangle.$ (2)
On the right-hand side of the equation, the cross section is expanded to
sensible Fock states noted by the subscript $n$. $F_{n}$, i.e., short distance
coefficients, which describe the process that produces intermediate $Q\bar{Q}$
in a short range before heavy quark and antiquark hadronization to the
physical meson state. Here we use initial partons to mark the short distance
coefficients for different subprocesses. $\langle
0|\mathcal{O}_{n}^{J/\psi}|0\rangle$ are the long distance matrix elements
that represent the hadronization $Q\bar{Q}$ evolutes to the CS final state by
emitting soft gluons. $\mathcal{O}_{n}^{J/\psi}$ are local four fermion
operators. The factor of $m_{c}^{d_{n}-4}$ is introduced to make $F_{n}$
dimensionless.
In this section, our calculation on the differential cross section for this
process in the NRQCD factorization formula is divided into three parts,
namely, kinematics, long distance matrix elements, and short distance
coefficients.
### II.1 Kinematics
We denote the three relative momenta between heavy quark and antiquark as
$2\vec{q}$, with $|\vec{q}|\sim{m_{c}v}$, in $J/\psi$ rest frame, where
$m_{c}$ is the mass of charm quark and $v$ is the three relative velocity of
quark or antiquark in this frame. Thus, the momenta for the quark and
antiquark are expressed asHe:2007te ; Ma:2000 ; fourmomenta
$\displaystyle p_{c}$ $\displaystyle=$ $\displaystyle(E_{q},\vec{q}),$
$\displaystyle p_{\bar{c}}$ $\displaystyle=$ $\displaystyle(E_{q},-\vec{q}).$
(3)
where $E_{q}=\sqrt{m_{c}^{2}+|\vec{q}|^{2}}$ is the rest energy of both the
quark and antiquark, and $2E_{q}$ is the invariable mass of $J/\psi$. When
boosting to an arbitrary frame,
$\displaystyle\begin{array}[]{l}p_{c}\rightarrow\frac{1}{2}P+q,\quad
p_{\bar{c}}\rightarrow\frac{1}{2}P-q.\end{array}$ (5)
where $P$ is the four momenta of $J/\psi$, and $q$ receives the boost from
$(0,\vec{q})$.
The Lorentz invariant Mandelstam variables are defined as
$s=(k_{1}+k_{2})^{2}=(P+k_{3})^{2},\\\ $
$t=(k_{1}-P)^{2}=(k_{2}-k_{3})^{2},\\\ $ $u=(k_{1}-k_{3})^{2}=(k_{2}-P)^{2}.$
with the relationship $s+t+u=P^{2}=4E_{q}^{2}$. Here, s is $|\vec{q}|^{2}$
independence. To expand $t$, $u$ in terms of $E_{q}(i.e.|\vec{q}|^{2})$, we
can first write down $t$, $u$ in the center of initial partons mass frame:
$\displaystyle t(|\vec{q}|)$ $\displaystyle=$
$\displaystyle-(s-4E_{q}^{2})(1-cos\theta)/2=\frac{s-4E_{q}^{2}}{s-4m_{c}^{2}}t(0),$
$\displaystyle u(|\vec{q}|)$ $\displaystyle=$
$\displaystyle-(s-4E_{q}^{2})(1+cos\theta)/2=\frac{s-4E_{q}^{2}}{s-4m_{c}^{2}}u(0),$
(6)
where $t(0),u(0)$ are Lorentz invariants of $|\vec{q}|^{2}$ independence and
satisfies $s+t(0)+u(0)=4m_{c}^{2}$. These relations between
$t(|\vec{q}|)\big{(}u(|\vec{q}|)\big{)}$ and $t(0)\big{(}u(0)\big{)}$ are also
satisfied when boosting to arbitrary frame. In our subsequent calculation and
result, we adopt $t$($u$) to represent $t(0)$($u(0)$) directly for
simplification.
The FeynArts feynarts package was used to generate Feynman diagrams and
amplitudes, and the FeynCalc Mertig:an package was used to handle amplitudes.
The numerical phase space was integrated with Fortran.
### II.2 Long Distance Matrix Elements
According to NRQCD factorization, the differential cross section of each
partonic subprocess up to next order in $v^{2}$ to CS state
${}^{3}S_{1}^{[1]}$ and CO states ${{}^{1}}S_{0}^{[8]}$,
${{}^{3}}S_{1}^{[8]}$, ${{}^{3}}P_{J}^{[8]}$, can be expressed as
$\displaystyle d\sigma$ $\displaystyle=$ $\displaystyle
d\sigma_{lo}[^{3}S_{1}^{[1]}]+d\sigma_{lo}[^{1}S_{0}^{[8]}]+d\sigma_{lo}[^{3}S_{1}^{[8]}]+d\sigma_{lo}[^{3}P_{J}^{[8]}]$
(7) $\displaystyle+$ $\displaystyle
d\sigma_{rc}[^{3}S_{1}^{[1]}]+d\sigma_{rc}[^{1}S_{0}^{[8]}]+d\sigma_{rc}[^{3}S_{1}^{[8]}]+d\sigma_{rc}[^{3}P_{J}^{[8]}].$
In this expression, relativistic correction parts, denoted as ”$rc$”, can
easily be distinguished from LO, denoted as ”$lo$”. Ref.$[1]$ corresponds to
CS, and Ref.$[8]$ corresponds to CO. In addition, each differential cross
section to different Fock states should be divided in short distance
coefficient part and LDMEs. We can introduce $F(^{2s+1}L_{J}^{[c]})$ to
express the short distance coefficient of the LO cross section, corresponding
to $G(^{2s+1}L_{J}^{[c]})$ for relativistic correction. Many LDMEs are
presented , all of which are denoted by $\langle
0|\mathcal{O}^{J/\psi}(^{2s+1}L_{J}^{[c]})|0\rangle$ and $\langle
0|\mathcal{P}^{J/\psi}(^{2s+1}L_{J}^{[c]})|0\rangle$ for the LO and
relativistic correction term respectively. The explicit expressions of the ten
four-fermion operators areBodwin:1994jh
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[1]})|0>$ $\displaystyle=$
$\displaystyle<0|\chi^{\dagger}\sigma^{i}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}\sigma^{i}\chi|0>,$
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[1]})|0>$ $\displaystyle=$
$\displaystyle<0|\frac{1}{2}[\chi^{\dagger}\sigma^{i}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}\sigma^{i}(-\frac{i}{2}\overleftrightarrow{\mathbf{D}})^{2}\chi+h.c.]|0>,$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{1}S_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\chi^{\dagger}{T^{a}}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}\chi|0>,$
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{1}S_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\frac{1}{2}[\chi^{\dagger}{T^{a}}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{\mathbf{D}})^{2}\chi+h.c.]|0>,$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\chi^{\dagger}{T^{a}}\sigma^{i}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}\sigma^{i}\chi|0>,$
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\frac{1}{2}[\chi^{\dagger}{T^{a}}\sigma^{i}\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}\sigma^{i}(-\frac{i}{2}\overleftrightarrow{\mathbf{D}})^{2}\chi+h.c.]|0>,$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle{1\over
3}<0|\chi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D}\cdot\sigma)\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D}\cdot{\sigma})\chi|0>,$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{1}^{[8]})|0>$ $\displaystyle=$
$\displaystyle{1\over
2}<0|\chi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D}\times\sigma)\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D}\times\sigma)\chi|0>,$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{2}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\chi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D^{(i}}\sigma^{j)})\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D^{(i}}\sigma^{j)})\chi|0>,$
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$ $\displaystyle=$
$\displaystyle<0|\frac{1}{2}[\chi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{D^{i}}\sigma^{j})\psi(a^{\dagger}_{\psi}a_{\psi})\psi^{\dagger}{T^{a}}(-\frac{i}{2}\overleftrightarrow{\mathbf{D}})^{2}(-\frac{i}{2}\overleftrightarrow{D^{i}}\sigma^{j})\chi+h.c.]|0>,$
(8)
where $\chi$ and $\psi$ are the Pauli spinors describing anticharm quark
creation and charm quark annihilation, respectively. $T$ is the $SU(3)$ color
matrix. $\sigma$ is the Pauli matrices and $\mathbf{D}$ is the gauge-covariant
derivative with
$\overleftrightarrow{\mathbf{D}}=\overrightarrow{\mathbf{D}}-\overleftarrow{\mathbf{D}}$.
$\overleftrightarrow{D^{(i}}\sigma^{j)}$ is used as the notation for the
symmetric traceless component of a tensor:
$\overleftrightarrow{D^{(i}}\sigma^{j)}=(\overleftrightarrow{D^{i}}\sigma^{j}+\overleftrightarrow{D^{i}}\sigma^{j})/2-\overleftrightarrow{D^{k}}\sigma^{k}\delta^{ij}/3$.
Here we have
$v^{2}=\frac{\langle
0|\mathcal{P}^{J/\psi}(^{2s+1}L_{J}^{[c]})|0\rangle}{m_{c}^{2}\langle
0|\mathcal{O}^{J/\psi}(^{2s+1}L_{J}^{[c]})|0\rangle}.$ (9)
It should be noted that
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(2J+1)(1+\mathcal{O}(v^{2}))<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>,$
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(2J+1)(1+\mathcal{O}(v^{2}))<0|\mathcal{P}^{J/\psi}({}^{3}P_{0}^{[8]})|0>$
$\displaystyle\sim$ $\displaystyle{\cal
O}(v^{2})<0|\mathcal{O}^{J/\psi}({}^{3}P_{J}^{[8]})|0>.$
To NLO in $v^{2}$, we can ignore ${\cal O}(v^{4})$ terms and set
$\displaystyle<0|\mathcal{P}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(2J+1)<0|\mathcal{P}^{J/\psi}({}^{3}P_{0}^{[8]})|0>.$
So there are four CO LDMEs for $P$-wave, four CO LDMEs for $S$-wave and two CS
LDMEs at NLO in $v^{2}$. The LDMEs of heavy quarkonium decay may be determined
by potential modelBodwin:2007fz ; Bodwin:2006dm , lattice
calculationsBodwin:1996tg , or phenomenological extraction from experimental
dataFan:2009zq ; Guo:2011tz . But it is very difficult to determine the
production of CO LDMEs. Recently, two groups fitted CO LDMEs
$<0|\mathcal{O}^{J/\psi}(^{2s+1}L_{J}^{[8]})|0>$ to NLO in $\alpha_{s}$. It is
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{1}S_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(8.90\pm 0.98)\times 10^{-2}~{}GeV^{3},$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(0.3\pm 0.12)\times 10^{-3}~{}GeV^{3},$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>/m_{c}^{2}$
$\displaystyle=$ $\displaystyle(0.56\pm 0.21)\times 10^{-2}~{}GeV^{3},$ (11)
with data of $J/\psi$ production and polarization at $p_{t}>7~{}GeV$ at
Tevatron in Ref.Ma:2010jj and
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{1}S_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(4.50\pm 0.72)\times 10^{-2}~{}GeV^{3},$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(3.12\pm 0.93)\times 10^{-3}~{}GeV^{3},$
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(-1.21\pm 0.35)\times 10^{-2}~{}GeV^{5},$ (12)
with data of $J/\psi$ production at $p_{t}>3~{}GeV$ at Tevatron and
$p_{T}>2.5~{}GeV$ at HERA in Ref.Butenschoen:2011yh . The two series CO LDMEs
are not consistent with each other. For the three CO $P$ wave LDMEs
$<0|\mathcal{O}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$, it is hard to determine. To
simplify the discussion of the numerical result, it is assumed that
$\displaystyle<0|\mathcal{O}^{J/\psi}({}^{3}P_{J}^{[8]})|0>$ $\displaystyle=$
$\displaystyle(2J+1)<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>.$ (13)
At the same time, we can estimate the relation between their order from the
Gremm-Kapustin relation Gremm:1997dq in the weak-coupling regime
$v^{2}=v_{1}^{2}=v_{8}^{2}=\frac{M_{J/\psi}-2m_{c}^{pole}}{2m_{c}^{QCD}},$
(14)
where $m_{c}^{QCD}$ is the mass of charm quark that appears in the NRQCD
actions and $m_{c}^{pole}$ is the pole mass of charm quark. This equation was
given only for CS in Ref.Gremm:1997dq . This is the same with
Ref.Bodwin:2003wh , and we can get $v_{1}^{2}=v_{8}^{2}$. If we select
$M_{J/\psi}=3.1~{}GeV$ and $m_{c}^{QCD}=m_{c}^{pole}=1.39~{}GeV$ , we can get
$v^{2}\sim 0.23$.
After those presses, there are three CO LDMEs in the numerical calculation.
### II.3 Short distance coefficients calculation
The short distance coefficients can be evaluated by matching the computations
of perturbative QCD and NRQCD:
$\displaystyle d\sigma\Big{|}_{pert~{}QCD}$
$\displaystyle=\sum_{n}\frac{F_{n}}{m_{c}^{d_{n}-4}}\langle
0|\mathcal{O}_{n}^{c\bar{c}}|0\rangle\Big{|}_{pert~{}NRQCD}.$ (15)
The covariant projection operator method should be adopted to compute the
expression on the left-hand side of the equation. Using this method, spin-
singlet and spin-triplet combinations of spinor bilinears in the amplitudes
can be written in covariant form. For the spin-singlet case,
$\displaystyle\sum_{s\bar{s}}v(s)\bar{u}(\bar{s})\langle\frac{1}{2},s;\frac{1}{2},\bar{s}|0,0\rangle$
$\displaystyle=\frac{1}{2\sqrt{2}(E_{q}+m)}(-\not{p}_{\bar{c}}+m_{c})\gamma_{5}\frac{\not{P}+2E_{q}}{2E_{q}}(\not{p}_{c}+m_{c}).$
(16)
For spin-triplet case, the expression is defined as
$\displaystyle\sum_{s\bar{s}}v(s)\bar{u}(\bar{s})\langle\frac{1}{2},s;\frac{1}{2},\bar{s}|1,S_{z}\rangle$
$\displaystyle=\frac{1}{2\sqrt{2}(E_{q}+m)}(-\not{p}_{\bar{c}}+m_{c})\not{\epsilon}\frac{\not{P}+2E_{q}}{2E_{q}}(\not{p}_{c}+m_{c}),$
(17)
where $\epsilon$ denotes the polarization vector of the spin-triplet state. In
our calculation, Dirac spinors are normalized as $\bar{u}u=-\bar{v}v=2m_{c}$.
The differential cross section of each state then satisfies:
$\displaystyle
d\sigma(^{(2s+1)}L_{J}^{[c]}){\sim}\bar{\sum}|\mathcal{M}(a+b{\rightarrow}(c\bar{c})(^{(2s+1)}L_{J}^{[c]})+d)|^{2}\langle
0|\mathcal{O}^{J/\psi}(^{2s+1}L_{J}^{[c]})|0\rangle,$ (18)
where $\bar{\sum}$ means sum over the final state color and polarization and
average over initial states. According to this expression and Eq.(9),
expanding the cross section to next leading order of $v^{2}$ is to expand the
amplitude squared on the right side of the above expression to
$\mathcal{O}(|\vec{q}|^{2})$.
Next, we prepare to expand the short distance coefficients to the next order
in $v^{2}$. First, we expand each Fock state amplitude, including the $S$-wave
and $P$-wave states, in terms of the relative momentum $|\vec{q}|$:
$\displaystyle\mathcal{M}(a+b{\rightarrow}(c\bar{c})(^{3}S_{1}^{[1,8]})+d)$
$\displaystyle=\epsilon_{\rho}(\mathcal{M}^{\rho}_{t}\Big{|}_{q=0}+\frac{1}{2}q^{\alpha}q^{\beta}\frac{\partial^{2}(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}^{\rho}_{t})}{\partial
q^{\alpha}\partial q^{\beta}}\Big{|}_{q=0})+\mathcal{O}(q^{4}).$ (19)
$\displaystyle\mathcal{M}(a+b{\rightarrow}(c\bar{c})(^{1}S_{0}^{[8]})+d)$
$\displaystyle=\mathcal{M}_{s}\Big{|}_{q=0}+\frac{1}{2}q^{\alpha}q^{\beta}\frac{\partial^{2}(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}_{s})}{\partial
q^{\alpha}\partial q^{\beta}}\Big{|}_{q=0}+\mathcal{O}(q^{4}).$ (20)
$\displaystyle\mathcal{M}(a+b{\rightarrow}(c\bar{c})(^{3}P_{J}^{[8]})+d)=\epsilon_{\rho}(s_{z})\epsilon_{\sigma}(L_{z})(\frac{\partial\mathcal{M}^{\rho}_{t}}{\partial{q^{\sigma}}}\Big{|}_{q=0}$
$\displaystyle+\frac{1}{6}q^{\alpha}q^{\beta}\frac{\partial^{3}(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}^{\rho}_{t})}{\partial
q^{\alpha}\partial
q^{\beta}\partial{q^{\sigma}}}\Big{|}_{q=0})+\mathcal{O}(q^{4}).$ (21)
The factor $\sqrt{\frac{m_{c}}{E_{q}}}$ comes from the relativistic
normalization of $c\bar{c}$ state. Odd power terms of four-momentum $q$ vanish
in either the $S$-wave or the $P$-wave amplitudes, where $\mathcal{M}_{t}$ and
$\mathcal{M}_{s}$ are inclusive production amplitudes to triplet and singlet
$c\bar{c}$, respectively.
$\mathcal{M}_{s}=\sum_{s\bar{s}}\sum_{ij}\langle\frac{1}{2},s;\frac{1}{2},\bar{s}|0,0\rangle\langle
3i;\bar{3j}|1,8a\rangle\mathcal{A}(a+b{\rightarrow}c^{i}+\bar{c}^{j}+d).$
$\mathcal{M}_{t}=\sum_{s\bar{s}}\sum_{ij}\langle\frac{1}{2},s;\frac{1}{2},\bar{s}|1,S_{z}\rangle\langle
3i;\bar{3j}|1,8a\rangle\mathcal{A}(a+b{\rightarrow}c^{i}+\bar{c}^{j}+d).$
In evaluating the amplitudes in power series in $|\vec{q}|$, it needs to be
integrated over the space angle to $\vec{q}$. We can obtain the following
replacements to extract the contribution of fixed power of $|\vec{q}|$:
For $S$-wave case:
$q^{\alpha}q^{\beta}{\rightarrow}\frac{1}{3}\lvert\vec{q}\rvert{{}^{2}}\Pi^{\alpha\beta}.$
(22)
For $P$-wave case:
$q^{\alpha}q^{\beta}q^{\sigma}{\rightarrow}\frac{1}{5}\lvert\vec{q}\rvert{{}^{3}}\big{[}\Pi^{\alpha\beta}\epsilon^{\sigma}(L_{z})+\Pi^{\alpha\sigma}\epsilon^{\beta}(L_{z})+\Pi^{\beta\sigma}\epsilon^{\alpha}(L_{z})\big{]},$
(23)
where $\Pi^{\mu\nu}=-g^{\mu\nu}+\frac{P^{\mu}P^{\nu}}{P^{2}}$ and
$\epsilon(L_{z})$ is the orbital polarization vector of $P$-wave states.
Subsequently, by multiplying the complex conjugate of the amplitude, the
amplitude squared up to the next order can be obtained:
$\displaystyle\sum|\mathcal{M}({}^{3}S_{1}^{[1,8]})|^{2}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{t}^{\rho}(0)\mathcal{M}_{t}^{{\lambda}*}(0)\sum_{s_{z}}\epsilon_{\rho}\epsilon^{*}_{\lambda}$
(24) $\displaystyle+$
$\displaystyle\frac{1}{3}|\vec{q}|^{2}\left[\left(\Pi^{\alpha\beta}\frac{\partial^{2}(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}_{t}^{\rho})}{\partial
q^{\alpha}\partial
q^{\beta}}\right)_{q=0}\mathcal{M}_{t}^{*{\lambda}}(0)\right](\sum_{s_{z}}\epsilon_{\rho}\epsilon^{*}_{\lambda})_{q=0}+\mathcal{O}(v^{4}).$
$\displaystyle\sum|\mathcal{M}({}^{1}S_{0}^{[8]})|^{2}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{s}(0)\mathcal{M}_{s}^{*}(0)+\frac{1}{3}|\vec{q}|^{2}\left[\left(\Pi^{\alpha\beta}\frac{\partial^{2}(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}_{s})}{\partial
q^{\alpha}\partial
q^{\beta}}\right)_{q=0}\mathcal{M}_{s}^{*}(0)\right]+\mathcal{O}(v^{4}).$ (25)
$\displaystyle\sum|\mathcal{M}({}^{3}P_{J}^{[8]})|^{2}$ $\displaystyle=$
$\displaystyle|\vec{q}|^{2}\frac{\partial\mathcal{M}_{t}^{\rho}}{\partial{q^{\alpha}}}\Big{|}_{q=0}\frac{\partial\mathcal{M}_{t}^{*{\lambda}}}{\partial{q^{\beta}}}\Big{|}_{q=0}\sum_{L_{z}}\epsilon_{\alpha}\epsilon^{*}_{\beta}\sum_{s_{z}}\epsilon_{\rho}\epsilon^{*}_{\lambda}$
(26) $\displaystyle+$
$\displaystyle\frac{1}{15}|\vec{q}|^{4}\bigg{[}\left(\Pi^{\sigma\tau}(\frac{\partial^{3}}{\partial{q^{\alpha}}\partial{q^{\sigma}}\partial{q^{\tau}}}+\frac{\partial^{3}}{\partial{q^{\sigma}}\partial{q^{\alpha}}\partial{q^{\tau}}}+\frac{\partial^{3}}{\partial{q^{\tau}}\partial{q^{\sigma}}\partial{q^{\alpha}}})(\sqrt{\frac{m_{c}}{E_{q}}}\mathcal{M}_{t}^{\rho})\right)\times$
$\displaystyle\frac{\partial\mathcal{M}_{t}^{*{\lambda}}}{\partial{q^{\beta}}}(\sum_{L_{z}}\epsilon_{\alpha}\epsilon^{*}_{\beta})(\sum_{s_{z}}\epsilon_{\rho}\epsilon^{*}_{\lambda})\bigg{]}_{q=0}+\mathcal{O}(v^{6}).$
Any term, which is in the order of $|\vec{q}|^{2}$, must not be missed to
obtain the correction up to the order of $v^{2}$. In the three expressions
above, the first term on the right side of each equation can be expressed in
terms of kinematics variables $s,t(|\vec{q}|),u(|\vec{q}|)$. Here
$t(|\vec{q}|),u(|\vec{q}|)$ is $|\vec{q}|$ dependence and should be expanded
by Eq.(II.1). The sum of terms in the order of $|\vec{q}|^{2}$ in the first
term as well as all the second term is the contribution of the next leading
order. Orbit polarization sum $\sum_{L_{z}}$ and spin-triplet polarization sum
$\sum_{s_{z}}$ are equal to $\Pi^{\rho\lambda}(\Pi^{\alpha\beta})$. According
to the expression of $\Pi$ mentioned above, the $q$ dependence of $\Pi$ only
appears in the denominator $P^{2}$ which equals to $4E_{q}^{2}$ and only
contains even powers of four momentum $q$. So in the computation of
unpolarized cross section to next order of $v^{2}$ as in Eqs.(24,26),
expanding the polarization vector in order of $v^{2}$ is to handle the sum
expression $\Pi$.
Therefore, the differential cross section in Eq.(7) takes the following form:
$\displaystyle d\hat{\sigma}(a+b\rightarrow J/\psi+d)$ $\displaystyle=$
$\displaystyle\Bigg{(}\frac{F({}^{3}S_{1}^{[1]})}{m_{c}^{2}}\langle
0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle+\frac{G({}^{3}S_{1}^{[1]})}{m_{c}^{4}}\langle
0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle+$ (27)
$\displaystyle\frac{F({}^{1}S_{0}^{[8]})}{m_{c}^{2}}\langle
0|\mathcal{O}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle+\frac{G({}^{1}S_{0}^{[8]})}{m_{c}^{4}}\langle
0|\mathcal{P}^{J/\psi}({}^{1}S_{0}^{[8]})|0\rangle+$
$\displaystyle\frac{F({}^{3}S_{1}^{[8]})}{m_{c}^{2}}\langle
0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle+\frac{G({}^{3}S_{1}^{[8]})}{m_{c}^{4}}\langle
0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[8]})|0\rangle+$
$\displaystyle\frac{F({}^{3}P_{0}^{[8]})}{m_{c}^{2}}\langle
0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0\rangle+\frac{G({}^{3}P_{0}^{[8]})}{m_{c}^{4}}\langle
0|\mathcal{P}^{J/\psi}({}^{3}P_{0}^{[8]})|0\rangle\Bigg{)}\times$
$\displaystyle\Big{(}1+\mathcal{O}(v^{4})\Big{)}.$
The explicit expressions of the short distance coefficients to the
relativistic correction of CO states ${{}^{1}}S_{0}^{[8]}$ and
${{}^{3}}S_{1}^{[8]}$ , ${{}^{3}}P_{J}^{[8]}$ for partonic processes
$gg{\rightarrow}{J/\psi}g$, $gq(\bar{q}){\rightarrow}{J/\psi}q(\bar{q})$ and
$q\bar{q}{\rightarrow}{J/\psi}g$ are relegated to the Appendix. The result of
our relativistic correction of ${}^{3}S_{1}^{[1]}$ is consistent with that of
Ref.Fan:2009zq and was not given in this paper.
## III numerical result and discussion
We adopt the gluon distribution function CTEQ6 PDFsPumplin:2002vw . And the
charm quark is set as $m_{c}=1.5~{}GeV$. The ratios of the short distance
coefficient between LO $F$ and its relativistic correction $G$ at the Tevatron
with $\sqrt{s}=1.96~{}TeV$ and at the LHC with $\sqrt{s}=7~{}TeV$ or
$\sqrt{s}=14~{}TeV$ are presented in Fig.1. The ratios of $R[n]=G[n]/F[n]$ at
the Tevatron and at the LHC are very close at large $p_{T}$. In the large
$p_{T}$ limit,
$\displaystyle-\frac{M^{2}}{u}\sim-\frac{M^{2}}{t}<\frac{M^{2}}{p_{T}^{2}}\sim
0,$ (28)
where $M$ is the $J/\psi$ mass. Then we can expand the short distance
coefficients with $M$. The ratios of first order in the expansion are
$\displaystyle R({}^{3}S_{1}^{[1]})\Big{|}_{p_{T}\gg
M}=\frac{G({}^{3}S_{1}^{[1]})}{F({}^{3}S_{1}^{[1]})}\Big{|}_{p_{T}\gg M}$
$\displaystyle\sim$ $\displaystyle\frac{1}{6}$ $\displaystyle
R({}^{1}S_{0}^{[8]})\Big{|}_{p_{T}\gg
M}=\frac{G({}^{1}S_{0}^{[8]})}{F({}^{1}S_{0}^{[8]})}\Big{|}_{p_{T}\gg M}$
$\displaystyle\sim$ $\displaystyle-\frac{5}{6}$ $\displaystyle
R({}^{3}S_{1}^{[8]})\Big{|}_{p_{T}\gg
M}=\frac{G({}^{3}S_{1}^{[8]})}{F({}^{3}S_{1}^{[8]})}\Big{|}_{p_{T}\gg M}$
$\displaystyle\sim$ $\displaystyle-\frac{11}{6}$ $\displaystyle
R({}^{3}P_{0}^{[8]})\Big{|}_{p_{T}\gg
M}=\frac{G({}^{3}P_{0}^{[8]})}{F({}^{3}P_{0}^{[8]})}\Big{|}_{p_{T}\gg M}$
$\displaystyle\sim$ $\displaystyle-\frac{31}{30}$ (29)
These asymptotic behaviors of the ratios to each state are same for all the
partonic subprocesses of $gg$, $gq(\bar{q})$ and $qq$. It is consistent with
the curves in Fig.1. The ratio $R({}^{3}S_{1}^{[1]})$ is consistent with
Ref.Fan:2009zq , and the ratio $R({}^{3}S_{1}^{[8]})$ is consistent with
Ref.Bodwin:2003wh .
Figure 1: The ratios of the short distance coefficient between LO $F$ and its
relativistic correction $G$ at the Tevatron with $\sqrt{s}=1.96~{}TeV$ and at
the LHC with $\sqrt{s}=7~{}TeV$ or $\sqrt{s}=14~{}TeV$.
As discussed in Sec. II, the LDMEs of relativistic correction are depressed by
approximately 0.23 to LO. If we fix LDMEs $\langle 0|\mathcal{O}|0\rangle$ and
estimate $\langle 0|\mathcal{P}|0\rangle$ through the velocity scaling rule
with adopting $v^{2}=0.23$, then the LO cross sections of CO subprocesses are
reduced by about a factor of $20\sim~{}40\%$ at large $p_{T}$ at both Tevatron
and LHC. In the CS case, the LO cross sections are enhanced by approximately
$4\%$ by the NLO relativistic corrections. 111In Ref.Fan:2009zq , the ratio of
the CS cross sections enhanced by NLO relativistic corrections is about $1\%$
. The difference comes from adopting the different LDMEs: $\langle
0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle=1.64~{}GeV^{3},~{}~{}~{}~{}~{}~{}~{}\langle
0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle=0.320~{}GeV^{5}.$ (30) Then
$\langle 0|\mathcal{P}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle/\langle
0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[1]})|0\rangle/m_{c}^{2}=0.087,$ (31)
which is much smaller than $v^{2}\approx 0.23$.
The QCD corrections of both CO and CS states had been calculated in Ma:2010jj
; Chao:2012iv ; Butenschoen:2011yh . Ratios of NLO $\mathcal{O}(v^{2})$,
$\mathcal{O}(\alpha_{s})$, and $\mathcal{O}(\alpha_{s},v^{2})$ to LO cross
sections of $J/\psi$ production at Tevatron are presented in Fig.2. Here
$v^{2}=0.23$, and QCD corrections are taken from Refs.Ma:2010jj ; Chao:2012iv
; Butenschoen:2011yh . The $K$ factor of NLO QCD corrections is very large for
${}^{3}P_{0}^{[8]}$ and ${}^{3}S_{1}^{[1]}$ at large $p_{T}$, and it is about
$1.3$ for ${}^{3}S_{1}^{[8]}$ and $1.5$ for ${}^{1}S_{0}^{[8]}$.
Figure 2: Ratios of NLO $\mathcal{O}(v^{2})$, $\mathcal{O}(\alpha_{s})$, and
$\mathcal{O}(\alpha_{s},v^{2})$ to LO cross sections of $J/\psi$ production at
Tevatron. Here $v^{2}=0.23$, and QCD corrections are taken form Refs.Ma:2010jj
; Chao:2012iv .
The ratio of ${}^{3}S_{1}^{[8]}$ is approximately $-11/6$. In the large
$p_{T}$ limit, the dominate contribution of this subprocess is $g^{*}\to
c\bar{c}({}^{3}S_{1}^{[8]})$. The propagator of virtual gluon $g^{*}$ is
proportional to $1/E_{q}^{2}$. This term offers a factor of $-2$ to the ratio
$R({}^{3}S_{1}^{[8]})$. And the factor of $-2$ at large $p_{T}$ is same for
the polarization of ${}^{3}S_{1}^{[8]}$ states. At the same time, the
${}^{1}S_{0}^{[8]}$ state is a scalar state and contributes to unpolarized
production of $J/\psi$, and the $K$ factor of NLO QCD corrections is much
larger than relativistic corrections for ${}^{3}P_{0}^{[8]}$ and
${}^{3}S_{1}^{[1]}$ at large $p_{T}$. So the $J/\psi$ polarization at large
$p_{T}$ is insensitive to the relativistic corrections.
If we fit the differential cross section of prompt $J/\psi$ production at
$p_{t}>7~{}GeV$ at the Tevatron arXiv:0704.0638 to NLO in $\alpha_{s}$ and
$v^{2}$Ma:2010jj , we can get CO LDMEs but with large errors for
${}^{3}S_{1}^{[8]}$ and ${}^{3}P_{J}^{[8]}$ states. In Ref.Ma:2010jj , they
considered two combined LDMEs to fit the data:
$\displaystyle
M_{0,r_{0}}^{J/\psi}=<0|\mathcal{O}^{J/\psi}({}^{1}S_{0}^{[8]})|0>+\frac{r_{0}}{m_{c}^{2}}<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>,$
$\displaystyle
M_{1,r_{1}}^{J/\psi}=<0|\mathcal{O}^{J/\psi}({}^{3}S_{1}^{[8]})|0>+\frac{r_{1}}{m_{c}^{2}}<0|\mathcal{O}^{J/\psi}({}^{3}P_{0}^{[8]})|0>.$
(32)
Here $r_{0},r_{1}$ determined from the short distance coefficient
decomposition holding within a small error
$\displaystyle
d\hat{\sigma}[^{3}P_{J}^{[8]}]=r_{0}d\hat{\sigma}[^{1}S_{0}^{[8]}]+r_{1}\hat{\sigma}[^{3}S_{1}^{[8]}].$
(33)
In Ref.Ma:2010jj , they found $r_{0}=3.9$ and $r_{1}=-0.56$ using the
NLO$(\alpha_{s})$ results. When considering relativistic corrections as well
as NLO$(\alpha_{s})$ data we find $r_{0}=3.64$ and $r_{1}=-0.84$. Then we can
fit CDF $J/\psi$ prompt production data to determine these two LDMEs as Fig. 3
showns. (Here, we do not consider the effect of the feed-down cross section
form $\chi_{cJ}$ and ${\psi}\prime$ to the fit):
$\displaystyle M_{0,3.64}^{J/\psi}=(11.0\pm 0.3)\times 10^{-2}GeV^{3},$
$\displaystyle M_{1,-0.84}^{J/\psi}=(0.16\pm 0.02)\times 10^{-2}GeV^{3},$ (34)
comparing with fitting results only considering NLO$(\alpha_{s})$ data
$\displaystyle M_{0,3.9}^{J/\psi}=(9.0\pm 0.3)\times 10^{-2}GeV^{3},$
$\displaystyle M_{1,-0.56}^{J/\psi}=(0.13\pm 0.02)\times 10^{-2}GeV^{3}.$ (35)
About $20\%$ difference is shown for either LDMEs between the two sets.
Figure 3: Transverse momentum distribution of prompt $J/\psi$ production at
Tevatron. By fitting the CDF experimental data we obtained the two sets of
combined LDMEs $M_{0,r0}^{J/\psi}$ and $M_{1,r1}^{J/\psi}$ using the results
of NLO$(\alpha_{s})$ and NLO$(\alpha_{s},v^{2})$ short distance coefficients,
respectively.
Complete NLO$(\alpha_{s})$ calculations show the LDMEs fitting the Tevatron
data agree with all the LHC data. However, it does not agree well at the small
$p_{T}$ region Ma:2010jj . The $K$ factor curves in Fig. 2 imply that
relativistic corrections suppress the trend of the $K$ factors of
NLO$(\alpha_{s})$ mainly at small $p_{T}$ region. To investigate the effect of
new fitting LDMEs to the total cross section at hadron colliders, especially
at small $p_{T}$ region, we compare the cross sections of NLO$(\alpha_{s})$
and NLO$(\alpha_{s},v^{2})$ at the LHC using the corresponding set of LDMEs
above, and the results are shown in Fig.4. NLO$(\alpha_{s},v^{2})$ results
suppressed by about $50\sim 20\%$ along with $p_{T}$ increasing comparing with
NLO$(\alpha_{s})$ results. But the calculations of relativistic correction of
direct production fail to explain the tend of experimental data at the small
$p_{T}$ region, and it is still an open problem. It is expected to solve the
problem by two ways. First, contribution from the feed-down of high excited
charmonia production process as $p+p(\bar{p}){\rightarrow}{\chi_{cJ}}+X$ and
$p+p(\bar{p}){\rightarrow}{\psi\prime}+X$ may account for $30\%$ to prompt
$J/\psi$ production. In this case, the calculations of relativistic correction
to feed-down parts are necessary. Second, recently, the calculation method of
resummation of relativistic correction had been presented by Bodwin, Lee and
Yu and applied to calculate the resummation of relativistic correction to
exclusive production $e^{+}e^{-}\to J/\psi\eta_{c}$ at $e^{+}e^{-}$ colliders
that payed an important contribution to total cross sectionBodwin:2007ga .
Wether contributions of resummation of relativistic correction may play an
important role, further calculations are needed.
Figure 4: Transverse momentum distribution of NLO$(\alpha_{s})$ and
NLO$(\alpha_{s},v^{2})$ to $J/\psi$ direct production. The LHC experimental
data can be found in Refs.Aaij:2011jh ; Aad:2011sp .
## IV SUMMARY
In summary, we calculate the relativistic correction terms to CO states for
$J/\psi$ production at the Tevatron and at the LHC. The short distance
coefficient ratios of relativistic correction to LO for CO states
${{}^{1}}S_{0}^{[8]}$, ${{}^{3}}S_{1}^{[8]}$ and ${{}^{3}}P_{J}^{[8]}$ at
large $p_{T}$ are approximately -5/6, -11/6, and -31/30, respectively, and it
is 1/6 for the color singlet-state ${{}^{3}}S_{1}^{[1]}$. If NLO long distance
matrix elements are estimated through the velocity scaling rule with adopting
$v^{2}=0.23$, the cross sections are reduced by about a factor of $20\sim
40\%$ at large $p_{T}$ to LO results of CO states at both the Tevatron and the
LHC. Compared with the relativistic corrections to the CS state, that LO cross
sections are enhanced by a factor of $4\%$. Thus the result may affect the
production of $J/\psi$ at hadronic colliders. Beacuse of the large results of
QCD corrections at large $p_{T}$ especially to ${}^{3}P_{J}^{[8]}$ states,
relativistic corrections are small, even ignored, along with $p_{T}$
increasing. But relativistic corrections can also affect the total cross
section with a considerable contribution. We computed the unpolarized cross
sections at the LHC with CO LDMEs extracted from the fit to $J/\psi$ direct
production at the Tevatron, and the results of NLO$(\alpha_{s},v^{2})$
suppress that of NLO$(\alpha_{s})$ by about $20\sim 50\%$ at different $p_{T}$
regions. These results indicate that relativistic corrections may play an
important role in $J/\psi$ production at the Tevatron and LHC.
###### Acknowledgements.
The authors would like to thank Professor K.T. Chao, Z.G. He, Y.Q. Ma, H.S.
Shao, and K. Wang for useful discussion and the data of NLO QCD corrections.
Y.J. Zhang also thanks J.P. Lansberg for the discussion of polarization and
B.Q. Li for the discussion of $v^{2}$. This work was supported by the National
Natural Science Foundation of China (Grants No.10805002, No.10875055, and
No.11075011), the Foundation for the Author of National Excellent Doctoral
Dissertation of China (Grants No. 2007B18 and No. 201020), the Project of
Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, Grant No.
KJCX2.YW.W10, and the Education Ministry of Liaoning Province.
## V Appendix: Short Distance Coefficients
The short distance coefficients of ${}^{1}S_{0}^{[8]}$ for
$gg{\rightarrow}{J/\psi}g$ subprocess were
$\displaystyle\frac{F_{gg}({}^{1}S_{0}^{[8]})}{m_{c}^{2}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\times$
$\displaystyle
640\Bigg{[}M^{12}\left(t^{2}+tu+u^{2}\right)-M^{10}\left(4t^{3}+7t^{2}u+7tu^{2}+4u^{3}\right)$
$\displaystyle+M^{8}\left(8t^{4}+21t^{3}u+27t^{2}u^{2}+21tu^{3}+8u^{4}\right)-M^{6}\left(10t^{5}+35t^{4}u+57t^{3}u^{2}+57t^{2}u^{3}+35tu^{4}+10u^{5}\right)$
$\displaystyle+M^{4}\left(8t^{6}+33t^{5}u+66t^{4}u^{2}+81t^{3}u^{3}+66t^{2}u^{4}+33tu^{5}+8u^{6}\right)$
$\displaystyle-M^{2}\left(t^{2}+tu+u^{2}\right)^{2}\left(4t^{3}+9t^{2}u+9tu^{2}+4u^{3}\right)+\left(t^{2}+tu+u^{2}\right)^{4}\Bigg{]}$
$\displaystyle\Bigg{/}\Bigg{[}M\left(M^{2}-t\right)^{2}t\left(M^{2}-u\right)^{2}\left(M^{2}-t-u\right)u(t+u)^{2}\Bigg{]},$
(36)
$\displaystyle\frac{G_{gg}({}^{1}S_{0}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}$
$\displaystyle
1280\Bigg{[}5tu(t+u)\left(t^{2}+tu+u^{2}\right)^{4}+12M^{18}\left(t^{2}+tu+u^{2}\right)-5M^{16}\left(11t^{3}+20t^{2}u+20tu^{2}+11u^{3}\right)$
$\displaystyle+M^{14}\left(95t^{4}+280t^{3}u+358t^{2}u^{2}+280tu^{3}+95u^{4}\right)$
$\displaystyle-3M^{12}\left(16t^{5}+95t^{4}u+175t^{3}u^{2}+175t^{2}u^{3}+95tu^{4}+16u^{5}\right)$
$\displaystyle-2M^{10}\left(45t^{6}+72t^{5}u+21t^{4}u^{2}-22t^{3}u^{3}+21t^{2}u^{4}+72tu^{5}+45u^{6}\right)$
$\displaystyle+M^{8}\left(198t^{7}+678t^{6}u+1141t^{5}u^{2}+1345t^{4}u^{3}+1345t^{3}u^{4}+1141t^{2}u^{5}+678tu^{6}+198u^{7}\right)$
$\displaystyle-M^{6}\left(180t^{8}+756t^{7}u+1583t^{6}u^{2}+2224t^{5}u^{3}+2446t^{4}u^{4}+2224t^{3}u^{5}+1583t^{2}u^{6}+756tu^{7}+180u^{8}\right)$
$\displaystyle+M^{4}(85t^{9}+408t^{8}u+1000t^{7}u^{2}+1637t^{6}u^{3}+2028t^{5}u^{4}+2028t^{4}u^{5}+1637t^{3}u^{6}+1000t^{2}u^{7}$
$\displaystyle+408tu^{8}+85u^{9})-M^{2}\left(t^{3}+2t^{2}u+2tu^{2}+u^{3}\right)^{2}\left(17t^{4}+30t^{3}u+30t^{2}u^{2}+30tu^{3}+17u^{4}\right)\Bigg{]}\Bigg{/}$
$\displaystyle\Bigg{[}3M^{3}\left(M^{2}-t\right)^{3}t\left(M^{2}-u\right)^{3}u(t+u)^{3}\left(-M^{2}+t+u\right)\Bigg{]}.$
(37)
The short distance coefficients of ${}^{3}S_{1}^{[8]}$ for
$gg{\rightarrow}{J/\psi}g$ subprocess were
$\displaystyle\frac{F_{gg}({}^{3}S_{1}^{[8]})}{m_{c}^{2}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}$
$\displaystyle
256\Bigg{[}27\left(t^{2}+tu+u^{2}\right)^{3}+19M^{8}\left(t^{2}+tu+u^{2}\right)-M^{6}\left(65t^{3}+111t^{2}u+111tu^{2}+65u^{3}\right)$
$\displaystyle+M^{4}\left(100t^{4}+227t^{3}u+300t^{2}u^{2}+227tu^{3}+100u^{4}\right)$
$\displaystyle-27M^{2}\left(3t^{5}+8t^{4}u+13t^{3}u^{2}+13t^{2}u^{3}+8tu^{4}+3u^{5}\right)\Bigg{]}$
$\displaystyle\Bigg{/}\Bigg{[}3M^{3}\left(M^{2}-t\right)^{2}\left(M^{2}-u\right)^{2}(t+u)^{2}\Bigg{]},$
(38) $\displaystyle\frac{G_{gg}({}^{3}S_{1}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}$
$\displaystyle(-512)\Bigg{[}M^{14}\left(87t^{2}+22tu+87u^{2}\right)+M^{12}\left(-14t^{3}+335t^{2}u+335tu^{2}-14u^{3}\right)$
$\displaystyle-2M^{10}\left(399t^{4}+1612t^{3}u+2020t^{2}u^{2}+1612tu^{3}+399u^{4}\right)$
$\displaystyle+M^{8}\left(2100t^{5}+8976t^{4}u+14497t^{3}u^{2}+14497t^{2}u^{3}+8976tu^{4}+2100u^{5}\right)$
$\displaystyle-M^{6}\left(2590t^{6}+12096t^{5}u+23855t^{4}u^{2}+29314t^{3}u^{3}+23855t^{2}u^{4}+12096tu^{5}+2590u^{6}\right)$
$\displaystyle+M^{4}\left(1620t^{7}+8498t^{6}u+19905t^{5}u^{2}+29152t^{4}u^{3}+29152t^{3}u^{4}+19905t^{2}u^{5}+8498tu^{6}+1620u^{7}\right)$
$\displaystyle-27M^{2}\left(15t^{8}+104t^{7}u+295t^{6}u^{2}+510t^{5}u^{3}+612t^{4}u^{4}+510t^{3}u^{5}+295t^{2}u^{6}+104tu^{7}+15u^{8}\right)$
$\displaystyle+297tu(t+u)\left(t^{2}+tu+u^{2}\right)^{3}\Bigg{]}\Bigg{/}\Bigg{[}9M^{5}\left(M^{2}-t\right)^{3}\left(M^{2}-u\right)^{3}(t+u)^{3}\Bigg{]}.$
(39)
The short distance coefficients of ${}^{3}P_{J}^{[8]}$ for
$gg{\rightarrow}{J/\psi}g$ subprocess were
$\displaystyle\frac{F_{gg}({}^{3}P_{J}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\times$
$\displaystyle
2560\bigg{[}7M^{16}\left(t^{3}+2t^{2}u+2tu^{2}+u^{3}\right)-M^{14}\left(35t^{4}+99t^{3}u+120t^{2}u^{2}+99tu^{3}+35u^{4}\right)$
$\displaystyle+M^{12}\left(84t^{5}+296t^{4}u+450t^{3}u^{2}+450t^{2}u^{3}+296tu^{4}+84u^{5}\right)$
$\displaystyle-3M^{10}\left(42t^{6}+171t^{5}u+304t^{4}u^{2}+362t^{3}u^{3}+304t^{2}u^{4}+171tu^{5}+42u^{6}\right)$
$\displaystyle+M^{8}\left(126t^{7}+577t^{6}u+1128t^{5}u^{2}+1513t^{4}u^{3}+1513t^{3}u^{4}+1128t^{2}u^{5}+577tu^{6}+126u^{7}\right)$
$\displaystyle-M^{6}\left(84t^{8}+432t^{7}u+905t^{6}u^{2}+1287t^{5}u^{3}+1436t^{4}u^{4}+1287t^{3}u^{5}+905t^{2}u^{6}+432tu^{7}+84u^{8}\right)$
$\displaystyle+M^{4}\left(35t^{9}+204t^{8}u+468t^{7}u^{2}+700t^{6}u^{3}+819t^{5}u^{4}+819t^{4}u^{5}+700t^{3}u^{6}+468t^{2}u^{7}+204tu^{8}+35u^{9}\right)$
$\displaystyle-M^{2}\left(t^{2}+tu+u^{2}\right)^{2}\left(7t^{6}+36t^{5}u+45t^{4}u^{2}+28t^{3}u^{3}+45t^{2}u^{4}+36tu^{5}+7u^{6}\right)$
$\displaystyle+3tu(t+u)\left(t^{2}+tu+u^{2}\right)^{4}\bigg{]}\bigg{/}\bigg{[}{M^{3}tu\left(M^{2}-t\right)^{3}\left(M^{2}-u\right)^{3}(t+u)^{3}\left(-M^{2}+t+u\right)}\bigg{]}.$
(40)
$\displaystyle\frac{G_{gg}({}^{3}P_{J}^{[8]})}{m_{c}^{6}}=\frac{1}{16\pi
s^{2}}\frac{1}{64}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\times$
$\displaystyle(-1024)\bigg{[}140M^{22}\left(t^{3}+2t^{2}u+2tu^{2}+u^{3}\right)-M^{20}\left(725t^{4}+2095t^{3}u+2596t^{2}u^{2}+2095tu^{3}+725u^{4}\right)$
$\displaystyle+6M^{18}\left(235t^{5}+978t^{4}u+1599t^{3}u^{2}+1599t^{2}u^{3}+978tu^{4}+235u^{5}\right)$
$\displaystyle-M^{16}\left(705t^{6}+6528t^{5}u+16050t^{4}u^{2}+20350t^{3}u^{3}+16050t^{2}u^{4}+6528tu^{5}+705u^{6}\right)$
$\displaystyle+M^{14}\left(-2190t^{7}-3022t^{6}u+5603t^{5}u^{2}+15689t^{4}u^{3}+15689t^{3}u^{4}+5603t^{2}u^{5}-3022tu^{6}-2190u^{7}\right)$
$\displaystyle+M^{12}(5400t^{8}+19278t^{7}u+25697t^{6}u^{2}+19598t^{5}u^{3}+14174t^{4}u^{4}$
$\displaystyle+19598t^{3}u^{5}+25697t^{2}u^{6}+19278tu^{7}+5400u^{8})$
$\displaystyle-M^{10}(6110t^{9}+28087t^{8}u+52760t^{7}u^{2}+62879t^{6}u^{3}+60308t^{5}u^{4}+60308t^{4}u^{5}$
$\displaystyle+62879t^{3}u^{6}+52760t^{2}u^{7}+28087tu^{8}+6110u^{9})$
$\displaystyle+M^{8}(4055t^{10}+22235t^{9}u+50834t^{8}u^{2}+74420t^{7}u^{3}+83867t^{6}u^{4}+84706t^{5}u^{5}$
$\displaystyle+83867t^{4}u^{6}+74420t^{3}u^{7}+50834t^{2}u^{8}+22235tu^{9}+4055u^{10})$
$\displaystyle-M^{6}(1530t^{11}+10029t^{10}u+27765t^{9}u^{2}+49691t^{8}u^{3}+67682t^{7}u^{4}+76683t^{6}u^{5}$
$\displaystyle+76683t^{5}u^{6}+67682t^{4}u^{7}+49691t^{3}u^{8}+27765t^{2}u^{9}+10029tu^{10}+1530u^{11})$
$\displaystyle+M^{4}(255t^{12}+2250t^{11}u+8158t^{10}u^{2}+18865t^{9}u^{3}+32387t^{8}u^{4}+43880t^{7}u^{5}+48446t^{6}u^{6}$
$\displaystyle+43880t^{5}u^{7}+32387t^{4}u^{8}+18865t^{3}u^{9}+8158t^{2}u^{10}+2250tu^{11}+255u^{12})$
$\displaystyle-M^{2}tu\left(t^{2}+tu+u^{2}\right)^{2}(150t^{7}+726t^{6}u+1575t^{5}u^{2}+2117t^{4}u^{3}+2117t^{3}u^{4}+1575t^{2}u^{5}+726tu^{6}$
$\displaystyle+150u^{7})+31t^{2}u^{2}(t+u)^{2}\left(t^{2}+tu+u^{2}\right)^{4}\bigg{]}$
$\displaystyle\bigg{/}\bigg{[}{M^{5}tu\left(M^{2}-t\right)^{4}\left(M^{2}-u\right)^{4}(t+u)^{4}\left(-M^{2}+t+u\right)}\bigg{]}.$
(41)
The short distance coefficients of ${}^{1}S_{0}^{[8]}$ for
$q\bar{q}{\rightarrow}{J/\psi}g$ subprocess were
$\frac{F_{q\bar{q}}({}^{1}S_{0}^{[8]})}{m_{c}^{2}}=-\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{160\left(t^{2}+u^{2}\right)}{3M(t+u)^{2}\left(-M^{2}+t+u\right)}.$
(42)
$\displaystyle\frac{G_{q\bar{q}}({}^{1}S_{0}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1600\left(t^{2}+u^{2}\right)}{9M^{3}(t+u)^{2}\left(-M^{2}+t+u\right)}.$
(43)
The short distance coefficients of ${}^{3}S_{1}^{[8]}$ for
$q\bar{q}{\rightarrow}{J/\psi}g$ subprocess were
$\displaystyle\frac{F_{q\bar{q}}({}^{3}S_{1}^{[8]})}{m_{c}^{2}}=-\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}\frac{64\left(4t^{2}-tu+4u^{2}\right)\left(2M^{4}-2M^{2}(t+u)+t^{2}+u^{2}\right)}{3M^{3}tu(t+u)^{2}}.$
(44)
$\displaystyle\frac{G_{q\bar{q}}({}^{3}S_{1}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}$
$\displaystyle(-128)\bigg{[}24M^{6}\left(4t^{2}-tu+4u^{2}\right)-14M^{4}\left(4t^{3}+3t^{2}u+3tu^{2}+4u^{3}\right)$
$\displaystyle-8M^{2}\left(5t^{4}+11t^{3}u+3t^{2}u^{2}+11tu^{3}+5u^{4}\right)+11\left(4t^{5}+3t^{4}u+7t^{3}u^{2}+7t^{2}u^{3}+3tu^{4}+4u^{5}\right)\bigg{]}$
$\displaystyle\bigg{/}\bigg{[}{9M^{5}tu(t+u)^{3}}\bigg{]}.$ (45)
The short distance coefficients of ${}^{3}P_{J}^{[8]}$ for
$q\bar{q}{\rightarrow}{J/\psi}g$ subprocess were:
$\displaystyle\frac{F_{q\bar{q}}({}^{3}P_{J}^{[8]})}{m_{c}^{4}}=-\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{640\left(8M^{4}(t+u)-4M^{2}\left(t^{2}+4tu+u^{2}\right)+3\left(t^{3}+t^{2}u+tu^{2}+u^{3}\right)\right)}{3M^{3}(t+u)^{3}\left(-M^{2}+t+u\right)}.$
(46)
$\displaystyle\frac{G_{q\bar{q}}({}^{3}P_{J}^{[8]})}{m_{c}^{6}}=\frac{1}{16\pi
s^{2}}\frac{1}{9}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}$
$\displaystyle
256\bigg{[}160M^{6}(t+u)-16M^{4}\left(5t^{2}+17tu+5u^{2}\right)+4M^{2}\left(t^{3}-11t^{2}u-11tu^{2}+u^{3}\right)$
$\displaystyle+31(t+u)^{2}\left(t^{2}+u^{2}\right)\bigg{]}\bigg{/}\bigg{[}{3M^{5}(t+u)^{4}\left(-M^{2}+t+u\right)}\bigg{]}.$
(47)
The short distance coefficients of ${}^{1}S_{0}^{[8]}$ for
$gq(\bar{q}){\rightarrow}{J/\psi}q(\bar{q})$ subprocess were
$\frac{F_{gq(\bar{q})}({}^{1}S_{0}^{[8]})}{m_{c}^{2}}=-\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{160\left(s^{2}+u^{2}\right)}{3M(s+u)^{2}\left(-M^{2}+s+u\right)}.$
(48)
$\displaystyle\frac{G_{gq(\bar{q})}({}^{1}S_{0}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{320\left(M^{2}\left(11s^{3}+23s^{2}u-su^{2}+11u^{3}\right)-5s\left(s^{3}+s^{2}u+su^{2}+u^{3}\right)\right)}{9M^{3}\left(M^{2}-s\right)(s+u)^{3}\left(M^{2}-s-u\right)}.$
(49)
The short distance coefficients of ${}^{3}S_{1}^{[8]}$ for
$gq(\bar{q}){\rightarrow}{J/\psi}q(\bar{q})$ subprocess were
$\displaystyle\frac{F_{gq(\bar{q})}({}^{3}S_{1}^{[8]})}{m_{c}^{2}}=-\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}\frac{64\left(4s^{2}-su+4u^{2}\right)\left(2M^{4}-2M^{2}(s+u)+s^{2}+u^{2}\right)}{3M^{3}su(s+u)^{2}}.$
(50)
$\displaystyle\frac{G_{gq(\bar{q})}({}^{3}S_{1}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{1}{3}$
$\displaystyle
128\bigg{[}2M^{6}\left(20s^{3}+69s^{2}u-39su^{2}+20u^{3}\right)-2M^{4}\left(40s^{4}+113s^{3}u+27s^{2}u^{2}+10su^{3}+20u^{4}\right)$
$\displaystyle+M^{2}\left(108s^{5}+193s^{4}u+41s^{3}u^{2}+225s^{2}u^{3}+su^{4}+20u^{5}\right)$
$\displaystyle-11s\left(4s^{5}+3s^{4}u+7s^{3}u^{2}+7s^{2}u^{3}+3su^{4}+4u^{5}\right)\bigg{]}\bigg{/}\bigg{[}{9M^{5}su\left(M^{2}-s\right)(s+u)^{3}}\bigg{]}.$
(51)
The short distance coefficients of ${}^{3}S_{J}^{[8]}$ for
$gq(\bar{q}){\rightarrow}{J/\psi}q(\bar{q})$ subprocess were
$\frac{F_{gq(\bar{q})}({}^{3}P_{J}^{[8]})}{m_{c}^{4}}=\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}\frac{640\left(8M^{4}(s+u)-4M^{2}\left(s^{2}+4su+u^{2}\right)+3\left(s^{3}+s^{2}u+su^{2}+u^{3}\right)\right)}{3M^{3}(s+u)^{3}\left(-M^{2}+s+u\right)}.$
(52)
$\displaystyle\frac{G_{gq(\bar{q})}({}^{3}P_{J}^{[8]})}{m_{c}^{6}}=-\frac{1}{16\pi
s^{2}}\frac{1}{24}\frac{1}{4}\frac{(4\pi\alpha_{s})^{3}}{N_{c}^{2}-1}$
$\displaystyle
256\bigg{[}8M^{6}\left(5s^{2}+26su+25u^{2}\right)+4M^{4}\left(s^{3}-23s^{2}u-111su^{2}-19u^{3}\right)$
$\displaystyle+M^{2}\left(57s^{4}+226s^{3}u+166s^{2}u^{2}+58su^{3}+61u^{4}\right)-31s(s+u)^{2}\left(s^{2}+u^{2}\right)\bigg{]}$
$\displaystyle\bigg{/}\bigg{[}{3M^{5}\left(M^{2}-s\right)(s+u)^{4}\left(-M^{2}+s+u\right)}\bigg{]}.$
(53)
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|
arxiv-papers
| 2012-03-01T15:12:04 |
2024-09-04T02:49:28.167669
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guang-Zhi Xu, Yi-Jie Li, Kui-Yong Liu, Yu-Jie Zhang",
"submitter": "Yu-Jie Zhang Dr.",
"url": "https://arxiv.org/abs/1203.0207"
}
|
1203.0449
|
# $K^{*0}$ and $\Sigma^{*}$ production in Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV and 62.4 GeV
Kai Zhang Department of Physics, Qufu Normal University, Shandong 273165,
People’s Republic of China Jun Song Department of Physics, Jining
University, Shandong 273155, People’s Republic of China Feng-lan Shao
shaofl@mail.sdu.edu.cn Department of Physics, Qufu Normal University, Shandong
273165, People’s Republic of China
###### Abstract
Applying a quark combination model for the hadronization of Quark Gluon Plasma
(QGP) and A Relativistic Transport (ART) model for the subsequent hadronic
rescattering process, we investigate the production of $K^{*0}$ and
$\Sigma^{*}$ resonances in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200
GeV and 62.4 GeV. The initial $K^{*0}$ produced via hadronization is higher
than the experimental data in the low $p_{T}$ region and is close to the data
at 2-3 GeV/c. We take into account the hadronic rescattering effects which
lead to a strong suppression of $K^{*0}$ with low $p_{T}$ , and find that the
$p_{T}$ spectrum of $K^{*0}$ can be well described. According to the
suppressed magnitude of $K^{*0}$ yield, the time span of hadronic rescattering
stage is estimated to be about 13 fm/c at 200 GeV and 5 fm/c at 62.4 GeV. The
$p_{T}$ spectrum of $\Sigma^{*}$ directly obtained by quark combination
hadronization in central Au+Au collisions at 200 GeV is in well agreement with
the experimental data, which shows a weak hadronic rescattering effects. The
elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au collisions at 200 GeV
and $p_{T}$ spectrum of $\Sigma^{*}$ at lower 62.4 GeV are predicted.
###### pacs:
25.75.Dw, 24.10.Lx, 25.75.Nq, 25.75.-q
## I Introduction
The short-lived resonances are efficient tools of probing the properties of
the hot and dense medium produced in relativistic heavy ion collisions. At
RHIC energies, QGP with extremely high energy density is created in primordial
collision stage stock08RHICrev , and the system undergoes a long time to
expand and cool Kolb0305084nuth . The lifetime of the resonance is about a few
fm/c, which is less than (or roughly the order of) the lifetime of the system
formed in heavy ion collisions. After QGP hadronization, but before the
interactions of hadrons cease, the initially produced resonances and stable
hadrons will undergo a hadronic rescattering stage. The resonance might be
destructed by rescattering with other hadrons and also be regenerated by the
collisions of other hadrons, and decay daughter particles of resonance are
kicked by other hadrons causing the signal loss. The physical properties of
resonances, e.g. their masses and widths, might be modified by the surrounding
medium Lutz02npa ; Shuryak03Medium . In addition, the yields and momentum
spectra of resonances might be changed. The experimentally reconstructed
resonances are the synthetic results of the hadronization and hadronic
rescattering effects.
RHIC and SPS experiments have provided rich data of production of resonances
such as $K^{*}$, $\Sigma^{*}$ in relativistic heavy ion collisions
KstarRHIC62_200GeV ; StrangeBresonance200Gev ; kstarAuAuVpp05y ; Kstar158GeV .
The relative yield ratios of resonances to stable hadrons are studied
experimentally, which also invokes many theoretical explanations Bleicher02 ;
vogel06 ; VogelAndBleicher05 . This progress greatly promote our understanding
of QGP hadronization mechanism tested against stable hadrons and the hadronic
rescattering dynamics, e.g. the time span of hadronic stage and cross sections
of various hadronic interaction channels Rafelski01 ; Marker02 .
The production mechanism of resonances at QGP hadronization is hard to
identify due to the entanglement of hadronization and rescattering effects.
RHIC data, e.g. the phenomena of v2 quark number scaling and high $p/\pi$
ratio etc, show that the production of various stable hadrons at QGP
hadronization is realized by the combination of constituent (anti-)quarks
Fries:2003prc ; Greco2003prc ; Hwa:2004prc . Recently, STAR experiments
observed that the v2 of $K^{*0}$, the same as stable particles, follows the
constituent quark number scaling rule in the intermediate $p_{T}$ region
KstarRHIC62_200GeV . This provides an evidence for the quark
(re-)combination/coalescence mechanism of resonance production at
hadronization in relativistic heavy ion collisions at RHIC. The K* meson and
$\Sigma^{*}$ baryon are of particular interests due to their very short
lifetime ( 4fm/c) and strange valence quark content. The experimental data of
their midrapidity yields and $p_{T}$ spectra are all available recently
KstarRHIC62_200GeV ; StrangeBresonance200Gev . A systematical study of this
pair of meson and baryon resonance should be able to further test the quark
combination mechanism of the hadron production in relativistic heavy ion
collisions.
In this paper, we apply a quark (re-)combination model for the hadronization
of hot and dense quark matter and ART model LbaoART95 ; LbaoART01 for the
hadronic rescattering process to study the $K^{*0}$ and $\Sigma^{*}$
production in central Au+Au collisions at $\sqrt{s_{NN}}=$200 GeV and 62.4
GeV. The investigation strategy is divided into two steps. Firstly, we compare
directly the hadronization results with experimental data to investigate the
proportion/magnitude of hadronization exhibited in the final observation.
Here, as a tool, we use the quark combination model developed by Shandong
group (SDQCM) QBXie1988PRD ; Shao2005prc to deal with QGP hadronization.
Secondly, we take into account effects of hadronic rescattering and study the
entanglement of hadronization and hadronic rescattering effects at RHIC 200
GeV and 62.4 GeV and time span of hadronic stage for the system produced at
high RHIC energies.
## II Initial $K^{*0}$ production via hadronization
The performance of quark (re-)combination mechanism on explaining the
production of various stable hadrons in the intermediate $p_{T}$ region in
relativistic heavy ion collisions is pretty well Fries:2003prc ; Greco2003prc
; Hwa:2004prc ; ClwKoCM_phi_Omega . The mechanism can also well describe
$p_{T}$-integrated yields and rapidity spectra of hadrons at RHIC and high SPS
energies Shao2005prc ; shao2007prc ; CEShao2009PRC ; JSong2009MPA . There are
several popular recombination models at RHIC. Quark recombination model
Fries:2003prc ; Hwa:2004prc and parton coalescence model Greco2003prc
inclusively describe the combination of quarks into hadrons. ALCOR alcor95
and SDQCM apply the exclusive description. The spirit of quark combination has
been extended to various transport, variation and statistic methods of hadron
production in relativistic heavy ion collisions RavaTS07 ; MHZpfDyCoal07 ;
Alaladyqcm08 ; Cassing09 ; Abir09 .
In this paper, we use SDQCM to treat the initial production of various
hadrons. Of all the “on market” combination models, SDQCM is unique for its
combination rule which guarantees that mesons and baryons exhaust the
probability of all the fates of the (anti)quarks in deconfined color-neutral
system at hadronization. The main idea of the combination rule is to line up
the (anti)quarks in a one-dimensional order in phase space, e.g., in rapidity,
and then let them combine into initial hadrons one by one according to this
order Shao2005prc . Three (anti)quarks or a quark-antiquark pair in the
neighborhood form a (anti)baryon or a meson, respectively. The exclusive
nature of the model make it convenient to predict the $K^{*0}$ and
$\Sigma^{*}$ production on the basis of the reproduction of the yields and
momentum spectra of various stable hadrons.
In the meson formation, the relative formation probability of the lowest lying
vector meson (V) to pseudo-scalar meson (P) with the same valance quark
composition is tuned by the parameter V/P ratio which can not be given from
first principles. Spin counting arguments would suggest a 3:1 mixture between
vector and pseudoscalar mesons. In Lund string fragmentation, the V/P ratio is
taken to be 1 for light mesons and 1.5 for strange mesons, which is based on
wave function overlap arguments And82a . The resulting $K^{*}/K$ yield ratio
is about 0.56 which is calculated using PYTHIA 6.4 with default settings
pythia6.4 . The data of $K^{*0}/K^{-}$ yield ratio in $pp$ collisions at RHIC
energies is about 0.35 kstarAuAuVpp05y . This will lead to a rough estimation
of the V/P ratio by the relationship V/(V+P) =0.35 after taking into account
$K^{*}$ decay, and the resulting V/P ratio is about 0.5. The SDQCM fit of the
value of $K^{*0}/K^{-}$ in $pp$ reactions gives V/P ratio of 0.45. The choice
of V/P ratio influence directly the predicted abundance of $K^{*0}$ at
hadronization, and thus influence the identification of the magnitude of
rescattering effects in hadronic stage in explaining the experimental data.
Here, we give the prediction of $K^{*0}$ at different V/P values to test its
influence quantitatively.
Figure 1: Panel (a): The $p_{T}$ spectra of stable hadrons at midrapidity in
central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. Panel (b): The $p_{T}$
spectrum of $K^{*0}$. Symbols are the experimental data abelev:152301 ;
Adams07hyperon ; Abelev07phiv2 ; KstarRHIC62_200GeV and lines are the results
of SDQCM. Panel (c): Yield ratio of $K^{*0}/K^{-}$ at midrapidity. Open
squares are results by hadronization for different V/P ratios. Experimental
data of $pp$ collisions and central Au+Au collisions are shown as dashed line
and band area KstarRHIC62_200GeV , respectively. Panel (d): Elliptic flow v2
of $K^{*0}$ as the function of $p_{T}$ in minimum bias Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Symbols are the experimental data KstarRHIC62_200GeV
and the solid line is the result of SDQCM. Dashed lines marked by $n=3$ and 4
are the guidance of the consequence of aggravating regeneration effects in
hadronic stage, which is from Ref. DongX04v2decay .
The left panel (a) in Fig.1 is the $p_{T}$ spectra of various stable hadrons
at midrapidity in central Au+Au at $\sqrt{s_{NN}}=$ 200 GeV. Symbols are the
experimental data and lines are results of SDQCM in Ref. CEShao2009PRC in
which the $p_{T}$ spectra of their anti-particles are also studied in detail.
V/P ratio is taken to be 3 in the calculation. The input of model is the
$p_{T}$ spectra of light and strange quarks just before hadronization, i.e.
$f_{q}(p_{T})$ and $f_{s}(p_{T})$, which are fixed by the data of $\pi^{-}$
and $K^{-}$. Clearly, the transverse momentum spectra of these stable hadrons
can be self-consistently explained by two quark $p_{T}$ spectra via
combination. This means that constituent quark degrees of freedom play a
dominated role in the production of these thermal hadrons.
The middle panel (b) in Fig.1 shows midrapidity $p_{T}$ spectrum of $K^{*0}$
produced by hadronization based on the left panel results and parameters
(quark spectra and V/P ratio). The V/P ratio is sensitive to the $K^{*0}$
yield and is relatively less sensitive to the slope of $K^{*0}$ $p_{T}$
spectrum. The $K^{*0}$ spectrum at other V/P values, e.g. 1 and 0.45, are all
presented. Here, we stress that even at V/P=1 and 0.45 the model can still
reproduce the $p_{T}$ and rapidity spectra of stable hadrons shown left. It is
because the V/P ratio does not alter the nature of hadron formation, and it
just changes the decay contributions of resonance to stable hadrons. We find
that the slope of $K^{*0}$ $p_{T}$ spectrum in 1.5-3.0 GeV/c is roughly
consistent with the data but in the low $p_{T}$ region the directly produced
$K^{*0}$ by combination is above the data even for V/P=0.45.
The right panel (c) in Fig.1 is the yield ratio $N(K^{*0})/N(K^{-})$ at
midrapidity for different V/P values. Because kaon and $K^{*}$ both contain
same valence quarks, their ratio roughly cancel the effect of strangeness
enhancement in relativistic heavy ion collisions and thus is sensitive to the
mechanism of hadron production. The data of $N(K^{*0})/N(K^{-})$ is $0.2\pm
0.03\pm 0.03$ in central Au+Au collisions and is $0.34\pm 0.01\pm 0.05$ in
minimum bias $pp$ reactions at $\sqrt{s_{NN}}=$ 200 GeV KstarRHIC62_200GeV ,
respectively. We find that the calculated ratio $N(K^{*0})/N(K^{-})$ at V/P =3
and 1 are higher than the data of Au+Au collisions which is shown as the band
area. The result of V/P=0.45 is consistent with the data of $pp$ reactions
shown as dashed line but is also higher than the data of Au+Au collisions. The
over-prediction of hadronization in the low $p_{T}$ region indicates the
necessity of hadronic rescattering for the suppression of $K^{*0}$ yield.
To further manifest the hadronization effect in final state observation, we
present in panel (d) in Fig.1 the elliptic flow v2 of $K^{*0}$ at midrapidity
in minimum bias Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. This result is
calculated using the extracted quark v2 in the previous study of v2 of various
stable hadrons in Ref. qcmv2 . V/P ratio does not influence the predicted v2
which is irrespective of particle abundance. One can see that the calculated
$K^{*0}$ v2, shown as solid line, is in well agreement with the experimental
data. The dashed lines are the parameterization of hadronic v2 as the function
of number of constituent quarks $n$ in Ref. DongX04v2decay . $n=3$ and 4 are
the guidance of the consequence of aggravating regeneration effects in
hadronic stage. The agreement of our result with the data means that the
$K^{*0}$ mesons observed experimentally mainly come from the hadronization.
## III Hadronic rescattering effects on $K^{*0}$ production
The subsequent hadronic rescattering stage after hadronization is simulated by
A Relativistic Transport (ART) model LbaoART95 ; LbaoART01 which includes
baryon-baryon, baryon-meson, and meson-meson elastic and inelastic
scatterings. We apply the code in AMPT event generator V2.25t3 LinAMPT05
which has provided proper extension to higher RHIC energies from original SPS
energies. The time span of hadronic stage is important for the hadronic
rescattering effects such as the suppression magnitude of $K^{*0}$ yield. In
transport theory, the rescattering time should be infinite in principle. In
practice, one would like to choose a finite rescattering time, and this
choice, to some extent, is similar to the decouple criteria of kinetic freeze-
out in hydrodynamic theory of heavy ion collisions. In this work, we treat the
rescattering time as an adjustable parameter and study how long the time span
of hadronic phase is favored by the experimental data when the initial
produced $K^{*0}$ is fixed.
Figure 2: Top panel: The survived ratio of $K^{*}$yield as the function of
hadronic rescattering time $t_{res}$ in central Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Bottom panel: The survived $K^{*0}$ as the function
of $p_{T}$ at different $t_{res}$.
Figure 3: Top panel: The final state $K^{*}$yield at midrapidity as the
function of hadronic rescattering time $t_{res}$ in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV. The symbols connected with lines are calculation
results corresponding three differen V/P ratios at hadronization. The
experimental data are shown as the band area KstarRHIC62_200GeV . Bottom
panel: The $p_{T}$ spectra of final state $K^{*}$ for three V/P values with
the corresponding rescattering times for yield reproduction in central Au+Au
collisions at 200 GeV. The experimental data are shown as symbols
KstarRHIC62_200GeV . The band area shows the result of $K^{*}$ just after
hadronization.
Fig.2 top panel shows the survived ratio of $K^{*}$ abundance
$N(K^{*})^{final}/N(K^{*})^{initial}$ after hadronic rescattering stage as the
function of rescattering time $t_{res}$. Here $N(K^{*})^{initial}$ is the
number of $K^{*}$ just after hadronization and $N(K^{*})^{final}$ is the
number of $K^{*}$ that can be experimentally reconstructed after hadronic
rescatterings. $K^{*}$ mesons are reconstructed from their hadronic decay
channels using pion-kaon invariant mass analysis, and the survived $K^{*}$
incorporates all effects of destruction, signal loss and regeneration. One can
see that the number of survived $K^{*}$ almost exponentially decreases with
the rescattering time. The time of $K^{*}$ number reducing by half is about 20
fm/c. Fig.2 bottom panel shows the number of survived $K^{*0}$ after hadronic
rescattering stage as the function of transverse momentum $p_{T}$ at different
rescattering time $t_{res}$. We find that the suppression of $K^{*0}$ caused
by hadronic rescattering effects is strong in low $p_{T}$ region. This is
qualitatively consistent with the result of UrQMD transport model
VogelAndBleicher05 ; Kstar158GeV .
Taking into account of effects of hadronic rescatterings on the yield
suppression and spectra distortion, we obtain the yield and $p_{T}$ spectrum
of final state $K^{*0}$ comparable to experimental data. The top panel in
Fig.3 shows the yield of final survived $K^{*0}$ at midrapidity as the
function of rescattering time $t_{res}$ in central Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. The directly produced $K^{*0}$ for different V/P
ratios at hadronization is taken to be the starting point of the hadronic
rescattering stage. For V/P=3 the rescattering time needed to reproduce the
experimental data is about 60 fm/c while for V/P=1 the needed time is about 30
fm/c. For V/P=0.45 favored by $pp$ $N(K^{*0})/N(K^{-})$ data the rescattering
time is about 13 fm/c. This value is consistent with the typical lifespan of
$13\pm 3$ fm/c in UrQMD simulation of hadronic rescattering stage Bleicher02 .
The estimation of time span between chemical and thermal freeze-out by a
thermal model using an additional pure rescattering phase is $2.5^{+1.5}_{-1}$
fm/c from the analysis of $K^{*0}/K^{-}$ yield ratio Rafelski01 ;
VogelAndBleicher06proc . The bottom panel in Fig.3 shows the $p_{T}$ spectrum
of final state $K^{*0}$ at midrapidity at three V/P values with the
corresponding rescattering times for yield reproduction in central Au+Au
collisions at 200 GeV. We find that the strong suppression of low $p_{T}$
$K^{*0}$ in hadronic rescattering process offset the over-predication of
initial $K^{*0}$ by hadronization shown as the band area, and this leads to an
obviously improved description of $p_{T}$ spectrum of $K^{*0}$.
The $K^{*0}$ data at different collision energies enable the further test of
hadronization mechanism and the investigation of the energy dependence of
hadronic rescattering effects. In Fig.4, we present the $p_{T}$ spectra of
various stable hadrons and $K^{*0}$ resonance at midrapidity in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. Symbols are the experimental data and
lines are the results of SDQCM. The $p_{T}$ spectra of constituent quarks at
hadronization are taken to be the thermal exponential pattern
$\exp(-m_{T}/T_{s})$. The slope parameter $T_{s}$ is taken to be 0.31 GeV for
strange quarks and 0.29 GeV for light quarks, which are smaller than those at
200 GeV CEShao2009PRC . The numbers and rapidity spectra of constituent quarks
and antiquarks are fixed by the experimental data of pion and kaon rapidity
spectra Arsene62gev_y_spectra . One can see that, similar to Au+Au 200 GeV,
the model results of various stable hadrons are in well agreement with the
data. Then we can predict the $p_{T}$ spectrum of initial $K^{*0}$ just after
hadronization, and the results are presented in the right panel in Fig.4. The
degree of the agreement between the model result and experimental data is also
similar to that in Au+Au 200 GeV. The calculated $K^{*0}$ yield densities in
low $p_{T}$ region exceed the data and become to close to the data as $p_{T}$
rises to 2-3 GeV/c. This indicates the influence of the hadronic rescattering
on $K^{*0}$ production is still significant at intermediate RHIC energy.
Figure 4: The $p_{T}$ spectra of stable hadrons (left panel), their anti-
hadrons (middle panel) and $K^{*0}$ resonance (right panel) at midrapidity in
central Au+Au collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. Symbols are the
experimental data piproton62gev ; multH62.4GeV and lines are results of
SDQCM.
Figure 5: Top panel: The final state $K^{*}$ yield at midrapidity as the
function of hadronic rescattering time $t_{res}$ in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 62.4 GeV. The symbols connected with lines are calculation
results corresponding three differen V/P ratios at hadronization. The
experimental data are shown as the band area KstarRHIC62_200GeV . Bottom
panel: The $p_{T}$ spectra of final state $K^{*}$ for three V/P values with
the corresponding rescattering times for yield reproduction in central Au+Au
collisions at 62.4 GeV. The experimental data are shown as symbols
KstarRHIC62_200GeV . The band area shows the result of $K^{*}$ just after
hadronization.
The top panel in Fig.5 shows the yield of final survived $K^{*0}$ at
midrapidity as the function of rescattering time $t_{res}$ in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. For V/P=3 the rescattering time
needed to reproduce the experimental data is about 20 fm/c while for V/P=1 the
needed time is about 14 fm/c and for V/P=0.45 favored by $pp$
N($K^{*}$)/N($K^{-}$ ) data the time is about 5 fm/c. It is found that the
needed time at 62.4 GeV is about half of that at 200 GeV.
The bottom panel in Fig.5 presents the $p_{T}$ spectrum of final state
$K^{*0}$ at midrapidity at three V/P values with the corresponding
rescattering times for yield reproduction in central Au+Au collisions at 62.4
GeV. The strong suppression in low $p_{T}$ region in hadronic rescattering
process offset the over-predication of initial $K^{*}$by hadronization shown
as the band area, and this leads to an improved description of $p_{T}$
spectrum of $K^{*0}$.
## IV $\Sigma^{*}$ production by hadronization at RHIC
The $\Sigma^{*}$ hyperon has a very short lifetime (4.5 fm/c). The measurement
of STAR Collaboration found that the $\Sigma^{*}/\Lambda$ yield ratio at
midrapidity in central Au+Au collisions is nearly the same as that in $pp$
collision at $\sqrt{s_{NN}}=$ 200 GeV StrangeBresonance200Gev . This indicates
that the net effects of rescattering loss and rescattering gain in hadronic
stage are very small. This poses serious constraint on cross sections of
various reaction channels of collision gain and collision loss. The small net
effect of hadronic rescattering has two possibilities. The first is that both
the effect of collision gain and that of collision loss on $\Sigma^{*}$ yield
are large but they offset with each other. The $p_{T}$ spectrum of
$\Sigma^{*}$ might change during hadronic rescattering in this case. The
second is that two components are all small and $p_{T}$ spectrum of
$\Sigma^{*}$ should not be changed dramatically compared with the result of
hadronization. Here, using the quark spectra fixed by stable hadrons in Fig.1,
we use SDQCM to study the yield and $p_{T}$ spectrum of $\Sigma^{*}$ at
midrapidity in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. The yield
density $dN/dy$ of $\Sigma^{*}+\overline{\Sigma}^{*}$ at midrapidity just
after hadronization is 9.0, and the data of STAR Collaboration is $9.3\pm
1.4\pm 1.2$ StrangeBresonance200Gev . Here $\Sigma^{*}$ represents
$\Sigma^{*+}+\Sigma^{*-}$. The result of $p_{T}$ spectrum of $\Sigma^{*}$ just
after hadronization is presented in left panel in Fig.6 and is compared with
the experimental data. We find that the agreement between hadronization
results and the data is well. The result favors the second case. We further
predict the elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au collisions
at 200 GeV and the result is shown in middle panel in Fig. 6. It is well known
that the elliptic flow is sensitive to the mechanism of hadron production. The
elliptic flow of hadrons formed via quark combination mechanism follows the
constituent quark number scaling rule. The phenomenological regeneration of
$\Sigma^{*}$ by $\Lambda+\pi\rightarrow\Sigma^{*}$ will result in higher v2.
To test the effects of hadronic rescattering on $\Sigma^{*}$ production at
RHIC energies, we further predict the $\Sigma^{*}$ $p_{T}$ spectrum in central
Au+Au collisions at 62.4 GeV. The yield density of $\Sigma^{*}$ at midrapidity
is 4.3 just after hadronization.
Figure 6: Left panel: The $p_{T}$ spectrum of $\Sigma^{*}$ in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 200 GeV. Symbols are the experimental data
StrangeBresonance200Gev and the line is the result of SDQCM. Middle panel:
The prediction of elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au
collisions at 200 GeV. Right panel: The prediction of $p_{T}$ spectrum of
$\Sigma^{*}$ in central Au+Au collisions at $\sqrt{s_{NN}}=$ 62.4 GeV.
## V summary
In this paper, we have used SDQCM model for the QGP hadronization and ART
model for the subsequent hadronic rescattering process to investigate the
production of $K^{*0}$ and $\Sigma^{*}$ resonances in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV and 62.4 GeV. The initial $K^{*0}$ produced by
quark combination mechanism is higher than the experimental data in the low
$p_{T}$ region and is close to data at 2-3 GeV/c. In the subsequent hadronic
rescattering stage, the number of $K^{*0}$ that can be reconstructed
experimentally exponentially decreases with the increasing rescattering time,
and the suppression of $K^{*0}$ yield focuses on low $p_{T}$ region, which
offsets the over-prediction of hadronization. Therefore the quark combination
hadronization plus hadronic rescattering can provide a well description of the
experimental data of $K^{*0}$ production. According to the suppression
magnitude of $K^{*0}$ yield, the time span of hadronic rescattering stage is
estimated. V/P ratio at hadronization is important for the extraction of
hadronic rescattering time from the data of $K^{*0}$ yield. For V/P =0.45
based on the data of $K^{*0}/K^{-}$ in $pp$ reaction, the time span of
hadronic rescattering stage is about 13 fm/c at 200 GeV and 5 fm/c at 62.4
GeV. Higher V/P ratio leads to the longer rescattering time. The yield density
and $p_{T}$ spectrum of $\Sigma^{*}$ directly from quark combination
hadronization in central Au+Au collisions at 200 GeV is found to be in well
agreement with the experimental data, which indicates a weak hadronic
rescattering effects. To make a further test, we predict the v2 of
$\Sigma^{*}$ in minimum bias Au+Au collisions at 200 GeV and $p_{T}$ spectrum
of $\Sigma^{*}$ at lower 62.4 GeV for the comparison with future STAR data .
## ACKNOWLEDGMENTS
The authors thank X. B. Xie, Z. T. Liang and G. Li for helpful discussions.
The work is supported in part by the National Natural Science Foundation of
China under grant 11175104 and 10947007, and by the Natural Science Foundation
of Shandong Province, China under grant ZR2011AM006.
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|
arxiv-papers
| 2012-03-02T13:07:14 |
2024-09-04T02:49:28.180269
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kai Zhang, Jun Song, and Feng-lan Shao",
"submitter": "Jun Song",
"url": "https://arxiv.org/abs/1203.0449"
}
|
1203.0452
|
# Angular momentum at null infinity in higher dimensions
Kentaro Tanabe Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan Tetsuya Shiromizu Shunichiro Kinoshita Department of
Physics, Kyoto University, Kyoto 606-8502, Japan
###### Abstract
We define the angular momentum at null infinity in higher dimensions. The
asymptotic symmetry at null infinity becomes the Poincaré group in higher
dimensions. This fact implies that the angular momentum can be defined without
any ambiguities such as supertranslation in four dimensions. Indeed we can
show that the angular momentum in our definition is transformed covariantly
with respect to the Poincaré group.
###### pacs:
04.20.-q, 04.20.Ha
††preprint: YITP-12-13
## I introduction
Motivated by the string theory and the scenario with large extra dimensions
such as the TeV scale gravity ArkaniHamed:1998rs ; Antoniadis:1998ig , the
gravitational theory in higher dimensions has been investigated PTP . Then it
has been realized that the higher dimensional gravity has much different
features from that in four dimensions. As one of such differences, there is an
issue of the asymptotic structure of the spacetime Hollands:2003ie ;
Hollands:2003xp ; Ishibashi:2007kb ; Tanabe:2011es ; Tanabe:2009va ;
Tanabe:2010rm ; Tanabe:2009xb . The asymptotically flat spacetime has two
asymptotic infinities: spatial infinity and null infinity. At spatial infinity
we can define the global conserved quantities of spacetime such as the mass
and angular momentum. In addition the multipole moments of spacetime is also
defined at spatial infinity Hansen ; Tanabe:2010ax . These multipole moments
can be used to classify the black hole solutions. At null infinity, the
asymptotic structure describes dynamical properties of the spacetime because
gravitational waves can reach at null infinity. Then the study of the
asymptotic structure at null infinity is indispensable when one considers the
dynamical phenomena such as the perturbation for black holes and the formation
of higher dimensional black holes in particle accelerators. As a fundamental
aspect of the general relativity, the notion of the asymptotic flatness at
null infinity is also necessary for the rigorous definition of black hole
Hawking:1973uf .
The asymptotic structure at null infinity in higher dimensions has been
investigated using the conformal method in Refs Hollands:2003ie ;
Hollands:2003xp ; Ishibashi:2007kb . In the conformal method, spacetime is
conformally embedded into the compact region of the another spacetime and the
null infinity is defined as the boundary of the spacetime. The asymptotic
structure at null infinity can be investigated using the introduced conformal
factor $\Omega\sim 1/r$ as a coordinate. Therein the null infinity is defined
on $\Omega=0$. However, there is one problem in this treatment. The
gravitational waves behave near null infinity with a half integer power of
$\Omega$ in odd dimensions. At first glance, this shows the non-smoothness of
the gravitational fields at null infinity in odd dimensions. Because of this
non-smooth behavior of the gravitational fields, using the conformal method,
we cannot define the asymptotic flatness at null infinity in odd dimensions.
In the Bondi coordinate method Refs. Tanabe:2011es ; Bondi:1962px ;
Sachs:1962wk ; Tanabe:2009va ; Tanabe:2010rm , on the other hand, we can
define the asymptotic flatness at null infinity in arbitrary higher dimensions
and safely investigate the asymptotic structure at null infinity. In the
analysis of the asymptotic structure, it was found that the Bondi mass always
decreases due to the gravitational waves and the asymptotic symmetry at null
infinity is the Poincaré group.
It is reminded that the asymptotic symmetry is not the Poincaré group in four
dimensions. The asymptotic symmetry in four dimensions is the semi-direct
group of the Lorentz group and supertranslation. The supertranslation has the
functional degree of freedom and then it is the infinite dimensional group.
This means that there are infinite directions of the translation which causes
the ambiguities in the definition of the angular momentum. Although there were
many efforts to define the angular momentum Prior:1977 ; Streubel:1978 ;
Winicour:1980 ; Geroch:1981ut ; Dray:1984 , there is no sharp definition
without any ambiguities in four dimensions.
In higher dimensions it was shown that the asymptotic symmetry at null
infinity is the Poincaré group Tanabe:2011es ; Tanabe:2009va . The
$n$-dimensional Poincaré group has the $n$ directions of the translation. In
this paper, we define the angular momentum at null infinity in higher
dimensions and shows that it has no ambiguities. In fact the angular momentum
is transformed covariantly with respect to the Poincaré group. Note that the
study of the angular momentum at null infinity was performed in five
dimensions Tanabe:2010rm . In this paper, we generalize this analysis to
arbitrary higher dimensions following our previous work of Ref. Tanabe:2011es
.
The organization of this paper is as follows. In the next section, we review
our previous work on null infinity Tanabe:2011es . Therein, using the Bondi
coordinates, we introduce the definition of null infinity in arbitrary
dimensions. We also discussed the asymptotic symmetry at the null infinities
briefly. In Sec. III, we define the Bondi angular momentum together with the
Bondi mass/momentum and show its radiation formulae using the Einstein
equations. In Sec. IV, it will be shown that the angular momentum defined here
is transformed covariantly under the transformation generated by the
asymptotic symmetry. Finally we give the summary and outlook in Sec. V.
## II Review of our previous work
In this section we review our previous work Tanabe:2011es . First we introduce
the Bondi coordinates adopted here and write down some components of the
Einstein equation explicitly. Solving them we specify the boundary condition
which gives us the definition of the null infinity. Then we discuss the
asymptotic symmetries at the null infinity.
### II.1 Bondi coordinates and Einstein equations
We introduce the Bondi coordinates in $n$ dimensions. First we assume the
function $u$ which satisfies $\hat{\nabla}_{a}u\hat{\nabla}^{a}u=0$ where
$\hat{\nabla}_{a}$ denotes the covariant derivative with respect to
$n$-dimensional metric $g_{ab}$. $u$ is used as the time coordinate. Next the
angular coordinate $x^{I}$ is defined as
$\hat{\nabla}^{a}u\hat{\nabla}_{a}x^{I}=g^{uI}=0$. We define the radial
coordinate $r$ as $\sqrt{\det{g}_{IJ}}=r^{n-2}\omega_{n-2}$ where
$\omega_{n-2}$ is the volume element of the unit $(n-2)$-dimensional sphere
$S^{n-2}$. Then the metric in the Bondi coordinates $x^{a}=(u,r,x^{I})$ can be
written as
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{ab}dx^{a}dx^{b}$ (1)
$\displaystyle=$ $\displaystyle-
Ae^{B}du^{2}-2e^{B}dudr+\gamma_{IJ}(dx^{I}+C^{I}du)(dx^{J}+C^{J}du).$
In this coordinate system, the null infinity is defined at $r=\infty$ and its
topology is ${\mathbf{R}}\times S^{n-2}$. For the convenience of our
discussion, we define $h_{IJ}$ as $\gamma_{IJ}=r^{2}h_{IJ}$ with the following
gauge condition
$\sqrt{\det{h_{IJ}}}\,=\,\omega_{n-2}.$ (2)
We provide the Einstein equations in the Bondi coordinates. The vacuum
Einstein equations can be decomposed into the constraint equation without the
$u$ derivative terms and evolution equations with the $u$ derivative terms.
The constraint equations are $\hat{R}_{rr}=0,\hat{R}_{aI}\gamma^{IJ}=0$ and
$\hat{R}_{IJ}\gamma^{IJ}=0$. $\hat{R}_{ab}$ is the Ricci tensor with respect
to $g_{ab}$. Using the formulae in Appendix A and Sec. II in Ref.
Tanabe:2011es , we can write the equation $\hat{R}_{rr}=0$ as
$B^{\prime}=\frac{r}{4(n-2)}h_{IJ}^{\prime}h_{KL}^{\prime}h^{IK}h^{JL},$ (3)
where the prime denotes the $r$ derivative. The equation
$\hat{R}_{rJ}\gamma^{IJ}=0$ yields
$\frac{1}{r^{n-2}}(r^{n}e^{-B}h_{IJ}{C^{J}}^{\prime})^{\prime}=-{}^{(h)}\nabla_{I}B^{\prime}+\frac{n-2}{r}{}^{(h)}\nabla_{I}B+{}^{(h)}\nabla^{J}h_{IJ}^{\prime},$
(4)
where ${}^{(h)}\nabla_{I}$ is the covariant derivative with respect to
$h_{IJ}$.
From $\hat{R}_{IJ}\gamma^{IJ}=0$, we obtain
$\displaystyle(n-2)\frac{(r^{n-3}A)^{\prime}}{r^{n-2}}$ $\displaystyle=$
$\displaystyle-{}^{(h)}\nabla_{I}{C^{I}}^{\prime}-\frac{2(n-2)}{r}{}^{(h)}\nabla_{I}C^{I}-\frac{r^{2}e^{-B}}{2}h_{IJ}{C^{I}}^{\prime}{C^{J}}^{\prime}$
(5)
$\displaystyle-\frac{e^{B}}{2r^{2}}h^{IJ}{}^{(h)}\nabla_{I}B{}^{(h)}\nabla_{J}B-\frac{e^{B}}{r^{2}}{}^{(h)}\nabla_{I}(h^{IJ}{}^{(h)}\nabla_{J}B)+\frac{e^{B}}{r^{2}}{}^{(h)}R,$
where ${}^{(h)}R$ is the Ricci scalar of $h_{IJ}$. Once $h_{IJ}$ is given on a
surface $u=u_{0}$, we can obtain the metric functions $A,B$ and $C^{I}$ by
solving the constraint equations (3), (4) and (5) on the surface.
The evolution equation is contained in
$\hat{R}_{ab}\gamma_{I}{}^{a}\gamma_{J}{}^{b}=0$ as
$\displaystyle
e^{-B}\Bigg{[}r^{2}\dot{h}_{IJ}^{{}^{\prime}}+\frac{n-2}{2}r\dot{h}_{IJ}-\frac{r^{2}}{2}\dot{h}_{IK}h^{{}^{\prime}}_{JL}h^{KL}-\frac{r^{2}}{2}\dot{h}_{JK}h^{{}^{\prime}}_{IL}h^{KL}\Bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{Ae^{-B}}{2}\Big{[}r^{2}h^{{}^{\prime\prime}}_{IJ}+(n-2)rh^{{}^{\prime}}_{IJ}+2(n-3)h_{IJ}{-}r^{2}h^{KL}h^{{}^{\prime}}_{IK}h^{{}^{\prime}}_{JL}\Big{]}-\frac{A^{{}^{\prime}}e^{-B}}{2}\Big{[}r^{2}h^{{}^{\prime}}_{IJ}+2rh_{IJ}\Big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{e^{-B}}{2}\Big{[}2r^{2}\mathcal{L}_{C}h^{{}^{\prime}}_{IJ}+(n-2)r\mathcal{L}_{C}h_{IJ}+r^{2}\mathcal{L}_{C^{{}^{\prime}}}h_{IJ}-{r^{2}}h^{{}^{\prime}}_{JL}h^{KL}\mathcal{L}_{C}h_{IK}$
$\displaystyle~{}~{}~{}~{}~{}~{}-{r^{2}}h^{{}^{\prime}}_{IL}h^{KL}\mathcal{L}_{C}h_{JK}+{}^{(h)}\nabla_{K}C^{K}(r^{2}h^{{}^{\prime}}_{IJ}+2rh_{IJ})\Big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{e^{-2B}}{2}r^{4}h_{IK}h_{JL}{C^{K}}^{\prime}{C^{L}}^{\prime}-{}^{(h)}\nabla_{I}{}^{(h)}\nabla_{J}B-\frac{1}{2}{}^{(h)}\nabla_{I}B{}^{(h)}\nabla_{J}B+{}^{(h)}R_{IJ}\,=\,0,$
(6)
where ${}^{(h)}R_{IJ}$ is the Ricci tensor with respect to $h_{IJ}$ and the
dot denotes the $u$ derivative. The evolution of $h_{IJ}$ in the Bondi
coordinates is determined by Eq. (6).
### II.2 Asymptotic flatness at null infinity
The asymptotic flatness at null infinity is defined by the boundary condition
at null infinity in the Bondi coordinates (1). In $n$ dimensions, the boundary
conditions for the asymptotic flatness at null infinity are
$h_{IJ}\,=\,\omega_{IJ}+O\left(\frac{1}{r^{n/2-1}}\right),$ (7)
where $\omega_{IJ}$ is the unit round metric on $S^{n-2}$. The boundary
conditions for the other metric functions are determined by the constraint
equations of Eqs. (3), (4) and (5) as 111The condition for $B$ will be relaxed
to $B=O(r^{-n/2})$ for non-vacuum cases Godazgar:2012zq . In our present
paper, we will focus on the vacuum cases. It is easy to extend our result to
non-vacuum cases.
$\displaystyle A\,=\,1+O\left(\frac{1}{r^{n/2-1}}\right),\quad
B\,=\,O\left(\frac{1}{r^{n-2}}\right),\quad
C^{I}\,=\,O\left(\frac{1}{r^{n/2}}\right).$ (8)
Let us see the above solving the constraint equations explicitly. We will use
some equations for later discussions. First, we expand $h_{IJ}$ near null
infinity as
$h_{IJ}\,=\,\omega_{IJ}+\sum_{k=0}\frac{h^{(k+1)}_{IJ}}{r^{n/2+k-1}},$ (9)
where the summation is taken over $k\in\bf{Z}$ in even dimensions and
$2k\in\bf{Z}$ in odd dimensions. The indices $I,J,\dots$ are raised and
lowered by $\omega_{IJ}$. From the gauge condition of Eq. (2), we find that
$h^{(k+1)}_{IJ}$ is traceless $\omega^{IJ}h^{(k+1)}_{IJ}=0$ for $k<n/2-1$ and,
for $k=n/2-1$,
$\omega^{IJ}h^{(n/2)}_{IJ}\,=\,\frac{1}{2}h^{(1)IJ}h^{(1)}_{IJ}.$ (10)
Solving the constraint equation (3), we have
$B\,=\,\frac{B^{(1)}}{r^{n-2}}+O\left(\frac{1}{r^{n-3/2}}\right),$ (11)
where
$B^{(1)}\,=\,-\frac{1}{16}\omega^{IK}\omega^{JL}h^{(1)}_{IJ}h^{(1)}_{KL}.$
(12)
From Eq. (4), $C^{I}$ is obtained as
$C^{I}\,=\,\sum_{k=0}^{k<n/2-1}\frac{C^{(k+1)I}}{r^{n/2+k}}+\frac{j^{I}}{r^{n-1}}+O\left(\frac{1}{r^{n-1/2}}\right),$
(13)
where, for $k<n/2-1$,
$C^{(k+1)I}\,=\,\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\nabla_{J}h^{(k+1)IJ}$ (14)
and $\nabla_{I}$ is the covariant derivative with respect to $\omega_{IJ}$.
$j^{I}$ is the integration function in the $r$ integration of Eq. (4). As seen
later, we can see that $j^{I}$ represents the angular momentum of the
spacetime at null infinity.
Integrating Eq. (5) we find
$A\,=\,1+\sum_{k=0}^{k<n/2-2}\frac{A^{(k+1)}}{r^{n/2+k-1}}-\frac{m}{r^{n-3}}+O(r^{-(n-5/2)}),$
(15)
where, for $k<n/2-2$,
$\displaystyle A^{(k+1)}\,=$
$\displaystyle\,-\frac{2(n+2k-4)}{(n-2k-4)(n+2k-2)}\nabla^{I}C_{I}^{(k+1)}$
(16) $\displaystyle\,=$
$\displaystyle\,-\frac{4(n+2k-4)}{(n+2k)(n-2k-2)(n-2k-4)}\nabla^{I}\nabla^{J}h^{(k+1)}_{IJ}.$
$m$ is the integration function and reflects the energy-momentum of the
spacetime at null infinity. For $k=n/2-2$, the left-hand sides of Eqs. (4) and
(5) vanish. Then the right-hand sides of Eqs. (4) and (5) provide the
following constraint equations
$\displaystyle\nabla_{I}C^{(n/2-1)I}\,=\,0,$ (17)
$\displaystyle\nabla^{I}\nabla^{J}h^{(n/2-1)}_{IJ}\,=\,0.$ (18)
In addition, for $k=n/2-1$, we have 222We found a minor error in Eq. (32) of
Ref. Tanabe:2011es , corresponding to Eq. (19) in the current paper. But it
does not affect the results/all equations presented there except for Eq. (32).
$\nabla^{J}h^{(n/2)}_{IJ}\,=\,2\nabla_{I}B^{(1)}{+\nabla_{J}\left(\frac{1}{2}h^{(1)}_{IK}h^{(1)JK}+\frac{1}{8}\omega_{I}{}^{J}h^{(1)}_{KL}h^{(1)KL}\right)}.$
(19)
We can solve the evolution equation of Eq. (6) as
$\displaystyle(k+1)\dot{h}^{(k+2)}_{IJ}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(n-2k-4\right)A^{(k+1)}\omega_{IJ}+\frac{1}{8}\Big{[}n^{2}-6n-(4k^{2}+4k-16)\Big{]}h^{(k+1)}_{IJ}$
(20)
$\displaystyle+\frac{1}{2}\left(-\nabla^{2}h^{(k+1)}_{IJ}+2\nabla_{(I}\nabla^{K}h^{(k+1)}_{J)K}\right)-\frac{1}{2}(n-2k-4)\nabla_{(I}C^{(k+1)}_{J)}-\nabla^{K}C^{(k+1)}_{K}\omega_{IJ}$
for $k<n/2-1$. The solutions for the higher order of $k\geq n/2-1$ are not
important in the following analysis. For the convenience of later discussions,
we derive the evolution equations for $A^{(k+1)}$ and $C^{(k+1)I}$.
Contracting $\nabla^{J}$ with Eq. (20), we obtain the evolution equation for
$C^{(k+1)I}$ as
$\displaystyle\frac{(k+1)(n+2k+2)}{2(n+2k)}\dot{C}^{(k+2)}_{I}$
$\displaystyle=$
$\displaystyle-\frac{n-4}{2(n+2k-4)}\nabla_{I}A^{(k+1)}-\frac{1}{4}\nabla^{2}C^{(k+1)}_{I}$
(21)
$\displaystyle+\frac{1}{16}\Big{[}n^{2}-6n-(4k^{2}+4k-12)\Big{]}C^{(k+1)}_{I}.$
Contracting $\nabla^{I}$ with Eq. (21) and using the solutions of the
constraint equations (14) and (16), we have the evolution equation for
$A^{(k+1)}$ as
$\dot{A}^{(k+2)}\,=\,-\frac{n+2k-2}{2(k+1)(n+2k+2)}\nabla^{2}A^{(k+1)}+\frac{(n+2k-2)^{2}(n-2k-4)}{8(k+1)(n+2k+2)}A^{(k+1)}.$
(22)
### II.3 Asymptotic symmetry
The asymptotic symmetry is the global symmetry at null infinity generated by
the coordinate transformations preserving the gauge and boundary conditions in
the Bondi coordinates (1). The variation of the metric $\delta g_{ab}$ due to
the coordinate transformation generated by $\xi^{a}$ is given by
$\delta g_{ab}\,=\,\hat{\nabla}_{a}\xi_{b}+\hat{\nabla}_{b}\xi_{a},$ (23)
where $\hat{\nabla}_{a}$ is the covariant derivative with respect to $g_{ab}$.
From Eqs. (1) and (2), the gauge conditions to be satisfied are
$\delta g_{rr}\,=\,0\,,\quad\delta g_{rI}\,=\,0\,,\quad g^{IJ}\delta
g_{IJ}\,=\,0.$ (24)
From Eqs. (7) and (8), the boundary conditions to be preserved by the
coordinate transformations are
$\delta g_{uu}\,=\,O(r^{-(n/2-1)})\,,\quad\delta
g_{uI}\,=\,O(r^{-(n/2-2)})\,,\quad\delta g_{IJ}\,=\,O(r^{n/2-3}).$ (25)
To satisfy Eq. (24), the generator of the asymptotic symmetry $\xi$ becomes
$\displaystyle\xi^{u}\,=\,f(u,x^{I}),$ (26)
$\displaystyle\xi^{I}\,=\,f^{I}(u,x^{I})+\int
dr\frac{e^{B}}{r^{2}}h^{IJ}\nabla_{J}f,$ (27)
$\displaystyle\xi^{r}\,=\,-\frac{r}{n-2}\left(C^{I}\nabla_{I}f+\nabla_{I}\xi^{I}\right).$
(28)
$f(u,x^{I})$ and $f^{I}(u,x^{I})$ are the integration functions in the $r$
integration of the equation $\delta g_{rr}=0$ and $\delta g_{rI}=0$. The
asymptotic symmetry is the group generated by $f$ and $f^{I}$.
The boundary conditions (25) give the equations which $f$ and $f^{I}$ should
satisfy as
$\displaystyle\partial_{u}f^{I}\,=\,0,$ (29)
$\displaystyle\nabla_{I}f_{J}+\nabla_{J}f_{I}\,=\,\frac{2\nabla_{K}f^{K}}{n-2}\omega_{IJ},\quad\nabla_{I}f^{I}=(n-2)\frac{\partial
f}{\partial u},$ (30)
$\displaystyle\nabla_{I}\nabla_{J}f\,=\,\frac{\nabla^{2}f}{n-2}\omega_{IJ}.$
(31)
Note that Eq. (31) is required only in $n>4$ dimensions. In the following, for
the moment, we discuss the asymptotic symmetry in $n>4$ dimensions. We will
comment on the four dimensional case later.
From Eq. (29), we find $f^{I}=f^{I}(x^{I})$. $f^{I}$ is the vector on
$S^{n-2}$ and Eq. (30) implies that $f^{I}$ generates the conformal isometry
on $S^{n-2}$. The conformal group of $S^{n-2}$ is $\mathrm{SO}(1,n-1)$, which
is the Lorentz group. Thus $f^{I}$ is the generator of the Lorentz group.
Integrating the trace part of Eq. (30), we obtain
$\displaystyle f\,=\,\frac{F(x^{I})}{n-2}u+\alpha(x^{I}),$ (32)
where $F\,\equiv\,\nabla_{I}f^{I}$ and $\alpha(x^{I})$ is the integration
function on $S^{n-2}$. Note that the transverse part $f^{\text{(tra)}I}$ of
$f^{I}$ which satisfies $\nabla_{I}f^{\text{(tra)}I}=0$ is nothing but the
Killing vector on $S^{n-2}$, that is, the generator of $\mathrm{SO}(n-1)$.
This Killing vector plays an important role in defining the angular-momentum
later.
Now Eq. (31) gives the equation which $\alpha$ should satisfy as
$\displaystyle\nabla_{I}\nabla_{J}\alpha\,=\,\frac{1}{n-2}\omega_{IJ}\nabla^{2}\alpha.$
(33)
The general solutions of this equation are the $l=0$ and $l=1$ modes of the
scalar harmonics on $S^{n-2}$. Note that the $l=1$ modes satisfy
$\nabla_{I}\nabla_{J}\alpha=-\alpha\omega_{IJ}$ too. From Eq. (31), we find
that $F(x^{I})$ should satisfy $\nabla^{2}F+(n-2)F=0$. The solutions of this
equation are the $l=1$ modes of the scalar harmonics on $S^{n-2}$. These
results mean that the functions $\alpha(x^{I})$ and $F(x^{I})$ are the
generators of the translation and Lorentz boost respectively, and $f$
represents the semi-direct property of the Lorentz group and translation. Then
it turns out that the asymptotic symmetry is the semi-direct group of the
Lorentz group and translation, which is the Poincaré group, in $n>4$
dimensions.
In four dimensions, Eqs. (29) and (30) are required while Eq. (31) is not.
Therefore $f^{I}$ generates the Lorentz group and $f$ can be written as Eq.
(32) in $n=4$ dimensions. However, there are no constraints on $\alpha$ in
four dimensions because of the absence of Eq. (31). Thus, $\alpha(x^{I})$ is
the arbitrary function on $S^{2}$ and generates so called supertranslation,
not translation. The asymptotic symmetry in four dimensions is the semi-direct
group of the Lorentz group and the supertranslation. This supertranslation
leads the ambiguity to the definition of the angular momentum at null infinity
in four dimensions.
## III Bondi angular momentum and radiation formula
In this section, we will define the Bondi angular momentum. In the pedagogical
aspect, we describe the definition of the Bondi mass too given in our previous
work Tanabe:2011es .
### III.1 Bondi mass and angular momentum
We define the Bondi mass and angular momentum. In the Bondi coordinates,
$g_{uu}$ and $g_{uI}$ can be expanded near null infinity as
$\displaystyle
g_{uu}\,=\,-1-\sum_{k=0}^{k<n/2-2}\frac{A^{(k+1)}}{r^{n/2+k-1}}+\frac{m(u,x^{I})}{r^{n-3}}+O(r^{-(n-5/2)})$
(34)
and
$\displaystyle
g_{uI}\,=\,\sum_{k=0}^{k<n/2-1}\frac{C_{I}^{(k+1)}}{r^{n/2+k-2}}+\frac{j_{I}(u,x^{I})+h^{(1)}_{IJ}C^{(1)J}}{r^{n-3}}+O(r^{-(n-5/2)}).$
(35)
The functions $m$ and $j_{I}$ are the integration functions and they are free
functions on the initial surface $u=u_{0}$.
The Bondi mass $M_{\text{Bondi}}$ and momentum $P^{i}_{\text{Bondi}}$ are
defined by Tanabe:2011es
$\displaystyle M_{\text{Bondi}}(u)$
$\displaystyle\equiv\frac{n-2}{16\pi}\int_{S^{n-2}}md\Omega,$ (36)
$\displaystyle P^{i}_{\text{Bondi}}(u)$
$\displaystyle\equiv\frac{n-2}{16\pi}\int_{S^{n-2}}m\hat{x}^{(i)}d\Omega,$
where $\hat{x}^{(i)}$ is the scalar function on $S^{n-2}$ satisfying
$\nabla_{I}\nabla_{J}\hat{x}^{(i)}+\omega_{IJ}\hat{x}^{(i)}\,=\,0$. These
functions are the $l=1$ modes of the scalar harmonic on $S^{n-2}$, which are
defined by $\hat{x}^{(i)}=x^{(i)}/\rho$ in the Cartesian coordinates
$\\{x^{(i)}\\}$ of the $(n-1)$-dimensional Euclidean flat space. Here
$S^{n-2}$ is embedded into the $(n-1)$-dimensional Euclidean flat space as
$\rho^{2}=\sum_{i=1}^{n-1}(x^{(i)})^{2}$. The indices $i$ represent the
directions of the translation. Note that $A^{(k+1)}$ for $k<n/2-2$ does not
contribute to the global quantities at null infinities because $A^{(k+1)}$ and
$\hat{x}^{(i)}A^{(k+1)}$ are written as the form of total derivative [see Eq.
(16)]. For the details, see Eq. (80) and Appendix B in Ref. Tanabe:2011es .
The Bondi energy-momentum vector
$P^{\mu}_{\text{Bondi}}=(M_{\text{Bondi}},P^{i}_{\text{Bondi}})$ is defined as
the $n$-dimensional vector at null infinity. In the definition of
$P^{\mu}_{\text{Bondi}}$, we introduce the $n$-dimensional vector by
$\hat{x}^{\mu}=(1,\hat{x}^{(i)})$. This vector can be naturally identified to
the bases in the $n$-dimensional Minkowski spacetime as mentioned in the next
section. Thus the Bondi energy-momentum vector $P^{\mu}_{\text{Bondi}}$ can be
also regarded as a vector in the Minkowski spacetime. In the following, the
Greek indices represent the index in Minkowski spacetime.
The Bondi angular momentum $J^{\text{Bondi}}_{(p)}$ is defined by
$J^{\text{Bondi}}_{(p)}\,=\,-\frac{n-1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}j_{I}d\Omega,$ (37)
where $\varphi^{I}_{(p)}$ is the Killing vector of the round metric
$\omega_{IJ}$ on $S^{n-2}$. $p$ labels the Killing vectors and $1\leq
p\leq(n-1)(n-2)/2$. Note that the Bondi angular momentum in five dimensions
were defined for the Killing vectors which commute mutually in Ref.
Tanabe:2010rm . In this paper, we generalized this to define the Bondi angular
momentum in arbitrary dimensions for all Killing vectors. The
$\lfloor\frac{n-1}{2}\rfloor$ independent angular momenta are, of course,
given by the mutually commuting Killing vectors.
Here we show that the first term in Eq. (35) does not contribute to the Bondi
angular momentum. This is because $\varphi^{I}_{(p)}C^{(k+1)}_{I}$ for
$k<n/2-1$ can be written as the total derivative as
$\displaystyle\varphi^{I}_{(p)}C^{(k+1)}_{I}$ $\displaystyle=$
$\displaystyle\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\varphi^{I}_{(p)}\nabla^{J}h^{(k+1)}_{IJ}$
(38) $\displaystyle=$
$\displaystyle\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\nabla^{J}\left(\varphi^{I}_{(p)}h^{(k+1)}_{IJ}\right),$
where we used Eq. (14) and the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$. The term of
$h_{IJ}^{(1)}C^{(1)J}$ in Eq. (35) is nothing, but it just comes from the
lowering of the index of the metric. Therefore, we will not think that it
contributes to the angular momentum. This will be also confirmed later when
one considers the transformation property generated by asymptotic symmetry at
null infinity (Sec. IV).
### III.2 Radiation formula
The functions $m$ and $j_{I}$ are free functions on the initial surface
$u=u_{0}$. The evolutions of these quantities are determined from the Einstein
equations. The Einstein equation $\hat{R}^{rr}=0$ (see Eq. (16) in Ref.
Tanabe:2011es ) can be expanded near null infinity as
$\hat{R}^{rr}\,=\,\sum_{k=0}^{k<n/2-2}\frac{(\hat{R}^{rr})^{(k+1)}}{r^{n/2+k-1}}+\frac{(\hat{R}^{rr})^{(n/2-1)}}{r^{n-3}}+O\left(\frac{1}{r^{n-5/2}}\right).$
(39)
The equations $(\hat{R}^{rr})^{(k+1)}=0$ for $k<n/2-2$ provide us Eq. (22)
again and has no new informations. This feature is guaranteed by the Bianchi
identity. The equation $(\hat{R}^{rr})^{(n/2-1)}=0$ describes the evolution of
the function $m$ as
$\displaystyle\dot{m}\,=\,-\frac{1}{2(n-2)}\dot{h}_{IJ}^{(1)}\dot{h}^{(1)IJ}+\frac{n-5}{n-2}\nabla^{I}C^{(n/2-2)}_{I}+\frac{1}{n-2}\nabla^{2}A^{(n/2-2)}.$
(40)
Integrating this equation on the unit $(n-2)$-dimensional sphere, we obtain
the Bondi mass-loss law
$\displaystyle\frac{d}{du}M_{\text{Bondi}}\,=\,-\frac{1}{32\pi}\int_{S^{n-2}}\dot{h}_{IJ}^{(1)}\dot{h}^{(1)IJ}d\Omega\leq
0.$ (41)
The above implies that the Bondi mass always decreases by radiating the
gravitational waves in any dimensions. In other words, the gravitational waves
carry the positive energy flux to null infinity in any dimensions.
The Einstein equations $\hat{R}^{rI}=0$ contain the evolution equations of
$j_{I}$. $\hat{R}^{rI}$ can be expanded near null infinity as (see Eq. (19) in
Ref. Tanabe:2011es )
$\varphi_{(p)I}\hat{R}^{rI}\,=\,\sum_{k=0}^{k<n/2-1}\frac{\varphi_{(p)I}(\hat{R}^{rI})^{(k+1)}}{r^{n/2+k-1}}+\frac{\varphi_{(p)I}(\hat{R}^{rI})^{(n/2)}}{r^{n-2}}+O\left(\frac{1}{r^{n-3/2}}\right).$
(42)
The equations $\varphi_{(p)I}(\hat{R}^{rI})^{(k+1)}=0$ for $k<n/2-1$ provide
us Eq. (21) again. It is also guaranteed by the Bianchi identity. The equation
$\varphi_{(p)I}(\hat{R}^{rI})^{(n/2)}=0$ presents the evolution equation of
$j_{I}$ as
$\displaystyle-(n-1)\varphi^{I}_{(p)}\dot{j}_{I}$ $\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}\partial_{u}(h^{(1)}_{IJ}\nabla_{K}h^{(1)JK})+h^{(1)JK}\nabla_{J}\dot{h}^{(1)}_{IK}+\nabla_{K}h^{(1)JK}\dot{h}^{(1)}_{IJ}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}$
(43)
$\displaystyle-\varphi^{I}_{(p)}\Big{[}\nabla_{I}m{-}\nabla_{I}\dot{B}^{(1)}{+}\nabla^{J}\dot{h}^{(n/2)}_{IJ}{+}(n-3)\nabla^{J}h^{(n/2-1)}_{IJ}\Big{]}+2\varphi^{I}_{(p)}\nabla^{J}\nabla_{(I}C^{(n/2-1)}_{J)}$
$\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla^{K}h^{(1)IJ}\dot{h}^{(1)}_{JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}$
$\displaystyle+\nabla_{I}\Bigg{[}\varphi^{J}_{(p)}\partial_{u}(h^{(1)}_{JK}h^{(1)IK})+\varphi^{I}_{(p)}(-m+\dot{B}^{(1)})-\varphi_{(p)J}\dot{h}^{(n/2)IJ}$
$\displaystyle-(n-3)\varphi_{(p)J}h^{(n/2-1)IJ}+\varphi^{J}_{(p)}\nabla^{I}C^{(n/2-1)}_{J}-C^{(n/2-1)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Bigg{]},$
where we used the Killing equation,
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$.
Then, we can obtain the radiation formula of the Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ as
$\displaystyle\frac{d}{du}J^{\text{Bondi}}_{(p)}\,=\,\frac{1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega.$
(44)
This equation shows that the Bondi angular momentum is changed when the
spacetime has time and angular dependences. The radiation formula (44) is
natural in this sense.
## IV Poincaré covariance
In this section we consider the transformation of our Bondi mass
$M_{\text{Bondi}}$ and angular momentum $J^{\text{Bondi}}_{(p)}$ generated by
the asymptotic symmetry. The validity of our definitions of the Bondi mass and
angular momentum will be supported by the fact that the Bondi mass and angular
momentum are transformed covariantly with respect to the asymptotic symmetry.
### IV.1 Poincaré covariance
Let us investigate the transformation rule of the Bondi energy-momentum
$P^{\mu}_{\text{Bondi}}$ and angular momentum $J^{\text{Bondi}}_{(p)}$ by the
asymptotic symmetry. In particular, we focus the cases with $f=\alpha$ and
$f^{I}=0$, which is the translation of the Poincaré group in $n>4$ dimensions.
As we mentioned in Sec. II.3, $\alpha(x^{I})$ can be decomposed into the $l=0$
and $l=1$ modes of the scalar harmonics on $S^{n-2}$. Using these harmonics as
bases $\hat{x}^{\mu}=(1,\hat{x}^{(i)})$, we can naturally introduce the
translational vector $\alpha_{\mu}$ in the $n$-dimensional Minkowski spacetime
defined by $\alpha(x^{I})=\alpha_{\mu}\hat{x}^{\mu}$. Moreover, because the
asymptotic symmetry at null infinity is the Poincaré group, we can identify
asymptotic structure at null infinity with the $n$-dimensional Minkowski
spacetime and then obtain a natural map between quantities at null infinity
and those of vector spaces in the Minkowski spacetime. Then we can discuss the
transformations of $P^{\mu}_{\text{Bondi}}$ and $J^{\text{Bondi}}_{(p)}$ by
the translational vector $\alpha_{\mu}$ in the Minkowski spacetime.
In general, the energy-momentum vector $P_{\mu}$ and angular momentum
$M_{\mu\nu}$ in the Minkowski spacetime are expected to be transformed by
translation of the Poincaré group as
$\displaystyle P_{\mu}$ $\displaystyle\rightarrow P_{\mu},$ (45)
$\displaystyle M_{\mu\nu}$ $\displaystyle\rightarrow
M_{\mu\nu}-2P_{[\mu}\alpha_{\nu]},$
where $\alpha_{\mu}$ is a translational vector. However, since the
gravitational waves carry the energy and angular momentum to null infinity,
the Bondi energy-momentum $P^{\mu}_{\text{Bondi}}$ and angular momentum
$J^{\text{Bondi}}_{(p)}$ are changed under the translation. Then, taking these
effects into account, $P^{\mu}_{\text{Bondi}}$ and $J^{\text{Bondi}}_{(p)}$
should be transformed as
$\displaystyle P_{\text{Bondi}}^{\mu}$ $\displaystyle\rightarrow
P_{\text{Bondi}}^{\mu}+\alpha_{\nu}\frac{d}{du}P^{\mu\nu}_{\text{Bondi}},$
(46) $\displaystyle M^{\text{Bondi}}_{\mu\nu}$ $\displaystyle\rightarrow
M^{\text{Bondi}}_{\mu\nu}-2P^{\text{Bondi}}_{[\mu}\alpha_{\nu]}+\alpha^{\rho}\frac{d}{du}M^{\text{Bondi}}_{\mu\nu\rho},$
instead of Eq. (45). Note that the each space-space component of
$M^{\text{Bondi}}_{\mu\nu}$ corresponds to $J^{\text{Bondi}}_{(p)}$. From now
on, we will confirm these equations. The last terms in each transformations
come from the effect of radiations and the concrete expressions will be given
later.
The generator of the translation $f=\alpha$ and $f^{I}=0$ can be expanded near
null infinity as
$\displaystyle\xi^{u}$ $\displaystyle=$ $\displaystyle\alpha(x^{I}),$ (47)
$\displaystyle\xi^{I}$ $\displaystyle=$
$\displaystyle-\frac{1}{r}\nabla^{I}\alpha+\sum_{k=0}^{k<n/2-1}\frac{2h^{(k+1)IJ}\nabla_{J}\alpha}{n+2k}\frac{1}{r^{n/2+k}}$
(48)
$\displaystyle-\frac{1}{n-1}\frac{1}{r^{n-1}}\left(B^{(1)}\nabla^{I}\alpha-h^{(n/2)IJ}\nabla_{J}\alpha{+}h^{(1)IL}h^{(1)J}_{L}\nabla_{J}\alpha\right)+O(r^{-(n-1/2)}),$
$\displaystyle\xi^{r}$ $\displaystyle=$
$\displaystyle\frac{\nabla^{2}\alpha}{n-2}-\sum_{k=0}^{k<n/2-1}\frac{2}{n+2k-2}\frac{{C^{(k+1)I}\nabla_{I}\alpha}}{r^{n/2+k-1}}+O(r^{-(n-2)}).$
(49)
### IV.2 Covariance of Bondi energy-momentum
Following Ref. Tanabe:2011es , we briefly sketch the argument to show the
covariance of the Bondi energy-momentum. The Bondi energy-momentum
$P^{\mu}_{\text{Bondi}}$ is defined from $g_{uu}$ as in Eq. (36). To find the
variation of $m$, we look at the variation $\delta g_{uu}$. $\delta g_{uu}$
can be expanded near null infinity as
$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle
2\hat{\nabla}_{u}\xi_{u}$ (50) $\displaystyle=$
$\displaystyle\sum_{k=0}^{k<n/2-2}\delta
g^{(k+1)}_{uu}r^{-(n/2+k-1)}+\frac{\delta m}{r^{n-3}}+O(r^{-(n-5/2)}),$
where
$\displaystyle\delta g^{(k+1)}_{uu}$ $\displaystyle=$
$\displaystyle\frac{2}{n+2k}[\nabla^{2}(\alpha A^{(k)})+(n-2)\alpha
A^{(k)}]+\frac{4}{(n+2k)(n-2k-2)}\nabla^{I}\nabla^{J}(\alpha\dot{h}^{(k+1)}_{IJ})$
(51)
$\displaystyle-\frac{2(n+2k-6)}{(n+2k)(n-2k-2)}[\nabla^{I}\nabla^{J}(\nabla_{I}\alpha
C^{(k)}_{J})+C^{(k)}_{I}\nabla^{I}\alpha]$
for $0\leq k<n/2-2$. $\delta m$ is given by
$\displaystyle\delta m$ $\displaystyle=$
$\displaystyle\,\alpha\dot{m}+\frac{2}{n-3}\nabla^{I}\alpha\dot{C}_{I}^{(n/2-1)}-(n-4)\alpha
A^{(n/2-2)}+\nabla^{I}\alpha\nabla_{I}A^{(n/2-2)}$ (52) $\displaystyle=$
$\displaystyle-\frac{\alpha}{2(n-2)}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}+\frac{1}{n-2}\Big{[}\nabla^{2}(\alpha
A^{(n/2-2)})+(n-2)\alpha A^{(n/2-2)}\Big{]}$
$\displaystyle-\frac{n-5}{n-2}\Big{[}\nabla^{I}\nabla^{J}(\nabla_{I}\alpha
C_{J}^{(n/2-2)})+C^{(n/2-2)}_{I}\nabla^{I}\alpha\Big{]},$
where we used Eqs. (16) and (22). From Eq. (51) and the fact that
$\hat{x}^{(i)}$ satisfies
$\nabla_{I}\nabla_{J}\hat{x}^{(i)}+\hat{x}^{(i)}\omega_{IJ}=0$, we see
$\displaystyle\int_{S^{n-2}}\delta g_{uu}^{(k+1)}\,=\,0$ (53)
and
$\displaystyle\int_{S^{n-2}}\hat{x}^{(i)}\delta g_{uu}^{(k+1)}\,=\,0$ (54)
for $k<n/2-2$. This means that $g_{uu}^{(k+1)}$ for $k<n/2-2$ does not
contribute to the global quantities in the transformed Bondi coordinates.
The variation of the Bondi energy-momentum $\delta P^{\mu}_{\text{Bondi}}$ can
be obtained by integrating Eq. (52) as
$\displaystyle\delta P^{\mu}_{\text{Bondi}}$ $\displaystyle=$
$\displaystyle\frac{n-2}{16\pi G}\int_{S^{n-2}}\hat{x}^{\mu}\delta md\Omega$
(55) $\displaystyle=$ $\displaystyle-\frac{1}{16\pi
G}\int_{S^{n-2}}\alpha\hat{x}^{\mu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega$
$\displaystyle=$ $\displaystyle-\alpha_{\nu}\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}^{\mu}\hat{x}^{\nu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega.$
In the last line of the above, we used $\alpha=\alpha_{\mu}\hat{x}^{\mu}$.
Then we regard the right-hand side of this equation as
$\alpha_{\nu}dP^{\mu\nu}_{\text{Bondi}}/du$ in Eq. (46)
$\displaystyle\frac{d}{du}P^{\mu\nu}_{\text{Bondi}}=-\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}^{\mu}\hat{x}^{\nu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega.$
(56)
This means that the Bondi energy-momentum defined is transformed covariantly
with respect to the Poincaré group. In particular, since the time-component
becomes $dP^{\mu 0}_{\text{Bondi}}/du=dP^{\mu}_{\text{Bondi}}/du$, we have
$P^{\mu}_{\text{Bondi}}\rightarrow
P_{\text{Bondi}}^{\mu}+\alpha\frac{d}{du}P_{\text{Bondi}}^{\mu},$ (57)
for the time-translation.
### IV.3 Poincaré covariance of Bondi angular momentum
Next we investigate the variation of the Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ by the translation. The Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ is identified with a space-space component of
$M^{\text{Bondi}}_{\mu\nu}$ in Eq. (46). Thus, in the following, we consider
the space-space components. Note that the time-space component of
$M^{\text{Bondi}}_{\mu\nu}$ represents the Lorentz boost. The $l=0$ mode of
$\alpha$ generates the time-translation and the $l=1$ modes generate the
translations in the spatial directions.
The Bondi angular momentum is defined using a part of $g_{uI}$ as Eq. (37).
The variation of the Bondi angular momentum is given by
$\displaystyle\delta J_{(p)}^{\text{Bondi}}=-\frac{n-1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}\delta j_{I}d\Omega.$ (58)
The variation $\delta g_{uI}$ can be expanded as
$\displaystyle\delta g_{uI}$ $\displaystyle=$
$\displaystyle\hat{\nabla}_{u}\xi_{I}+\hat{\nabla}_{I}\xi_{u}$ (59)
$\displaystyle=:$ $\displaystyle\sum_{k=0}^{k<n/2-1}\frac{\delta
g^{(k+1)}_{uI}}{r^{n/2+k-2}}+\frac{\delta
g^{(n/2)}_{uI}}{r^{n-3}}+O(r^{-(n-5/2)}),$
where
$\displaystyle\delta g_{uI}^{(k+1)}$ $\displaystyle=$
$\displaystyle-A^{(k)}\nabla_{I}\alpha+\frac{2}{n+2k-4}\nabla_{I}(C^{(k)J}\nabla_{J}\alpha)-C^{(k)J}\nabla_{I}\nabla_{J}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(k)}_{I}$
(60)
$\displaystyle-\frac{n+2k-6}{2(n-2)}C^{(k)}_{I}\nabla^{2}\alpha+\alpha\dot{C}_{I}^{(k+1)}+\frac{2}{n+2k}\dot{h}^{(k+1)}_{IJ}\nabla^{J}\alpha$
for $k<n/2-1$. Then we find
$\displaystyle\varphi^{I}_{(p)}\delta g_{uI}^{(k+1)}$ $\displaystyle=$
$\displaystyle\nabla_{I}\Bigg{[}\frac{2}{n+2k-4}\varphi^{I}_{(p)}C^{(k)J}\nabla_{J}\alpha-\frac{4(n-4)}{(n+2k)(n+2k-6)}\varphi^{I}_{(p)}\alpha
A^{(k)}$ (61)
$\displaystyle~{}~{}+\frac{2\Big{[}n^{2}+(4k-10)n+(4k^{2}-12k+16)\Big{]}}{(n+2k)(n-2k-2)(n+2k-4)}C^{(k)I}\varphi^{J}_{(p)}\nabla_{J}\alpha$
$\displaystyle~{}~{}-\frac{2}{n+2k}\Big{[}\alpha\varphi_{(p)J}\nabla^{I}C^{(k)J}-\varphi^{J}_{(p)}C^{(k)}_{J}\nabla^{I}\alpha-\alpha
C^{(k)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Big{]}-\varphi_{(p)}^{J}C_{J}^{(k)}\nabla^{I}\alpha$
$\displaystyle~{}~{}+\frac{n+2k-4}{n+2k}\alpha\varphi_{(p)J}h^{(k)IJ}+\frac{2(n+2k-6)}{(n+2k)(n-2k-2)}h^{(k)IJ}\nabla^{K}\alpha\nabla_{K}\varphi_{(p)J}$
$\displaystyle~{}~{}+\frac{n^{2}-8n-4k^{2}+20}{(n-2)(n+2k)(n-2k-2)}\varphi_{(p)J}h^{(k)IJ}\nabla^{2}\alpha\Bigg{]},$
where we used Eqs. (14), (16) and (21). Also we used the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$ and
$\nabla_{I}\nabla_{J}\alpha=\omega_{IJ}\nabla^{2}\alpha/(n-2)$. Thus we could
confirm again that $g_{uI}^{(k+1)}$ for $k<n/2-1$ does not contribute to the
global quantities because it can be written by the total derivative as Eq.
(61). For $k=n/2-1$, on the other hand, the variation becomes
$\displaystyle\delta g_{uI}^{(n/2)}$ $\displaystyle=$
$\displaystyle\delta(j_{I}+h^{(1)}_{IJ}C^{(1)J})$ (62) $\displaystyle=$
$\displaystyle\alpha\partial_{u}(j_{I}+h^{(1)}_{IJ}C^{(1)J})+\frac{2}{n}h^{(1)}_{IJ}\partial_{u}h^{(1)JL}\nabla_{L}\alpha+{\frac{1}{n-3}}\nabla_{I}(C^{(n/2-1)}_{J}\nabla^{J}\alpha)$
$\displaystyle+m\nabla_{I}\alpha-C^{(n/2-1)J}\nabla_{I}\nabla_{J}\alpha-\frac{1}{n-1}\left(\partial_{u}B^{(1)}\nabla_{I}\alpha-\dot{h}^{(n/2)}_{IJ}\nabla^{J}\alpha{+}\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha\right)$
$\displaystyle-\frac{n-4}{n-2}C^{(n/2-1)}_{I}\nabla^{2}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(n/2-1)}_{I}.$
We must evaluate $\delta j_{I}$ to see the variation of the Bondi angular
momentum. Therefore, we should subtract the variation
$\delta(h^{(1)}_{IJ}C^{(1)J})$ from Eq. (62). The variation $\delta C^{(1)I}$
is given by
$\displaystyle\delta
C^{(1)I}\,=\,\frac{2}{n}\nabla_{J}\alpha\dot{h}^{(1)IJ}+\alpha\dot{C}^{(1)I},$
(63)
from Eq. (60) for $k=0$. Since $\delta g_{IJ}$ is
$\displaystyle\delta g_{IJ}$ $\displaystyle=$
$\displaystyle\hat{\nabla}_{I}\xi_{J}+\hat{\nabla}_{J}\xi_{I}$ (64)
$\displaystyle=$ $\displaystyle
r^{2}\left(\frac{\alpha\dot{h}^{(1)}_{IJ}}{r^{n/2-1}}+O(r^{-(n/2-1/2)})\right),$
we find $\delta h^{(1)}_{IJ}=\alpha\dot{h}^{(1)}_{IJ}$. Then we have
$\delta(h^{(1)}_{IJ}C^{(1)J})\,=\,{\alpha\left(\dot{h}^{(1)}_{IJ}C^{(1)J}+h^{(1)}_{IJ}\dot{C}^{(1)J}\right)+\frac{2}{n}h^{(1)}_{IJ}\dot{h}^{(1)JK}\nabla_{K}\alpha.}$
(65)
Subtracting the above from Eq. (62), we obtain
$\displaystyle\varphi^{I}_{(p)}\delta j_{I}$ $\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}\alpha\dot{j}_{I}+\frac{1}{n-3}\nabla_{I}(C^{(n/2-1)}_{J}\nabla^{J}\alpha)+m\nabla_{I}\alpha-C^{(n/2-1)J}\nabla_{I}\nabla_{J}\alpha$
(66)
$\displaystyle~{}~{}-\frac{1}{n-1}\left(\dot{B}^{(1)}\nabla_{I}\alpha-\dot{h}^{(n/2)}_{IJ}\nabla^{J}\alpha+\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha\right)$
$\displaystyle~{}~{}-\frac{n-4}{n-2}C^{(n/2-1)}_{I}\nabla^{2}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(n/2-1)}_{I}\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{n-1}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}+\frac{n-2}{n-1}m\varphi^{I}_{(p)}\nabla_{I}\alpha$
$\displaystyle+\nabla_{I}\Bigg{[}-\frac{1}{n-1}\Big{[}\alpha\varphi^{I}_{(p)}(\dot{B}^{(1)}-m)-\alpha\varphi_{(p)J}\dot{h}^{(n/2)IJ}+\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha$
$\displaystyle+\alpha\varphi^{J}_{(p)}\nabla^{I}C^{(n/2-1)}_{J}-\alpha
C^{(n/2-1)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Big{]}+\frac{1}{n-3}\varphi^{I}_{(p)}C^{(n/2-1)J}\nabla_{J}\alpha-\frac{n-2}{n-1}\varphi^{J}C^{(n/2-1)}_{J}\nabla^{I}\alpha$
$\displaystyle+\frac{n-3}{n-1}\Big{[}\varphi_{(p)J}h^{(n/2-1)IJ}\alpha+\frac{1}{(n-2)^{2}}\varphi_{(p)J}h^{(n/2-1)IJ}\nabla^{2}\alpha+\frac{n-3}{n-2}h^{(n/2-1)IJ}\nabla_{K}\varphi_{(p)J}\nabla^{K}\alpha\Big{]}\Bigg{]},$
where we used Eq. (43), the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$ and
$\nabla_{I}\nabla_{J}\alpha=\omega_{IJ}\nabla^{2}\alpha/(n-2)$.
Using Eq. (66), the variation of the Bondi angular momentum $\delta
J^{\text{Bondi}}_{(p)}$ becomes
$\displaystyle\delta J^{\text{Bondi}}_{(p)}$ $\displaystyle=$
$\displaystyle-\frac{n-1}{16\pi G}\int_{S^{n-2}}\varphi^{I}_{(p)}\delta
j_{I}d\Omega$ (67) $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G}\int_{S^{n-2}}\alpha\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega$
$\displaystyle-\frac{n-2}{16\pi
G}\int_{S^{n-2}}m\varphi^{I}_{(p)}\nabla_{I}\alpha d\Omega.$
Note that the total derivative terms in Eq. (66) do not contribute to $\delta
J^{\text{Bondi}}_{(p)}$.
From the result of Eq. (67), we can show that Eq. (46) holds as follows. We
note that a rotational Killing vector $\varphi^{I}_{(p)}$ can be rewritten as
$\varphi^{I}_{(p)}=\varphi_{ij}\hat{x}^{(i)}\nabla^{I}\hat{x}^{(j)}$ where
$\varphi_{ij}$ is a constant anti-symmetric tensor with $(n-1)(n-2)/2$
independent components. Using the relation
$\nabla_{I}\hat{x}^{(i)}\nabla^{I}\hat{x}^{(j)}=\delta^{ij}-\hat{x}^{(i)}\hat{x}^{(j)}$,
we have
$\varphi^{I}_{(p)}\nabla_{I}\alpha=\varphi_{ij}\alpha^{j}\hat{x}^{(i)}$. Since
$J^{\text{Bondi}}_{(p)}$ is expressed by
$J^{\text{Bondi}}_{(p)}=\varphi^{ij}M^{\text{Bondi}}_{ij}$, Eq. (67) yields
$\varphi^{ij}M^{\text{Bondi}}_{ij}\rightarrow\varphi^{ij}\Big{[}M^{\text{Bondi}}_{ij}-2P^{\text{Bondi}}_{[i}\alpha_{j]}+\alpha^{\mu}\frac{d}{du}M^{\text{Bondi}}_{ij\mu}\Big{]},$
(68)
where we wrote
$\varphi^{ij}\frac{d}{du}M^{\text{Bondi}}_{ij\mu}=\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}_{\mu}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega.$
(69)
Note that the indices $\mu$, $\nu$, $\dots$ and $i$, $j$, $\dots$ are raised
and lowered by the $n$-dimensional Minkowski metric and the
$(n-1)$-dimensional Euclidean flat metric, respectively. Consequently, we
could show that the Bondi angular momentum $J^{\text{Bondi}}_{(p)}$ is
transformed covariantly as Eq. (46).
Here we have a comment on the four dimensional cases. Because there is no
condition on $\alpha$ in four dimensions, we cannot obtain the expressions
corresponding to Eqs. (66) and (67). Therefore the Bondi angular momentum is
not transformed as Eq. (68) in four dimensions. In fact, the variation of the
Bondi angular momentum has additional contributions from supertranslations,
which are given by $l>1$ modes of spherical harmonics in $\alpha$. This is
called the supertranslation ambiguity of the angular momentum at null
infinity. Hence, we cannot have well-defined notion of the angular momentum at
null infinity in four dimensions.
## V summary and outlook
In this paper we defined the Bondi angular momentum at null infinity in
arbitrary higher dimensions and showed its covariant property with respect to
the asymptotic symmetry at null infinity. The asymptotic symmetry becomes the
Poincaré group in higher dimensions than four. This means that we can choose
the $n$ directions of the translation without any ambiguities at null
infinity. Then the angular momentum with the rotational axis can be defined.
In four dimensions, on the other hand, the asymptotic symmetry at null
infinities has the supertranslation, not the translation. The supertranslation
has the infinite directions of the translation. Hence there are ambiguities of
the choice of the rotational axis. This effects of the freedom in the
definition of the rotational axis due to the supertranslation cannot be
distinguished from the contributions of the variation of angular momentum by
gravitational waves. This is the reason why we cannot define the angular
momentum at null infinity in four dimensions.
As one of applications of our analysis, there is the investigation of the
peeling theorem in higher dimensions Bondi:1962px ; Sachs:1962wk ;
Godazgar:2012zq . The peeling property has played an important role in the
study of the gravity in four dimensions, such as the stability analysis of
black holes and construction of the exact solutions. We expect that the
peeling theorem is useful in higher dimensions too. Using our results, general
higher dimensional spacetimes with gravitational waves are classified by the
decaying rate of the Weyl tensor or some geometric quantities. The effort for
this direction has been reported Godazgar:2012zq .
## Acknowledgment
KT is supported by JSPS Grant-in-Aid for Scientific Research (No. 21-2105).
This work is supported in part by MEXT thorough Grant-in-Aid for Scientific
Research (A) No. 21244033 (TS) and Grant-in-Aid for Creative Scientific
Research No. 19GS0219 (TS and SK). This work is also supported in part by MEXT
through Grant-in-Aid for the Global COE Program “The Next Generation of
Physics, Spun from Universality and Emergence” at Kyoto University.
## References
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* (3) R. Emparan and H. S. Reall, Living Rev. Rel. 11,6 (2008); K. Maeda, T. Shiromizu and T. Tanaka(Eds.), “Higher Dimensional Black Holes”, Progress of Theoretical Physics Supplement No. 189(2011).
* (4) K. Tanabe, N. Tanahashi and T. Shiromizu, J. Math. Phys. 50, 072502 (2009) [arXiv:0902.1583 [gr-qc]].
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* (7) A. Ishibashi, Class. Quant. Grav. 25, 165004 (2008) [arXiv:0712.4348 [gr-qc]].
* (8) K. Tanabe, N. Tanahashi and T. Shiromizu, J. Math. Phys. 51, 062502 (2010)
* (9) K. Tanabe, N. Tanahashi and T. Shiromizu, J. Math. Phys. 52, 032501 (2011) arXiv:1010.1664 [gr-qc].
* (10) K. Tanabe, S. Kinoshita and T. Shiromizu, Phys. Rev. D 84, 044055 (2011) [arXiv:1104.0303 [gr-qc]].
* (11) R. O. Hansen, J. Math. Phys. 15, 46 (1974).
* (12) K. Tanabe, S. Ohashi and T. Shiromizu, Phys. Rev. D 82, 104042 (2010) [arXiv:1009.1486 [gr-qc]].
* (13) S. W. Hawking and G. F. R. Ellis, The Large scale structure of space-time, (Cambridge Univ. Press, Cambridge, 1973).
* (14) H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, Proc. Roy. Soc. Lond. A 269, 21 (1962).
* (15) R. K. Sachs, Proc. Roy. Soc. Lond. A 270, 103 (1962).
* (16) C. R. Prior, Proc. Roy. Soc. Lond. A 354, 379 (1977).
* (17) M. Streubel, Gen. Rel. Grav. 9, 551 (1978).
* (18) J. H. Winicour, “Angular momentum in general relativity,” in A. Held, editor, “General Relativity and Gravitation,” volume 2 (1980).
* (19) R. P. Geroch and J. Winicour, J. Math. Phys. 22, 803 (1981).
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|
arxiv-papers
| 2012-03-02T13:10:11 |
2024-09-04T02:49:28.186805
|
{
"license": "Public Domain",
"authors": "Kentaro Tanabe, Tetsuya Shiromizu, Shunichiro Kinoshita",
"submitter": "Kentaro Tanabe",
"url": "https://arxiv.org/abs/1203.0452"
}
|
1203.0595
|
# Entanglement and nonclassicality of photon-added two-mode squeezed thermal
state
Li-Yun Hu1,2,†, Fang Jia1 and Zhi-Ming Zhang2,∗ E-mail:
hlyun2008@126.com.E-mail: zmzhang@scnu.edu.cn 1Department of physics, Jiangxi
Normal University, Nanchang 330022, China
2Laboratory of Nanophotonic Functional Materials and Devices, SIPSE & LQIT,
South China Normal University, Guangzhou 510006, China
${\dagger}$Email: hlyun@jsnu.edu.cn; $\ast$Email: zmzhang@scnu.edu.cn.
###### Abstract
We introduce a kind of entangled state—photon-addition two-mode squeezed
thermal state (TMSTS) by adding photons to each mode of the TMSTS. Using the
P-representation of thermal state, the compact expression of the normalization
factor is derived, a Jacobi polynomial. The nonclassicality is investigated by
exploring especially the negativity of Wigner function. The entanglement is
discussed by using Shchukin-Vogel criteria. It is shown that the photon-
addtion to the TMSTS may be more effective for the entanglement enhancement
than the photon-subtraction from the TMSTS. In addition, the quantum
teleportation is also examined, which shows that symmetrical photon-added
TMSTS may be more useful for quantum teleportation than the non-symmetric
case.
PACS number(s): 42.50.Dv, 03.65.Wj, 03.67.Mn
## I Introduction
Quantum entanglement with continuous-variable is an essential resource in
quantum information processing 1 , such as teleportation, dense coding, and
quantum cloning. In a quantum optics laboratory, a Gaussian two-mode squeezed
vacuum state is ofen used as entangled resource, which cannot be distilled
only by Gaussian local operators and classical communications due to the
limitation from the no-go theorem 2 ; 3 ; 4 . To satisfy the requirement of
quantum information protocols for long-distance communication, there have been
suggestions and realizations for engineering the quantum state, which are
plausible ways to conditionally manipulate a nonclassical state of an optical
field by subtracting or adding photons from/to a Gaussian field 5 ; 6 ; 7 ; 8
; 8a ; 9 ; 10 . Actually, the photon addition and subtraction have been
successfully demonstrated experimentally for probing quantum commutation rules
by Parigi et al. 11 .
In order to increase quantum entanglement, two-mode photon-subtraction
squeezed vacuum states (TPSSV) have received more attention from both
experimentalists and theoreticians 5 ; 9 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ;
19 ; 20 ; 21 . Olivares et al. 12 considered the photon subtraction using
on–off photo detectors and showed improvement of quantum teleportation,
depending on the various parameters involved. Kitagawa et al. 13 , on the
other hand, investigated the degree of entanglement for the TPSSV by using an
on–off photon detector. Using an operation with single photon counts,
Ourjoumtsev et al. 14 ; 15 demonstrated experimentally that entanglement
between Gaussian entangled states can be increased by subtracting only one
photon from two-mode squeezed vacuum states. In addition, Lee et. al 21
proposed a coherent superposition of photon subtraction and addition to
enhance quantum entanglement of two-mode Gaussian sate. It is shown that,
especially for the small-squeezing regime, the effects of coherent operation
are more prominent than those of the mere photon subtraction and the photon
addition.
Recently, we proposed the any photon-added squeezed thermal state
theoretically, and investigated its nonclassicality by exploring the sub-
Poissonian and negative Wigner function (WF) 22 . The results show that the WF
of single photon-added squeezed thermal state (PASTS) always has negative
values at the phase space center. The decoherence effect on the PASTS is
examined by the analytical expression of WF. It is found that a longer
threshold value of decay time is included in single PASTS than in single-
photon subtraction squeezed thermal state (STS). In this paper, as a natural
extension, we shall introduce a kind of nonclassical state—photon-addition
two-mode STS (PA-TMSTS), generated by adding photons to each mode of two-mode
STS (TMSTS) which can be considered as a generalized bipartite Gaussian state.
Then we shall investigate the entanglement and nonclassical properties.
This paper is organized as follows. In Sec. II we introduce the PA-TMSTS. By
using the P-representation of density operator of thermal state, we derive the
normal ordering and anti-normal form of the TMSTS, which is convenient to
obtain distribution function, such as Q-function and WF. Then a compact
expression for the normalization factor of the PA-TMSTS, which is a Jacobi
polynomial of squeezing parameter $r$ and mean number $\bar{n}$ of thermal
state. In Sec III, we present the nonclassical properties of the PA-TMSTS in
terms of cross-correlation function, distribution of photon number,
antibunching effect and the negativity of its WF. It is shown that the WF lost
its Gaussian property in phase space due to the presence of two-variable
Hermite polynomials and the WF of single PA-TMSTS always has its negative
region at the center of phase space. Then, in Secs. IV and V are devoted to
discussing the entanglement properties of the PA-TMSTS by Shchukin-Vogel
criteria and the quantum teleportation. The conclusions are involved in Sec.
VI.
## II Photon-addition two-mode squeezed thermal state (PA-TMSTS)
As Agarwal et al 23 . introduced the excitations on a coherent state by
repeated application of the photon creation operator on the coherent state, we
introduce theoretically the photon-addition two-mode squeezed thermal state
(PA-TMSTS).
For two-mode case, the photon-added scheme can be presented by the mapping
$\rho\rightarrow a^{{\dagger}m}b^{{\dagger}n}\rho a^{m}b^{{\dagger}n}$. Here
we introduce the PA-TMSTS, which can be generated by repeatedly operating the
photon creation operator $a^{\dagger}$ and $b^{\dagger}$ on a two-mode
squeezed thermal state (TMSTS), so its density operator is
$\rho^{SA}\equiv N_{m,n}^{-1}{}a^{\dagger m}b^{\dagger
n}S\left(r\right)\rho_{th1}\rho_{th2}S^{\dagger}\left(r\right)a^{m}b^{n},$ (1)
where $m,n$ are the added photon number to each mode (non-negative integers),
and $N_{m,n}$ is the normalization of the PA-TMSTS to be determined by
$\mathtt{tr}\rho^{SA}=1$, and $S(r)=\exp[r(a^{\dagger}b^{\dagger}-ab)]$ is the
two-mode squeezing operator with squeezing parameter $r$. Here $\rho_{th1,2}$
is a density operator of single-mode thermal state,
$\rho_{th1,2}=\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{\left(\bar{n}+1\right)^{n+1}}\left|n\right\rangle\left\langle
n\right|,$ (2)
where $\bar{n}$ is the average photon number of thermal state $\rho_{thj}$
($j=1,2$). For simplicity, we assume the average photon number of $\rho_{thj}$
($j=1,2$) to be identical. In addition, the P-representation of density
operator $\rho_{thj}$ can be expanded as 24
$\rho_{thj}=\frac{1}{\bar{n}}\int\frac{d^{2}\alpha}{\pi}e^{-\frac{1}{\bar{n}}\left|\alpha\right|^{2}}\left|\alpha\right\rangle\left\langle\alpha\right|,$
(3)
which is useful for later calculation and here $\left|\alpha\right\rangle$ is
the coherent state.
### II.1 Normal ordering and anti-normal form of the TMSTS
In order to simplify our calculation, here we shall derive the normally
ordering form of the TMSTS. For this purpose, we examine the two-mode squeezed
coherent states $S\left|\alpha,\beta\right\rangle$
($\left|\alpha,\beta\right\rangle=\left|\alpha\right\rangle\otimes\left|\beta\right\rangle$).
Note that
$\left|\alpha\right\rangle=\exp[-\frac{1}{2}\left|\alpha\right|^{2}+\alpha
a^{\dagger}]\left|0\right\rangle$ and the following transformation relations
25 ; 26 :
$\displaystyle S(r)a^{\dagger}S^{\dagger}(r)$ $\displaystyle=a^{\dagger}\cosh
r-b\sinh r,$ $\displaystyle S(r)b^{\dagger}S^{\dagger}(r)$
$\displaystyle=b^{\dagger}\cosh r-a\sinh r,$ (4)
we see
$\displaystyle S\left|\alpha,\beta\right\rangle$ $\displaystyle=$
$\displaystyle\text{sech}r\exp\left[-\frac{1}{2}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})\right]$
(5) $\displaystyle\times\exp\left[\alpha\left(a^{\dagger}\cosh r-b\sinh
r\right)\right]$ $\displaystyle\times\exp[\beta\left(b^{\dagger}\cosh r-a\sinh
r\right)]$ $\displaystyle\times\exp\left[a^{\dagger}b^{\dagger}\tanh
r\right]\left|00\right\rangle,$
where $S(r)\left|00\right\rangle=$sech$r\exp\left[a^{\dagger}b^{\dagger}\tanh
r\right]\left|00\right\rangle$ is used.
Further noting $e^{\tau a}a^{\dagger}e^{-\tau a}=a^{\dagger}+\tau,$ and for
operators $A,B$ satisfying the conditions
$\left[A,[A,B]\right]=\left[B,[A,B]\right]=0,$ we have
$e^{A+B}=e^{A}e^{B}e^{-[A,B]/2}=e^{B}e^{A}e^{[A,B]/2},$ thus Eq.(5) can be put
into the following form
$\displaystyle S\left|\alpha,\beta\right\rangle$ $\displaystyle=$
$\displaystyle\text{sech}r\exp\left[-\frac{1}{2}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})-\alpha\beta\tanh
r\right]$ (6)
$\displaystyle\times\exp\left[\left(a^{\dagger}\alpha+b^{\dagger}\beta\right)\text{sech}r+a^{\dagger}b^{\dagger}\tanh
r\right]\left|00\right\rangle.$
Thus inserting Eq.(6) into Eq.(3) and using the vacuum projector
$\left|00\right\rangle\left\langle
00\right|=\colon\exp[-a^{\dagger}a-b^{\dagger}b]\colon$ (where $\colon\colon$
denotes the normally ordering) as well as the IWOP technique 27 ; 28 , we can
obtain
$\displaystyle\rho^{S}$ $\displaystyle\equiv$ $\displaystyle
S\rho_{th1}\rho_{th2}S^{\dagger}$ (7) $\displaystyle=$
$\displaystyle\frac{1}{\bar{n}^{2}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}e^{-\frac{1}{\bar{n}}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})}S\left|\alpha,\beta\right\rangle\left\langle\alpha,\beta\right|S^{\dagger}$
$\displaystyle=$ $\displaystyle
A_{1}\colon\exp\left[A_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-A_{3}\left(a^{\dagger}a+b^{\dagger}b\right)\right]\colon,$
where we have set
$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{\text{sech}^{2}r}{\left(\bar{n}+1\right)^{2}-\bar{n}^{2}\tanh^{2}r},$
$\displaystyle A_{2}$ $\displaystyle=$
$\displaystyle\frac{\left(2\bar{n}+1\right)\sinh r\cosh
r}{\left(2\bar{n}+\allowbreak 1\right)\cosh^{2}r+\bar{n}^{2}},$ $\displaystyle
A_{3}$ $\displaystyle=$ $\displaystyle\frac{\allowbreak\cosh^{2}r+\bar{n}\cosh
2r}{\left(2\bar{n}+\allowbreak 1\right)\cosh^{2}r+\bar{n}^{2}},$ (8)
and used the integration formula 29
$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\zeta<0.$ (9)
Eq.(7) is just the normally ordering form of TMSTS to be used to realize our
calculations below.
In addition, using Eqs.(7), (9) and the formula converting any single-mode
operator $\hat{O}$ into its anti-normal ordering form 30 ,
$\hat{O}=\vdots\int\frac{d^{2}z}{\pi}\left\langle-z\right|\hat{O}\left|z\right\rangle
e^{|z|^{2}+z^{\ast}a-za^{\dagger}+a^{\dagger}a}\vdots,$ (10)
where $\left|z\right\rangle$ is the coherent state, and the symbol $\vdots$
$\vdots$ denotes antinormal ordering, (note that the order of Bose operators
$a$ and $a^{\dagger}$ within $\vdots$ $\vdots$ can be permuted), one can
obtain the anti-normal ordering form of the TMSTS,
$\rho^{S}=\tilde{A}_{1}\vdots\exp\left[\tilde{A}_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-\tilde{A}_{3}\left(a^{\dagger}a+b^{\dagger}b\right)\right]\vdots,$
(11)
where we have set
$\displaystyle\tilde{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\bar{n}+1\right)^{2}-\left(2\bar{n}+1\right)\cosh^{2}r},$
$\displaystyle\tilde{A}_{2}$ $\displaystyle=$
$\displaystyle\frac{\left(2n+1\right)\sinh r\cosh
r}{\left(\bar{n}+1\right)^{2}-\left(2\bar{n}+1\right)\cosh^{2}r},$
$\displaystyle\tilde{A}_{3}$ $\displaystyle=$
$\displaystyle\frac{\sinh^{2}r+n\cosh
2r}{\left(\bar{n}+1\right)^{2}-\left(2\bar{n}+1\right)\cosh^{2}r}.$ (12)
Eq.(11) implies that the P function $P(\alpha,\beta)$ of the TMSTS is
$P(\alpha,\beta)=\tilde{A}_{1}\exp\left[\tilde{A}_{2}\left(\alpha^{\ast}\beta^{\ast}+\alpha\beta\right)-\tilde{A}_{3}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})\right],$
(13)
which leads to the P representation of density operator
$S\rho_{th1}\rho_{th2}S^{\dagger}$ i.e.,
$\rho^{S}=\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}P(\alpha,\beta)\left|\alpha,\beta\right\rangle\left\langle\alpha,\beta\right|.$
(14)
In particular, for the case without squeezing, $r=0,$ then Eqs.(7) and (11)
just reduce to, respectively,
$\displaystyle\rho^{S}\left(r=0\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\bar{n}+1\right)^{2}}\colon\exp\left[-\frac{a^{\dagger}a+b^{\dagger}b}{\bar{n}+1}\right]\colon$
(15) $\displaystyle=$
$\displaystyle\frac{1}{\bar{n}^{2}}\vdots\exp\left[-\frac{1}{\bar{n}}\left(a^{\dagger}a+b^{\dagger}b\right)\right]\vdots,$
as expected 24 . It is interesting to notice that, for the case of
$\bar{n}=0$, corresponding to the two-mode squeezed vaccum state (TMSVS),
Eqs.(7) and (11) become
$\displaystyle\rho^{S}\left(\bar{n}=0\right)$ (16) $\displaystyle=$
$\displaystyle\text{sech}^{2}r\colon\exp\left[\left(a^{\dagger}b^{\dagger}+ab\right)\tanh
r-\left(a^{\dagger}a+b^{\dagger}b\right)\right]\colon$ $\displaystyle=$
$\displaystyle-\text{csch}^{2}r\vdots\exp\left[a^{\dagger}a+b^{\dagger}b-\left(a^{\dagger}b^{\dagger}+ab\right)\coth
r\right]\vdots,$
which are just the normal ordering form and anti-normal ordering form of the
TMSVS. The second equation in Eq.(16) seems a new result. Here, we should
mention that the normal (anti-)normal ordering forms of the TMSTS are useful
to higher-order squeezing and photon statistics 31 ; 32 for the TMSTS.
### II.2 Normalization of the PA-TMSTS
To fully describe a quantum state, its normalization is usually necessary.
Using Eq.(7), the PA-TMSTS reads as
$\rho^{SA}=\frac{A_{1}}{N_{m,n}}\colon a^{\dagger m}b^{\dagger
n}e^{A_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-A_{3}\left(a^{\dagger}a+b^{\dagger}b\right)}a^{m}b^{n}\colon.$
(17)
Thus using the completeness relation of coherent state $\int d^{2}\alpha
d^{2}\beta\left|\alpha,\beta\right\rangle\left\langle\alpha,\beta\right|/\pi^{2}=1$
and Eq.(9), the normalization factor $N_{m,n}$ is given by (Appendix A)
$N_{m,n}=\left.\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime n}}e^{B_{1}\left(\tau
t+\tau^{\prime}t^{\prime}\right)+B_{2}\left(\tau\tau^{\prime}+tt^{\prime}\right)}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0},$
(18)
where
$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle\cosh^{2}r+\bar{n}\cosh
2r,$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle\left(2\bar{n}+1\right)\sinh r\cosh r.$ (19)
Here we introduce a new expression of generating function for Jacobi
polynomials in form (Proof see Appendix B)
$\displaystyle\left.\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}e^{A\left(\tau^{\prime}t^{\prime}+\tau
t\right)+B\left(\tau\tau^{\prime}+t^{\prime}t\right)}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}$
$\displaystyle=m!n!\left\\{\begin{array}[]{cc}A^{n-m}\left(B^{2}-A^{2}\right)^{m}P_{m}^{(n-m,0)}\left(\frac{B^{2}+A^{2}}{B^{2}-A^{2}}\right)&m\leqslant
n\\\ &\\\
A^{m-n}\left(B^{2}-A^{2}\right)^{n}P_{n}^{(m-n,0)}\left(\frac{B^{2}+A^{2}}{B^{2}-A^{2}}\right)&n\leqslant
m\end{array}\right.,$ (23)
thus the normalization factor $N_{m,n}$ can be put into (without loss of
generality assuming $m\leqslant n$)
$N_{m,n}=m!n!B_{1}^{n-m}\omega^{m}P_{m}^{(0,n-m)}\left(\frac{\upsilon}{\omega}\right),$
(24)
where we have used the property of the Jacobi polynomials
$P_{m}^{(\alpha,\beta)}(-x)=(-1)^{m}P_{m}^{(\beta,\alpha)}(x),$and
$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\bar{n}^{2}+\left(2\bar{n}+1\right)\cosh^{2}r,$
$\displaystyle\upsilon$ $\displaystyle=$
$\displaystyle\bar{n}\left(\bar{n}+1\right)\cosh
4r+\left(\allowbreak\bar{n}+\cosh^{2}r\right)\cosh 2r.$ (25)
Eq.(24) indicates that the normalization factor is related to the Jacobi
polynomials, which is important for further studying analytically the
statistical properties of the PA-TMSTS. Note Eq.(24) exhibits the exchanging
symmetry.
It is clear that, when $m=n=0,$ Eq.(24) just reduces to the TMSTS due to
$P_{0}^{(0,0)}\left(x\right)=1$; while for $n\neq 0$ and $m=0,$ noticing
$P_{0}^{(0,n)}\left(x\right)=1,$ Eq.(24) becomes $N_{0,n}=n!B_{1}^{n}$. For
the case $m=n$, $N_{m,m}$ is related to Legendre polynomial of the parameter
$\frac{\upsilon}{\omega}$, because of $P_{n}^{(0,0)}(x)=P_{n}(x),$
$P_{0}(x)=1$. In addition, when $\bar{n}=0$ leading to
$\omega=B_{1}=\cosh^{2}r\ $and $\frac{\upsilon}{\omega}=\cosh 2r,$ then
Eq.(24) reads
$N_{m,n}\left(\bar{n}=0\right)=m!n!\cosh^{2n}rP_{m}^{(0,n-m)}\left(\cosh
2r\right),$ (26)
which is just the normalization of two-mode photon-added squeezed vacuum state
33 .
## III Nonclassical properties of the PA-TMSTS
In this section, we shall discuss the nonclassical properties of the PA-TMSTS
in terms of cross-correlation function, photon statistics, anti-bunching
effect and the negativity of its WF.
### III.1 Cross-correlation function of the PA-TMSTS
The cross-correlation between the two modes reflects correlation between
photons in two different modes, which plays a key role in rendering many two-
mode radiations nonclassically. From Eqs. (17) and (24) we can easily
calculate the average photon number in the PA-TMSTS,
$\left\langle
a^{\dagger}a\right\rangle=\frac{N_{m+1,n}}{N_{m,n}}-1,\left\langle
b^{\dagger}b\right\rangle=\frac{N_{m,n+1}}{N_{m,n}}-1,$ (27)
and
$\left\langle
a^{\dagger}b^{\dagger}ab\right\rangle=\frac{N_{m+1,n+1}-N_{m+1,n}-N_{m,n+1}}{N_{m,n}}+1.$
(28)
Thus the cross-correlation function $g_{m,n}$ can be obtained by 34
$\displaystyle g_{m,n}(r)$ $\displaystyle=$ $\displaystyle\frac{\left\langle
a^{\dagger}b^{\dagger}ab\right\rangle}{\left\langle
a^{\dagger}a\right\rangle\left\langle b^{\dagger}b\right\rangle}-1$ (29)
$\displaystyle=$
$\displaystyle\frac{N_{m+1,n+1}N_{m,n}-N_{m,n+1}N_{m+1,n}}{\left(N_{m,n+1}-N_{m,n}\right)\left(N_{m+1,n}-N_{m,n}\right)}.$
The positivity of the cross-correlation function $g_{m,n}$ refers to
correlations between the two modes. In particular, when $m=n=0$ corresponding
to the TMSTS, noticing $N_{0,0}=1,N_{0,1}=N_{1,0}=B_{1}$, and
$N_{1,1}=\upsilon$, then Eq.(29) reduces to
$g_{0,0}(r)=\left(2\bar{n}+1\right)^{2}\sinh^{2}r\cosh^{2}r/\left(B_{1}-1\right)^{2},$which
implies that the parameter $g_{0,0}(r)$ is always positive for any $\bar{n}$
and non-zero squeezing ($B_{1}\neq 1$). Further, for the case of $\bar{n}=0,$
$g_{0,0}(r)=\coth^{2}r,$ which is just the correlation function of the TMSVS;
while for $r=0,$ i.e., the TMSTS, $g_{0,0}(0)=0$, so there is no correlation
between two thermal states, as expected. On the other hand, when $m=0,n=1$,
noticing $N_{1,2}=B_{1}\left(3\upsilon-\omega\right),N_{0,2}=2B_{1}^{2}$, and
$P_{1}^{(0,1)}\left(x\right)=(3x-1)/2,$ then Eq.(29) becomes
$g_{0,1}(r)=\left(\upsilon-\omega\right)B_{1}/[\left(2B_{1}-1\right)\left(\upsilon-
B_{1}\right)]$. Noticing that $\upsilon-B_{1}>0$ and
$\left(2B_{1}-1\right)>0$, and
$\upsilon-\omega=\frac{1}{2}\left(2\bar{n}+1\right)^{2}\sinh^{2}2r\geqslant
0,$ so $g_{0,1}(r)\ $is always positive.
Figure 1: (Color online) Cross-correlation function between the two modes
${\small a}$ and ${\small b}$ as a function of ${\small r}$ for different
parameters (m,n) and ${\small\bar{n}=0.01.}$
In order to see clearly the variation of $g_{m,n}$-parameter, we plot the
graph of $g_{m,n}$ as the function of $r$ for some different ($m,n$) and
$\bar{n}$ values. It is shown that $g_{m,n}$ are always larger than zero, thus
there exist correlations between the two modes. This implies that the
nonclassicality is enhanced by adding photon to squeezed state. For given
($m,n$) and $\bar{n}$ values, $g_{m,n}$ increases as $r$ increasing; while
$g_{m,n}$ decreases as $\bar{n}$ decreasing for a given ($m,n$) value. It is
interesting to notice that for single-photon-addition TMSTS, the $g_{m,n}$
parameter presents its maximum value, which implies that single-photon-
addition TMSTS may possess a stronger nonclassicality than the other TMSTSs.
To compare the further nonclassicality of quantum states for a different
number added case, the measurments based on the volume of the negative part of
the Wigner function 35 , on the nonclassical depth 36 , and on the
entanglement potential 37 , Vogel’s noncalssicality criterion 38 and the
Klyshko criterion 39 may be other alternative methods.
### III.2 Distribution of photon number of the PA-TMSTS
In order to obtain the photon number distribution (PND) of the PA-TMSTS, we
begin with evaluating the PND of TMSTS. For two-mode case described by density
operator $\rho^{S}$, the PND is defined by
$\mathcal{P}(m_{a},n_{b})=\left\langle
m_{a},n_{b}\right|\rho^{S}\left|m_{a},n_{b}\right\rangle.$ Employing the non-
normalized coherent state $\left|\alpha\right\rangle=\exp[\alpha
a^{{\dagger}}]\left|0\right\rangle$ leading to
$\left|n\right\rangle=\frac{1}{\sqrt{n!}}\left.\frac{d^{n}}{d\alpha^{n}}\left|\alpha\right\rangle\right|_{\alpha=0}$
$\left(\left\langle\beta\right.\left|\alpha\right\rangle=e^{\alpha\beta^{\ast}}\right)$,
as well as the normal ordering form of $\rho^{S}$ in Eq.(7), the probability
of finding $\left(m_{a},n_{b}\right)$ photons in the two-mode field is given
by
$\displaystyle\mathcal{P}(m_{a},n_{b})$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{m_{a}!n_{b}!}\frac{d^{2m_{a}+2n_{b}}}{d\alpha^{m_{a}}d\alpha^{\ast
m_{a}}d\beta^{n_{b}}d\beta^{\ast n_{b}}}$ (30)
$\displaystyle\times\left.e^{\left(1-A_{3}\right)\left(\alpha^{\ast}\alpha+\beta^{\ast}\beta\right)+A_{2}\left(\alpha\beta+\alpha^{\ast}\beta^{\ast}\right)}\right|_{\alpha,\beta,\alpha^{\ast},\beta^{\ast}=0}$
$\displaystyle=$ $\displaystyle
A_{1}\left[\allowbreak\bar{n}\left(\bar{n}+1\right)\right]^{n_{b}-m_{a}}\frac{\mu\allowbreak^{m_{a}}}{\nu^{n_{a}}}P_{m_{a}}^{(n_{b}-m_{a},0)}\left(\chi\right),$
where in the last step, we have used the new formula in Eq.(23), and
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle\left(2\bar{n}+\allowbreak
1\right)\cosh^{2}r+\bar{n}^{2},$ $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\left(2\bar{n}+1\right)\cosh^{2}r-\left(\bar{n}+1\right)^{2},$
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{\left(\left(2\bar{n}+1\right)\sinh
2r\right)^{2}+4\allowbreak\bar{n}^{2}\left(\bar{n}+1\right)^{2}}{\left(\left(2\bar{n}+1\right)\sinh
2r\right)^{2}-4\allowbreak\bar{n}^{2}\left(\bar{n}+1\right)^{2}}.$ (31)
Thus the PND of TMSTS is also related to Jacobi polynomials of the parameter
$\chi$. In particular, when $\bar{n}\rightarrow 0$ leading to $\chi\rightarrow
1$, corresponding to the two-mode squeezed vacuum, Eq.(30) reduces to
$\displaystyle\mathcal{P}_{\bar{n}\rightarrow 0}(m_{a},n_{b})$
$\displaystyle=\lim_{\bar{n}\rightarrow
0}\frac{m_{a}!n_{b}!}{\left[\bar{n}^{2}+\left(2\bar{n}+1\right)\cosh^{2}r\right]^{m_{a}+n_{b}+1}}$
$\displaystyle\times\sum_{l=0}^{\min[m_{a},n_{a}]}\frac{\left(\allowbreak\allowbreak
2\bar{n}+1\right)^{2l}\left[\bar{n}\left(\bar{n}+1\right)\right]^{m_{a}+n_{b}-2l}\sinh^{2l}2r}{2^{2l}\left(l!\right)^{2}\left(n_{b}-l\right)!\left(m_{a}-l\right)!}$
$\displaystyle=\frac{\tanh^{2m_{a}}r}{\cosh^{2}r}\delta_{m_{a},n_{b}},$ (32)
which is just the PND of two-mode squeezed vacuum state 23 . On the other
hand, when $r\rightarrow 0$ corresponding to the case of two-mode thermal
state, leading to
$\chi\rightarrow-1,\nu\rightarrow\left(\bar{n}+1\right)^{2},\mu\rightarrow-\bar{n}^{2},A_{1}\rightarrow
1/\left(\bar{n}+1\right)^{2}$ and noting
$P_{m_{a}}^{(n_{b}-m_{a},0)}(-1)=(-1)^{m_{a}},$ thus Eq.(30) becomes
$\mathcal{P}_{r\rightarrow
0}(m_{a},n_{b})=\allowbreak\frac{\bar{n}^{n_{b}}}{\left(\bar{n}+1\right)^{n_{b}+1}}\frac{\bar{n}^{m_{a}}}{\left(\bar{n}+1\right)^{m_{a}+1}},$
(33)
which is just the product of PNDs of two thermal fields, as expected.
Using the result (30) and noticing
$a^{m}b^{n}\left|m_{a},n_{b}\right\rangle=\sqrt{m_{a}!n_{b}!/(m_{a}-m)!(n_{b}-n)!}\left|m_{a}-m,n_{b}-n\right\rangle$,
we can directly obtain the PND
$\mathcal{\bar{P}}^{SA}(m_{a},n_{b})\equiv\left\langle
m_{a},n_{b}\right|\rho^{SA}\left|m_{a},n_{b}\right\rangle$ of the PA-TMSTS as
$\displaystyle\mathcal{\bar{P}}^{SA}(m_{a},n_{b})$ (34) $\displaystyle=$
$\displaystyle\frac{N_{m,n}^{-1}m_{a}!n_{b}!}{(m_{a}-m)!(n_{b}-n)!}$
$\displaystyle\times\left\langle
m_{a}-m,n_{b}-n\right|\rho^{S}\left|m_{a}-m,n_{b}-n\right\rangle$
$\displaystyle=$
$\displaystyle\frac{N_{m,n}^{-1}m_{a}!n_{b}!}{(m_{a}-m)!(n_{b}-n)!}\mathcal{P}(m_{a}-m,n_{b}-n).$
Eq.(34) is a Jacobi polynomial with a condition $m_{a}\geqslant m$ and
$n_{b}\geqslant n$ which shows that the photon-number ($m_{a},n_{b}$) involved
in PA-TMSTS are always no-less than the photon-number ($m,n$) operated on the
TMSTS, and there is no photon distribution when $m_{a}<m$ and $n_{b}<n$. Here
we should point out that this result (30) can be applied directly to calculate
the PND of some other non-Gaussian states generated by subtracting photons
from (or adding photons to) two-mode squeezed thermal states, such as
$a^{m}b^{n}\rho^{S}a^{\dagger m}b^{\dagger n},$ and
$a^{m}b^{{\dagger}n}\rho^{S}b^{n}a^{{\dagger}m}$.
### III.3 Antibunching effect of the PA-TMSTS
Next we will discuss the antibunching for the PA-TMSTS. The criterion for the
existence of antibunching in two-mode radiation is given by 40
$R_{ab}\equiv\frac{\left\langle a^{\dagger 2}a^{2}\right\rangle+\left\langle
b^{\dagger 2}b^{2}\right\rangle}{2\left\langle
a^{\dagger}ab^{\dagger}b\right\rangle}-1<0.$ (35)
In a similar way to Eq.(29) we have
$\displaystyle\left\langle a^{\dagger 2}a^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{N_{m+2,n}}{N_{m,n}}-4\frac{N_{m+1,n}}{N_{m,n}}+2,$
$\displaystyle\left\langle b^{\dagger 2}b^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{N_{m,n+2}}{N_{m,n}}-4\frac{N_{m,n+1}}{N_{m,n}}+2.$ (36)
Thus, for the state $\rho^{SA}$, substituting Eqs.(36), (28) and (24) into
Eq.(35), yields
$\displaystyle R_{ab}$ $\displaystyle=$
$\displaystyle\frac{N_{m+2,n}+N_{m,n+2}+2\left(\Omega-
N_{m+1,n+1}\right)}{2\left(N_{m+1,n+1}+\Omega\right)},$
$\displaystyle\left(\Omega=N_{m,n}-N_{m+1,n}-N_{m,n+1}\right).$
In particular, when $m=n=0$ (corresponding to the TMSTS) leading to
$N_{0,0}=1,N_{0,1}=N_{1,0}=B_{1}$, and $N_{1,1}=\upsilon$,
$N_{0,2}=N_{2,0}=B_{1}^{2}$, thus Eq.(III.3) becomes
$R_{ab,m=n=0}=-\frac{\left(2\bar{n}+1\right)\left(4\cosh
2r+\left(2\bar{n}+1\right)\sinh^{2}2r\right)}{\left(2\bar{n}+1\right)\left[\left(2\bar{n}+1\right)\cosh
4r-2\cosh 2r\right]+1}.$ (38)
From Eq.(38), it is easily seen that $R_{ab,m=n=0}<0$ for any $\bar{n}$ and
non-zero $r$ values. In addition, when $m=n,$ the PA-TMSTS can always be
antibunching for a small value $\bar{n}$ (see Fig.2(a)). However, for any
parameter values $m,n(m\neq n)$, the case is not true. The $R_{ab}$ parameter
as a function of $r$ and $m,n$ is plotted in Fig. 2. It is easy to see that,
for a given $m$ the PA-TMSTS presents the antibunching effect when the
squeezing parameter $r$ exceeds to a certain threshold value. For instance,
when $m=0\ $and $n=1$ then $R_{ab}\ $may be less than zero with $r>0.1$
thereabout ($\bar{n}=0.01$). The value $R_{ab}$ parameter increases with
$\bar{n}$ increasing.
Figure 2: (Color online) ${\small R}_{ab}$ as a function of ${\small r}$ for
different parameters (m,n) and ${\small\bar{n}=0.01.}$
### III.4 Wigner function of PA-TMSTS
For further discussing the nonclassicality of PA-TMSTS, we examine its Wigner
function (WF) whose partial negativity implies the highly nonclassical
properties of quantum states. In this section, we derive the analytical
expression of WF for the PA-TMSTS. The normally ordering form of the PA-TMSTS
shall be used to realize our purpose.
For the two-mode case, the WF $W\left(\alpha,\beta\right)$ associated with a
quantum state $\rho$ can be derived as follows 42 :
$\displaystyle W\left(\alpha,\beta\right)$ $\displaystyle=$ $\displaystyle
e^{2(\left|\alpha\right|^{2}+\left|\beta\right|^{2})}\int\frac{\mathtt{d}^{2}z_{1}\mathtt{d}^{2}z_{2}}{\pi^{4}}\left\langle-
z_{1},-z_{2}\right|\rho\left|z_{1},z_{2}\right\rangle$ (39)
$\displaystyle\times\exp\left[2\left(\alpha
z_{1}^{\ast}-\alpha^{\ast}z_{1}\right)+2\left(\beta
z_{2}^{\ast}-\beta^{\ast}z_{2}\right)\right],$
where
$\left|z_{1},z_{2}\right\rangle=\left|z_{1}\right\rangle\left|z_{2}\right\rangle$
is the two-mode coherent state.
Substituting Eq.(17) into Eq.(39), we can finally obtain the WF of the PA-
TMSTS (see Appendix C),
$W_{m,n}\left(\alpha,\beta\right)=W_{0}\left(\alpha,\beta\right)F_{m,n}\left(\alpha,\beta\right),$
(40)
where $W_{0}\left(\alpha,\beta\right)$ is the WF of TMSTS,
$\displaystyle W_{0}\left(\alpha,\beta\right)$ $\displaystyle=$
$\displaystyle\frac{\pi^{-2}}{\left(2\bar{n}+1\right)^{2}}\exp\left\\{-2\frac{\cosh
2r}{2\bar{n}+1}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})\right.$ (41)
$\displaystyle+\allowbreak\left.2\frac{\sinh
2r}{2\bar{n}+1}\left(\beta\alpha+\alpha^{\ast}\beta^{\ast}\right)\right\\},$
and
$\displaystyle F_{m,n}\left(\alpha,\beta\right)$ $\displaystyle=$
$\displaystyle\frac{K_{3}^{m+n}}{N_{m,n}}\sum_{l=0}^{m}\sum_{j=0}^{n}\frac{\left(m!n!\right)^{2}\left(-K_{1}/K_{3}\right)^{l+j}}{l!j!\left[\left(m-l\right)!\left(n-j\right)!\right]^{2}}$
(42)
$\displaystyle\times\left|H_{m-l,n-j}\left(\frac{R_{1}}{i\sqrt{K_{3}}},\frac{R_{3}}{i\sqrt{K_{3}}}\right)\right|^{2},$
where we have set
$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle
2\left(K_{1}\allowbreak\alpha-
K_{3}\beta^{\ast}\right),R_{3}=2\left(K_{1}\beta-K_{3}\alpha^{\ast}\right),$
$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\frac{\bar{n}+\cosh^{2}r}{2\bar{n}+1},K_{3}=\frac{\sinh r\cosh
r}{2\bar{n}+1}.$ (43)
Equation (40) is just the analytical expression of the WF for the PA-TMSTS, a
real function as expected. It is obvious that the WF lost its Gaussian
property in phase space due to the presence of two-variable Hermite
polynomials $H_{m-l,n-j}\left(x,y\right)$.
From Eq.(42), we see that when $m=n=0$ corresponding to the TMSTS,
$F_{0,0}=1,$ and
$W_{m,n}\left(\alpha,\beta\right)=W_{0}\left(\alpha,\beta\right)$; whereas for
the case of $m=0$ and $n\neq 0$, noticing $H_{0,n}\left(x,y\right)=y^{n}$ and
$N_{0,n}=n!B_{1}^{n},$ Eq.(40) reduces to
$W_{0,n}\left(\alpha,\beta\right)=W_{0}\left(\alpha,\beta\right)\left(-K_{1}/B_{1}\right)^{n}L_{n}(\left|R_{3}\right|^{2}/K_{1}),$
(44)
where $L_{n}$ is the $n$-order Laguerre polynomial. Eq. (44) is just the WF of
the PA-TMSTS generated by single-mode photon addition, which becomes the WF of
the negative binomial state $S(r)\left|0,n\right\rangle$ with $\bar{n}=0$
[JOSAB, ??]. In particular, for the case of single photon-addition, $n=1$, it
is found that
$W_{0,1}\left(0,0\right)=-\frac{\left(\bar{n}+\cosh^{2}r\right)/\left(2\bar{n}+1\right)^{3}}{\left(\cosh^{2}r+\bar{n}\cosh
2r\right)\pi^{2}},$ (45)
which implies that the WF of single PA-TMSTS always has its negative region at
the phase space center $\alpha=\beta=0$. The maximum value of
$\left|W_{0,1}\right|$ decreases with the increasement of $\bar{n}$ and $r$
but not disappears, which can be seen clearly from Fig3,4. (a) and (b).
Further, there are more visible negative region than the WF for the case of
$m=n=1$. And the negative region will be absence for the latter with the
increasing $\bar{n}$ value (see Fig3,4. (c) and (d)). In addition, from Figs 3
and 4, the squeezing in one of quadratures is clear, which can be seen as an
evidence of nonclassicality of the state. For a given value $m$ and several
different values $n$ ($\neq m$), the WF distributions are presented in Fig.5,
from which it is interesting to notice that there are around
$\left|m-n\right|$ wave valleys and $\left|m-n\right|+1$ wave peaks.
Figure 3: (Color online) The Wigner function W($\alpha,\beta$) in phase space
($Q_{+},P_{+}$) for several different parameter values $\left(m,n\right)$ and
$\bar{n}$ with $r=0.3.$ (a) m=0,n=1,$\bar{n}=0.2$; (b) m=0,n=1,$\bar{n}=1$;
(c) m=n=1,$\bar{n}=0.2$ and (d) m=n=1,$\bar{n}=1.$ Figure 4: (Color online)
The Wigner function W($\alpha,\beta$) in phase space ($Q_{-},P_{-}$) for
several different parameter values $\left(m,n\right)$ and $\bar{n}$ with
$r=0.3.$ (a) m=0,n=1,$\bar{n}=0.2$; (b) m=0,n=1,$\bar{n}=1$; (c)
m=n=1,$\bar{n}=0.2$ and (d) m=n=1,$\bar{n}=1.$ Figure 5: (Color online) The
Wigner function W($\alpha,\beta$) in phase space ($Q_{+},P_{+}$) for several
different parameter values $\left(m,n\right)$ with $\bar{n}=0.2$ and $r=0.3.$
(a) m=0,n=2; (b) m=1,n=2; (c) m=1,n=3, and (d) m=2,n=3.
## IV Entanglement properties of the PA-TMSTS
It is well known that photon subtraction/addition can be applied to improve
entanglement between Gaussian states 14 ; 43 , loophole-free tests of Bell’s
inequality 44 , and quantum computing 18 . In this section, we examine the
entanglement properties of PA-TMSTS only with single and two photon-addition.
Here, for a bipartite continuous variable state, we shall take the Shchukin-
Vogel (SV) 38 criteria to describe the inseparability of PA-TMSTS.
According to the SV criteria, the sufficient condition of inseparability is
$SV_{m,n}\equiv\left\langle
a^{{\dagger}}a-\frac{1}{2}\right\rangle\left\langle
b^{{\dagger}}b-\frac{1}{2}\right\rangle-\left\langle
a^{{\dagger}}b^{{\dagger}}\right\rangle\left\langle ab\right\rangle<0.$ (46)
In a similar way to derive the normalization factor Eq.(24), using Eqs.(17)
and (18), we have
$\left\langle
a^{{\dagger}}b^{{\dagger}}\right\rangle=\frac{N_{m,m+1,n,n+1}}{N_{m,n}},\text{
}\left\langle ab\right\rangle=\frac{N_{m+1,m,n+1,n}}{N_{m,n}},$ (47)
where we have set
$\displaystyle N_{l,p,q,r}$ $\displaystyle\equiv$
$\displaystyle\left.\frac{\partial^{l+p+q+r}}{\partial\tau^{l}\partial
t^{p}\partial\tau^{\prime q}\partial t^{\prime
r}}e^{\left(\tau^{\prime}t^{\prime}+\tau
t\right)B_{1}+\left(\tau\tau^{\prime}+tt^{\prime}\right)B_{2}}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}$
(48) $\displaystyle=$
$\displaystyle\sum_{s=0}\frac{l!r!q!p!B_{1}^{l-q+2r}B_{2}^{q-r}\left(B_{2}^{2}/B_{1}^{2}\right)^{s}\delta_{p+q,l+r}}{s!\left(q-r+s\right)!\left(r-s\right)!\left(l+r-q-s\right)!}.$
Thus $SV$ is given by
$\displaystyle SV_{m,n}$ $\displaystyle=$
$\displaystyle\left(\frac{N_{m+1,n}}{N_{m,n}}-\frac{3}{2}\right)\left(\frac{N_{m,n+1}}{N_{m,n}}-\frac{3}{2}\right)$
(49)
$\displaystyle-\frac{N_{m,m+1,n,n+1}}{N_{m,n}}\frac{N_{m+1,m,n+1,n}}{N_{m,n}}.$
Next, we examine two special cases. For the case of $m=0,n=1$, using Eqs.(24)
and (48), as well as noticing $N_{0,1,1,2}=N_{1,0,2,1}=2B_{1}B_{2}$, Eq.(49)
becomes
$SV_{0,1}=\left(\frac{\upsilon}{B_{1}}-\frac{3}{2}\right)\left(2B_{1}-\frac{3}{2}\right)-4B_{2}^{2}.$
(50)
While for the case of $m=n=1$, it is shown that
($N_{1,2,1,2}=N_{2,1,2,1}=2\left(\allowbreak
2B_{1}^{2}+B_{2}^{2}\right)B_{2}$)
$SV_{1,1}=\left(B_{1}\left(3-\frac{\omega}{\upsilon}\right)-\frac{3}{2}\right)^{2}-4\frac{\left(\allowbreak
2B_{1}^{2}+B_{2}^{2}\right)^{2}B_{2}^{2}}{\upsilon^{2}}.$ (51)
In particular, when $\bar{n}=0$, i.e., the single PA-TMSVS, Eq.(50) is always
negative for any $r>0,$ as expected (also see Fig.6 (a)). In general, it is
difficult to obtain the explicit expressions of the sufficient condition of
inseparability for the above cases. Here, we appeal to the number calculation
shown in Fig.6. It is shown that for single PA-TMSTS with a smaller average
photon number $\bar{n}$, the condition $SV_{0,1}<0$ can always be satisfied
only if $r>0$; while for a larger $\bar{n}$ then the condition $SV_{0,1}<0$ is
satisfied only when the squeezing parameter $r$ exceeds a certain threshold
value $r_{a}$. However, it is very interesting to notice that for the photon-
subtraction TMSTS, there is a threshold value $r_{c}$ for any $\bar{n}$, i.e.,
$r>r_{c}\equiv\frac{1}{2}\ln(2\bar{n}+1)$ 45 , which is different from the
case of single PA-TMSTS. For instance, for $\bar{n}=1,$ the two threshold
values are $r_{a}\approx 0.31$ and $r_{c}\approx 0.55$. This comparision may
imply that the photon-addition to the TMSTS can be more effective for the
entanglement enhancement than the photon-subtraction from the TMSTS. On the
other hand, for the case of the PA-TMSTS with $m=n=1$ (see Fig.6 (b)), it is
found that a certain threshold is needed for satisfying this condition
$SV_{1,1}<0,$ which is also smaller than that of the photon-subtraction TMSTS.
Figure 6: (Color online) The sufficient condition of inseparability as the
function of ${\small r}$ and $\bar{n},$ for (a) m=0 and n=1; (b) m=n=1.
## V Quantum teleportation with PA-TMSTS
As mentioned above, photon-subtraction from or photon-addition to bipartite
Gaussian states can be used to improve the entanglement. In this section, we
investigate the quantum teleportation with PA-TMSTS, especially for the cases
$m=0,n=1$ and $m=n=1$. The role of teleportation in the CV quantum information
is analyzed in the review Ref.46 .
Here, we consider the QT by using PA-TMSTS as entangled resource. Using the
normal ordering form Eq.(17) and noticing the displacement operator
$D_{a}\left(\alpha\right)=e^{\left|\alpha\right|^{2}/2}e^{-\alpha^{\ast}a}e^{\alpha
a^{\dagger}}$, then the characteristic function (CF) of PA-TMSTS is given by
(see Appendix D)
$\displaystyle\chi_{E}\left(\alpha,\beta\right)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{m,n}}e^{-(B_{1}-\frac{1}{2})(\left|\alpha\right|^{2}+\left|\beta\right|^{2})+B_{2}\left(\alpha\beta+\alpha^{\ast}\beta^{\ast}\right)}$
(52)
$\displaystyle\times\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial\tau^{\prime
n}\partial t^{m}\partial t^{\prime
n}}e^{B_{1}\left(\tau^{\prime}t^{\prime}+t\tau\right)+B_{2}\left(t^{\prime}t+\tau^{\prime}\tau\right)}$
$\displaystyle\times e^{t\left(\alpha B_{1}-\allowbreak
B_{2}\beta^{\ast}\right)+\tau\left(\beta B_{2}-B_{1}\alpha^{\ast}\right)}$
$\displaystyle\times\left.e^{\tau^{\prime}\left(\alpha
B_{2}-B_{1}\beta^{\ast}\right)+t^{\prime}\left(\beta\allowbreak
B_{1}-B_{2}\alpha^{\ast}\right)}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0},$
where for further calculation the differential form of $\chi_{E}$ is kept.
To quantify the performance of a QT protocol, the fidelity of QT is commonly
used as a measure, $\mathcal{F=}\mathtt{tr}\left(\rho_{in}\rho_{out}\right)$,
a overlap between a pure input state $\rho_{in}$ and the output (teleported,
mixed) state $\rho_{out}$. For a CV system, a teleportation protocol has been
given in terms of the CFs of the quantum states involved (input, source and
teleported (output) states) 47 . It is shown that the CF
$\chi_{out}\left(\eta\right)$ of the output state has a remarkably factorized
form
$\chi_{out}\left(\eta\right)=\chi_{in}\left(\eta\right)\chi_{E}\left(\eta^{\ast},\eta\right),$
(53)
where $\chi_{in}\left(\eta\right)$ and $\chi_{E}\left(\eta^{\ast},\eta\right)$
are the CFs of the input state and the entangled source, respectively. Then
the fidelity of QT of CVs reads 47
$\mathcal{F=}\int\frac{d^{2}\eta}{\pi}\chi_{in}\left(\eta\right)\chi_{out}\left(-\eta\right).$
(54)
Here, we consider the Braunstein and Kimble protocol 48 of QT for single-mode
coherent-input states $\left|\gamma\right\rangle$. Note that the fidelity is
independent of amplitude of the coherent state, thus for simplicity we take
$\gamma=0$, then we have only to calculate the fidelity of the vacuum input
state with the CF $\chi_{in}\left(\eta\right)=\exp[-\left|\eta\right|^{2}/2]$.
On substituting these CFs into Eq.(54), we worked out the fidelity for
teleporting a coherent state by using the PA-TMSTS as an entangled resource,
$\displaystyle\mathcal{F}_{m,n}^{\bar{n}}$ $\displaystyle=$
$\displaystyle\frac{(m+n)!}{B_{1}-B_{2}}\frac{\left(B_{1}+B_{2}\right)^{m+n}}{2^{m+n+1}N_{m,n}}$
(55) $\displaystyle=$
$\displaystyle\frac{\left[\allowbreak\left(2\bar{n}+1\right)e^{2r}+\allowbreak
1\right]^{m+n}}{\allowbreak\left(2\bar{n}+1\right)e^{-2r}+1}\frac{(m+n)!}{2^{2m+2n}N_{m,n}}.$
It can be seen that the fidelity is not only dependent on the parameter $r$,
the average photon-number $\bar{n}$, but also on the photon number
$\left(m,n\right)$ added to each mode of the TMSTS. In particular, when
$m=n=0,$ Eq.(55) just reduces to
$\mathcal{F}_{0,0}^{\bar{n}}=\frac{1}{\allowbreak\left(2\bar{n}+1\right)e^{-2r}+1},$
(56)
which leads to the condition $r>\frac{1}{2}\ln\left(2\bar{n}+1\right)$ for
satisfying the effective QT with $\mathcal{F}>\frac{1}{2}$ which is the
classical limit. In addition, for the case of $\bar{n}=0$, i.e., the photon-
added TMSVS, Eq.(55) becomes
$\mathcal{F}_{m,n}^{0}=\frac{\left(\allowbreak e^{2r}+\allowbreak
1\right)^{n+m}}{\allowbreak e^{-2r}+1}\frac{(m+n)!}{2^{2m+2n}N_{m,n}}.$ (57)
Further, when $\left(m,n\right)=\left(0,0\right),\left(1,1\right)$ and
$\left(0,1\right)$, Eq.(56) just reduce, respectively, to
$\displaystyle\mathcal{F}_{0,0}^{0}$ $\displaystyle\mathcal{=}$
$\displaystyle(1+\tanh r)/2,$ $\displaystyle\mathcal{F}_{1,1}^{0}$
$\displaystyle=$ $\displaystyle\frac{(1+\tanh r)^{3}}{4(1+\tanh^{2}r)},$
$\displaystyle\mathcal{F}_{0,1}^{0}$ $\displaystyle=$
$\displaystyle\frac{1+\tanh r}{4\left(1-\tanh r\right)}\text{sech}^{2}r.$ (58)
The two expressions $\mathcal{F}_{0,0}^{0}$ and $\mathcal{F}_{1,1}^{0}$ are
agreement with Eqs.(15) and (17) in Ref. 49 .
In Fig. 7, for some given $\left(m,n\right)$ values, the fidelity of
teleporting the coherent state is shown as a function of $r$ by using the PA-
TMSTS as the entangled resource. It is shown that the fidelity with this
resource is smaller than that with TMSTS, although the PA-TMSTS posesses
larger entanglement 49 . In addition, for the symmetrical case $m=n$, when the
squeezing parameter $r$ exceeds a certain threshold value, the fidelity
increases with a increasing $m$ (see Fig.7(c)); while for non-symmetric case
$m\neq n$, the fidelity decreases with increasing $n\ $(see Fig.7(b)). For the
former, the threshold value $r$ decreases with increasing $m\left(=n\right)$;
the case is not true for the latter. This indicate that the symmetrical PA-
TMSTS may be more effective for QT than the non-symmetric case.
Figure 7: (Color online) The fidelity as the function of ${\small r}$ for
several different (m,n) values and $\bar{n}{\small=0.01}.$
## VI Conclusions
In this paper, we introduce the PA-TMSTS and investigate its entanglement and
nonclassicality. By using the coherent state representation of thermal state,
the normally and antinormally ordering forms of the TMSTS are obtained. Based
on this, the normalization factor of the PA-TMSTS is derived, which is related
to the Jacobi polynomials of the squeezing parameter $r$ and average photon
number $\bar{n}$ of the thermal state. Then we discuss the nonclassical
properties by using cross-correlation function, distribution of photon number,
antibunching effect and the negativity of its WF. It is found that the WF lost
its Gaussian property in phase space due to the presence of two-variable
Hermite polynomials and the WF of single PA-TMSTS always has its negative
region at the center of phase space. Further, there are more visible negative
region than the WF for the case of $m=n=1$. And the negative region will be
absence for the latter with the increasing $\bar{n}$ value. The entanglement
properties of the PA-TMSTS by Shchukin-Vogel criteria and the quantum
teleportation. It is shown that the photon-addtion to the TMSTS can be more
effective for the entanglement enhancement than the photon-subtraction from
the TMSTS; And using the PA-TMSTS as an entangled resource, the fidelity for
teleporting a coherent state is not only dependent on the parameter $r$, the
average photon-number $\bar{n}$, but also on the photon number
$\left(m,n\right)$ added to each mode of the TMSTS. From this point, the
symmetrical PA-TMSTS may be more effective for quantum teleportation than the
non-symmetric case.
Acknowledgments: This work was supported by the NSFC (Grant No. 60978009),
the Major Research Plan of the NSFC (Grant No. 91121023), and the “973”
Project (Grant No. 2011CBA00200), and the Natural Science Foundation of
Jiangxi Province of China (No. 2010GQW0027) as well as the Sponsored Program
for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.
Appendix A: Derivation of Eq.(18)
According to the normalization condition, $\mathtt{tr}\rho^{SA}=1,$ we have
$\displaystyle N_{m,n}$ $\displaystyle=A_{1}\mathtt{tr}\left[\colon a^{\dagger
m}b^{\dagger
n}e^{A_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-A_{3}\left(a^{\dagger}a+b^{\dagger}b\right)}a^{m}b^{n}\colon\right]$
$\displaystyle=\frac{A_{1}\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime n}}\mathtt{tr}\left[\colon
e^{A_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-A_{3}\left(a^{\dagger}a+b^{\dagger}b\right)+\tau
a^{\dagger}+ta+\tau^{\prime}b^{\dagger}+t^{\prime}b}\colon\right].$ (A1)
Using the completeness relation of coherent state $\int d^{2}\alpha
d^{2}\beta\left|\alpha,\beta\right\rangle\left\langle\alpha,\beta\right|/\pi^{2}=1$
and Eq.(9), Eq.(A1)
$\displaystyle N_{m,n}$ $\displaystyle=A_{1}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\left|\alpha\right|^{2m}\left|\beta\right|^{2n}e^{A_{2}\left(\alpha^{\ast}\beta^{\ast}+\alpha\beta\right)-A_{3}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})}$
$\displaystyle=\frac{A_{1}\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime n}}\int\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}\exp\left[-A_{3}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})\right.$
$\displaystyle\left.+\left(A_{2}\beta+\tau\right)\alpha+\left(A_{2}\beta^{\ast}+t\right)\alpha^{\ast}+\tau^{\prime}\beta+t^{\prime}\beta^{\ast}\right]_{t,\tau,t^{\prime},\tau^{\prime}=0}.$
(A2)
Using Eq.(9), (A2) becomes
$\displaystyle N_{m,n}$
$\displaystyle=\frac{A_{1}\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}\frac{1}{A_{3}}\int\frac{d^{2}\beta}{\pi}\exp\left[-\frac{A_{3}^{2}-A_{2}^{2}}{A_{3}}\left|\beta\right|^{2}\right.$
$\displaystyle+\left.\left(\frac{A_{2}t}{A_{3}}+\tau^{\prime}\right)\beta+\left(\frac{A_{2}\tau}{A_{3}}+t^{\prime}\right)\beta^{\ast}+\frac{\tau
t}{A_{3}}\right]_{t,\tau,t^{\prime},\tau^{\prime}=0}$ $\displaystyle=\text{LHS
of Eq.(\ref{t18})},$ (A3)
where we have used $A_{1}/(A_{3}^{2}-A_{2}^{2})=1$ and
$B_{2}=A_{2}\allowbreak/(A_{3}^{2}-A_{2}^{2})=\left(2\bar{n}+1\right)\sinh
r\cosh r$ as well as $B_{1}=A_{3}/(A_{3}^{2}-A_{2}^{2})$.
Appendix B: New expression of generating function for Jacobi polynomials
In this appendix, we shall prove Eq.(23). Rewriting
$H\equiv\left.\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}e^{A\left(\tau^{\prime}t^{\prime}+\tau
t\right)+B\left(\tau\tau^{\prime}+t^{\prime}t\right)}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}.$
(B1)
Expanding the exponential items, we see
$\displaystyle H$
$\displaystyle=\sum_{l,j,k,s=0}^{\infty}\frac{A^{l+j}B^{s+k}}{l!j!k!s!}\left.\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}\tau^{k+j}t^{j+s}\tau^{\prime l+k}t^{\prime
l+s}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}$
$\displaystyle=\sum_{s=0}^{\min\left[m,n\right]}\frac{\left(m!n!\right)^{2}A^{n+m}}{s!s!\left(n-s\right)!\left(m-s\right)!}\left(\frac{B^{2}}{A^{2}}\right)^{s}.$
(B2)
Comparing Eq.(B2) with the standard expression of Jacobi polynomials,
$P_{m}^{(\alpha,\beta)}(x)=\left(\frac{x-1}{2}\right)^{m}\sum_{k=0}^{m}\left(\begin{array}[]{c}m+\alpha\\\
k\end{array}\right)\left(\begin{array}[]{c}m+\beta\\\
m-k\end{array}\right)\left(\frac{x+1}{x-1}\right)^{k},$ (B3)
we can find that taking $m\leqslant n$ and
$y=\left(B^{2}+A^{2}\right)/\left(B^{2}-A^{2}\right)$,
$\displaystyle H$
$\displaystyle=\sum_{s=0}^{m}\frac{\left(m!n!\right)^{2}A^{n+m}}{s!s!\left(n-s\right)!\left(m-s\right)!}\left(\frac{B^{2}}{A^{2}}\right)^{s}$
$\displaystyle=m!n!\left(\frac{y-1}{2}\right)^{-m}A^{m+n}\left\\{\left(\frac{y-1}{2}\right)^{m}\right.$
$\displaystyle\times\left.\sum_{k=0}^{\min[m,n]}\frac{m!n!}{k!k!\left(n-k\right)!\left(m-k\right)!}\left(\frac{y+1}{y-1}\right)^{k}\right\\}$
$\displaystyle=m!n!A^{n-m}\left(B^{2}-A^{2}\right)^{m}P_{m}^{(n-m,0)}\left(y\right).$
(B4)
In a similar way, for $n\leqslant m$, we also have
$H=m!n!A^{m-n}\left(B^{2}-A^{2}\right)^{n}P_{n}^{(m-n,0)}\left(y\right).$ (B5)
Thus we finish the proof of Eq.(23).
In addition, when $m=n$, Eq.(23) becomes
$\displaystyle\left.\frac{\partial^{4m}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime m}\partial t^{\prime
m}}e^{A\left(\tau^{\prime}t^{\prime}+\tau
t\right)+B\left(\tau\tau^{\prime}+t^{\prime}t\right)}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}$
$\displaystyle=\left.\frac{\partial^{4m}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime m}\partial t^{\prime
m}}e^{\left(At^{\prime}+B\tau\right)\tau^{\prime}+\left(A\tau+Bt^{\prime}\right)t}\right|_{t,\tau,t^{\prime},\tau^{\prime}=0}$
$\displaystyle=\left.\frac{\partial^{4m}}{\partial\tau^{m}\partial
t^{m}}\left[\left(At+B\tau\right)\left(A\tau+Bt\right)\right]^{m}\right|_{\tau,t^{\prime}=0}$
$\displaystyle=\left(m!\right)^{2}\left(B^{2}-A^{2}\right)^{m}P_{m}\left(\frac{B^{2}+A^{2}}{B^{2}-A^{2}}\right),$
(B6)
where $P_{m}\left(x\right)$ is the $m$th Legendre polynomials. Eq.(B6) is just
a new formula for the generating function of Legendre polynomials $P_{m}(x)$,
which is different from the new form found in Ref.50 . In fact, one can check
Eq. (B6) by expanding directly the whole exponential items and comparing with
the standard expression of Legendre polynomials.
Appendix C: Derivation of Eq.(40)
Substituting Eq.(17) into Eq.(39) and usiing Eq.(9), we have
$\displaystyle W\left(\alpha,\beta\right)$
$\displaystyle=A_{1}N_{m,n}^{-1}e^{2(\left|\alpha\right|^{2}+\left|\beta\right|^{2})}\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{4}}$
$\displaystyle\times\exp\left\\{-\left(2-A_{3}\right)\left|z_{1}\right|^{2}-\left(2-A_{3}\right)\left|z_{2}\right|^{2}\right.$
$\displaystyle+\left(t-2\alpha^{\ast}+A_{2}z_{2}\right)z_{1}+\left(2\alpha+\tau+A_{2}z_{2}^{\ast}\right)z_{1}^{\ast}$
$\displaystyle\left.\left.+2\left(\beta
z_{2}^{\ast}-\beta^{\ast}z_{2}\right)+\tau^{\prime}z_{2}^{\ast}+t^{\prime}z_{2}\right\\}\right|_{t,t^{\prime},\tau,\tau^{\prime}=0}$
$\displaystyle=W_{0}\left(\alpha,\beta\right)F_{m,n}\left(\alpha,\beta\right),$
(C1)
where $W_{0}\left(\alpha,\beta\right)$ is defined in Eq.(41), and
$\displaystyle F_{m,n}\left(\alpha,\beta\right)$
$\displaystyle=\frac{\left(-1\right)^{m+n}}{N_{m,n}}\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime n}}$ $\displaystyle\times
e^{R_{1}t+R_{2}\tau+R_{3}t^{\prime}+R_{4}\tau^{\prime}}$
$\displaystyle\times\left.e^{K_{1}\left(\tau
t+\tau^{\prime}t^{\prime}\right)+K_{3}\left(\tau\tau^{\prime}+tt^{\prime}\right)}\right|_{t,t^{\prime},\tau,\tau^{\prime}=0},$
(C2)
and
$\displaystyle R_{1}$ $\displaystyle=2\left(K_{1}\allowbreak\alpha-
K_{3}\beta^{\ast}\right)=-R_{2}^{\ast},$ $\displaystyle R_{3}$
$\displaystyle=2\left(K_{1}\beta-K_{3}\alpha^{\ast}\right)=-R_{4}^{\ast},$
(C3)
as well as
$K_{1}=\frac{\bar{n}+\cosh^{2}r}{2\bar{n}+1},K_{3}=\frac{\sinh r\cosh
r}{2n+1}.$ (C4)
Expanding the partial exponential items in Eq.(C2), then Eq.(C2) becomes
$\displaystyle F_{m,n}\left(\alpha,\beta\right)$
$\displaystyle=\frac{\left(-1\right)^{m+n}}{N_{m,n}}\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}\sum_{l,j=0}^{\infty}\frac{K_{1}^{l+j}}{l!j!}$
$\displaystyle\times\left.\left(\tau
t\right)^{l}\left(\tau^{\prime}t^{\prime}\right)^{j}e^{R_{1}t+R_{3}t^{\prime}-R_{1}^{\ast}\tau-
R_{3}^{\ast}\tau^{\prime}+K_{3}\left(\tau\tau^{\prime}+tt^{\prime}\right)}\right|_{t,t^{\prime},\tau,\tau^{\prime}=0}$
$\displaystyle=\frac{\left(-1\right)^{m+n}}{N_{m,n}}\sum_{l,j=0}^{\infty}\frac{K_{1}^{l+j}}{l!j!}\frac{\partial^{2l+2j}}{\partial\left(-R_{1}^{\ast}\right)^{l}\partial
R_{1}^{l}\partial\left(-R_{3}^{\ast}\right)^{j}\partial R_{3}^{j}}$
$\displaystyle\left.\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial
t^{m}\partial\tau^{\prime n}\partial t^{\prime
n}}e^{R_{1}t+R_{3}t^{\prime}-R_{1}^{\ast}\tau-
R_{3}^{\ast}\tau^{\prime}+K_{3}\left(\tau\tau^{\prime}+tt^{\prime}\right)}\right|_{t,t^{\prime},\tau,\tau^{\prime}=0}.$
(C5)
Further using the generating function of two-variable Hermite polynomials,
$\displaystyle\left.\frac{\partial^{m}}{\partial\tau^{m}}\frac{\partial^{n}}{\partial\upsilon^{n}}e^{-A\tau\upsilon+B\tau+C\upsilon}\right|_{\tau=\upsilon=0}$
$\displaystyle=(\sqrt{A})^{m+n}H_{m,n}\left(\frac{B}{\sqrt{A}},\frac{C}{\sqrt{A}}\right),$
(C6)
Eq.(C5) can be put into the following form
$\displaystyle F_{m,n}\left(\alpha,\beta\right)$
$\displaystyle=\frac{K_{3}^{m+n}}{N_{m,n}}\sum_{l,j=0}^{\infty}\frac{K_{1}^{l+j}}{l!j!}\frac{\partial^{2l+2j}}{\partial\left(-R_{1}^{\ast}\right)^{l}\partial
R_{1}^{l}\partial\left(-R_{3}^{\ast}\right)^{j}\partial R_{3}^{j}}$
$\displaystyle\times
H_{m,n}\left(\frac{R_{1}}{\sqrt{-K_{3}}},\frac{R_{3}}{\sqrt{-K_{3}}}\right)$
$\displaystyle\times
H_{m,n}\left(\frac{-R_{1}^{\ast}}{\sqrt{-K_{3}}},\frac{-R_{3}^{\ast}}{\sqrt{-K_{3}}}\right).$
(C7)
Using the relation
$\frac{\partial^{l+k}}{\partial x^{l}\partial
y^{k}}H_{m,n}\left(x,y\right)=\frac{m!n!}{\left(m-l\right)!\left(n-k\right)!}H_{m-l,n-k}\left(x,y\right),$
(C8)
thus we can obtain Eq.(42).
Appendix D: Derivation of Eq.(52)
Using the displacement operator
$D_{a}\left(\alpha\right)=e^{\left|\alpha\right|^{2}/2}e^{-\alpha^{\ast}a}e^{\alpha
a^{\dagger}}$ and
$D_{b}\left(\beta\right)=e^{\left|\beta\right|^{2}/2}e^{-\beta^{\ast}b}e^{\beta
b^{\dagger}}$ as well as the normally ordering form of PA-TMSTS (17), the CF
of PA-TMSTS is given by
$\displaystyle\chi_{E}\left(\alpha,\beta\right)\left.=\right.\mathtt{tr}\left[D_{a}\left(\alpha\right)D_{b}\left(\beta\right)\rho^{SA}\right]$
$\displaystyle=\frac{A_{1}e^{(\left|\beta\right|^{2}+\left|\alpha\right|^{2})/2}}{N_{m,n}}\frac{\partial^{2m+2n}}{\partial\alpha^{m}\partial\beta^{n}\partial\left(-\alpha^{\ast}\right)^{m}\partial\left(-\beta^{\ast}\right)^{n}}$
$\displaystyle\times\mathtt{tr}\left[\colon e^{\alpha a^{\dagger}+\beta
b^{\dagger}-\alpha^{\ast}a-\beta^{\ast}b+A_{2}\left(a^{\dagger}b^{\dagger}+ab\right)-A_{3}\left(a^{\dagger}a+b^{\dagger}b\right)}\colon\right].$
(D1)
In a similar way to derive Eq.(18), using Eqs.(A1) and (18), one can directly
obtain
$\displaystyle\chi_{E}\left(\alpha,\beta\right)$
$\displaystyle=\frac{e^{(\left|\beta\right|^{2}+\left|\alpha\right|^{2})/2}}{N_{m,n}}\frac{\partial^{2m+2n}}{\partial\alpha^{m}\partial\beta^{n}\partial\left(-\alpha^{\ast}\right)^{m}\partial\left(-\beta^{\ast}\right)^{n}}$
$\displaystyle\times
e^{B_{1}\left(\alpha\left(-\alpha^{\ast}\right)+\beta\left(-\beta^{\ast}\right)\right)+B_{2}\left(\alpha\beta+\left(-\alpha^{\ast}\right)\left(-\beta^{\ast}\right)\right)}.$
(D2)
Taking the following transformations
$\displaystyle\alpha$
$\displaystyle\rightarrow\alpha+\tau,-\alpha^{\ast}\rightarrow
t-\alpha^{\ast},$ $\displaystyle\beta$
$\displaystyle\rightarrow\beta+\tau^{\prime},-\beta^{\ast}\rightarrow
t^{\prime}-\beta^{\ast},$ (D3)
which leads to
$\displaystyle
e^{B_{1}\left(\alpha\left(-\alpha^{\ast}\right)+\beta\left(-\beta^{\ast}\right)\right)+B_{2}\left(\alpha\beta+\left(-\alpha^{\ast}\right)\left(-\beta^{\ast}\right)\right)}$
$\displaystyle\rightarrow\exp\left[-B_{1}(\left|\alpha\right|^{2}+\left|\beta\right|^{2})+B_{2}\left(\alpha\beta+\alpha^{\ast}\beta^{\ast}\right)\right]$
$\displaystyle\times\exp\left[B_{1}\left(\tau^{\prime}t^{\prime}+t\tau\right)+B_{2}\left(t^{\prime}t+\tau^{\prime}\tau\right)\right]$
$\displaystyle\times\exp\left[t\left(\alpha B_{1}-\allowbreak
B_{2}\beta^{\ast}\right)+\tau\left(\beta
B_{2}-B_{1}\alpha^{\ast}\right)\right]$
$\displaystyle\times\exp\left[\tau^{\prime}\left(\alpha
B_{2}-B_{1}\beta^{\ast}\right)+t^{\prime}\left(\beta\allowbreak
B_{1}-B_{2}\alpha^{\ast}\right)\right],$ (D4)
thus Eq.(D2) becomes Eq.(52).
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* (47) P. Marian and T. A. Marian, Phys. Rev. A 74, 042306 (2006).
* (48) S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998).
* (49) Y. Yang and F. L. Li, Phys. Rev. A 80, 022315 (2009).
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|
arxiv-papers
| 2012-03-03T01:11:58 |
2024-09-04T02:49:28.204711
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-Yun Hu, Fang Jia, and Zhi-Ming Zhang",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1203.0595"
}
|
1203.0687
|
# Magnetic States at Short Distances
Horace W. Crater1∗ 000∗Email address: hcrater@utsi.edu and Cheuk-Yin Wong2,3†
000†Email address: wongc@ornl.gov 1The University of Tennessee Space
Institute, Tullahoma, Tennessee 37388
2Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
3Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 1The
University of Tennessee Space Institute, Tullahoma, Tennessee 37388
2Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
3Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 The
University of Tennessee Space Institute, Tullahoma, Tennessee 37388
Department of Physics and Astronomy, University of Tennessee, Knoxville, TN
37996
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
###### Abstract
The magnetic interactions between a fermion and an antifermion of opposite
electric or color charges in the ${}^{1}S_{0}^{-+}$ and ${}^{3}P_{0}^{++}$
states with $J=0$ are very attractive and singular near the origin and may
allow the formation of new bound and resonance states at short distances. In
the two body Dirac equations formulated in constraint dynamics, the short-
distance attraction for these states for point particles leads to a
quasipotential that behaves near the origin as $-\alpha^{2}/r^{2}$, where
$\alpha$ is the coupling constant. Representing this quasipotential at short
distances as $\lambda(\lambda+1)/r^{2}$ with
$\lambda=(-1+\sqrt{1-4\alpha^{2}})/2$, both ${}^{1}S_{0}^{-+}$ and
${}^{3}P_{0}^{++}$ states admit two types of eigenstates with drastically
different behaviors for the radial wave function $u=r\psi$. One type of
states, with $u$ growing as $r^{\lambda+1}$ at small $r$, will be called usual
states. The other type of states with $u$ growing as $r^{-\lambda}$ will be
called peculiar states. Both of the usual and peculiar eigenstates have
admissible behaviors at short distances. Remarkably, the solutions for both
sets of ${}^{1}S_{0}$ states can be written out analytically. The usual bound
${}^{1}S_{0}$ states possess attributes the same as those one usually
encounters in QED and QCD, with bound state energies explicitly agreeing with
the standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar bound ${}^{1}S_{0}$ states, yet to be observed, not only have
different behaviors at the origin, but also distinctly different bound state
properties (and scattering phase shifts). For the peculiar ${}^{1}S_{0}$
ground state of fermion-antifermion pair with fermion rest mass $m$, the root-
mean-square radius is approximately $1/m$, binding energy is approximately
$(2-\sqrt{2})m$, and rest mass approximately $\sqrt{2}m$. On the other hand,
the $(n+1)$${}^{1}S_{0}$ peculiar state with principal quantum number $(n+1)$
is nearly degenerate in energy and approximately equal in size with the
$n$${}^{1}S_{0}$ usual states. For the ${}^{3}P_{0}$ states, the usual
solutions lead to the standard bound state energies and no resonance, but
resonances have been found for the peculiar states whose energies depend on
the description of the internal structure of the charges, the mass of the
constituent, and the coupling constant. The existence of both usual and
peculiar eigenstates in the same system leads to the non-self-adjoint property
of the mass operator and two non-orthogonal complete sets. As both sets of
states are physically admissible, the mass operator can be made self-adjoint
with a single complete set of admissible states by introducing a new
peculiarity quantum number and an enlarged Hilbert space that contains both
the usual and peculiar states in different peculiarity sectors. Whether or not
these newly-uncovered quantum-mechanically acceptable peculiar ${}^{1}S_{0}$
bound states and ${}^{3}P_{0}$ resonances for point fermion-antifermion
systems correspond to physical states remains to be further investigated.
###### pacs:
25.75.-q 25.75.Dw
## I INTRODUCTION
It is well known that for some combinations of the spin configurations and
orbital motion the magnetic interaction can be strongly attractive and
singular111 A potential is quantum mechanically singular if it is more
attractive than $-1/4r^{2}$ at the origin in the context of ($-d^{2}/dr^{2}$
$-1/4r^{2})$. See Case . at short distances Bar77 ; Bar81 ; Won86 . We can
illustrate this by a classical example as shown schematically in Fig. 1(a)
where a positive charge $q^{+}$ is making a circular orbit about a fixed
negative charge $q^{-}$ whose spin ${\ \hbox{\boldmath${s}$}}(q^{-})$ is
pointing in a direction opposite to the orbital angular momentum of $q^{+}$
Won86 . In the external field problem, (e.g., Fermi’s treatment of hyperfine
structure), the charged particle $q^{-}$ at rest with a magnetic moment
$\hbox{\boldmath${\mu}$}(q^{-})$ generates a vector potential
${\hbox{\boldmath${A}$}}={\hbox{\boldmath${\mu}$}}(q^{-})\times{\hbox{\boldmath${r}$}}/r^{3}$
which acts on the other particle, $q^{+}$. Such a “magnetic” interaction can
be very attractive when the spins and the orbital angular momentum are
oppositely aligned, as shown in the configuration of ($q^{+}q^{-}$) in Fig. 1,
where the vector potential ${A}$, arising from the $q^{-}$ magnetic dipole
moment ${\hbox{\boldmath${\mu}$}}(q^{-})$, is parallel to the $q^{+}$ orbital
momentum ${p}$. The interaction
$(-\hbox{\boldmath${p}$}\cdot\hbox{\boldmath${A}$})$ from $q^{-}$ acting on
$q^{+}$ is attractive and is proportional to
$[\hbox{\boldmath${L}$}(q^{+})\cdot\hbox{\boldmath${s}$}(q^{-})]/r^{3}$ that
is quite singular in nature. At short distances it may overwhelm the
centrifugal barrier that is proportional to $1/r^{2}$. Similarly, the
interaction from $q^{+}$ acting on $q^{-}$ will be likewise attractive and
singular if the spin of the ${\hbox{\boldmath${s}$}}(q+)$ is parallel to the
electron spin ${\ \hbox{\boldmath${s}$}}(q-)$ and pointing in the same
direction, resulting in the total spin of the $q^{+}q^{-}$ system aligning
opposite to the orbital angular momentum, as in the ${}^{3}P_{0}^{++}$ state
with $S=1,L=1$, $J=0$, $P=+1$, and $C=+1$.
The ${}^{3}P_{0}^{++}$ state is not the only state with a strong magnetic
interaction. One can envisage classically another spin configuration, the
${}^{1}S_{0}^{-+}$ state, that also has attractive and singular magnetic
interactions. As illustrated schematically in Fig. 1(b), a fermion $q^{-}$
with an electric or color charge interacts with an antifermion $q^{+}$ of
opposite electric or color charge with spins ${\
\hbox{\boldmath${s}$}}(q^{-})$ and ${\ \hbox{\boldmath${s}$}}(q^{+})$ pointing
in a opposite directions in the ${}^{1}S_{0}^{-+}$ state configuration. With
the spins opposite to each other, the magnetic moments of $q^{-}$ and $q^{+}$
are parallel to each other. The interaction between the magnetic moments is
Jac62
$H_{\mathrm{int}}=-({8\pi}/{3}){\hbox{\boldmath${\mu}$}}_{q^{-}}\cdot{\hbox{\boldmath${\mu}$}}_{q^{+}}\delta(\hbox{\boldmath${r}$}),$
which is attractive and singular at short distances. The strong and singular
magnetic interaction may overcome other repulsive interactions and may allow
the formation of bound states of the fermion and antifermion system at short
distances. For brevity of notation, the quantum numbers $P$ and $C$ in and
${}^{1}S_{0}^{-+}$ and ${}^{3}P_{0}^{++}$ will be understood.
Figure 1: (a) The schematic picture of the ${}^{3}P_{0}$ state spin
configuration and the orbital motion of a negative charge $q^{-}$ and a
positive charge $q^{+}$ that can lead to a strong magnetic attraction at short
distances. Here, ${\hbox{\boldmath${\mu}$}}(q^{\pm})$ is the magnetic moment
of the charge $q^{\pm}$ arising from its spin
$\hbox{\boldmath${s}$}(q^{\pm})$. (b) The schematic picture of the spin
configurations of $q^{-}$ and $q^{+}$ in the ${}^{1}S_{0}$ state.
Previously, one of us (CYW), in collaboration with R. L. Becker, studied the
$(e^{+}e^{-})$ system using the Kemmer-Fermi-Yang equation Kem37 with
interactions consisting of the Coulomb interaction and the vector (magnetic)
interaction,
${\hbox{\boldmath${A}$}}_{i}={\hbox{\boldmath${\mu}$}}_{j}\times({\hbox{\boldmath${r}$}}_{i}-{\hbox{\boldmath${r}$}}_{j})/|{\hbox{\boldmath${r}$}}_{i}-{\hbox{\boldmath${r}$}}_{j}|^{3}$,
in connection with a possible scalar ${}^{3}P_{0}$ magnetic resonance Won86 .
The interest was to investigate whether there could be a resonance at the mass
of 1.579 MeV that might explain the anomalous positron peak in heavy-ion
collisions near the Coulomb barrier Sch83 . The experimental evidence for the
anomalous positron peak later turned out to be negative when greater
statistics were accumulated Ahm95 . Nevertheless, it remains of interest to
study the behavior of the two-body system at short distances and see how the
attractive magnetic interaction in the ${}^{3}P_{0}$ state may reveal itself
in some observable properties.
While the use of the Kemmer-Fermi-Yang equation with a two-body magnetic
interaction is useful to motivate an approximate description Won86 , a
consistent relativistic description of the two-body interaction at short
distances can be found in the relativistic two body Dirac equations (TBDE)
formulated in Dirac’s constraint dynamics dirac ; cnstr ; cra82 ; cww . These
relativistic two body Dirac equations give a good description to the entire
meson mass spectrum (excluding most flavor-mixed mesons) with constituent
world-scalar and vector potentials depending on just two or three invariant
functions, in previous relativistic quark-model calculations crater2 ; tmlk ;
unusual .
The application of the TBDE equations to two-body bound and resonance states
in quantum electrodynamics has intrinsic merits. In Ref. bckr , the properties
of these TBDE equations that made them work so well for the relativistic quark
model were investigated by solving them nonperturbatively (i.e. analytically
or numerically) in quantum electrodynamics (QED), where order $\alpha^{4}$
perturbative solutions are well known. The two coupled Dirac equations in the
constraint formalism depend on Lorentz-covariant potentials between the two
constituents and act on a 16-component wave function. An exact Pauli reduction
led to a second-order relativistic Schrödinger-like equation for a reduced
four-component wave function with an effective interaction containing all the
dependencies on spin, orbital angular momentum, and tensor operators. We were
able to solve the TBDE nonperturbatively (analytically or numerically) as well
as perturbatively because the spin dependent short-distance components of the
effective interaction are not singular cww ; bckr . The situation is very
different from the approximate Fermi-Breit forms, which contain singular
potentials and necessitate the introduction of arbitrary short-distance cut-
off parameters. The spin dependence of the relativistic potentials in the
exact Schrödinger-like equation arises naturally from the relativistic
reduction procedure and it incorporates detailed minimal interaction and
dynamical recoil effects, characteristic of field theory. We shall also use
the term “quasipotential” to represent this effective, non-singular
interaction.
To obtain the interaction used in the TBDE formalism, we first determined the
relativistic quasipotential to the lowest order in $\alpha$ for the Schr
ödinger-like equation in Ref. bckr by comparing the effective interaction
with the interaction derived from the Bethe-Salpeter equation. This, in turn
led to an invariant Coulomb-like potential $A(r)=-\alpha/r$, where $\alpha$ is
the coupling constant. Insertion of this information into the minimal
interaction structures of the two body Dirac equations then completely
determined all aspects (spin-dependent as well as spin-independent) of the
interaction. (In decay we gave a procedure to construct the full 16-component
solution to our coupled first-order Dirac equations from a solution of the
second-order equation for the reduced wave function.)
Next, we showed that both the quantum mechanical perturbative and the TBDE
non-perturbative treatments (i.e. analytic or numerical) yield the standard
spectral results for QED and related interactions through order $\alpha^{4}$ .
Such an agreement depends crucially on the inclusion of the coupling between
various components of our 16-component Dirac wave functions and on the short-
distance behavior of the relativistic quasipotential in the associated
Schrödinger-like equation. We then examined the speculations Won86 whether
the quasipotentials (including the angular momentum barrier) for some states
in the $e^{+}e^{-}$ system may become attractive enough at short distances to
yield a pure QED resonance corresponding to the anomalous positron peaks in
heavy-ion collisions Sch83 . For the ${}^{3}P_{0}$ state we found that, even
though the quasipotential becomes attractive and overwhelms the centrifugal
barrier at short distances, the spatial extension of the attraction is not
large enough to hold a resonance at the energy of 1.579 MeV bckr . This result
contradicted predictions of such states by other authors Spe91 based on
numerical solutions of three-dimensional truncations of the Bethe-Salpeter
equation, for which the entire QED bound state spectrum has been treated
successfully through order $\alpha^{4}$ only by perturbation theory.
In this paper we return to this problem of the magnetic resonance and magnetic
states, not motivated so much by new experimental data as by a discovery of an
additional peculiar solution of the TBDE overlooked in the earlier work in
Ref. bckr . Our examination of the two body Dirac equations reveals that at
short distance for both ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states, the magnetic
interactions is indeed quite strong. As a consequence, they counterbalance
other repulsive interactions to result in a quasipotential for these states
that behaves as $-\alpha^{2}/r^{2}$ at short distances.
In standard quantum mechanics for central interactions including the angular
momentum barrier $L(L+1)/r^{2}$ for states with $L\neq 0$ at short distances,
one generally retains only one of the two solutions for the radial part of the
wave function, $u=r\psi$, the one that grows with distance as ($\sim
r^{L+1})$, dropping the other solution($\sim r^{-L})$ as being too singular.
If we likewise represent the quasipotential as $\lambda(\lambda+1)/r^{2}$ with
$\lambda=(-1+\sqrt{(1-4\alpha^{2}})/2$, it leads to a short-distance solution
that behaves as $r^{\lambda+1}$, which we call the usual solution, in addition
to a solution, whose radial part grows as $r^{-\lambda}$, which we call the
peculiar solution. However, both usual and peculiar states have quantum-
mechanically acceptable behaviors at short distances, as the wave functions at
short-distances are square-integrable.
In the case of the spin singlet ${}^{1}S_{0}$ states, the eigenstates and
eigenenergies can be obtained analytically and are found to encompass both
usual and peculiar states. We find usual bound states with attributes the same
as those one usually encounters in QED and QCD, explicitly agreeing with the
standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar ${}^{1}S_{0}$ ground state of a fermion-antifermion pair with a
fermion rest mass $m$ has a root-mean-square radius approximately $1/m$, a
binding energies approximately $B_{p}$$\sim$$(2-\sqrt{2})m$, and a rest mass
approximately $\sqrt{2}m$. However, the $(n+1)$ th ${}^{1}S_{0}$ peculiar
state is nearly degenerate in energy and approximately equal in size with the
$n$th usual ${}^{1}S_{0}$.
The existence of both usual and peculiar eigenstates in the same system brings
with them conceptual and mathematical problems of the non-self-adjoint
property of the mass operator and the over-completeness of the set of
eigenstates. We resolve these problems by the introduction of a new quantum
number, the peculiarity quantum number, that makes the mass operator self-
adjoint and the combined set of usual and peculiar states a complete set in an
enlarged Hilbert space.
In the case of the ${}^{3}P_{0}$ states, both of the usual and peculiar
solutions reflect the overwhelmed centrifugal barrier and so differ
substantially from the $r^{L+1}$ and $r^{-L}$ behaviors at short distances
respectively. As a peculiar state radial wave function $u$ rises from the zero
value at the origin as $r^{-\lambda}\sim r^{\alpha^{2}}$, the strongly
attractive magnetic interaction has the tendency of bending the wave function
in such a way to allow for the possibility of a resonance. Furthermore, as the
quasipotential obtained through the relativistic reduction is sensitively
energy dependent, we can explore the behavior of the two-body system over a
larger domain of energies. We find that the usual solutions lead to no
resonant behavior, but the peculiar solution can lead to a ${}^{3}P_{0}$
resonance whose phase shift changes by $\pi$ at an appropriate energy,
depending on the description of the internal structure of the charges, the
mass of the constituent, and the coupling constant.
This paper is organized as follows. In Sec. II we give a review of the two
body Dirac equations of constraint dynamics. For those readers who are already
familiar with the constraint approach we refer them to the TBDE given in Eq.
(14) and their Schrödinger-like Pauli reduction given in Eq. (17). We
specialize to electromagnetic-like interactions only in this paper. We give in
Sec. III the single-component radial forms of Eq. (17) relevant to this paper.
In Sec. IV we examine both solutions for the ${}^{1}S_{0}$ states. In addition
to examining new bound state solutions, we show how the ${}^{1}S_{0}$ wave
functions for positive energies (and their corresponding phase shifts) can be
determined analytically in terms of Coulomb wave functions for noninteger
angular momentum. This is done for both the usual and peculiar solutions. We
explain why and how we introduce of a new quantum number, which we call the
peculiarity quantum number, to solve the problems of the non-self-adjoint
property of the mass operator and the over-completeness of the set of
eigenstates. In Sections V we examine the short distance behaviors for the
${}^{3}P_{0}$ state for the usual and peculiar solutions. In Sec. VI we
discuss the variable phase shift formalism of Calogero cal and outline how we
use it for our phase shift analysis. Since the short distance behavior of the
${}^{3}P_{0}$ quasipotential is the same as that of ${}^{1}S_{0}$
quasipotential, we can use those same ${}^{1}S_{0}$ Coulomb wave functions as
reference wave functions in that region to compute phase shifts. There is,
however, an additional term (proportional to
$\delta({\hbox{\boldmath${r}$})}$) that does not appear in the extreme short
distance region for the ${}^{1}S_{0}$ quasipotential. Even though this term
does not contribute in the case of the phase shift for the usual solution, its
contribution to the phase shift calculations for the peculiar solution must be
considered. In Sec. VI we discuss our numerical results and in Sec. VII our
conclusions. Various technical results are presented in the appendices. In
Appendix A we give an outline of the details on the relation between the two-
body Dirac equations and their Pauli reduced Schrödinger forms. In Appendix B
we present the radial forms of those Schrödinger-like equations for a general
angular momentum coupling. In Appendix C we present details of the
${}^{1}S_{0}$ usual and peculiar bound states. In Appendix D we review the
connections between the Coulomb wave functions for noninteger angular momentum
index. Appendix E presents a review of the variable phase method of Calogero
cal for our problem.
## II TWO BODY DIRAC EQUATIONS
We briefly review the two body Dirac equations of constraint dynamics cra82 ;
cww ; crater2 ; tmlk ; unusual ; saz86 providing a covariant three
dimensional truncation of the Bethe Salpeter equation for the two body system.
Sazdjian saz85 ; saz92 ; saz97 has shown that the Bethe-Salpeter equation can
be algebraically transformed into two independent equations. The first yields
a covariant three dimensional eigenvalue equation which for spinless particles
takes the form
$\biggl{(}\mathcal{H}_{10}+\mathcal{H}_{20}+2\Phi\biggr{)}\Psi(x_{1},x_{2})=0,$
(1)
where $\mathcal{H}_{i0}=p_{i}^{2}+m_{i}^{2}$ . The quasipotential $\Phi$ is a
modified geometric series in the Bethe-Salpeter kernel $K$ such that in lowest
order in $K$
$\Phi=\pi i\delta(P\cdot p)K,$ (2)
where $P=p_{1}+p_{2}$ is the total momentum, $p=\mu_{2}p_{1}-\mu_{1}p_{2}$ is
the relative momentum, $w$ is the invariant total center of momentum (c.m.)
energy with $P^{2}=-w^{2}$. The $\mu_{i}$ must be chosen so that the relative
coordinate $x=x_{1}-x_{2}$ and $p$ are canonically conjugate, i.e.
$\mu_{1}+\mu_{2}=1$. The second equation, Eq. (2), overcomes the difficulty of
treating the relative time in the center of momentum system by setting an
invariant condition on the relative momentum $p$,
$(\mathcal{H}_{10}-\mathcal{H}_{20})\Psi(x_{1},x_{2})=0=2P\cdot
p\Psi(x_{1},x_{2}).$ (3)
Note that this implies
$p^{\mu}\Psi=p_{\perp}^{\mu}\Psi\equiv(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})p_{\nu}\Psi$
in which $\hat{P}^{\mu}=P^{\mu}/w$ is a time like unit vector
$(\hat{P}^{2}=-1)$ in the direction of the total momentum222 We use the metric
$\eta^{11}=\eta^{22}=\eta^{33}=-\eta^{00}=1$..
One can further combine the sum and the difference of Eqs. (1) and (3) to
obtain a set of two relativistic equations one for each particle with each
equation specifying two generalized mass-shell constraints
$\mathcal{H}_{i}\Psi(x_{1},x_{2})=(p_{i}^{2}+m_{i}^{2}+\Phi)\Psi(x_{1},x_{2})=0,~{}i=1,2,$
(4)
including the interaction with the other particle. These constraint equations
are just those of Dirac’s Hamiltonian constraint dynamics for spinless
particles dirac ; cnstr ; cra84s . In order for Eq. (4) to have consistent
solutions, Dirac’s constraint dynamics stipulate that these two constraints
must be compatible among themselves,
$[\mathcal{H}_{1},\mathcal{H}_{2}]\Psi=0$, that is, they must be first class
constraints. This requires that the quasipotential $\Phi$ satisfy
$[p_{1}^{2}-p_{2}^{2},\Phi]\Psi=0$. Working out the commutator shows that for
this to be true in general, $\Phi$ must depend on the relative coordinate
$x=x_{1}-x_{2}$ only through its component, $x_{\perp},$ perpendicular to $P,$
$x_{\perp}^{\mu}=(\eta^{\mu\nu}+\hat{P}^{\mu}\hat{P}^{\nu})(x_{1}-x_{2})_{\nu}.$
(5)
The invariant $x_{\perp}^{2}\equiv r^{2}$ becomes $\mathbf{r}^{2}$ in the c.m.
frame. Since the total momentum is conserved, the single component wave
function $\Psi~{}$in coordinate space is a product of a plane wave eigenstate
of $P$ and an internal part $\psi(x_{\perp})$ cra87 333 We use the same symbol
$P$ for the eigenvalue so that the $w$ dependence of $m_{w}$ and
$\varepsilon_{w}$ in Eq. (6) is regarded as an eigenvalue dependence. The wave
function $\Psi$ can be viewed either as a relativistic 2-body wave function
(similar in interpretation to the Dirac wave function) or, if a close
connection to field theory is required, related directly to the Bethe Salpeter
wave function $\chi{\hbox{\boldmath${~{}}$}}$by saz92 $\Psi=-\pi
i\delta(P\cdot p)\mathcal{\ H}_{10}\chi=-\pi i\delta(P\cdot
p)\mathcal{H}_{20}\chi$..
We find a plausible structure for the quasipotential $\Phi$ by observing that
the one-body Klein-Gordon equation
$(p^{2}+m^{2})\psi=({\hbox{\boldmath${p}$}}^{2}-\varepsilon^{2}+m^{2})\psi=0$
takes the form
$({\hbox{\boldmath${p}$}}^{2}-\varepsilon^{2}+m^{2}+2mS+S^{2}+2\varepsilon
A-A^{2})\psi=0~{}$when one introduces a scalar interaction and time-like
vector interaction via the minimum substitutions $m\rightarrow m+S~{}$and
$\varepsilon\rightarrow\varepsilon-A$. In the two-body case, separate
classical fw and quantum field theory saz97 arguments show that when one
includes world scalar and vector interactions then $\Phi$ depends on two
underlying invariant functions $S(r)$ and $A(r)$ ($r=\sqrt{x_{\perp}^{2}}$)
through the two body Klein-Gordon-like potential form with the same general
structure, that is
$\Phi=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}.$ (6)
Those field theories further yield the c.m. energy dependent forms
$m_{w}=m_{1}m_{2}/w,$ (7)
and
$\varepsilon_{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w,$ (8)
ones that Tododov cnstr ; tod71 introduced as the relativistic reduced mass
and effective particle energy for the two-body system. Similar to what happens
in the nonrelativistic two-body problem, in the relativistic case we have the
motion of this effective particle taking place as if it were in an external
field (here generated by $S$ and $A$). The two kinematical variables (7) and
(8) are related to one another by the Einstein condition
$\varepsilon_{w}^{2}-m_{w}^{2}=b^{2}(w),$ (9)
where the invariant
$b^{2}(w)\equiv(w^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})/4w^{2},$
(10)
is the c.m. value of the square of the relative momentum expressed as a
function of $w$. One also has
$b^{2}(w)=\varepsilon_{1}^{2}-m_{1}^{2}=\varepsilon_{2}^{2}-m_{2}^{2},$ (11)
in which $\varepsilon_{1}$ and $\varepsilon_{2}$ are the invariant c.m.
energies of the individual particles satisfying
$\ \varepsilon_{1}+\varepsilon_{2}=w,\
\varepsilon_{1}-\varepsilon_{2}=(m_{1}^{2}-m_{2}^{2})/w.$ (12)
In terms of these invariants, the relative momentum appearing in Eq. (2) and
(3) is given by
$p^{\mu}=(\varepsilon_{2}p_{1}^{\mu}-\varepsilon_{1}p_{2}^{\mu})/w\mathrm{,}$
(13)
so that $\mu_{1}+\mu_{2}=(\varepsilon_{1}+\varepsilon_{2})/w=1$. In tod the
forms for these two-body effective kinematic variables are given sound
justifications based solely on relativistic kinematics, supplementing the
dynamical arguments of fw and saz97 .
This covariant and useful three-dimensional truncation of the Bethe-Salpeter
equation has been extended to the case of a two-fermion system where the two
constraint equations become the two body Dirac equations (TBDE) cra82 ; cra82
; cww ; crater2 ; tmlk ; unusual
$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle\equiv\gamma_{51}(\gamma_{1}\cdot(p_{1}-\tilde{A}_{1})+m_{1}+\tilde{S}_{1})\Psi=0,$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle\equiv\gamma_{52}(\gamma_{2}\cdot(p_{2}-\tilde{A}_{2})+m_{2}+\tilde{S}_{2})\Psi=0.$
(14)
Here $\Psi$ is a sixteen component wave function consisting of an external
plane wave part that is an eigenstate of $P$ and an internal part
$\psi=\psi(x_{\perp})$. The vector potential$\ \tilde{A}_{i}^{\mu}$ was taken
to be an electromagnetic-like four-vector potential with the time-like and
space-like portions both arising from a single invariant function $A(r)$. 444
In particular, in a perturbative context that would mean that these aspects of
$\tilde{A}_{i}^{\mu}$ were regarded as arising from a Feynman gauge vertex
coupling of a form proportional to $\gamma_{1}^{\mu}\gamma_{2\mu}A$ . The
tilde on these four-vector potentials indicate that they are not only position
dependent but also spin dependent by way of the gamma matrices. The operators
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ must commute or at the very least
$[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0$ since they operate on the same wave
function555 The $\gamma_{5}$ matrices for each of the two particles are
designated by $\gamma_{5i}$ $i=1,2$. The reason for putting these matrices out
front of the whole expression is that including them facilitates the proof of
the compatibility condition, see cra82 ; cra87 .. This compatibility condition
gives restrictions on the spin dependencies of the vector and scalar
potentials,
$\tilde{A}_{i}^{\mu}=\tilde{A}_{i}^{\mu}(A(r),p_{\perp},\hat{P},w,\gamma_{1},\gamma_{2}),$
(15)
in addition to requiring that they depend on the invariant separation
$r\equiv\sqrt{x_{\perp}^{2}}$ through the invariant $A(r)$. The covariant
constraint (3) can also be shown to follow from Eq. (14). We give the explicit
connections between $\tilde{A}_{i}^{\mu}$ and the invariant $A(r)$ in Appendix
A. (A similar dependence occurs for $\tilde{S}_{i}$ on $S(r).$) The general
structural dependence on $A(r)$ and $S(r)$ and the spin dependence of
$\tilde{A}_{i}^{\mu},\tilde{S}_{i}$ is a consequence of the compatibility
condition $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0.$
The Pauli reduction of these coupled Dirac equations lead to a covariant Schr
ödinger-like equation for the relative motion with an explicit spin-dependent
potential $\Phi$,
${\bigg{(}}p_{\perp}^{2}+\Phi(S(r),A(r),p_{\perp},\hat{P},w,\sigma_{1},\sigma_{2}){\bigg{)}}\psi_{+}=b^{2}(w)\psi_{+},$
(16)
with $b^{2}(w)$ playing the role of the eigenvalue666 Due to the dependence of
$\Phi$ on $w,$ this is a nonlinear eigenvalue equation.. This eigenvalue
equation can then be solved for the four-component effective particle spinor
wave function $\psi_{+}$ related to the sixteen component spinor
$\psi(x_{\perp})$ in appendix A.
In Appendix A we outline the steps needed to obtain the explicit c.m. form of
Eq. (16). That form is liu , saz94 , crater2 ; tmlk ; unusual
$\displaystyle\\{$
$\displaystyle{\hbox{\boldmath${p}$}}^{2}+\Phi({\hbox{\boldmath${r}$},}m_{1},m_{2},w,{\hbox{\boldmath${\sigma}$}}_{1},{\hbox{\boldmath${\sigma}$}}_{2})~{}\\}\psi_{+}$
(17) $\displaystyle=$
$\displaystyle\\{{\hbox{\boldmath${p}$}}^{2}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle+{\hbox{\boldmath${L}$}\cdot(\hbox{\boldmath${\sigma}$}}_{1}{+\hbox{\boldmath${\sigma}$}}_{2}{)}\Phi_{SO}+{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}L\cdot(\sigma}_{1}{\hbox{\boldmath${+}$}\sigma}_{2}{\hbox{\boldmath${)}$}}\Phi_{SOT}$
$\displaystyle+{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\sigma}_{2}\Phi_{SS}+(3{\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}\sigma}_{2}{\hbox{\boldmath${\
}$}\cdot\hat{r}-\hbox{\boldmath${\sigma}$}}_{1}{\hbox{\boldmath${\cdot}$}\sigma}_{2})\Phi_{T}$
$\displaystyle+{\hbox{\boldmath${L}$}\cdot(\sigma}_{1}{\hbox{\boldmath${-}$}\sigma}_{2}{\hbox{\boldmath${)}$}\Phi}_{SOD}+i{\hbox{\boldmath${L}$}\cdot\sigma}_{1}{\hbox{\boldmath${\times}$}\sigma}_{2}\Phi_{SOX}\\}\psi_{+}$
$\displaystyle=$ $\displaystyle b^{2}\psi_{+},$
where the detailed forms of the separate quasipotentials $\Phi_{i}$ are given
in Appendix A. The subscripts of most of the quasipotentials are self
explanatory777 The subscript on quasipotential $\Phi_{D}$ refers to Darwin. It
consist of what are called Darwin terms, those that are the two-body analogue
of terms that accompany the spin-orbit term in the one-body Pauli reduction of
the ordinary one-body Dirac equation, and ones related by canonical
transformations to Darwin interactions fw ; sch73 , momentum dependent terms
arising from retardation effects. The subscripts on the other quasipotentials
refer respectively to $SO$ (spin-orbit), $SOD$ (spin-orbit difference), $SOX$
(spin-orbit cross terms), $SS$ (spin-spin), $T$ (tensor), $SOT$ (spin-orbit-
tensor).. After the eigenvalue $b^{2}$ of (17) is obtained, the invariant mass
of the composite two-body system $w$ can then be obtained by inverting Eq.
(10). It is given explicitly by
$w=\sqrt{b^{2}+m_{1}^{2}}+\sqrt{b^{2}+m_{2}^{2}}.$ (18)
For this reason we call the operator that appears to the left of Eq. (17) the
invariant mass operator. The structure of the linear and quadratic terms in
Eq. (17) as well as the Darwin and spin-orbit terms, are plausible in light of
the discussion given above Eq. (6), and in light of the static limit Dirac
structures that come about from the Pauli reduction of the Dirac equation.
Their appearance as well as that of the remaining spin structures are direct
outcomes of the Pauli reductions of the simultaneous TBDE Eq. (14). In this
paper we take the scalar interaction $S(r)=0$.
## III TBDE SINGLE COMPONENT WAVE EQUATIONS.
The 4 component two-body wave function $\psi_{+}$ of the above Pauli-form (
17) of the TBDE can be conveniently represented by spin-singlet $S=0$ and
spin-triplet $S=1$ components with quantum numbers $\\{J,L,S\\}$ and basis
wave functions
$\langle{\hbox{\boldmath${r}$}}|wJLS\rangle\equiv\psi_{JLS}({\hbox{\boldmath${r}$}})=\frac{u_{JLS}(r)}{r}Y_{JM}(\mathbf{\hat{r})}.$
(19)
In general, the singlet and triplet states are coupled. However, we see from
Appendix B that for the case of equal masses and certain angular momentum
states, the spin singlet and spin triplet components decouple, and the TBDE
reduce to a single component equation.
Specifically, for the spin-singlet $S=0$ state with $J=L$, (the ${}^{1}J_{J}$
state), the TBDE is
$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}{-}3\Phi_{SS}\\}u_{JJ0}=b^{2}u_{JJ0},$
(20)
where, using the results in Appendix B, the magnetic interaction $-3\Phi_{SS}$
is
$\displaystyle-3\Phi_{SS}$ $\displaystyle=$
$\displaystyle-3\Phi_{SS}(A,A^{\prime},{\nabla}^{2}A)=-3(\frac{1}{r^{2}}-\frac{3}{2r}\left(\frac{A^{\prime}}{w-2A}\right))((\frac{1}{\sqrt{1-2A/w}}+\sqrt{1-2A/w})-2)$
$\displaystyle-\frac{3}{2r}\left(\frac{A^{\prime}}{w-2A}\right)(\frac{1}{\sqrt{1-2A/w}}-\sqrt{1-2A/w})\mathcal{-}\frac{21}{2}\left(\frac{A^{\prime}}{w-2A}\right)^{2}-\frac{3{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}$
$\displaystyle=$ $\displaystyle-\Phi_{D}(A,A^{\prime},{\nabla}^{2}A),$
which is attractive and singular, as we discussed in the Introduction. At
large distances and for $A=-\alpha/r$ potential,
${\hbox{\boldmath${\nabla}$}}^{2}A=4\pi\alpha\delta({\hbox{\boldmath${r}$}})$
and the spin-spin interaction indeed becomes a singular interaction as
described in Jac62 . In addition to the magnetic spin-spin interaction, there
is also the repulsive Darwin quasipotential $\Phi_{D}$. In the ${}^{1}J_{J}$
state, the attractive magnetic spin-spin quasipotential in the spin-singlet
configuration exactly cancels the repulsive Darwin quasipotential,
${-}3\Phi_{SS}+\Phi_{D}=0.$ (22)
As a result of this remarkable cancellation, the eigenvalue equation for the
${}^{1}J_{J}$ state in Eq. (III) becomes simply
$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}\\}u_{JJ0}=b^{2}u_{JJ0}.$
(23)
Of all spin-singlet states, only in the ${}^{1}S_{0}$ states ($J=L=0)$ do the
effects of the quasipotential and the absence of a centrifugal barrier make
the combined quasipotential strongly attractive at short-distances. This, of
course, would not happen were it not for the highly attractive spin-spin
interaction discussed in the Introduction and in Eq. (III). Among the spin-
singlet states with different $J$ quantum numbers, we shall therefore focus
our attention only on the ${}^{1}S_{0}$ states.
For the spin-triplet $S=1$ states, there are two states with single component
radial equations. The first is the ${}^{3}J_{J}$ state whose radial equation
takes the form ($J\geq 1$)
$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{2A^{\prime}}{r\left(w-2A\right)}+3\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}u_{J1J}=b^{2}u_{JJ1}.$
(24)
The second is the ${}^{3}P_{0}$ equation which takes the form
$\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{2{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}u_{011}=b^{2}u_{011}.$
(25)
Of the two spin-triplet cases, only in the ${}^{3}P_{0}$ states ($J=0,$
$L=1)~{}$ do the combined effects of the quasipotentials become so strongly
attractive at short-distances that they overwhelm the presence of the
centrifugal barrier. As discussed in the Introduction, this is due to the
highly attractive spin-orbit interaction (“magnetic” interaction) when the
total spin and the orbital angular momentum are oppositely aligned. In that
case, the competing effects of both the short-distance attraction and the
presence of the potential barrier raise the question whether the attraction is
strong enough to hold a resonance state in the continuum. Among the spin-
triplet states with different $J$ and $L$ quantum numbers, we shall therefore
focus our attention only on the ${}^{3}P_{0}$ states.
In the last term of the quasipotential in Eq. (25), the quantity $\nabla^{2}A$
is related to the particle charge density, $\rho(\hbox{\boldmath${r}$})$, seen
by each of the two particles by
$\nabla^{2}A(\hbox{\boldmath${r}$})=4\pi\alpha\rho(\hbox{\boldmath${r}$}).$
(26)
Therefore, the equation for the two-body relative wave function for the
${}^{3}P_{0}$ state becomes
$\biggl{\\{}-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{8\pi\alpha\rho(\hbox{\boldmath${r}$})}{w-2A}\biggr{\\}}u_{011}=b^{2}u_{011}.$
(27)
As we shall see in this case, the attractive magnetic interaction overwhelms
the centrifugal barrier, allowing the wave function to reach the short-
distance region where the particle charge density $\rho(r)$, if any, can be
exposed for scrutiny. This is in contrast to the situation for states in which
the centrifugal barrier dominates the short-distance region. In that case, the
centrifugal barrier will prevent the wave function from reaching the short-
distance region and the particle charge density will not make as a significant
difference in observable quantities888 Such would also be the case for
${}^{1}S_{0}$ states in which, due to the cancellation in Eq. (22), the
dependence on $\rho(r)$ is only indirect or implicit through the altered form
for $A(r)$.. We obtain the important result that the ${}^{3}P_{0}$
quasipotential depends explicitly on the particle charge density
$\rho(\hbox{\boldmath${r}$})$ at short distances. As a consequence, some
observable quantities may depend more critically on the nature of the particle
charge distribution and the forces binding the charge elements together.
For the ${}^{3}P_{0}$ state, it is convenient to separate out the centrifugal
barrier $2/r^{2}$ and the quasipotential $\Phi$ to write the above equation as
$\biggl{\\{}-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+\Phi(\hbox{\boldmath${r}$})\biggr{\\}}u_{011}=b^{2}u_{011},$
(28)
where
$\Phi(\hbox{\boldmath${r}$})=2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{8\pi\alpha\rho(\hbox{\boldmath${r}$})}{w-2A}.$
(29)
In our early work bckr , we limited our attention to energy regions around
1.579 MeV for the $(e^{+}e^{-})$ system and searched for ${}^{3}P_{0}$
resonances whose wave functions start from the origin in the usual way. We
found no resonance states. We return to this problem again including now an
additional (peculiar) solution of the TBDE that was overlooked in the earlier
work but has quantum-mechanically acceptable behaviors at short distances.
In Eqs. (27)), both the gauge field $A(r)$ and the gauge field source
$\rho(r)$ appear in the equation of motion for the wave function in the
${}^{3}P_{0}$ state. The appearance of the fermion charge source distribution
$\rho(r)$ brings into focus the question whether it is sufficient to describe
the magnetic interaction in the ${}^{3}P_{0}$ state completely within quantum
electrodynamics or quantum chromodynamics. Electrons in QED and quarks in QCD
are taken to be point particles with no structure. It may be necessary to go
beyond these field theories, to include additional auxiliary interactions that
hold the charge elements together, in order to properly describe the internal
structure of these particles. If these auxiliary interactions act on the
charged elements of the fermion to hold them together, they can also act on
the charged elements of the antifermion charge and will affect the
${}^{3}P_{0}$ wave function in the interior region of the charge distribution
$\rho(r)$.
The nature of these auxiliary forces holding the charged elements together is
completely unknown, although there have been many attempts to carry out such
an investigation. For example, in the Dirac’s model of an electron, a surface
tension from an unknown axillary interaction is invoked to hold the electric
charged elements of an electron together Dir48 ; Dir51 ; Dir52 ; Dir53 ; Dir62
; Dir65 . However, our knowledge on the internal structures of electrons and
quarks remain very uncertain. We shall return to examine how such a lack of
knowledge of the internal structures of these elementary quanta leads to
uncertainties in the ${}^{3}P_{0}$ magnetic resonance states in Sec. VC.
## IV SOLUTIONS OF THE TWO BODY DIRAC EQUATIONS FOR THE ${}^{1}S_{0}$ STATE
### IV.1 The ${}^{1}S_{0}$ quasipotential
We first consider the case of the ${}^{1}S_{0}$ state of a point fermion-
antifermion pair with electric or color charges interact through an
electromagnetic-type interaction arising from the exchange of a single photon
or gluon. The single photon annihilation diagram does not contribute because
the ${}^{1}S_{0}$ state is a charge parity even state. We thus have
$A=-\frac{\alpha}{r}.$ (30)
For brevity of notation in this subsection, we shall abbreviate the radial
wave function $u_{0JJ}$=$u_{000}$ as $u$. Equation (20) for $u$ becomes
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u.$
(31)
with a short distance ($r<<\alpha/2\varepsilon_{w})$ behavior given by
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{\alpha^{2}}{r^{2}}\right\\}u=0,$ (32)
with solutions
$\displaystyle u_{+}$ $\displaystyle\sim$ $\displaystyle r^{\lambda+1},$
$\displaystyle u_{-}$ $\displaystyle\sim$ $\displaystyle r^{-\lambda},$
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle(-1+\sqrt{1-4\alpha^{2}})/2,$ (33)
or
$u_{\pm}\sim r^{(1\pm\sqrt{1-4a^{2}})/2},$ (34)
both of which approach zero as $r$ approaches zero. With these behaviors, the
probability
$\psi_{\pm}^{2}d^{3}r=\frac{u_{\pm}^{2}}{r^{2}}r^{2}drd\Omega=u_{\pm}^{2}drd\Omega=r^{(1\pm\sqrt{1-4a^{2}})}drd\Omega,$
(35)
is finite for both signs. We call $u_{+}$ the usual solution, and it behaves
as $r^{\lambda+1}\sim r^{1-\alpha^{2}}$ for small $\alpha$. We call $u_{-}$
the peculiar solution, and it behaves as $r^{-\lambda}\sim r^{\alpha^{2}}$for
small $\alpha$. Both of these behaviors are physically acceptable near the
origin in the sense of (i) $u(0)$$\rightarrow 0$, and (ii) being square-
integrable in the neighborhood of $r\sim 0$. We note that if the sign in front
of $\alpha^{2}$ were positive or if we had non-zero angular momentum such that
$L(L+1)-\alpha^{2}>0$ then the second or peculiar set of solutions are not
physically admissible states.
In khel one finds a thorough discussion on the proper boundary condition for
the radial wave function of the Schrödinger equation at the origin. They
discuss several conditions that appear in the literature: (1) Continuity of
$R=u/r$ at $r=0,$ requiring $u(0)=0$. (2) A finite differential probability in
the spherical slice $(r,r+dr$), that is $R^{2}r^{2}dr<\infty$ requiring
$u(r)\rightarrow r^{s+1},s>-1$ and again $u(0)=0$. (3) Requiring a finite
total probability inside a sphere of small radius $a$ which allows a more
singular behavior, namely $u(r)\rightarrow r^{-1/2+\varepsilon}$ where
$\varepsilon>0$ is a small positive constant, which would also include a
finite behavior of the norm. (4) Requiring time independence of the norm
leading to $u(r)\rightarrow cr^{-s+1}$, $s<1$ which again leads to $u(0)=0.$
Reference khel furthermore shows that the radial Schrödinger equation
[$-d^{2}/dr^{2}+l(l+1)/r^{2}+2mV(r)]u(r)=2mEu(r)$ is compatible with the full
Schrödinger equation
($-\nabla^{2}+2mV(r))\frac{u(r)}{r}Y_{lm}=2mE\frac{u(r)}{r}Y_{lm}$ if and only
if the condition$~{}u(0)=0$ is satisfied. This $u(0)=0$ condition is clearly
satisfied for both solutions in Eq. (34).
In Schiff’s Quantum Mechanics schiff , a solution similar to the peculiar one
discussed here is examined for the case of the Klein-Gordon equation for the
Coulomb system. He argues that what we call the peculiar solution can be
discarded since the source of the Coulomb attraction is a finite sized nucleus
of radius $r_{0}$. In particular, he states that for $r<r_{0}$ for which the
potential is finite all the way to the origin, matching at $r_{0}$ would rule
out the peculiar solution. In our case, with point particles, the potential
does not satisfy this condition.
#### IV.1.1 ${}^{1}S_{0}$ Bound States
The solutions of the ${}^{1}S_{0}$ bound states can be obtained analytically.
In Appendix C we show how we can obtain the two sets of ${}^{1}S_{0}$ bound
state solutions that correspond to the usual and peculiar short distance
behaviors. The respective sets of eigenvalues and normalized eigenfunctions
for the state with total invariant c.m. energy (mass) $w_{\pm n}$ and the
principle quantum number $n$ is
$\displaystyle w_{\pm n}$ $\displaystyle=$ $\displaystyle
m\sqrt{2+2/\sqrt{1+{\alpha^{2}}/{(}n\pm\sqrt{1/4-\alpha^{2}}-1/2{)^{2}}}},$
$\displaystyle u_{\pm n}(r)$ $\displaystyle=$
$\displaystyle\left[\left(\frac{4\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}\right)^{2}\frac{n_{r}!}{2n_{\pm}^{\prime}(n_{\pm}^{\prime}+\lambda_{\pm})!}\right]^{1/2}\exp(-\frac{\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}})\left(\frac{2\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}\right)^{\lambda_{\pm}+1}L_{n_{r}}^{2\lambda_{\pm}+1}(\frac{2\varepsilon_{w_{\pm}}\alpha
r}{n_{\pm}^{\prime}}),$ (36)
where $n_{\pm}^{\prime}=n_{r}+\lambda_{\pm}+1=n+\lambda_{\pm}$ and
$\displaystyle\varepsilon_{w_{\pm}}=(w_{\pm}^{2}-2m^{2})/2w_{\pm}.$ (37)
For the usual states $u_{+n}$, the bound state eigenvalues $w_{+n}$ agree with
standard QED perturbative results through order $\alpha^{4}$,
$w_{+n}=2m-m{\alpha^{2}}/{4}n^{2}-m\alpha^{4}/2n^{3}(1-11/32/n)+O(\alpha^{6}),~{}n=1,2,3,...$
(38)
For the set of peculiar states $u_{-n}$, note that the peculiar ground state
$u_{-1}$ with $n=1$ has eigenenergy (mass)
$w_{-1}=m\sqrt{2+2/\sqrt{1+{\alpha^{2}}/({1/2}-\sqrt{1/4-\alpha^{2}}{)^{2}}}}\sim\sqrt{2}m\sqrt{1+\alpha},$
(39)
which represents very tight binding, with a binding energy on the order 300
keV for an $e^{+}e^{-}$ state and a root-mean-square radius on the order of a
Compton wave length instead of an angstrom. In particular we find (see
Appendix C)
$\sqrt{\langle r^{2}\rangle_{-1}}\rightarrow\frac{1}{m}.$ (40)
We note further the anti-intuitive behavior of the peculiar ground state
energy (mass), increasing with increasing coupling constant $\alpha$ instead
of decreasing. The excited states are quite near to the usual bound states. We
find the following pattern for those excited peculiar states
$w_{-n}=2m-m{\alpha^{2}}/{4(}n-1)^{2}+m\alpha^{4}/2(n-1)^{3}\left(1+11/32(n-1)\right)+O(\alpha^{6});\text{
}n=2,3,4,...$ (41)
In the nonrelativistic limit, where terms of order $\alpha^{4}$ are ignored we
find that the states are degenerate with the $n$th usual state identical to
the $(n+1)$th peculiar state. If we include the $\alpha^{4}$ corrections then
we find that
$w_{+n}-w_{-(n+1)}=-m\alpha^{4}/n^{3}.$ (42)
For all of the usual states and the remaining peculiar states we have
$\displaystyle\langle r^{2}\rangle_{+n}$ $\displaystyle=$
$\displaystyle\frac{(n+\lambda_{+})^{2}}{6\left(\varepsilon_{w_{+n}}\alpha\right)^{2}}[(n+\lambda_{+})^{2}+5\alpha^{2}+3]\text{,~{}}n=1,2,3...,$
$\displaystyle\langle r^{2}\rangle_{-n}$ $\displaystyle=$
$\displaystyle\frac{(n+\lambda_{-})^{2}}{6\left(\varepsilon_{w_{-n}}\alpha\right)^{2}}[(n+\lambda_{-})^{2}+5\alpha^{2}+3]\text{,~{}}n=2,3...,$
(43)
so that the size in the $(n+1)$th peculiar state is nearly the same as with
the $n$th usual state.
As shown in the Appendix C, the two sets of solutions, are not orthogonal with
respect to one another. For example, the two $n=1$ wave functions have the
respective forms
$\displaystyle u_{+}(r)$ $\displaystyle=$ $\displaystyle
c_{+}r^{\lambda_{+}+1}\exp(-\kappa_{+}\varepsilon_{w_{+}}\alpha r),$
$\displaystyle\kappa_{+}$ $\displaystyle=$
$\displaystyle\frac{2}{1+\sqrt{1-4\alpha^{2}}}=\frac{1}{\lambda_{+}+1},$
$\displaystyle u_{-}(r)$ $\displaystyle=$ $\displaystyle
c_{-}r^{\lambda_{-}+1}\exp(-\kappa_{-}\varepsilon_{w_{-}}\alpha r),$
$\displaystyle\kappa_{-}$ $\displaystyle=$
$\displaystyle\frac{2}{1-\sqrt{1-4\alpha^{2}}}=\frac{1}{\lambda_{-}+1},$ (44)
where for brevity of notation, we have omitted the principal quantum number
designation in $u\pm$ for the of the ground state. Clearly since they are both
zero node solutions we have
$\langle u_{-}|u_{+}\rangle=\int_{0}^{\infty}dru_{+}(r)u_{-}(r)\neq 0.$ (45)
How do we reconcile this with the expected orthogonality of the eigenfunctions
of a self-adjoint operator corresponding to different eigenvalues. In the
present context, the naive self-adjoint property requires that
$\langle
u_{+}|(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}})|u_{-}\rangle=\langle
u_{-}|(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}})|u_{+}\rangle.$
(46)
This boils down to
$\int_{0}^{\infty}dxu_{+}\frac{d^{2}u_{-}}{dx^{2}}=\int_{0}^{\infty}dxu_{-}\frac{d^{2}u_{+}}{dx^{2}}.$
(47)
Let us integrate by parts. Then we have
$\displaystyle\int_{0}^{\infty}dxu_{+}\frac{d^{2}u_{-}}{dx^{2}}$
$\displaystyle=$
$\displaystyle\left(u_{+}\frac{du_{-}}{dx}\right)\biggr{|}_{0}^{\infty}-\int_{0}^{\infty}dx\frac{du_{+}}{dx}\frac{du_{-}}{dx}$
$\displaystyle=$
$\displaystyle\left(u_{-}\frac{du_{+}}{dx}\right)\biggr{|}_{0}^{\infty}-\int_{0}^{\infty}dx\frac{du_{+}}{dx}\frac{du_{-}}{dx}.$
We thus have a self-adjoint operator if
$\left(u_{+}\frac{du_{-}}{dx}\right)\biggr{|}_{0}^{\infty}=\left(u_{-}\frac{du_{+}}{dx}\right)\biggr{|}_{0}^{\infty}.$
(49)
Now clearly these vanish at the upper end points. Since we have that
$\displaystyle\frac{du_{+}}{dx}$ $\displaystyle=$ $\displaystyle
u_{+}(\frac{\lambda_{+}+1}{x}-\frac{1}{\lambda_{+}+1}),$
$\displaystyle\frac{du_{-}}{dx}$ $\displaystyle=$ $\displaystyle
u_{-}(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1}).$ (50)
at the lower end point the LHS of Eq. (49) is
$\displaystyle\underset{x\rightarrow
0}{\lim}x^{\lambda_{+}+1}\exp(-x/(\lambda_{+}+1))x^{\lambda_{-}+1}\exp(-x/(\lambda_{-}+1))(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1})$
(51) $\displaystyle=$ $\displaystyle\underset{x\rightarrow
0}{\lim}x(\frac{\lambda_{-}+1}{x}-\frac{1}{\lambda_{-}+1})=\lambda_{-}+1$
whereas at the lower end point the RHS is $\lambda_{+}+1\neq\lambda_{-}+1$.
Thus, the second derivative is not self-adjoint in this context! This accounts
for the non-orthogonality of the usual and peculiar ground states in Eq. (45).
In general beginning with a set of usual and peculiar wave functions
$\\{u_{+n},u_{-n}\\}$ such that
$\displaystyle\langle u_{+n}|u_{+n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}},$ $\displaystyle\langle
u_{-n}|u_{-n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}},$ $\displaystyle\langle
u_{-n}|u_{+n}\rangle$ $\displaystyle\equiv$ $\displaystyle
b_{nn^{\prime}}=b_{n^{\prime}n},$ (52)
we find with
$\displaystyle H$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\left(\varepsilon_{w}\alpha\right)^{2}}\left[-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right]=\left[-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}}\right],$
$\displaystyle Hu_{\pm n}$ $\displaystyle=$ $\displaystyle h_{\pm n}u_{\pm
n},$ $\displaystyle h_{\pm n}$ $\displaystyle\equiv$
$\displaystyle-\kappa_{\pm n}^{2}=-1/(\lambda_{\pm}+n)^{2},~{}n=1,2,...$ (53)
where $x=\varepsilon_{w}\alpha r$, that
$\displaystyle\langle u_{+n}|H|u_{+n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}}h_{+n}$ $\displaystyle\langle
u_{-n}|H|u_{-n^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle\delta_{nn^{\prime}}h_{-n},$ $\displaystyle\langle
u_{-n}|H|u_{+n^{\prime}}\rangle$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{+n^{\prime}}$ $\displaystyle\langle
u_{+n}|H|u_{-n^{\prime}}\rangle$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{-n^{\prime}}\neq\langle u_{-n^{\prime}}|H|u_{+n}\rangle.$
(54)
In the first two terms it does not matter whether the $H$ operators operate to
the left or the right. In the last two cases we explicitly have $H$ operating
to the right. To emphasize that we write them as
$\displaystyle\langle u_{-n}|(H|u_{+n^{\prime}}\rangle)$ $\displaystyle=$
$\displaystyle b_{nn^{\prime}}h_{+n^{\prime}},$ $\displaystyle\langle
u_{+n}|(H|u_{-n^{\prime}}\rangle)$ $\displaystyle=$ $\displaystyle
b_{nn^{\prime}}h_{-n^{\prime}}.$ (55)
It is evident that with both sets of basis, $H$ is not self-adjoint since
$\langle u_{-n}|(H|u_{+n^{\prime}}\rangle)\neq(\langle
u_{-n}|H)|u_{+n}\rangle$ and $\langle
u_{+n}|(H|u_{-n^{\prime}}\rangle)\neq(\langle
u_{+n}|H)|u_{-n^{\prime}}\rangle$.
Let us see where the non-orthogonality leads us if we treat both basis on an
equal footing. In that case a general wave function for the ${}^{1}S_{0}$
system would be expanded as999 Strictly speaking we should include the
continuum states. See section below for discussion of those states. For the
purpose here the use of discrete states is sufficient.
$\Psi=\sum_{n_{+}}c_{+n}u_{+n}+\sum_{n_{-}}c_{-n}u_{-n},$ (56)
and applying the variational principle to
$\langle H\rangle=\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle},$
(57)
and defining (we show just a finite $n\times n$ portion of the matrices)
$\displaystyle\mathbf{B}$ $\displaystyle\mathbf{=}$
$\displaystyle\begin{bmatrix}b_{11}&b_{12}&...&b_{1n}\\\
b_{21}&b_{22}&...&b_{2n}\\\ ...&...&...&...\\\
b_{n1}&b_{n2}&...&b_{nn}\end{bmatrix},$ $\displaystyle\mathbf{H}_{+}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}h_{+1}&0&...&0\\\
0&h_{+2}&...&0\\\ ...&...&...&...\\\ 0&0&...&h_{+n}\end{bmatrix},$
$\displaystyle\mathbf{H}_{-}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}h_{-1}&0&...&0\\\ 0&h_{-2}&...&0\\\
...&...&...&...\\\ 0&0&...&h_{-n}\end{bmatrix},$ (58)
then in block form we would have the eigenvalues equation
$\begin{bmatrix}\mathbf{H}_{+}&\mathbf{B\mathbf{H}}_{-}\\\
\mathbf{B\mathbf{H}_{+}}&\mathbf{H}_{-}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}=-\kappa^{2}\begin{bmatrix}\mathbf{1}&\mathbf{B}\\\
\mathbf{B}&\mathbf{1}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}$ (59)
(Note that the way this stands , the matrix on the left is not self-adjoint.)
Multiplying both sides on the right by
$\begin{bmatrix}\mathbf{1}&\mathbf{B}\\\
\mathbf{B}&\mathbf{1}\end{bmatrix}^{-1}=\begin{bmatrix}\mathbf{(1-B}^{2})^{-1}&-\mathbf{B(1-B}^{2})^{-1}\\\
-\mathbf{B(1-B}^{2})^{-1}&\mathbf{(1-B}^{2})^{-1}\end{bmatrix}$ (60)
we obtain
$\begin{bmatrix}\mathbf{H}_{+}&\mathbf{0}\\\
\mathbf{0}&\mathbf{H}_{-}\end{bmatrix}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}=-\kappa^{2}\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{c}_{-}\end{bmatrix}.$ (61)
It is clear that the eigenvectors corresponding to the eigenvalue sets of
$-\kappa_{+}^{2}$ and $-\kappa_{-}^{2}$ are of the form
$\begin{bmatrix}\mathbf{c}_{+}\\\
\mathbf{0}\end{bmatrix},\begin{bmatrix}\mathbf{0}\\\
\mathbf{c}_{-}\end{bmatrix}.$ (62)
From Eq. (36). one recalls that the two sets of basis functions
$\\{u_{+n},u_{-n}\\}$ have distinctly different behaviors at the origin,
corresponding to the usual and peculiar solutions. In particular
$\displaystyle u_{+n}(x)$ $\displaystyle=$ $\displaystyle
c_{+n}x^{\lambda_{+}+1}\exp(-x)L_{n_{r}}^{2\lambda_{+}+1}(x),$ $\displaystyle
u_{-n}(x)$ $\displaystyle=$ $\displaystyle
c_{+n}x^{\lambda_{-}+1}\exp(-x)L_{n_{r}}^{2\lambda_{-}+1}(x),$ (63)
These generalized Laguerre polynomials are orthonormal with respect to
different weight functions $x^{\lambda_{\pm}+1}\exp(-x)$. They would each
correspond to a complete set. Together they would constitute an over-complete
set. However, that does not imply that Eq. (56) is incorrect as it allows for
the function $\Psi(x)$ to be a linear combination of functions with two
distinct behaviors at the origin. Nevertheless, the set up here is a bit
clumsy with questions of completeness and the non-self-adjoint property
remaining.
It should be realized that for the given quasipotential of the type
$-\alpha^{2}/r^{2}$ at short distances that is at hand, both the set of usual
states and the peculiar states are physically admissible states. There does
not appear to be reasons to exclude one set as being unphysical, if one is
given the attractive interaction near the origin as it is. We note however
that the peculiar states with the $r^{-\lambda}$ behavior at the origin are
excluded from existence if coefficient $\lambda(\lambda+1)$ for the $1/r^{2}$
term is greater than zero since that would lead to a $u(r)$ that is singular
at the origin. Only for interactions with sufficient attraction at the origin
(so that $-1/4\leq$ $\lambda(\lambda+1)<0)$ can these states be pulled into
existence and appear as eigenstates in the physically acceptable sheet, with
regular non-singular radial wave functions at the origin. It is desirable to
find ways to admit both types of physical states into a larger Hilbert space
to accommodate both sets of states with the mass operator to be self-adjoint
and the states to be part of a complete set. It is reasonable to assign a
quantum number which we call “peculiarity” for a states emerging into the
physical sheet in this way as physically acceptable states. The introduction
of the peculiarity quantum number enlarges the Hilbert space, allows the mass
operator to be self-adjoint, and the set of physically allowed states become a
complete set, as we shall demonstrate.
We introduce a new peculiarity observable $\hat{\zeta}$ with the quantum
number peculiarity $\zeta$ such that
$\displaystyle\hat{\zeta}\chi_{+}$ $\displaystyle=$
$\displaystyle\zeta\chi_{+}~{}~{}\mathrm{with~{}eigenvalue~{}}\zeta=+1,$
$\displaystyle\hat{\zeta}\chi_{-}$ $\displaystyle=$
$\displaystyle\zeta\chi_{-}~{}~{}\mathrm{with~{}eigenvalue~{}}\zeta=-1,$ (64)
with the corresponding spinor wave function $\chi_{\zeta}$ assigned to the
states so that a usual state is represented by the peculiarity spinor
$\chi_{+}$,
$\chi_{+}=\begin{pmatrix}1\\\ 0\end{pmatrix},$ (65)
and a peculiar state is represented by the peculiarity spinor $\chi_{-}$,
$\chi_{-}=\begin{pmatrix}0\\\ 1\end{pmatrix}.$ (66)
With this introduction, a general wave function can be expanded in terms of
the complete set of basis functions $\\{u_{+n},u_{-n}\\}$ as
$\Psi=\sum_{\zeta n}a_{\zeta n}u_{\zeta n}\chi_{\zeta},$ (67)
where $n$ represent all the spin and spatial quantum numbers of the state and
$\zeta$ the peculiarity quantum number. The variational principle applied to
$\langle H\rangle=\frac{\langle\Psi|H|\Psi\rangle}{\langle\Psi|\Psi\rangle},$
(68)
would lead to
$\displaystyle Hu_{+n}\chi_{+}$ $\displaystyle=$
$\displaystyle-\kappa_{+n}^{2}u_{+n}\chi_{+},$ $\displaystyle Hu_{-n}\chi_{-}$
$\displaystyle=$ $\displaystyle-\kappa_{-n}^{2}u_{-n}\chi_{-}.$ (69)
It is clear that in this context the usual and peculiar wave functions are
orthogonal, $H$ is self-adjoint, and the basis states are complete. That is,
$\langle i|j\rangle=\langle
u_{\zeta_{i}n_{i}}|u_{\zeta_{j}n_{j}}\rangle\equiv\int_{0}^{\infty}dru_{\zeta_{i}n_{i}}\chi_{\zeta_{i}}u_{\zeta_{j}n_{j}}\chi_{\zeta_{j}}=\delta_{\zeta_{i}\zeta_{j}}\delta_{n_{i}n_{j}}=\delta_{ij}$
(70)
and so the set of basis functions $\\{u_{+n}\zeta_{+},u_{-n}\zeta_{-}\\}$,
containing both the usual states and peculiar states in the enlarged Hilbert
space, form a complete set. We also have
$\displaystyle\langle i|H|j\rangle$ $\displaystyle=$ $\displaystyle\langle
u_{\zeta_{i}n_{i}}|H|u_{\zeta_{j}n_{j}}\rangle\equiv\int_{0}^{\infty}dru_{\zeta_{i}n_{i}}\chi_{\zeta_{i}}Hu_{\zeta_{j}n_{j}}\chi_{\zeta_{j}}$
(71) $\displaystyle=$ $\displaystyle
h_{\zeta_{i}}\delta_{\zeta_{i}\zeta_{j}}\delta_{n_{i}n_{j}}=\langle
u_{\zeta_{j}n_{j}}|H|u_{\zeta_{i}n_{i}}\rangle$ $\displaystyle=$
$\displaystyle\langle j|H|i\rangle,$
so that the mass operator $H$ in this enlarged Hilbert space is self-adjoint.
We see that the introduction of the peculiarity quantum number resolves the
problem of over-completeness property of the basis states and the non-self-
adjoint property of the mass operator.
#### IV.1.2 ${}^{1}S_{0}$ Scattering States
The ${}^{1}S_{0}$ state equation
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w(r)}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u$
(72)
has the same form as the nonrelativistic Schrödinger equation for Coulomb
interaction
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2m\alpha}{r}+\frac{L(L+1)}{r^{2}}\right\\}u=2mE\bar{u}=k^{2}u,$
(73)
except that the standard angular momentum term with $L(L+1)$ now take on the
value of $(-\alpha^{2})$. The two solutions the above equation are given by
the regular $F_{L}$ and irregular $G_{L}$ Coulomb wave functions,
$\displaystyle\bar{u}$ $\displaystyle=aF_{L}(\eta,kr)+cG_{L}(\eta,kr),$
$\displaystyle\eta$ $\displaystyle=-\frac{m\alpha}{k},$ (74)
with only the regular Coulomb wave function having an acceptable behavior at
the origin. The long distance behaviors of the regular and irregular solutions
are
$\displaystyle F_{L}(\eta,kr$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\sin(kr-\eta\log
2kr+\sigma_{L}-L\pi/2),$ $\displaystyle G_{L}(\eta,kr$
$\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(kr-\eta\log
2kr+\sigma_{L}-L\pi/2),$ (75)
in which $\sigma_{L}$ is the Coulomb phase shift given by
$\sigma_{L}=\arg(\Gamma(L+1+i\eta).$ (76)
Now we can solve Eq. (72) exactly for $b^{2}>0$ by analytically continuing the
above solutions to an arbitrary (non-integer) angular momentum $\lambda$ and
making a few obvious replacements by analogy,
$\displaystyle\bar{u}$
$\displaystyle=aF_{\lambda}(\eta,br)+cG_{\lambda}(\eta,br),$
$\displaystyle\lambda(\lambda+1)$ $\displaystyle=-\alpha^{2},$
$\displaystyle\eta$ $\displaystyle=-\frac{\varepsilon_{w}\alpha}{b}.$ (77)
Using the expressions for the analytically continued Coulomb wave functions to
non-integer $\lambda~{}$klein we will presently see that we have solutions
given by the $F$ and $G$ functions in Eqs. (83) and (84) below. We emphasize
that both solutions have an acceptable behavior at the origin. Since $\lambda$
is not an integer, one can replace the irregular solution
$G_{\lambda}(\eta,br)$ by $F_{-\lambda-1}(\eta,br)$.101010 The reason that
$G_{L}$ is used in place of $F_{-L-1}$ for $L$ integer is that the latter is
not linearly independent of $F_{L}$ in that case. It is melded together with
$F_{L}$ to produce $G_{L}$ by a limited process analogous to how the Neumann
function is obtained from the Bessel functions. For $\lambda\neq$ integer,
$F_{\lambda}$ and $F_{-\lambda-1}$ are linearly independent. In particular, as
shown in Appendix D, in terms of the confluent hypergeometric function
$M(a,b;z)$
$F_{\lambda}(\rho)=C_{\lambda}(\eta)\rho^{\lambda+1}\exp(-i\rho)M(\lambda+1-i\eta,2\lambda+2;2i\rho),$
(78)
one has with
$x(\lambda,\eta)\equiv(\lambda+\frac{1}{2})\pi+\sigma_{-\lambda-1}(\eta)-\sigma_{\lambda}(\eta),$
(79)
that
$G_{\lambda}(\rho)=\frac{F_{-\lambda-1}(\rho)-\cos
x(\lambda,\eta)F_{\lambda}(\rho)}{\sin x(\lambda,\eta)},$ (80)
a linear combination of $F_{\lambda}(\rho)$ and $F_{-\lambda-1}(\rho)$. In
others words, Eq. (77) can be written as
$\bar{u}=dF_{\lambda}(\eta,br)+eF_{-\lambda-1}(\eta,br),$ (81)
where
$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1+\sqrt{1-4\alpha^{2}})\equiv\lambda_{+},$ (82)
$\displaystyle-\lambda-1$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1-\sqrt{1-4\alpha^{2}})\equiv\lambda_{-},$
corresponding to the separate $\zeta=\pm 1$ sectors. As with the solutions in
Eq. (77), $F_{\lambda}(\eta,br)$ and $F_{-\lambda-1}(\eta,br)$ have acceptable
behaviors at the origin corresponding to Eq. (34). Their respective long
distance behaviors are given by
$\displaystyle F_{\lambda}(\eta,br$
$\displaystyle\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{\lambda_{+}}-\lambda_{+}\pi/2),$ $\displaystyle
F_{-\lambda-1}(\eta,br$
$\displaystyle\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{\lambda_{-}}-\lambda_{-}\pi/2).$ (83)
Alternatively we can use the related $\ G$ functions to determine the
behaviors
$\displaystyle G_{\lambda}(\eta,br$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(br-\eta\log
2br+\sigma_{\lambda_{+}}-\lambda_{+}\pi/2),$ $\displaystyle
G_{-\lambda-1}(\eta,br$ $\displaystyle\rightarrow$
$\displaystyle\infty)\rightarrow\mathrm{const}\times\cos(br-\eta\log
2br+\sigma_{\lambda_{-}}-\lambda_{-}\pi/2).$ (84)
The respective total Coulomb phase shifts for Eq. (72) are the phase shifts
for the usual and peculiar solutions over and above those due to any angular
barrier part (absent here). They are given by
$\displaystyle\delta_{\lambda_{\pm}}$
$\displaystyle=\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2,$
$\displaystyle\sigma_{\lambda_{\pm}}$
$\displaystyle=\arg(\Gamma(\lambda_{\pm}+1+i\eta),$ (85)
in which
$\displaystyle\arg\Gamma(\lambda_{\pm}+1+i\eta)$
$\displaystyle=\eta\psi(\lambda_{\pm}+1)+\sum_{n=0}^{\infty}\left(\frac{\eta}{\lambda_{\pm}+1+n}-\arctan(\frac{\eta}{\lambda_{\pm}+1+n})\right),$
with the digamma function given by
$\psi(\lambda_{\pm}+1)=-\gamma+\lambda_{\pm}\zeta(2)-\lambda_{\pm}^{2}\sum_{n=1}^{\infty}\frac{1}{n^{2}(n+\lambda_{\pm})}.$
(87)
The (modified) Coulomb phase shift
$\sigma_{\lambda_{\pm}}$$-\lambda_{\pm}\pi/2$ is that for the Coulomb
$2\varepsilon_{w}A$ plus $-A^{2}$ term alone. (Again, the $\pm$ sign
corresponds to the two sectors $\zeta=\pm 1$, with usual ($+)$ and peculiar
($-)~{}$boundary conditions given in Eq. (34).) Without the $-A^{2}$ term the
phase shift would be simply $\sigma_{0}$.
## V SOLUTIONS OF THE TWO BODY DIRAC EQUATIONS FOR THE ${}^{3}P_{0}$ STATE
### V.1 The ${}^{3}P_{0}$ quasipotential
We now consider the case of the ${}^{3}P_{0}$ state of a fermion-antifermion
pair with electric or color charges interacting through an electromagnetic-
type interaction arising from the exchange of a single photon or gluon. As
with the ${}^{1}S_{0}$ state, the single photon annihilation diagram does not
contribute because the ${}^{3}P_{0}$ state is a charge parity even state.
Then, the two terms in Eq. (25) that precede the $\nabla^{2}A$ term precisely
cancel the barrier term $2/r^{2}$ at very short distances to give the equation
for the radial wave function
$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}+\frac{8\pi\alpha
r\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha}\right\\}u=b^{2}u.$ (88)
The cancellation of terms takes place in the following way. In Eq. (25), the
three terms beyond $-A^{2}$ arise from a combination of spin-orbit, spin-spin,
tensor and spin-orbit tensor interactions. From a detailed examination of Eq.
(168) in the Appendix B, we can see that the spin-orbit and tensor terms gives
rise to the first “magnetic interaction” term on the right hand side of Eq.
(168) that has a strongly attractive $-8\alpha/wr^{3}$ attractive part down to
distances on the order of $2\alpha/w$ after which this magnetic interaction
approaches $-4/r^{2}$. The dominance of the attractive magnetic interaction at
short distances that can overwhelm the centrifugal barrier is in agreement
with the simple intuitive classical picture presented in the Introduction. The
second term on the right-hand side of Eq. (168), arising from a combination of
Darwin, spin-spin and tensor terms, has a stronger repulsive
$8\alpha^{2}/w^{2}r^{4}~{}$ part down to distances on the order of $2\alpha/w$
after which it approaches $+2/r^{2}$. Together they tend to exactly cancel the
angular momentum barrier term $+2/r^{2}$ at very short distances. In addition
to the repulsive interaction containing $\delta(\hbox{\boldmath${r}$})$
arising from the assumption that the electron and positron are point
particles, the quasipotential behaves as $-\alpha^{2}/r^{2}$ at short-
distances, separated from the outside long-distance region by a barrier. The
interaction containing the delta function comes from a combination of Darwin,
spin-spin, and tensor terms. Three fourths of the repulsive term containing
$\delta(\hbox{\boldmath${r}$})$ comes from the Darwin piece while one fourth
from the combination of the spin-spin and tensor parts. For brevity of
nomenclature we shall just call it the delta function term.
One of us (HWC) examined in a previous work atk the effects on bound state
energies due to a repulsive $\delta(\hbox{\boldmath${r}$})$ interaction by
itself, without additional radial dependence. It was found that for wave
functions $\psi$ that do not vanish at the origin and for potentials that are
less singular than $1/r^{2}$, the exact effects on the eigenvalue of including
a repulsive delta function do not agree with the results of perturbation
theory in the limit of weak coupling, when the delta function potential is
modeled as the limit of a sequence of spherically symmetric square wells. In
particular it is shown that the repulsive delta function, viewed as the limit
of square well potentials, produces no effects at all on bound state energies.
In our case here the appearance of the $\delta({\hbox{\boldmath${r}$}})$
potential differs from this reference in two aspects however. First of all the
$\delta({\hbox{\boldmath${r}$}})$ appears in conjunction with
$r/(wr+2\alpha)$, softening its repulsive effects. Secondly, the wave function
$\psi=u/r$ for the solution without the delta function term diverges at the
origin both for what we call the usual solution and what we call the peculiar
solution. If the null effects on bound state energies and phase shifts seen in
atk should occur in our case as well, this, however, does not lead to a
problem with perturbative agreement with the spectral results.
In the case of weak potentials where the denominator $(wr+2\alpha)$ is
replaced by $wr$, we have shown previously in bckr that the remaining terms
in Eq. (88) without the delta function term, when treated nonperturbatively,
would produce numerically the same spectral results for the ${}^{3}P_{0}$
state as the inclusion of the repulsive $\delta(\hbox{\boldmath${r}$})$
interaction treated perturbatively. The agreement of the perturbative
treatment with the delta function term for weak coupling with the
nonperturbative treatment containing no delta function term justifies the
first approximate analysis of ignoring the delta function term and treating
the remainder of the equation nonperturbatively in the following subsection.
### V.2 Usual and Peculiar Solutions for the ${}^{3}P_{0}$ State
The wave equation (88) for the ${}^{3}P_{0}$ state without the delta function
term becomes
$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u,$
(89)
with a short distance ($r<<2\alpha/w)$ form
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{\alpha^{2}}{r^{2}}\right\\}u=0,$ (90)
the same as with the ${}^{1}S_{0}$ states. Thus, the ${}^{3}P_{0}$ states also
have the same types of solutions as the ${}^{1}S_{0}$ states, with radial wave
functions near the origin as given in Eqs. (33)-(34). Thus, there are usual
${}^{3}P_{0}$ states with peculiarity $1$, and peculiar ${}^{3}P_{0}$ states
with peculiarity $-1$.
Note that both the usual and the peculiar solutions $u_{\pm}$ $\sim
r^{(1\pm\sqrt{1-4a^{2}})/2}$ arise from the strong magnetic interaction that
significantly modifies the qualitative behavior of the interaction at short
distances, when the total spin and the orbital angular momentum are oppositely
aligned in the ${}^{3}P_{0}$ state. If the strong magnetic interaction is
absent, the $2/(r+2\alpha/w)^{2}$ term in Eq. (88) would be $2/r^{2}$, and the
wave function near the origin would be
$u_{\pm}=ar^{(1\pm\sqrt{3^{2}-4a^{2}})/2},$ (91)
with
$\psi_{\pm}^{2}d^{3}r=r^{[(1\pm\sqrt{3^{2}-4a^{2}})]}drd\Omega.$ (92)
In that case, as stated below Eq. (35), only the usual $u_{+}$ solution is
quantum-mechanically admissible, while the $u_{-}$ state becomes singular at
short distances. Such a comparison shows that the peculiar solution $u_{-}$ is
not present when there is no strongly attractive magnetic interaction at short
distances or more generally for $J\neq 0.$
### V.3 The $\delta$ function term and the charge distribution
The discussions in the above subsection pertain to the quasipotential without
the delta function term. We now examine the full Eq. (25) for both the usual
and peculiar solutions with the $\delta({\hbox{\boldmath${r}$}})$ term
included. Consider first the perturbative treatment of taking the interaction
containing $\delta(\hbox{\boldmath${r}$})$ as a perturbation. We evaluate the
expectation value of the interaction term containing
$\delta(\hbox{\boldmath${r}$})$. Even though both usual and peculiar solutions
have a diverging $\psi_{\pm}({\hbox{\boldmath${r}$}})$ near the origin they
each are allowed as a probability amplitude since the probability
$\int_{\Delta V}\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}d^{3}r$
for an arbitrarily small volume $\Delta V$ about the origin would be finite,
in addition to the essential boundary condition $u_{\pm}(0)=0$. With
$\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}$ near the origin having
the behavior of $r^{(-1\pm\sqrt{1-4a^{2}})}$, the expectation value of
$\delta(\hbox{\boldmath${r}$})/(w-2A)$, after performing the angular
integration, is
$\displaystyle\int
d^{3}r\frac{r\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha}\left|\psi_{\pm}({\hbox{\boldmath${r}$}})\right|^{2}$
$\displaystyle\rightarrow$ $\displaystyle\int
d^{3}rr^{\pm\sqrt{1-4a^{2}}}\frac{\delta({\hbox{\boldmath${r}$}})}{wr+2\alpha},$
(93) $\displaystyle\rightarrow$ $\displaystyle\int
drr^{\pm\sqrt{1-4a^{2}}}\frac{\delta(r)}{(wr+2\alpha)},$
which is zero for the plus sign for the usual solution but diverges for the
minus sign for the peculiar solution.
The results of Eq. (93) for the usual solution explains our previous agreement
between (i) the perturbative treatment with the delta function term for weak
coupling and (ii) the nonperturbative treatment without the delta function
term bckr . The agreement arises because in Ref. bckr we limited our
attention only to the usual solution for which the expectation value of the
delta function term is zero.
The results of Eq. (93) for the peculiar solution indicates that the delta
function term cannot be treated as a perturbation in the present formulation,
as such a treatment will lead to a diverging energy. The delta function term
arises from the charge distribution of the interacting particles, as it is
related to the Laplacian of the gauge field, $\nabla^{2}A$, as given in Eq.
(26). A proper non-perturbative treatment of the problem of the peculiar
solution states requires the knowledge of the wave function at very short
distances. Therefore, it will require not only the knowledge of the structure
of the charge distribution but also the necessary auxiliary interactions at
even shorter distances that are needed to bind the charge elements of the
distribution together. The auxiliary interactions will affect the solutions of
the two-body wave functions at very short distances and the states of the
peculiar solution. At the present moment, we have little knowledge of the
structure of elementary charges, much less the auxiliary forces that would
bind the charge distribution together at very short distances.
The structure of the charge distribution of elementary particles at very short
distances is basically an experimental question. As the strong magnetic
interaction allows the two interacting particles to probe the short-distance
region, it is therefore useful to investigate quantities that may reveal
information on the structure of the charge distribution. While many
possibilities can be opened for examination, we shall examine the following
possibilities in the present manuscript:
(i) We shall first examine the case in which the (unknown) auxiliary
interaction that binds the charge elements of the elementary particles
together and the repulsive interaction arising from the charge density
$\rho(r)$ counteract in such a way that the total interaction at short
distances would still be dominated by the $-\alpha^{2}/r^{2}$ term. Under such
a circumstance, the effects of the auxiliary interaction would cause the delta
function term term in Eq. (88) to make no contribution at short distances.
Keeping the dominant terms, the equation of motion for the wave function
becomes Eq. (89) without the delta function term. It also must be recognized
that for the usual solution, the perturbative effect of the delta function
term (in which we ignore the effect or the potential in the denominator
$w-2A$) is accounted for by a nonperturbative (numerical) treatment of the
entire $\Phi$ without the delta function term. So, our treatment of the delta
function term in this case parallels that used in our earlier spectroscopy
calculations bckr .
(ii) We examine subsequently the case when the auxiliary interaction that
holds the charge element together leaves the gauge field $A(r)$ unchanged
while the delta function term in Eq. (88) is modified by treating the delta
function as the limit of a set of Gaussian distributions with different
widths.
(iii) We examine two additional models completely within QED (or QCD) with an
assumed basic charge distribution that generates the gauge field also in the
region interior to the charge distribution. However, the auxiliary
interactions that hold the charge together and that can interact with the
other antifermion are altogether neglected. It should be recognized that
within pure QED (or QCD), with the neglect of the auxiliary interactions that
hold the charge elements together, the charge distribution cannot be a stable
configuration.
In the next section we describe the method we use to indicate the presence or
absence of a resonance in the ${}^{3}P_{0}$ system.
## VI PHASE SHIFT ANALYSIS
In our study of the ${}^{3}P_{0}$ state for both the usual and peculiar
solutions, we wish to find out whether or not there is an energy that will
lead to a $\pi/2$ phase shift for a given $\alpha$ and constituent mass $m$ .
Equation (28) for the ${}^{3}P_{0}$ state is a Schrödinger-like equation of
the form
$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{L(L+1)}{r^{2}}+\Phi(r)\right\\}u(r)=b^{2}u(r).$
(94)
We calculate the phase shift for this problem by the variable phase method of
Calogero cal . We first describe this method generally (see Appendix E for a
more detailed review) and then later in this section describe its application
to the ${}^{3}P_{0}$ state. We take $W(r)$ to include not only the
quasipotential $\Phi(r)$ but also the angular momentum barrier.
$W(r)=\frac{L(L+1)}{r^{2}}+\Phi(r).$ (95)
Thus our equation has the form
$\left\\{-\frac{d^{2}}{dr^{2}}+W(r)\right\\}u(r)=b^{2}u(r).$ (96)
The Calogero method relies on introducing a reference potential $\bar{W}(r)$
that can be solved exactly, with two independent solutions $u_{1}$ and
$u_{2}$,
$\left\\{-\frac{d^{2}}{dr^{2}}+\bar{W}(r)\right\\}u_{i}(r)=b^{2}u_{i}(r),~{}i=1,2.$
(97)
There are many ways to choose the reference potential $\bar{W}(r)$. To display
the general idea, we consider the case in which $W(r)$ is short range. In that
case the phase shift $\delta_{L}$ is defined by
$u(r\rightarrow\infty)\rightarrow\sin(br-L\pi/2+\delta_{L}).$ (98)
The Calogero method uses two different types of $\bar{W}(r)$. In the first,
$\bar{W}(r)\equiv\bar{W}_{I}(r)$, the reference potential has the same long
and short distance behavior as $W(r)$. In the second $\bar{W}(r)\equiv
W_{II}(r)$, the reference potential does not have the same long and short
distance behavior as $W(r)$ but is especially simple.
We consider first Type I reference potential, $\bar{W}_{I}(r)=L(L+1)/r^{2}$,
the angular momentum barrier potential, for which the reference wave functions
$\bar{u}_{1}(r)$ and $\bar{u}_{2}(r)$ are the well known spherical Bessel
functions $\hat{\jmath}_{L}(br)$ and $\hat{n}_{L}(br)$ cal . The solution
$\bar{u}_{1}(r)$ is taken to be the regular solution, having the same short
distance behavior as $u(r),$ in particular, $\bar{u}_{1}(r\rightarrow 0)=0.$
The solution $\bar{u}_{2}(r)$ is taken to be the irregular solution,
$\bar{u}_{2}(r\rightarrow 0)\neq 0$. Those functions together with their long
distance behaviors are given by
$\displaystyle\bar{u}_{1}(r)$
$\displaystyle=\hat{\jmath}_{L}(br)\rightarrow\mathrm{const}\sin(br-L\pi/2),$
$\displaystyle\bar{u}_{2}(r)$
$\displaystyle=-\hat{n}_{L}(br)\rightarrow\mathrm{const}\cos(br-L\pi/2).$ (99)
We introduce the amplitude function $\alpha(r)$ and phase shift function
$\delta_{L}(r)$ to represent the wave function solutions for the $W(r)$
potential, $u(r)$ and $u^{\prime}(r)$, as
$\displaystyle u(r)$
$\displaystyle=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}(r)+\sin\delta_{L}(r)\bar{u}_{2}(r)),$
$\displaystyle u^{\prime}(r)$
$\displaystyle=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}^{\prime}(r)+\sin\delta_{L}(r)\bar{u}_{2}^{\prime}(r)).$
(100)
This leads to the following equation for the phase shift function (see
Appendix E)
$\tan\delta_{L}(r)=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))}.$
(101)
Further manipulations lead to the differential equation for $\delta_{L}(r)$
given by
$\delta_{L}^{\prime}(r)=-\frac{[W(r)-\bar{W}(r)]}{b}\biggl{[}\hat{\jmath}_{L}(br)\cos\delta_{L}(r)-\hat{n}_{L}(br)\sin\delta_{L}(r)\biggr{]}^{2},$
(102)
To find the connection to the phase shift $\delta_{L}$ note that from Eq. (
100) and (99),
$\displaystyle u(r$
$\displaystyle\rightarrow\infty)=\mathrm{const}\left\\{\cos\delta_{L}(r\rightarrow\infty)\sin(br-L\pi/2)+\sin\delta_{L}(r\rightarrow\infty)\cos(br-L\pi/2)\right\\}$
$\displaystyle=\mathrm{const}\times\sin(br-L\pi/2+\delta_{L}(\infty)),$ (103)
and so comparison with (98) gives the solution of the phase shift $\delta_{L}$
for the $W(r)$ potential as
$\delta_{L}=\delta_{L}(\infty).$ (104)
Thus, the second order linear different equation becomes a first order non-
linear equation whose solution at $r\rightarrow\infty$ gives the phase shifts
of the scattering problem with the $W(r)$ effective potential. The boundary
condition of $\delta_{L}(0)=0$ follows from Eq. (101) when one chooses
$\bar{u}_{1}(r)$ to have the same behavior as $u(r)$ as $r\rightarrow 0.$
We consider next type II of the short-range reference potentials
$\bar{W}_{II}(r)$ which do not need to have the same long distance behavior as
$W(r)$ as long as the Schrödinger Eq. (97) containing the reference potential
$\bar{W}_{II}(r)$ has an exact solution. For example we may choose
$\bar{W}_{II}(r)=0.$ Then the two exact reference solutions of Eq. (97) are
simply
$\displaystyle\bar{u}_{1}(r)$ $\displaystyle=\sin(br),$
$\displaystyle\bar{u}_{2}(r)$ $\displaystyle=\cos(br).$ (105)
One defines a phase shift function $\gamma_{L}(r)$ as in equation (100) so
that
$\displaystyle u(r$
$\displaystyle\rightarrow\infty)=\mathrm{const}\times\\{\cos\gamma_{L}(r\rightarrow\infty)\sin(br)+\sin\gamma_{L}(r\rightarrow\infty)\cos(br)\\}$
$\displaystyle=\mathrm{const}\times\sin(br+\gamma_{L}(\infty)).$ (106)
Comparison with (98) gives
$\delta_{L}=\gamma_{L}(\infty)+\frac{L\pi}{2}.$ (107)
Since the angular momentum barrier is excluded from the equations for
$\bar{u}_{i}(r)$ one finds that the phase shift equation for integrating the
phase shift function $\gamma_{L}(r)$ includes the repulsive barrier term in
$W$ [Eq. (34)],
$\displaystyle\gamma_{L}^{\prime}(r)$
$\displaystyle=-\frac{W(r)}{b}\biggl{[}\cos\gamma_{L}(r)\sin(br)+\sin\gamma_{L}(r)\cos(br)\biggr{]}^{2}$
$\displaystyle=-\frac{W(r)}{b}\sin^{2}(br+\gamma_{L}(r)).$ (108)
Note that because of the $L(L+1)/r^{2}$ behavior of
$W(r)-\bar{W}_{II}(r)(r)=W(r)$, which dominates at large distances, one will
have to integrate quite far to obtain convergence for $\gamma_{L}(r)$.111111
Alternatively Calogero gives a formula for avoiding integrating to large
distances to build up a centrifugal phase shift. (See cal , p 92). For this
case of $\bar{W}_{II}(r)=0$, one has an equation similar to (101) with
$\delta_{L}(r)$ replaced by $\gamma_{L}(r)$. Thus even though $\bar{u}_{1}(r)$
has a different behavior than $u(r)$, we still have the boundary condition
$\gamma_{L}(0)=0.$ Eq. (107) compensates for the $-L\pi/2$ effective phase
shift due to the barrier term in $W(r)$ in Eq. ( 108). We also have the
additional boundary condition of (see Appendix E)
$\gamma_{L}^{\prime}(0)=-\frac{bL}{L+1}.$ (109)
We now turn our attention to the ${}^{3}P_{0}$ system, in particular Eq. (28)
for a general $\Phi(r)$. In this application of the Calogero method we choose
a reference potential $\bar{W}(r)\equiv\bar{W}_{III}(r)$ that in a sense is a
hybrid of the two types of reference potentials considered above. Since Eq.
(28) contains a long range Coulomb interaction $-2\varepsilon_{w}\alpha/r$ we
must include that interaction into our choice for $\bar{W}_{III}(r)$. If it
did not have the same behavior as $W(r)$ at large distances we would have to
have a way of subtracting an infinite Coulomb phase shift, $\log 2br$ . So, in
this way our application is similar to the first type $\bar{W}_{I}$ (r)above.
We also include the $-\alpha^{2}/r^{2}$ term in $\bar{W}_{III}(r)$ because as
seen in Eqs. (89), (90) and (34) the solution displays the desired short
distance peculiar as well as usual behaviors. We do not include the angular
momentum barrier term $2/r^{2}$ however, as this would prohibit a treatment of
the peculiar solution (see comments below Eq. (92)). Thus we choose
$\displaystyle\bar{W}_{III}(r)$ $\displaystyle=$
$\displaystyle-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}},$
$\displaystyle W(r)$ $\displaystyle=$ $\displaystyle\frac{2}{r^{2}}+\Phi(r),$
(110)
where $\Phi(r)$ is given in Eq. (29). In this way $\bar{W}_{III}(r)$ has some
similarities to the second type $\bar{W}_{II}(r)$ discussed above. Our choice
for $\bar{W}_{III}(r)$ permits the two exact solutions
$\bar{u}_{1}(r),\bar{u}_{2}(r)$ of Eq. (97) which becomes that of the
${}^{1}S_{0}$ state
$\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w(r)}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}\bar{u}=b^{2}\bar{u}.$
(111)
Now to determine the phase shift for the actual ${}^{3}P_{0}$ state we return
to the conditions defined in Eq. (110). Then the full solution has the
asymptotic form
$u(r\rightarrow\infty)\rightarrow\mathrm{const}\times\sin(br-\eta\log
2br+\sigma_{1}-\pi/2+\delta_{1}).$ (112)
The appearance of $\sigma_{1}$ and $\delta_{1}$ includes the effects of the
angular momentum barrier term $2/r^{2}$ in the presence of the Coulomb
interaction. In Appendix E, using
$\displaystyle\bar{u}_{1}(r)$ $\displaystyle=$ $\displaystyle
F_{\lambda}(\eta,br),$ $\displaystyle\bar{u}_{2}(r)$ $\displaystyle=$
$\displaystyle G_{\lambda}(\eta,br),$ (113)
we show that the full ${}^{3}P_{0}$ phase shift $\delta$ is given by
$\delta=\delta_{1}+\sigma_{1}=\gamma_{\pm}(\infty)+\sigma_{\lambda_{\pm}}+(1-\lambda_{\pm})\pi/2,$
(114)
where (in analogy to the proof of Eq. (108) with $\bar{W}\neq 0)~{}$
$\gamma_{\pm}(r)$satisfies the nonlinear equation
$\gamma_{\pm}^{\prime}(r)=-\frac{W(r)-\bar{W}_{III}(r)}{b}\biggl{[}\cos\gamma_{\pm}(r)F_{\lambda_{\pm}}+\sin\gamma_{\pm}(r)G_{\lambda_{\pm}}\biggr{]}^{2},$
(115)
subject to the boundary condition that $\gamma_{\pm}(0)=0$ (see Appendix E).
The functions $F_{\lambda_{\pm}}$ and $G_{\lambda_{\pm}}$ are the regular and
irregular Coulomb wave functions corresponding to the negative effective
centrifugal barrier $-\alpha^{2}/r^{2}$. Again, because of the $2/r^{2}$
behavior of $W(r)-\bar{W}_{III}(r)$ which takes over at large distances, one
will have to integrate quite far to obtain convergence for $\gamma_{\pm}(r).$
We consider numerical solutions for both the usual solution with
$\lambda_{+}=(-1+\sqrt{1-4\alpha^{2}})/2$, and the peculiar solution, with
$\lambda_{-}=(-1-\sqrt{1-4\alpha^{2}})/2.~{}$In the next section we discuss
the results obtained in the numerical integration of the phase shift equation
( 115) for different behaviors at very short distances.
## VII NUMERICAL RESULTS FOR ${}^{3}P_{0}$ RESONANCES
### VII.1 The case without the delta function term
With the above general formalism, we can begin to examine states in the
quasipotential of Eq. (88) first without the delta function term. The
Schrödinger equation for the ${}^{3}P_{0}$ state becomes Eq. (89). In order to
gain an idea on the attractive magnetic interaction at short distances for
this ${}^{3}P_{0}$ state, we plot in Fig. 2 the corresponding quasipotential
including the angular momentum barrier,
$W(r)=\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\epsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}},$
for $w=27.85$ MeV, $\alpha=1/137$ and a constituent electron mass of 0.511
MeV. One observes that at short distances $W(r)$ becomes very attractive and
behaves as $-\alpha^{2}/r^{2}$. There is a barrier in the region between 10-2
to 10-1 GeV-1. Such a potential becomes singular at $r\rightarrow 0$ when
$\alpha$ exceeds 1/2 Case .
Figure 2: The effective potential
$W(r)=2/(r+2\alpha/w)^{2}-2\epsilon_{w}\alpha/r-\alpha^{2}/r^{2}$ for the for
the $(e^{+}e^{-})$ system in the ${}^{3}P_{0}$ state with $\alpha=1/137$ and
$w=27.85$ MeV.
We calculate the phase shift as a function of energy using the boundary
condition $\gamma_{\pm}(0)$=0, including the dependence of the potential as a
function of energy. For the usual solution ($\zeta=+1)$, our results for the
QED $e^{-}e^{+}$ system in the ${}^{3}P_{0}$ state with $\alpha=1/137$ and
$m=0.511$ MeV show no evidence whatsoever for resonances for all c.m. energies
tested (from about 1 MeV to about 100 MeV). The magnitude of the phase shifts
are of the order of $\pi/100$.
Table 1: Variation of the resonant energy as a function of the quark mass for a fixed $\alpha_{s}=0.11$. quark | mass | $w_{R}$ | |
---|---|---|---|---
up | 3 MeV | 27 MeV | |
down | 5 MeV | 45 MeV | |
strange | 135 MeV | 1220 MeV | |
charm | 1.5 GeV | 13.6 GeV | |
bottom | 4.5 GeV | 40.8 GeV | |
top | 175 GeV | 1590 GeV | |
For the peculiar solution ($\zeta=-1)~{}$with the wave functions starting with
a less positive slope, the attraction at short distances is able to bend the
wave function downward to result in a very sharp resonance at about $27.85$
MeV. In Figure 3(a) we plot the phase shift $\delta=\delta_{1}+\sigma_{1}$ as
a function of the c.m. energy $w$ and $\sin^{2}\delta$ versus $w$ in Fig. 3
(b). We start the integration at the origin and extend to about 1 angstrom. As
one observes, the phase shift undergoes a transition from near zero to $\pi$.
The resonance has a full width at half maximum of 15 KeV. We also include a
plot of the wave function in Fig. 4 from the origin up to about 1000 GeV-1.
The wave function rises as $r^{(1-\sqrt{1-4\alpha^{2}})/2}$ near the origin,
and appears nearly flat at $r\sim 10^{-3}$ GeV-1, and it slowly decreases near
the barrier. It oscillates when it emerges from the barrier at $r\sim 2\times
10^{-2}$ GeV-1.
Figure 3: The phase shift as a function of $w$ for the $(e^{+}e^{-})$ system
with $\alpha=1/137$ and $w=27.85$ MeV. Figure 4: The wave function $u$ of the
peculiar resonance at $w=27.85$ MeV for $\alpha=1/137$ and $m=0.511$ MeV.
Having observed a resonance for the QED interaction with the $e^{+}$ and
$e^{-}$ constituents, we turn our attention to quarks and antiquarks
interacting with a color-coulomb type interaction with an effective coupling
constant $\alpha_{s}$. We focus here only on the $\zeta=-1$ sector. In the
color-singlet $(q\bar{q})$ states of interest, the effective interaction is
then $\alpha_{\mathrm{eff}}=4\alpha_{s}/3$. To get an idea of the order of
energy for these quark-antiquark two-body resonance states, we calculate the
resonance energies for the typical case of $\alpha_{s}=0.11$ For this value,
the resonance energy varies nearly linearly with quark mass. The largest
energy resonances occur with the largest quark masses. In Table I we present
the resonance energies $w_{R}$ for the families of quarks from the up quark to
the top quark. It should be pointed out that these resonance values take into
account only the Coulomb-like portion of $A(r)=-(4/3)\alpha_{s}/r$ and ignores
any affects on the resonance values of the confining part of the potential.
To examine how the resonance energies varies with the coupling constant, we
have found that for fixed mass (e.g. 0.511 MeV) the resonance energy $w_{R}$
increases as the coupling parameter decreases until the coupling constant
gets to be on the order of $0.01$, when $w_{R}$ starts decreasing again.
### VII.2 The case of representing the delta function by a Gaussian function
For the second case for the ${}^{3}P_{0}$ state given in Eqs. (88) and ( 96)
using (115), we take
$W(r)=\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}+\frac{8\pi\alpha
r\delta({\hbox{\boldmath${r}$}})}{\left(wr+2\alpha\right)},$ (116)
in which we model the three dimensional delta function by
$\delta({\hbox{\boldmath${r}$}})\rightarrow\delta_{\sigma}({\hbox{\boldmath${r}$}})=\frac{\exp(-r^{2}/2\sigma^{2})}{(2\pi)^{3/2}\sigma^{3}}.$
(117)
Table 2: Variation of resonance energy with the width of the Gaussian distribution $\sqrt{2}\sigma(\mathrm{fm})$ | $w$(GeV)
---|---
$1000$ | $0.0279$
$100$ | $0.0278$
$10$ | $0.0279$
$1$ | $0.0398$
$0.1$ | $0.314$
$0.01$ | $3.13$
$0.001$ | $31.3$
$0.0001$ | $313$
$0.00001$ | $3130$
$0.000001$ | $31300$
$0.0000001$ | $313000$
In this treatment, we keep the point charge source term for the $A(r)$ so that
$A(r)=-\alpha/r$. What we are attempting to do is just present a mathematical
representation of the delta function that will allow a numerical solution. The
reference potential $\bar{W}_{III}(r)$ is the same as without the delta
function. With this modeling of the delta function we start off our Runge-
Kutta integration of Eq. (115) with $\gamma_{-}(0)=0$ since
$W(r)-\bar{W}_{III}(r)$ is $w^{2}2\alpha^{2}$ at the origin just as without
the delta function term. The function $\delta_{\sigma}$ does not alter the
extreme short distance behavior since it is multiplied by $r$ and vanishes at
the origin. We obtain the resonance energy results as given in Table II. It is
obvious that for small $r_{0}$ we obtain a limiting behavior of $w=3.13$ GeV-
fm/$\sqrt{2}\sigma$. There is however, a difference between what we are doing
here and what was done in atk . There the delta function was just regarded as
given, not related to other parts of the potential. Here that is not the case.
The delta function arose from the Laplacian of $A(r).$ There may therefore be
some ambiguity of, in effect, modeling $\nabla^{2}A$ in one part of the
quasipotential while leaving $A(r)$ unaffected in the other part. That leads
us then to the third case.
### VII.3 The case of representing the charge distribution by a continuous
function
In Eq. (25), if one replaces $A(r)$ by
$A=\left(\frac{\alpha}{r}-\frac{\alpha}{r_{0}}\right)\frac{1}{1+\exp\\{(r-r_{0})/\delta
r_{0}\\}}-\frac{\alpha}{r},$ (118)
or alternatively as
$A(r)=\begin{cases}\frac{\alpha
r^{2}}{2r_{0}^{3}}-\frac{3\alpha}{2r_{0}}&\mathrm{~{}for~{}}r\leq
r_{0}\cr-\frac{\alpha}{r}&\mathrm{~{}for~{}}r\geq r_{0}.\cr\end{cases}.$ (119)
then our numerical solutions show that there is no ${}^{3}P_{0}$ resonance for
the peculiar solution for both cases, resulting in a phase shift of $\pi$ all
the way down to threshold ($w=2m$). Both of these corresponds to smeared
charge distributions from $\nabla^{2}A$ but neither have auxiliary
interactions at short distances that would bind the elements of the charge
distribution together. The reason no resonance is produced in this case is
that in the interior of the charge distribution ($r<r_{0}$), the angular
momentum barrier in Eq. (25) comes out from under the dominance of the
magnetic interaction terms as $A(r)$ tends to a finite constant. By following
steps similar to those used to determine $\gamma_{-}(0)$ in Appendix E for the
point charge one can show this results in an initial value for $\gamma_{-}(0)$
defined by
$\tan\gamma_{-}(0)=\tan x(\lambda,\eta).$ (120)
This is positive and even though small ($\sim 0.007$) is large enough to
prevent the formation of a resonance. Note that this differs from the previous
section in that here we are giving a physical connection
$\nabla^{2}A=4\pi\alpha\rho(\hbox{\boldmath${r}$})$ between the smeared delta
function and the invariant potential $A(r)$, whereas in the previous section
we simply mathematically modeled the delta function in isolation.
## VIII DISCUSSION AND CONCLUSION
Magnetic interactions in the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states are very
attractive and singular at short distances. In the two-body Dirac equation
formulated in constraint dynamics, the magnetic interactions lead to
quasipotentials that behave as $-\alpha^{2}/r^{2}$ near the origin and admit
two different types of states. At short distances, the radial wave functions
$u(r)$ of the usual states, grow as $r^{\lambda+1}$, while the radial wave
functions of the peculiar states grow as $r^{-\lambda}$, where
$\lambda=(-1+\sqrt{1-4\alpha^{2}})/2$. They have drastically different
properties.
The existence of usual and peculiar states for the same fermion-antifermion
system poses conceptual and mathematical problems. If we keep both sets of
states in the same Hilbert space, then each set is complete by itself, but the
two sets of states are not orthogonal to each other. Our system is thus over-
complete. Furthermore, the matrix element of $H$ (the scaled invariant mass
operator for these states) between states of one type and state of the other
type are not symmetric and the $H$ operator is not self-adjoint.
Given our quasipotential of the type $-\alpha^{2}/r^{2}$ at short distances
for the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states, both the usual and peculiar
states are physically admissible. There do not appear to be compelling reasons
to exclude one of the two sets as being unphysical, if one is given the
attractive interaction $-\alpha^{2}/r^{2}$ near the origin as it is. It is
desirable to find ways to admit both types of states as physical states while
maintaining the self-adjoint property of the mass operator and the
completeness property of the set of basis states.
We are therefore motivated to introduce a quantum number $\zeta$, which we
call “peculiarity”, to specify the usual or peculiar properties of a state.
The peculiarity quantum number $\zeta$ is 1 for usual states which have
properties the same as those one usually encounters in QED and QCD. The
peculiarity quantum number is $-1$ for peculiar states which intrudes into the
physical region, when the interaction near the origin becomes very attractive,
such as the $\lambda(\lambda+1)/r^{2}$ interaction with
$-1/4\leq\lambda(\lambda+1)<0$. The introduction of the peculiarity quantum
number enlarges the Hilbert space, makes the mass operator self-adjoint, and
the enlarged physical basis states containing both usual and peculiar states
in a complete set. It is also clear from our discussions that to maintain the
self-adjoint property of the mass operator and to have a single complete set,
the presence of the peculiarity quantum number will be a general phenomenon,
when the mass operator contains very attractive interactions at short
distances such that there are more than one set of eigenstates satisfying the
boundary conditions at the origin.
It should be emphasized that the quasipotential $-\alpha^{2}/r^{2}$ has been
obtained under the assumption of a point fermion and a point antifermion for
which the gauge field potential between them is $A(r)=-\alpha/r$. The point
nature of an electron may be a good experimental concept as the lower limits
on the QED cut-off parameter $\Lambda_{\mathrm{cut}}$ with the present day
high energy accelerators exceeds the value of 250 GeV, suggesting that the
electron, muon, and tauon, behave as point particles down to 10-3 fm. The
asymptotic freedom is a good description for the interaction of quarks at
short distances. It may appear that point charge particles may be a reasonable
description. On the other hand, a finite structure of the electron or quarks
may modify significantly the short-distance attractive interactions so
substantially that the peculiar states may be pushed out of existence. The
experimental search of the peculiar states, which follows from the point
charge potential, can provide a probe of the point nature of these particles
and the interaction at short distances.
Our first focus on the attractive magnetic interaction is for the
${}^{1}S_{0}$ states, where the spins of the fermion and antifermion of
opposite electric or color charges are oppositely aligned. The usual bound
${}^{1}S_{0}$ states possess attributes the same as those one usually
encounters in QED and QCD, with bound state energies explicitly agreeing with
the standard perturbative results through order $\alpha^{4}$. In contrast, the
peculiar bound ${}^{1}S_{0}$ states, yet to be observed, not only have
different behaviors at the origin, but also distinctly different bound state
properties (and scattering phase shifts). For the peculiar ${}^{1}S_{0}$
ground state of a fermion-antifermion pair with fermion rest mass $m$, the
root-mean-square radius is approximately $1/m$, binding energies approximately
$(2-\sqrt{2})m$, and a rest mass approximately $\sqrt{2}m$. On the other hand,
the $(n+1)$${}^{1}S_{0}$ peculiar state with principal quantum number $(n+1)$
is nearly degenerate in energy and approximately equal in size with the
$n$${}^{1}S_{0}$ usual states.
Our second focus is for the ${}^{3}P_{0}$ state where the total spin and the
orbital angular momentum are oppositely aligned. The magnetic interaction
overwhelms the centrifugal repulsion at short distances and the wave function
admits a peculiar solution that grows with radial distances as $u\sim
r^{(1-\sqrt{1-4\alpha^{2}})/2}$. The particle charge density $\rho(r)$ and
auxiliary interactions that bind the charge elements together can be exposed
for scrutiny. As the structures of elementary particles are basically
experimental questions, it is useful to utilize the magnetic interaction to
probe such charge distributions at very short distances. While many
possibilities can be opened for examination, we have investigated only a few
possibilities in the present manuscript.
The ${}^{3}P_{0}$ quasipotential contains a term proportional to
$\delta(\hbox{\boldmath${r}$})$ . As the delta function term does not
contribute to the usual QED ${}^{3}P_{0}$ bound state energies, it was
plausible to ignore it as one of our explored possibilities. In that case, we
find that there is a magnetic ${}^{3}P_{0}$ resonance at 27.85 MeV for the
peculiar solution of the $(e^{+}e^{-})$ system. For various $(q\bar{q})$
systems of different flavors, we find magnetic ${}^{3}P_{0}$ resonances at
energies of the peculiar solution ranging from many tens of MeV to thousands
of GeV. It is interesting to note that these ${}^{3}P_{0}^{++}$ resonances
have the same quantum number as the vacuum.
In another one of our explored possibilities, if we mathematically model the
delta function at short distances by a sequence of Gaussians of different
widths without changing the gauge field $A(r)=-\alpha/r,$ then a completely
different behavior for the resonance energies ensues as they occur at
different energies, depending on the width of the Gaussian. In the third of
our explored possibilities, if we replace the delta function by a charge
distribution that also alters the gauge field $A(r)$, we obtain no resonance
at all.
Because of 1) the limited knowledge of the unknown auxiliary interactions and
charge distributions at very short distances, not to mention possible
alterations on the angular momentum barrier itself, and 2) the ambiguity of
treating the delta function in isolation nonperturbatively, and 3) the fact
that the delta function term does not contribute to the ${}^{3}P_{0}$ usual
bound state solution, we speculate that the first case may provided a more
reliable representation of the physics. It furthermore makes a clear
prediction of a QED resonance in a region that has not been investigated.
While we have studied the resonance ${}^{3}P_{0}$ states, future work calls
for the investigation of possible ${}^{3}P_{0}$ peculiar bound states where
the attractive interaction near the origin may allow the formation of bound
states. The presence of a delta function repulsion at the origin will also
lead to difficulties and problems similar to the ones we encounter here with
the ${}^{3}P_{0}$ peculiar resonances.
Fermion-antifermion states as we know them experimentally belong to the usual
states. Peculiar states have not been observed. Can the peculiar states be
observed? How do the usual and peculiar states interplay between them? Will
there be transitions between the usual states and peculiar states? Clearly,
the stability of peculiar states first and foremost depends on the strong
attraction near the origin, which in turn depends on the point-like nature of
the elementary particles. As we discussed earlier, substantial modification of
the attractive interaction at the origin may push the peculiar states out of
existence. Only for interactions with sufficient attraction at the origin can
the peculiar states be pulled into existence and appear as eigenstates in the
physically acceptable sheet, with non-singular radial wave functions at the
origin. This is true for both ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states. From
such a perspective, we expect that interactions at short distances have
important bearings on the existence or non-existence of the peculiar states,
and presumably also on the transition between the usual and peculiar states.
However, the interactions at short distances that may allow the peculiar
states to be stable and may effect transitions between states with different
peculiarity quantum numbers (flipping the peculiarity spinor) are not yet
known. They can only be obtained by careful experimental investigations. The
first task of such investigations should be to locate these peculiar states in
high-energy experiments where interactions at short-distance may be involved
and these strong interactions at short distances may lead us to probe short-
distance transition from the usual to the peculiar states. These new
${}^{1}S_{0}$ peculiar bound states correspond to a very tightly bound state
and a set of $(n+1$)th excited states nearly degenerate with the $n$th usual
states. It will also be of interest to search for these states as a result of
some tunneling process between the usual and peculiar states, relying on the
small probability of the usual states to explore short-distance regions where
the interaction at short distances may induce a transition from a usual state
to a peculiar state. The fact that peculiar states of $(n+1)$th${}^{1}S_{0}$
state is nearly degenerate with the usual $n$th ${}^{1}S_{0}$ state may
facilitate such a tunneling transition. Whether or not these quantum-
mechanically acceptable resonances correspond to physical states remains to be
further investigated. Future experimental as well as theoretical work on this
interesting topic will be of great interest in shedding light on the question
whether magnetic bound states and resonances play any role in the states of
fermion-antifermion systems.
Future work should include the effects of the weak interactions, in particular
the exchange of the $Z^{0}$ boson. Since the mass of the $Z^{0}$ is about 92.5
GeV the range is on the order of $10^{-2}$ GeV-1. The exchange of this
particle corresponds to not only a vector interaction but also a pseudovector
interaction. The coupling corresponding to the vector portion is wein
$\displaystyle e^{\ast}$ $\displaystyle\equiv$
$\displaystyle+\frac{g^{2}-g^{\prime 2}}{4\sqrt{g^{2}+g^{\prime
2}}}+\frac{g^{\prime}}{2},$ $\displaystyle g$ $\displaystyle=$
$\displaystyle-\frac{e}{\sin\theta},$ $\displaystyle g^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{e}{\cos\theta},$ (121)
and so
$\displaystyle e^{\ast}$ $\displaystyle=$ $\displaystyle
e[\frac{\frac{1}{\sin^{2}\theta}-\frac{1}{\cos^{2}\theta}}{4\sqrt{\frac{1}{\sin^{2}\theta}+\frac{1}{\cos^{2}\theta}}}-\frac{1}{2\cos\theta}]$
(122) $\displaystyle=$ $\displaystyle e[\frac{\cos 2\theta}{2\sin
2\theta}-\frac{1}{2\cos\theta}].$
With $\sin^{2}\theta\sim 0.23$ we find that
$e^{\ast}\sim-0.25e$ (123)
so, its coupling appears with the same sign as that of the photon. Since
$\alpha^{\ast}=e^{\ast 2}\sim 0.063$ Its effect should be small but not
negligible. There is also the question of the effects of the pseudovector
interaction, not discussed in this appendix but in jmath ,long .
Finally, there are however important mathematical and conceptual issues
associated with these two-body fermion-antifermion system at short distances
that require future careful considerations. In standard QED theory, the charge
and mass of a single charged object due to vacuum polarization and self energy
corrections need to be renormalized or regularized to render them finite for
comparison with observables. For the case with two-body magnetic bound and
resonance states, for example, how are the two-body Green’s functions
regularized, with internal lines off mass shell in a way that reflects the
Dirac constraints? How do such regularizations modify the short distance two-
body interaction? Can the regularization affects the magnetic interaction at
short distances so substantially that the peculiar states no longer survive to
intrude into the physical states? Are these peculiar states stable against
fluctuation of the vacuum in quantum field theory. These are some of the many
interesting questions associated with the two-body problem raised by the
possibility of magnetic states under consideration.
## Appendix A Details of the equivalent Relativistic Schrödinger Equation
### A.1
Connections between TBDE and the equivalent Relativistic Schrödinger equation
[Eq. (17)]
Here we present an outline of some details of Eq. (14) and its Pauli-
Schrödinger reduction given in full elsewhere (see cra87 ; jmath ; long ; liu
).This appendix and the one following it are specializations of Appendices A
and B given in tmlk . Each of the two Dirac equations in (14) has a form
similar to a single particle Dirac equation in an external four-vector and
scalar potential but here acting on sixteen component wave function $\Psi$
which is the product of an external part being a plane wave eigenstate of
$P~{}$multiplying the internal wave function $\psi$
$\psi=\begin{bmatrix}\psi_{1}\\\ \psi_{2}\\\ \psi_{3}\\\
\psi_{4}\end{bmatrix}.$ (124)
The four $\psi_{i}$ are each four-component spinor wave functions. To obtain
the actual general spin dependent forms of those $\tilde{A}_{i}^{\mu}$
potentials (including scalar interactions in general) which were required by
the compatibility condition $[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0$ was a
most perplexing problem, involving the discovery of underlying supersymmetries
in the case of scalar and time-like vector interactions cra82 ,cra87 .
Extending those external potential forms to more general covariant
interactions necessitated an entirely different approach leading to what is
called the hyperbolic form of the TBDE. Their most general form for compatible
TBDE is
$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle=(\cosh(\Delta){\hbox{\boldmath${S}$}}_{1}+\sinh(\Delta){\hbox{\boldmath${S}$}}_{2})\psi=0\mathrm{,}$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle=(\cosh(\Delta){\hbox{\boldmath${S}$}}_{2}+\sinh(\Delta){\hbox{\boldmath${S}$}}_{1})\psi=0,$
(125)
where $\Delta$ represents any invariant interaction singly or in combination.
It has a matrix structure in addition to coordinate dependence. Depending on
that matrix structure we have either covariant vector, scalar or more general
covariant tensor interactions jmath . The operators
${\hbox{\boldmath${S}$}}_{1}$ and ${\hbox{\boldmath${S}$}}_{2}$ are auxiliary
constraints satisfying
$\displaystyle{\hbox{\boldmath${S}$}}_{1}\psi$
$\displaystyle\equiv(\mathcal{S}_{10}\cosh(\Delta)+\mathcal{S}_{20}\sinh(\Delta)~{})\psi=0,$
$\displaystyle{\hbox{\boldmath${S}$}}_{2}\psi$
$\displaystyle\equiv(\mathcal{S}_{20}\cosh(\Delta)+\mathcal{S}_{10}\sinh(\Delta)~{})\psi=0,$
(126)
in which the $\mathcal{S}_{i0}$ are the free Dirac operators
$\mathcal{S}_{i0}=\frac{i}{\sqrt{2}}\gamma_{5i}(\gamma_{i}\cdot p_{i}+m_{i}).$
(127)
This, in turn leads to the two compatibility conditions cww ; jmath ; saz86
$[\mathcal{S}_{1},\mathcal{S}_{2}]\psi=0,$ (128)
and
$[{\hbox{\boldmath${S}$}}_{1},{\hbox{\boldmath${S}$}}_{2}]\psi=0,$ (129)
provided that $\ \Delta(x)=\Delta(x_{\perp}).$ These compatibility conditions
do not restrict the gamma matrix structure of $\Delta$. That matrix structure
is determined by the type of vertex-vertex structure we wish to incorporate in
the interaction. The three types of invariant interactions $\Delta$ that was
used in the relativistic quark model based on this approach (as most recently
discussed in unusual ,tmlk ) are
$\displaystyle\Delta_{\mathcal{L}}(x_{\perp})$
$\displaystyle=-1_{1}1_{2}\frac{\mathcal{L}(x_{\perp})}{2}\mathcal{O}_{1},\
\mathcal{O}_{1}=-\gamma_{51}\gamma_{52},~{}~{}~{}\text{ scalar}\mathrm{,}$
$\displaystyle\Delta_{\mathcal{J}}(x_{\perp})$
$\displaystyle=\beta_{1}\beta_{2}\frac{\mathcal{J}(x_{\perp})}{2}\mathcal{O}_{1},~{}~{}~{}\text{time-
like\ vector}\mathrm{,}$ $\displaystyle\Delta_{\mathcal{G}}(x_{\perp})$
$\displaystyle=\gamma_{1\perp}\cdot\gamma_{2\perp}\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1},~{}~{}\text{space-
like\ vector}{,}$ (130)
where
$\displaystyle\gamma_{5i}$
$\displaystyle=\gamma_{i}^{0}\gamma_{i}^{1}\gamma_{i}^{2}\gamma_{i}^{3},$
$\displaystyle\beta_{i}$ $\displaystyle=-\gamma_{i}\cdot\hat{P}.$ (131)
For general independent scalar, time-like vector, and space-like vector
interactions we have
$\Delta(x_{\perp})=\Delta_{\mathcal{L}}+\Delta_{\mathcal{J}}+\Delta_{\mathcal{G}}.$
(132)
The special case of an electromagnetic-like interaction (in the Feynman gauge)
applied in this paper and in bckr corresponds to $\mathcal{J}=-\mathcal{G}$
or
$\displaystyle\Delta_{\mathcal{J}}+\Delta_{\mathcal{G}}$
$\displaystyle\equiv\Delta_{\mathcal{EM}}=(-\gamma_{1}\cdot\hat{P}\gamma_{2}\cdot\hat{P}+\gamma_{1\perp}\cdot\gamma_{2\perp})\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1}$
$\displaystyle=\gamma_{1}\cdot\gamma_{2}\frac{\mathcal{G}(x_{\perp})}{2}\mathcal{O}_{1}.$
(133)
and for scalar and electromagnetic interaction,
$\Delta(x_{\perp})=\Delta_{\mathcal{L}}+\Delta_{\mathcal{EM}}.$ (134)
This leads to121212 In short, one inserts Eq. (126) into (125) and brings the
free Dirac operator (127) to the right of the matrix hyperbolic functions.
Using commutators and $\cosh^{2}\Delta-\sinh^{2}\Delta=1$ one arrives at Eq.
(135). The structure of these equations are very much the same as that of a
Dirac equation for each of the two particles, with $M_{i}$ and $E_{i}$ playing
the roles that $m+S$ and $\varepsilon-A$ do in the single particle Dirac
equation. Over and above the usual kinetic part, the spin-dependent
modifications involving $G\mathcal{P}_{i}$ and the last set of derivative
terms are two-body recoil effects essential for the compatibility
(consistency) of the two equations jmath ; long
$\displaystyle\mathcal{S}_{1}\psi$
$\displaystyle=\big{(}-G\beta_{1}\Sigma_{1}\cdot\mathcal{P}_{2}+E_{1}\beta_{1}\gamma_{51}+M_{1}\gamma_{51}-G\frac{i}{2}\Sigma_{2}\cdot\partial(\mathcal{L}\beta_{2}\mathcal{-J}\beta_{1})\gamma_{51}\gamma_{52}\big{)}\psi=0,$
$\displaystyle\mathcal{S}_{2}\psi$
$\displaystyle=\big{(}G\beta_{2}\Sigma_{2}\cdot\mathcal{P}_{1}+E_{2}\beta_{2}\gamma_{52}+M_{2}\gamma_{52}+G\frac{i}{2}\Sigma_{1}\cdot\partial(\mathcal{L}\beta_{1}\mathcal{-J}\beta_{2})\gamma_{51}\gamma_{52}\big{)}\psi=0,$
(135)
in which $\partial_{\mu}=\partial/\partial x^{\mu}.$ With
$\displaystyle G$ $\displaystyle=\exp\mathcal{G},$
$\displaystyle\mathcal{P}_{i}$ $\displaystyle\equiv
p_{\perp}-\frac{i}{2}\Sigma_{i}\cdot\partial\mathcal{G}\Sigma_{i}.$ (136)
The connections between what we call the vertex invariants
$\mathcal{L},\mathcal{J},\mathcal{G}$ and the mass and energy potentials
$M_{i},E_{i}$ are
$\displaystyle M_{1}$ $\displaystyle=m_{1}\ \cosh\mathcal{L}\
+m_{2}\sinh\mathcal{L},$ $\displaystyle M_{2}$ $\displaystyle=m_{2}\
\cosh\mathcal{L}\ +m_{1}\ \sinh\mathcal{L},$ $\displaystyle E_{1}$
$\displaystyle=\varepsilon_{1}\ \cosh\mathcal{J}\
+\varepsilon_{2}\sinh\mathcal{J},$ $\displaystyle E_{2}$
$\displaystyle=\varepsilon_{2}\
\cosh\mathcal{J}+\varepsilon_{1}\sinh\mathcal{J}.$ (137)
Eq. (135) depends on standard Pauli-Dirac representation of gamma matrices in
block forms (see Eq. (2.28) in crater2 for their explicit forms) and
where131313 Just as $x^{\mu}$ is a four vector, so is $P^{\mu}.$ Thus, the
time-like and space-like interactions in Eq. (130) become
$\gamma_{1}^{0}\gamma_{2}^{0}$ and
${\hbox{\boldmath${\gamma}$}}_{1}\cdot{\hbox{\boldmath${\gamma}$}}_{2}$ only
in the c.m. system due to the fact that from Eq. (131),
$\beta_{i}=\gamma_{i}^{0}$ only in the c.m. frame. Likewise,
$\Sigma_{i}^{\mu}=(0,{\hbox{\boldmath${\Sigma}$})}$ only in the c.m. frame
just as is $x_{\perp}^{\mu}=(0,{\hbox{\boldmath${r}$})}$ in that frame only.
$\Sigma_{i}=\gamma_{5i}\beta_{i}\gamma_{\perp i}.$ (138)
### A.2
Vector potentials $\tilde{A}_{i}^{\mu}$ in terms of the invariant $A(r)$
Comparing Eq. (135) with Eq. (14) we find that the spin-dependent
electromagnetic-like vector interactions of Eq. (14) are cra87 ; bckr
$\displaystyle\tilde{A}_{1}^{\mu}$
$\displaystyle=\big{(}(\varepsilon_{1}-E_{1})\big{)}\hat{P}^{\mu}+(1-G)p_{\perp}^{\mu}-\frac{i}{2}\partial
G\cdot\gamma_{2}\gamma_{2}^{\mu},$ $\displaystyle A_{2}^{\mu}$
$\displaystyle=\big{(}(\varepsilon_{2}-E_{2})\big{)}\hat{P}^{\mu}-(1-G)p_{\perp}^{\mu}+\frac{i}{2}\partial
G\cdot\gamma_{1}\gamma_{1}^{\mu},$ (139)
Note that the first portion of the vector potentials is time-like (parallel to
$\hat{P}^{\mu})$ while the next two portions are space-like (transverse to
$\hat{P}^{\mu})$. The spin-dependent scalar potentials $\tilde{S}_{i}$ are
$\displaystyle\tilde{S}_{1}$
$\displaystyle=M_{1}-m_{1}-\frac{i}{2}G\gamma_{2}\cdot\partial\mathcal{L},$
$\displaystyle\tilde{S}_{2}$
$\displaystyle=M_{2}-m_{2}+\frac{i}{2}G\gamma_{1}\cdot{\partial}\mathcal{\
L}{.}$ (140)
We have chosen a parametrization for the vertex invariants
$\mathcal{L},~{}\mathcal{J}=-\mathcal{G}$ that takes advantage of the Todorov
effective external potential forms and at the same time will display the
correct static limit form for the Pauli reduction. The logic of the choice for
these parametrizations is strengthened by the fact that for classical fw or
quantum field theories saz97 for separate scalar and time-like vector
interactions one can show that the spin independent part of the quasipotential
$\Phi~{}$ involves the difference of squares of the invariant mass and energy
potentials
$M_{i}^{2}=m_{i}^{2}+2m_{w}S+S^{2};\
E_{i}^{2}=\varepsilon_{i}^{2}-2\varepsilon_{w}A+A^{2},$ (141)
so that
$M_{i}^{2}-E_{i}^{2}=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w).$ (142)
Eqs. (14) and (135) involve combined scalar and electromagnetic-like vector
interactions (without the separate time-like interactions this amounts to
working in the Feynman gauge with the simplest relation between space- and
time-like parts, see Eqs. (133), (134), and cra88 ; crater2 ). In that case
the mass and energy potentials in place of Eq. (141) are respectively
$\displaystyle M_{i}^{2}$
$\displaystyle=m_{i}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2}),$
$\displaystyle E_{i}^{2}$
$\displaystyle=\exp(2\mathcal{G(A))(}\left(\varepsilon_{i}-A)^{2}\right),$
$\displaystyle M_{i}^{2}-E_{i}^{2}$
$\displaystyle=\exp(2\mathcal{G(A))[}2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w]$
(143)
so that from Eq. (137),
$\displaystyle\exp(\mathcal{L})$
$\displaystyle=\exp(\mathcal{L}(S,A))=\frac{\sqrt{m_{1}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}+\sqrt{m_{2}^{2}+\exp(2\mathcal{G)(}2m_{w}S\mathcal{+}S^{2})}}{m_{1}+m_{2}},\
$ (144) $\displaystyle\exp(\mathcal{J})$ $\displaystyle=\exp(-\mathcal{G)}$
with
$\exp(2\mathcal{G(}A\mathcal{))=}\frac{1}{(1-2A/w)}\equiv G^{2},$ (145)
or
$\displaystyle-\mathcal{G}$
$\displaystyle\mathcal{=}\frac{1}{2}\log(1-2A/w)=\log\frac{E_{1}+E_{2}}{w},$
(146)
and the spin-independent minimal coupling appears like
$\Phi_{SI}=2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}.$ (147)
### A.3 Interaction terms in the equivalent Relativistic Schrödinger Equation
[Eq. (17)]
The Klein-Gordon like potential energy terms appearing in the Pauli form ( 17)
arise from (see Eq. (143))
$M_{i}^{2}-E_{i}^{2}=\exp(2\mathcal{G)[}2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}-b^{2}(w)].$
(148)
To obtain the simple Pauli form of Eq. (16) and the subsequent detailed form
in Eq. (17) involves steps similar to those used in the Pauli reduction of the
single particle Dirac equation unusual but with the combinations
$\phi_{\pm}=\psi_{1}\pm\psi_{4}$ and $\chi_{\pm}=\psi_{2}\pm\psi_{3}$ instead
of the upper and lower components of the single particle wave function. This
reduces the Pauli forms to 4 uncoupled 4 component relativistic Schrödinger
equations saz94 ; long ; crater2 ; liu . We work in the c.m. frame in which
$\hat{P}=(1,{\ \hbox{\boldmath${0}$})}$ and
$\hat{r}=(0,{\hbox{\boldmath${\hat{r}}$}).}$ We also define four component
wave functions $\psi_{\pm},\eta_{\pm}$ by liu
$\displaystyle\phi_{\pm}$
$\displaystyle=\exp(\mathcal{F}+\mathcal{K}\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}})\psi_{\pm}=(\exp\mathcal{F})(\cosh\mathcal{K}+\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}}\sinh\mathcal{K})\psi_{\pm},$
$\displaystyle\chi_{\pm}$
$\displaystyle=\exp(\mathcal{F}+\mathcal{K}\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}})\eta_{\pm}=(\exp\mathcal{F})(\cosh\mathcal{K}+\bm{\sigma}_{1}{\hbox{\boldmath${\cdot}$}\hat{r}}\bm{\sigma}_{2}{\hbox{\boldmath${\cdot}$}\hat{r}}\sinh\mathcal{K})\eta_{\pm},$
(149)
in which
$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{2}\log\frac{\mathcal{D}}{\varepsilon_{2}m_{1}+\varepsilon_{1}m_{2}}-\mathcal{G},$
$\displaystyle\mathcal{D}$ $\displaystyle\mathcal{=}E_{2}M_{1}+E_{1}M_{2},$
$\displaystyle\mathcal{K}$
$\displaystyle=\frac{(\mathcal{L}+\mathcal{G})}{2}.$ (150)
The substitution (149) has the convenient property that in the resultant bound
state equation, the coefficients of the first order relative momentum terms
vanish.
Using the results in liu and unusual we obtain for the general case of
unequal masses the relativistic Schrödinger equation ( 17) that is a detailed
c.m. form of Eq. (16). In that equations we have introduced the
abbreviations141414 Minor misprints of the equations below have appeared in
appendices in unusual and tmlk . The ones presented here are corrected.
$\displaystyle\Phi_{D}$
$\displaystyle=-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\mathcal{F}^{\prime 2}+\mathcal{K}^{\prime
2}+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{r}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}+m(r),$
$\displaystyle\Phi_{SO}$
$\displaystyle=-\frac{\mathcal{F}^{\prime}}{r}-\frac{(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\frac{\mathcal{K}^{\prime}\sinh 2\mathcal{\ K}}{r},$
$\displaystyle\Phi_{SOD}$ $\displaystyle=(l^{\prime}\cosh
2\mathcal{K}-q^{\prime}\sinh 2\mathcal{K}),$ $\displaystyle\Phi_{SOX}$
$\displaystyle=(q^{\prime}\cosh 2\mathcal{K}+l^{\prime}\sinh 2\mathcal{K}),$
$\displaystyle\Phi_{SS}$
$\displaystyle=\kappa(r)+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{3r}-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{3r}+\frac{2\mathcal{F}^{\prime}\mathcal{K}^{\prime}}{3}-\frac{{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}}{3},$
$\displaystyle\Phi_{T}$
$\displaystyle=\frac{1}{3}[n(r)+\frac{(3\mathcal{F}^{\prime}-\mathcal{K}^{\prime}+3/r)\sinh
2\mathcal{K}}{r}+\frac{(\mathcal{F}^{\prime}-3\mathcal{\ \ \
K}^{\prime}+1/r)(\cosh 2\mathcal{K}-1)}{r}+2\mathcal{F}^{\prime}\mathcal{\
K}^{\prime}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}],$
$\displaystyle\Phi_{SOT}$ $\displaystyle=-\mathcal{K}^{\prime}\frac{\cosh
2\mathcal{K}-1}{r}-\frac{\mathcal{K}^{\prime}}{r}+\frac{(\mathcal{F}^{\prime}+1/r)\sinh
2\mathcal{K}}{r},$ (151)
where
$\displaystyle n(r)$
$\displaystyle=\nabla^{2}\mathcal{K}-\frac{1}{2}\nabla^{2}\mathcal{G}+\frac{3(\mathcal{G}-2\mathcal{K})^{\prime}}{2r}+\mathcal{F}^{\prime}\mathcal{G}^{\prime}-2\mathcal{F}^{\prime}\mathcal{K}^{\prime},$
$\displaystyle\kappa(r)$
$\displaystyle=\frac{1}{3}\nabla^{2}(\mathcal{G}+\mathcal{K})-\frac{1}{2}\mathcal{G}^{\prime
2}-\frac{2\mathcal{F}^{\prime}(\mathcal{G}+\mathcal{K})^{\prime}}{3},$
$\displaystyle m(r)$
$\displaystyle=-\frac{1}{2}\nabla^{2}\mathcal{G+}\frac{3}{4}\mathcal{G}^{\prime
2}-\mathcal{K}^{\prime 2}+\mathcal{G}^{\prime}\mathcal{F}^{\prime},$
$\displaystyle l^{\prime}$
$\displaystyle=-\frac{\mathcal{(L-G)}^{\prime}}{2r}\frac{E_{2}M_{2}-E_{1}M_{1}}{E_{2}M_{1}+E_{1}M_{2}},$
$\displaystyle q^{\prime}$
$\displaystyle=\frac{\mathcal{(L-G)}^{\prime}}{2r}\frac{E_{1}M_{2}-E_{2}M_{1}}{E_{2}M_{1}+E_{1}M_{2}}.$
(152)
(The prime symbol stands for $d/dr,$ and the explicit forms of the derivatives
are given in Eq. (153) ). For $L=J$ states, the hyperbolic terms cancel and
the spin-orbit difference terms in general produce spin mixing except for
equal masses or $J=0$. For ease of use we have listed below the explicit forms
that appear in the above $\Phi$s in Eqs. (151) -(152) in terms of the general
invariant potentials $A(r)$ and $S(r).~{}$ The radial components of Eq. (17)
are given in Appendix B.
### A.4
Explicit expressions for terms in the relativistic Schrödinger Equation (17)
from $A(r)$ and $S(r)$
Given the functions $A(r)$ and $S(r)$ for the interaction, users of the
relativistic Schrödinger equation (17) will find it convenient to have an
explicit expression in an order that would be useful for programing the terms
in the associated equation (151). We use the definitions above given in Eqs.
(143 )-(145), and (150). In order that the terms in Eq. (151) be reduced to
expressions involving just $A(r),~{}$and $S(r)$ and their derivatives, we list
the following formulae
$\displaystyle\mathcal{F}^{\prime}$
$\displaystyle=\frac{(\mathcal{L}^{\prime}-\mathcal{G}^{\prime})(E_{2}M_{2}+E_{1}M_{1})}{2(E_{2}M_{1}+E_{1}M_{2})}-\mathcal{G}^{\prime},$
$\displaystyle\mathcal{G}^{\prime}$ $\displaystyle=\frac{A^{\prime}}{w-2A},$
$\displaystyle\mathcal{L}^{\prime}$
$\displaystyle=\frac{M_{1}^{\prime}}{M_{2}}=\frac{M_{2}^{\prime}}{M_{1}}=\frac{w}{M_{1}M_{2}}\left(\frac{S^{\prime}(m_{w}+S)}{w-2A}+\frac{(2m_{w}S+S^{2})A^{\prime}}{(w-2A)^{2}}\right),$
$\displaystyle.\mathcal{K}^{\prime}$
$\displaystyle=\frac{(\mathcal{L}^{\prime}+\mathcal{G}^{\prime})}{2}.$ (153)
Also needed are
$\displaystyle\cosh 2\mathcal{K}$
$\displaystyle=\frac{1}{2}\left(\frac{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}{(m_{1}+m_{2})(E_{1}+E_{2})}+\frac{(m_{1}+m_{2})(E_{1}+E_{2})}{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}\right),$
$\displaystyle\sinh 2\mathcal{K}$
$\displaystyle=\frac{1}{2}\left(\frac{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}{(m_{1}+m_{2})(E_{1}+E_{2})}-\frac{(m_{1}+m_{2})(E_{1}+E_{2})}{(\varepsilon_{1}+\varepsilon_{2})(M_{1}+M_{2})}\right),$
(154)
and
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}$
$\displaystyle=\frac{({\hbox{\boldmath${\nabla}$}}^{2}\mathcal{L}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{G})(E_{2}M_{2}+E_{1}M_{1})}{2(E_{2}M_{1}+E_{1}M_{2})}-(\mathcal{\
\
L}^{\prime}-\mathcal{G}^{\prime})^{2}\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{2\left(E_{2}M_{1}+E_{1}M_{2}\right)^{2}}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{G},$
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{L}$
$\displaystyle=-\frac{\mathcal{L}^{\prime
2}(M_{1}^{2}+M_{2}^{2})}{M_{1}M_{2}}$
$\displaystyle+\frac{w}{M_{1}M_{2}}\left(\frac{{\hbox{\boldmath${\nabla}$}}^{2}S(m_{w}+S)+S^{\prime
2}}{w-2A}+\frac{4S^{\prime}(m_{w}+S)A^{\prime}+(2m_{w}S+S^{2}){\hbox{\boldmath${\nabla}$}}^{2}A}{(w-2A)^{2}}+\frac{4(2m_{w}S+S^{2})A^{\prime
2}}{(w-2A)^{3}}\right),$
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{G}$
$\displaystyle=\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}+2\mathcal{G}^{\prime
2}.$ (155)
The expressions for $\kappa(r),m(r),$ and $n(r)$ that appear in Eqs. ( 151))
are given in Eqs. (152). They can be evaluated using the above expressions
plus
${\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}=\frac{{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{L}+{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{G}}{2}.$
(156)
The only remaining parts of Eq. (151) that need expressing are those for
$l^{\prime}$ and $q^{\prime}.$ Using Eq. (150) they can be obtained in terms
of the above formulae.
## Appendix B Radial Equations
The following are radial eigenvalue equations liu ; unusual corresponding to
Eq. (17) . For a general singlet ${}^{1}J_{J}$ wave function
$u_{LSJ}=u_{J0J}\equiv u_{0}$ coupled to a general triplet ${}^{3}J_{J}$ wave
function $u_{J1J}\equiv u_{1}$, the wave equation
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}{-}3\Phi_{SS}\\}u_{0}$
$\displaystyle+2\sqrt{J(J+1)}(\Phi_{SOD}-\Phi_{SOX})u_{1}$
$\displaystyle=b^{2}u_{0},$ (157)
is coupled to
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle-2\Phi_{SO}+\Phi_{SS}+2\Phi_{T}-2\Phi_{SOT}\\}u_{1}+2\sqrt{J(J+1)}(\Phi_{SOD}+\Phi_{SOX})u_{0}$
$\displaystyle=b^{2}u_{1}.$ (158)
For a general $S=1,$ $J=L+1$ wave function $u_{J-11J}\equiv u_{+}~{}$coupled
to a general $S=1,$ $J=L-1~{}$wave function $u_{J+11J}\equiv u_{-}$ the
equation
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J-1)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle+2(J-1)\Phi_{SO}+\Phi_{SS}+\frac{2(J-1)}{2J+1}(\Phi_{SOT}-\Phi_{T})\\}u_{+}$
$\displaystyle+\frac{2\sqrt{J(J+1)}}{2J+1}\\{3\Phi_{T}-2(J+2)\Phi_{SOT}\\}u_{-}$
$\displaystyle=b^{2}u_{+},$ (159)
is coupled to
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{(J+1)(J+2)}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle-2(J+2)\Phi_{SO}+\Phi_{SS}+\frac{2(J+2)}{2J+1}(\Phi_{SOT}-\Phi_{T})\\}u_{-}$
$\displaystyle+\frac{2\sqrt{J(J+1)}}{2J+1}\\{3\Phi_{T}+2(J-1)\Phi_{SOT}\\}u_{+}$
$\displaystyle=b^{2}u_{-}.$ (160)
For the uncoupled ${}^{3}P_{0}$ states the single equation is
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2m_{w}S+S^{2}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle-4\Phi_{SO}+\Phi_{SS}+4(\Phi_{SOT}-\Phi_{T})\\}u_{-}=b^{2}u_{-}.$
(161)
### B.1 Specialization to vector interactions, equal masses and $J=0$.
In this case we need only consider the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states.
The corresponding equations are
$\\{-\frac{d^{2}}{dr^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}{-}3\Phi_{SS}\\}u_{0}=b^{2}u_{0},$
(162)
and
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}$
$\displaystyle-4\Phi_{SO}+\Phi_{SS}+4(\Phi_{SOT}-\Phi_{T})\\}u_{-}=b^{2}u_{-}.$
(163)
We consider the explicit forms for the quasipotentials given above that appear
in these equations for the case of vector interactions only, for $J=0$ and
equal masses. In that case we have
$\displaystyle\mathcal{F}^{\prime}$
$\displaystyle=-\frac{3\mathcal{G}^{\prime}}{2},$
$\displaystyle\mathcal{G}^{\prime}$ $\displaystyle=\frac{A^{\prime}}{w-2A},$
$\displaystyle\mathcal{L}^{\prime}$ $\displaystyle=0,$
$\displaystyle\mathcal{J}^{\prime}$
$\displaystyle=-\mathcal{G}^{\prime}=-\frac{A^{\prime}}{w-2A},$
$\displaystyle.\mathcal{K}^{\prime}$
$\displaystyle=\frac{(\mathcal{L}^{\prime}-\mathcal{J}^{\prime})}{2}=\frac{\mathcal{G}^{\prime}}{2}.$
(164)
Also needed are
$\displaystyle\cosh 2\mathcal{K}$
$\displaystyle=\cosh\mathcal{G}=\frac{1}{2}(\frac{1}{\sqrt{1-2A/w}}+\sqrt{1-2A/w}),$
$\displaystyle\sinh 2\mathcal{K}$
$\displaystyle=-\sinh\mathcal{G=-}\frac{1}{2}(\frac{1}{\sqrt{1-2A/w}}-\sqrt{1-2A/w}),$
(165)
and
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}$
$\displaystyle=-\frac{3}{2}{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{G},$
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{L}$ $\displaystyle=0,$
$\displaystyle{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{J}$
$\displaystyle\mathcal{=}\mathcal{-}{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{\
G}=-\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}-2\mathcal{G}^{\prime 2}.$
(166)
In that case we have that the combination for the ${}^{1}S_{0}$ equation is
$\displaystyle\Phi_{D}-3\Phi_{SS}$
$\displaystyle=-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\mathcal{F}^{\prime 2}+\mathcal{K}^{\prime
2}+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{r}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}+m(r)$
$\displaystyle-3\kappa(r)-\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{r}+\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}-2\mathcal{F}^{\prime}\mathcal{K}^{\prime}+{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}$
$\displaystyle=\nabla^{2}(-\mathcal{F+K-}\frac{\mathcal{G}}{2}-\mathcal{G-K)+F}^{\prime
2}+\frac{9}{4}\mathcal{G}^{\prime
2}+3\mathcal{F}^{\prime}\mathcal{G}^{\prime}$ $\displaystyle=0,$ (167)
while the combination that appears in the ${}^{3}P_{0}$ equation is
$\displaystyle\Phi_{D}-4\Phi_{SO}+\Phi_{SS}+4(\Phi_{SOT}-\Phi_{T})$
$\displaystyle=-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+\mathcal{F}^{\prime 2}+\mathcal{K}^{\prime
2}+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{r}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{F}+m(r)$
$\displaystyle+\frac{4\mathcal{F}^{\prime}}{r}+\frac{4(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}-\frac{4\mathcal{K}^{\prime}\sinh 2\mathcal{K}}{r}$
$\displaystyle+\kappa(r)+\frac{2\mathcal{K}^{\prime}\sinh
2\mathcal{K}}{3r}-\frac{2(\mathcal{F}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{3r}+\frac{2\mathcal{F}^{\prime}\mathcal{K}^{\prime}}{3}-\frac{{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}}{3}$
$\displaystyle-4\mathcal{K}^{\prime}\frac{\cosh
2\mathcal{K}-1}{r}-\frac{4\mathcal{K}^{\prime}}{r}+\frac{4(\mathcal{F}^{\prime}+1/r)\sinh
2\mathcal{K}}{r}$
$\displaystyle-\frac{4}{3}[n(r)+\frac{(3\mathcal{F}^{\prime}-\mathcal{K}^{\prime}+3/r)\sinh
2\mathcal{K}}{r}+\frac{(\mathcal{F}^{\prime}-3\mathcal{K}^{\prime}+1/r)(\cosh
2\mathcal{K}-1)}{r}+2\mathcal{F}^{\prime}\mathcal{K}^{\prime}-{\hbox{\boldmath${\nabla}$}}^{2}\mathcal{K}]$
$\displaystyle=-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{2{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}.$
(168)
Thus we have the two $J=0$ single component equations reducing to
$\\{-\frac{d^{2}}{dr^{2}}+2\varepsilon_{w}A-A^{2}\\}u_{0}=b^{2}u_{0},$ (169)
and
$\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{2{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}=b^{2}u_{-}.$
(170)
We consider the case in which
$\displaystyle A$ $\displaystyle=$ $\displaystyle-\frac{\alpha}{r},$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{r^{2}},$ $\displaystyle\nabla^{2}A$
$\displaystyle=$ $\displaystyle 4\pi\delta(\mathbf{r).}$ (171)
In that case
$\displaystyle-\frac{8A^{\prime}}{r\left(w-2A\right)}$ $\displaystyle=$
$\displaystyle-\frac{8\alpha}{r^{2}\left(wr+2\alpha\right)}\underset{r\rightarrow
0}{\rightarrow}-\frac{4}{r^{2}},$
$\displaystyle+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{8}{r^{2}}\left(\frac{\alpha}{wr+2\alpha}\right)^{2}\underset{r\rightarrow
0}{\rightarrow}+\frac{2}{r^{2}}.$ (172)
This displays explicitly how the spin-orbit and other effects completely
overwhelm the angular momentum barrier leaving a nonsingular potential at the
origin In particular, combining with $2/r^{2}$ we obtain
$\frac{2}{r^{2}}-\frac{8A^{\prime}}{r\left(w-2A\right)}+8\left(\frac{A^{\prime}}{w-2A}\right)^{2}=\frac{2}{(r+2\alpha/w)^{2}}.$
(173)
From this we obtain Eq. (88).
### B.2
Specialization to vector interactions, equal masses, and $J=L>0$.
In this case we need only consider the ${}^{1}J_{J}$ and ${}^{3}J_{J}$ states.
The corresponding equations are
$\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}{-}3\Phi_{SS}\\}u_{0}=b^{2}u_{0},$
(174)
and
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{J(J+1)}{r^{2}}+2\varepsilon_{w}A-A^{2}+\Phi_{D}-2\Phi_{SO}+\Phi_{SS}+2\Phi_{T}-2\Phi_{SOT}\\}u_{1}$
$\displaystyle=b^{2}u_{1}.$ (175)
The first equation simplifies as before $\Phi_{D}{=}3\Phi_{SS}$ while for the
second equation we have
$\displaystyle\Phi_{D}-2\Phi_{SO}+\Phi_{SS}+2\Phi_{T}-2\Phi_{SOT}$
$\displaystyle=\frac{2\mathcal{G}^{\prime}}{r}+\nabla^{2}\mathcal{G-G}^{\prime
2}$
$\displaystyle=-\frac{2}{r}\frac{A^{\prime}}{w-2A}+3\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}.$
(176)
Hence, our two $J=1$ uncoupled equations become
$\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}\\}u_{0}=b^{2}u_{0},$
(177)
and
$\displaystyle\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{r^{2}}+2\varepsilon_{w}A-A^{2}-\frac{1}{r}\frac{A^{\prime}}{w-2A}+\frac{3}{2}\left(\frac{A^{\prime}}{w-2A}\right)^{2}+\frac{1}{2}\frac{{\hbox{\boldmath${\nabla}$}}^{2}A}{w-2A}\\}u_{1}$
$\displaystyle=b^{2}u_{1}.$ (178)
## Appendix C
Solutions of Eq. (31) for Usual and Peculiar ${}^{1}S_{0}$ Bound States
Let us use the Coulomb variable $r=x/\varepsilon_{w}\alpha$ so that our
${}^{1}S_{0}$ equation becomes
$\displaystyle Hu$ $\displaystyle\equiv$
$\displaystyle(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}})u=\frac{(\varepsilon_{w}^{2}-m_{w}^{2})}{\varepsilon_{w}^{2}\alpha^{2}}u\equiv-\kappa^{2}u,$
$\displaystyle u$ $\displaystyle=$ $\displaystyle
x^{\lambda+1}v(x)\exp(-\kappa x),$ (179)
in which the two solutions for $\lambda$ are
$\displaystyle\lambda_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1+\sqrt{1-4\alpha^{2}}),$
$\displaystyle\lambda_{-}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-1-\sqrt{1-4\alpha^{2}}).$ (180)
corresponding to the usual and peculiar solutions respectively. Then our
equation becomes
$-v^{\prime\prime}+2v^{\prime}\kappa-\frac{2(\lambda+1)v^{\prime}}{x}+\frac{2\kappa(\lambda+1)v}{x}-\frac{2}{x}v=0,$
(181)
Let
$v=\sum_{n_{r}=0}^{\infty}v_{n_{r}}x^{n_{r}},$ (182)
and we obtain
$v_{n_{r}+1}=\frac{(2\kappa
n_{r}-2+2\kappa(\lambda+1))}{(n_{r}+1)(n_{r}+2(\lambda+1))}v_{n_{r}},$ (183)
For bound states we have
$\kappa=\frac{1}{n_{r}+\lambda+1},~{}n_{r}=0,1,2,..$ (184)
We let
$n^{\prime}=n_{r}+\lambda+1.$ (185)
If $\lambda$ were an integer then this would be the principle quantum number
$n$. We write
$(-\frac{d^{2}}{dx^{2}}-\frac{2}{x}-\frac{\alpha^{2}}{x^{2}}+\kappa^{2})u=0,$
(186)
as
$(\frac{d^{2}}{dy^{2}}+\frac{1}{y\kappa}+\frac{\alpha^{2}}{y^{2}}-\frac{1}{4})u=0,$
(187)
where $x=y/\left(2\kappa\right),$ so that arf
$u(y)=\exp(-y/2)y^{\lambda+1}L_{n_{r}}^{2\lambda+1}(y)$ (188)
Let
$r=\frac{x}{\varepsilon_{w}\alpha}=\frac{y}{2\kappa\varepsilon_{w}\alpha},$
(189)
and so our radial wave function is
$u(r)=k\exp(-\frac{\varepsilon_{w}\alpha
r}{n^{\prime}})\left(\frac{2\varepsilon_{w}\alpha
r}{n^{\prime}}\right)^{\lambda+1}L_{n_{r}}^{2\lambda+1}(\frac{2\varepsilon_{w}\alpha
r}{n^{\prime}}).$ (190)
The corresponding hydrogenic radial wave function is
$u(r)=k\exp(-\frac{r}{na_{0}})\left(\frac{2}{na_{0}}\right)^{L+1}L_{n_{r}}^{2L+1}(\frac{2r}{na_{0}}).$
(191)
Using the result arf for the hydrogenic wave function
$\langle r^{2}\rangle=\frac{a_{0}^{2}n^{2}}{6}[n^{2}-5L(L+1)+3]$ (192)
and identifying $L(L+1)\rightarrow-\alpha^{2}$, $n\rightarrow n^{\prime}$
,$a_{0}\rightarrow 1/(\varepsilon_{w}\alpha)$ we see that for our states
$\langle r^{2}\rangle=\frac{n^{\prime
2}}{6\left(\varepsilon_{w}\alpha\right)^{2}}[n^{\prime 2}+5\alpha^{2}+3].$
(193)
Our total c.m. energy eigenvalues come from
$\displaystyle\frac{(\varepsilon_{w}^{2}-m_{w}^{2})}{\varepsilon_{w}^{2}\alpha^{2}}$
$\displaystyle=$ $\displaystyle-\kappa^{2}=-\frac{1}{n^{\prime 2}}$
$\displaystyle\varepsilon_{w}^{2}(1+\frac{\alpha^{2}}{n^{\prime 2}})$
$\displaystyle=$ $\displaystyle m_{w}^{2},$ $\displaystyle\varepsilon_{w}$
$\displaystyle=$
$\displaystyle\pm\frac{m_{w}}{\sqrt{(1+\frac{\alpha^{2}}{n^{\prime 2}})}}.$
$\displaystyle n^{\prime}$ $\displaystyle=$ $\displaystyle
n_{r}+\lambda+1,~{}n_{r}=0,1,...$ (194)
In the static limit case for which $m_{2}>>m_{1}$ we use $w=m_{2}+\varepsilon$
in which $\varepsilon<<m_{2}$ includes the rest mass and binding energy of
particle 1. Then
$\displaystyle m_{w}$ $\displaystyle=$
$\displaystyle\frac{m_{1}m_{2}}{m_{2}+\varepsilon}\rightarrow m_{1},$
$\displaystyle\varepsilon_{w}$ $\displaystyle=$
$\displaystyle\frac{m_{2}^{2}+2\varepsilon
m_{2}+\varepsilon^{2}-m_{1}^{2}-m_{2}^{2}}{2m_{1}m_{2}}$ (195)
$\displaystyle\rightarrow$ $\displaystyle\frac{2\varepsilon
m_{2}+\varepsilon^{2}-m_{1}^{2}}{2m_{1}m_{2}}\rightarrow\varepsilon.$
In that case the above solution would be for the binding energy
$\varepsilon=\pm\frac{m}{\sqrt{(1+\frac{\alpha^{2}}{n^{\prime 2}})}}.$ (196)
Since we do not include negative energies we dispense with the lower sign.
Let us solve for the total c.m. energy in the case of equal masses
$m_{1}=m_{2}\equiv m$,
$\displaystyle\frac{\varepsilon_{w}}{m_{w}}$ $\displaystyle=$
$\displaystyle\frac{w^{2}-2m^{2}}{2m^{2}}=f(\alpha)\equiv\frac{1}{\sqrt{(1+\frac{\alpha^{2}}{n^{\prime
2}})}},$ $\displaystyle w^{2}$ $\displaystyle=$ $\displaystyle
2m^{2}(1+f(\alpha)).$ (197)
Thus the solutions are
$\displaystyle w_{\pm}$ $\displaystyle=$
$\displaystyle\sqrt{2}m\sqrt{1+\frac{1}{\sqrt{(1+\frac{\alpha^{2}}{\left(n_{r}+\lambda_{\pm}+1\right)^{2}})}}}$
Since $L=0$ we take our principle quantum number to be $n=n_{r}+1$. This leads
to the results in the text for the spectrum. The value of $\langle
r^{2}\rangle$ for the peculiar ground state is
$\displaystyle\langle r^{2}\rangle_{-}$ $\displaystyle=$
$\displaystyle\frac{n^{\prime
2}}{6\left(\varepsilon_{w}\alpha\right)^{2}}[n^{\prime 2}+5\alpha^{2}+3]$
(199) $\displaystyle=$
$\displaystyle\frac{(1-\sqrt{1-4\alpha^{2}})^{2}}{8\left(\varepsilon_{w}\alpha\right)^{2}}[\frac{1}{4}(1-\sqrt{1-4\alpha^{2}})^{2}+5\alpha^{2}+3]$
$\displaystyle\rightarrow$
$\displaystyle\frac{\alpha^{2}}{2\varepsilon_{w}^{2}}\rightarrow\frac{\alpha^{2}}{2(m\alpha/\sqrt{2})^{2}}=\frac{1}{m^{2}}$
so that $\sqrt{\langle r^{2}\rangle_{-}}$ is the electron Compton radius. For
all of the usual states and the remaining peculiar states they have the
following forms
$\displaystyle\langle r^{2}\rangle_{+}$ $\displaystyle=$
$\displaystyle\frac{n_{+}^{\prime
2}}{6\left(\varepsilon_{w_{+}}\alpha\right)^{2}}[n_{+}^{\prime
2}+5\alpha^{2}+3]=\frac{(n+\lambda_{+})^{2}}{6\left(\varepsilon_{w_{+}}\alpha\right)^{2}}[(n+\lambda_{+})^{2}+5\alpha^{2}+3]\text{,~{}}n=1,2,3...,$
$\displaystyle\langle r^{2}\rangle_{-}$ $\displaystyle=$
$\displaystyle\frac{n_{-}^{\prime
2}}{6\left(\varepsilon_{w_{-}}\alpha\right)^{2}}[n_{-}^{\prime
2}+5\alpha^{2}+3]=\frac{(n+\lambda_{-})^{2}}{6\left(\varepsilon_{w_{+}}\alpha\right)^{2}}[(n+\lambda_{-})^{2}+5\alpha^{2}+3]\text{,~{}}n=2,3...,$
(200)
and we see that the size of the $nth$ usual state is very nearly the same as
the size of the $n+1st$ peculiar state.
In light of this one might wonder how the excited peculiar states (which have
the size of angtroms) can be orthogonal to the peculiar ground state, that has
size of a Compton wave length. As an example, as seen from Eq. ( 183) the
first node of the first excited state occurs at
$\displaystyle x$ $\displaystyle=$
$\displaystyle(\lambda_{-}+1)(\lambda_{-}+2)\sim\alpha^{2},$ $\displaystyle r$
$\displaystyle\sim$
$\displaystyle\frac{\alpha}{\varepsilon_{w}}\sim\frac{\sqrt{2}}{m}.$ (201)
which is on the order of 560 fermis.
## Appendix D
The Connection between $F_{\lambda}(\eta,br)$ and $G_{\lambda}(\eta,br)$
We begin with whit ; hum ; abram
$F_{\lambda}(\rho)=C_{\lambda}(\eta)\rho^{\lambda+1}\exp(-i\rho)M(\lambda+1-i\eta,2\lambda+2;2i\rho),$
(202)
and
$\displaystyle G_{\lambda}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left|\Gamma(\lambda+1+i\eta)\right|\exp(\pi\eta/2)[\frac{\exp(i\pi\lambda/2)}{\Gamma(\lambda+1+i\eta)}W_{i\eta},_{\lambda+1/2}(2i\rho)$
(203)
$\displaystyle+\frac{\exp(-i\pi\lambda/2)}{\Gamma(\lambda+1-i\eta)}W_{-i\eta},_{\lambda+1/2}(-2i\rho)].$
We introduce the Coulomb phase shift
$\displaystyle\sigma_{\lambda}(\eta)$ $\displaystyle=$
$\displaystyle\frac{1}{2i}[\log(\Gamma(\lambda+1+i\eta)-\log(\Gamma(\lambda+1-i\eta)],$
$\displaystyle\Gamma(\lambda+1+i\eta)$ $\displaystyle=$
$\displaystyle\left|\Gamma(\lambda+1+i\eta)\right|\exp(i\sigma_{\lambda}(\eta)),$
(204)
and so
$\displaystyle G_{\lambda}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\exp(-i[\sigma_{\lambda}(\eta)-(\lambda-i\eta)\pi/2)W_{i\eta},_{\lambda+1/2}(2i\rho)+\exp(i[\sigma_{\lambda}(\eta)-(\lambda-i\eta)\pi/2)W_{-i\eta},_{\lambda+1/2}(-2i\rho)]$
(205) $\displaystyle\equiv$
$\displaystyle\frac{1}{2}[\psi_{-}(\lambda,\eta,\rho)+\psi_{+}(\lambda,\eta,\rho)],$
where
$\displaystyle\psi_{+}(\lambda,\eta,\rho)$ $\displaystyle=$
$\displaystyle\exp(i[\sigma_{\lambda}(\eta)-(\lambda-i\eta)\pi/2])W_{-i\eta},_{\lambda+1/2}(-2i\rho),$
$\displaystyle\psi_{-}(\lambda,\eta,\rho)$ $\displaystyle=$
$\displaystyle\exp(-i[\sigma_{\lambda}(\eta)-(\lambda-i\eta)\pi/2])W_{i\eta},_{\lambda+1/2}(2i\rho),$
(206)
and since $\lambda,\eta,\rho$ are all real
$\psi_{-}(\lambda,\eta,\rho)=\psi_{+}^{\ast}(\lambda,\eta,\rho).$ (207)
Also we have
$\displaystyle F_{\lambda}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2i}[\psi_{+}(\lambda,\eta,\rho)-\psi_{-}(\lambda,\eta,\rho)],$
(208) $\displaystyle\psi_{\pm}(\lambda,\eta,\rho)$ $\displaystyle=$
$\displaystyle G_{\lambda}(\rho)\pm iF_{\lambda}(\rho).$
Note that since the Whittaker function $W_{\kappa,\upsilon}(z)$ is an even
function of $\mu$ we have that
$\displaystyle\psi_{+}(-\lambda-1,\eta,\rho)$ $\displaystyle=$
$\displaystyle\exp(i[\sigma_{-\lambda-1}(\eta)-(-\lambda-1-i\eta)\pi/2])W_{-i\eta},_{-\lambda-1/2}(-2i\rho)$
(209) $\displaystyle=$
$\displaystyle\exp(ix(\lambda,\eta))\exp(i[\sigma_{\lambda}(\eta)-(\lambda-i\eta)\pi/2])W_{-i\eta},_{\lambda+1/2}(-2i\rho)$
$\displaystyle=$
$\displaystyle\exp(ix(\lambda,\eta))\psi_{+}(\lambda,\eta,\rho),$
where
$x(\lambda,\eta)=(\lambda+\frac{1}{2})\pi+\sigma_{-\lambda-1}(\eta)-\sigma_{\lambda}(\eta).$
(210)
Similarly
$\psi_{-}(-\lambda-1,\eta,\rho)=\exp(-ix(\lambda,\eta))\psi_{-}(\lambda,\eta,\rho).$
(211)
As a result of this we have
$\displaystyle F_{-\lambda-1}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2i}[\psi_{+}(-\lambda-1,\eta,\rho)-\psi_{-}(-\lambda-1,\eta,\rho)]$
(212) $\displaystyle=$
$\displaystyle\frac{1}{2i}[\exp(ix(\lambda,\eta))\psi_{+}(\lambda,\eta,\rho)-\exp(-ix(\lambda,\eta))\psi_{-}(\lambda,\eta,\rho)]$
$\displaystyle=$
$\displaystyle\frac{1}{2i}[\exp(ix(\lambda,\eta))\left[G_{\lambda}(\rho)+iF_{\lambda}(\rho)\right]-\exp(-ix(\lambda,\eta))\left[G_{\lambda}(\rho)-iF_{\lambda}(\rho)\right]$
$\displaystyle=$ $\displaystyle\cos x(\lambda,\eta)F_{\lambda}(\rho)+\sin
x(\lambda,\eta)G_{\lambda}(\rho),$
and thus
$G_{\lambda}(\rho)=\frac{F_{-\lambda-1}(\rho)-\cos
x(\lambda,\eta)F_{\lambda}(\rho)}{\sin x(\lambda,\eta)}.$ (213)
## Appendix E The Variable Phase Method of Calogero
Here we outline the variable phase method, first applied to short range
potentials and then to long range potentials. We begin with the short range
potentials. We consider the following two sets of differential equations
$\displaystyle u^{\prime\prime}+(b^{2}-W)u$ $\displaystyle=0,$
$\displaystyle\bar{u}_{i}^{\prime\prime}+(b^{2}-\bar{W}_{I})\bar{u}_{i}$
$\displaystyle=0,~{}i=1,2,$ $\displaystyle\bar{u}_{1}(0)$
$\displaystyle=u(0)=0,$ $\displaystyle\bar{W}_{I}$
$\displaystyle=\frac{L(L+1)}{r^{2}},$ (214)
where $W(r)$ is a short range potential less singular at the origin than
$const/r^{2}$ and
$\displaystyle\bar{u}_{1}(r)$ $\displaystyle=\hat{\jmath}_{L}(br)\rightarrow
const\sin(br-L\pi/2),$ $\displaystyle\bar{u}_{2}(r)$
$\displaystyle=-\hat{n}_{L}(br)\rightarrow const\cos(br-L\pi/2).$ (215)
Let
$\displaystyle u(r)$
$\displaystyle=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}(r)+\sin\delta_{L}(r)\bar{u}_{2}(r))$
$\displaystyle u(r$
$\displaystyle\rightarrow\infty)=const(\cos\delta_{L}(r\rightarrow\infty)\sin(br-L\pi/2)$
$\displaystyle+\sin\delta_{L}(r$
$\displaystyle\rightarrow\infty)\cos(br-L\pi/2)$
$\displaystyle=const\sin(br-L\pi/2+\delta_{L}(\infty))\rightarrow\sin(br-L\pi/2+\delta_{L}),$
(216)
and so
$\delta_{L}=\delta_{L}(\infty).$ (217)
To find the differential equation that $\delta_{L}(r)$ satisfies, define
$u^{\prime}(r)=\alpha(r)(\cos\delta_{L}(r)\bar{u}_{1}^{\prime}(r)+\sin\delta_{L}(r)\bar{u}_{2}^{\prime}(r)),$
(218)
and so
$\displaystyle\frac{u^{\prime}(r)}{u(r)}$
$\displaystyle=\frac{(\cos\delta_{L}(r)\bar{u}_{1}^{\prime}(r)+\sin\delta_{L}(r)\bar{u}_{2}^{\prime}(r))}{(\cos\delta_{L}(r)\bar{u}_{1}(r)+\sin\delta_{L}(r)\bar{u}_{2}(r))}=\frac{(\bar{u}_{1}^{\prime}(r)+\tan\delta_{L}(r)\bar{u}_{2}^{\prime}(r))}{(\bar{u}_{1}(r)+\tan\delta_{L}(r)\bar{u}_{2}(r))},$
$\displaystyle\tan\delta_{L}(r)$
$\displaystyle=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))}.$
(219)
Then
$\displaystyle\delta_{L}^{\prime}(r)\sec^{2}\delta_{L}(r)$
$\displaystyle=\delta_{L}^{\prime}(r)(1+\tan^{2}\delta_{L}(r))$
$\displaystyle=\frac{\left(\bar{u}_{2}u^{\prime}-\bar{u}_{2}^{\prime}u\right)(\bar{u}_{1}^{\prime\prime}u-\bar{u}_{1}u^{\prime\prime})-\left(\bar{u}_{1}^{\prime}u-\bar{u}_{1}u^{\prime}\right)(\bar{u}_{2}u^{\prime\prime}-\bar{u}_{2}^{\prime\prime}u)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))^{2}}$
$\displaystyle=-\frac{(W-\bar{W}_{I})u^{2}b}{(\bar{u}_{2}u^{\prime}-\bar{u}_{2}^{\prime}u)^{2}},$
(220)
where we have used the Wronskian relation
$\bar{u}_{2}\bar{u}_{1}^{\prime}-\bar{u}_{2}^{\prime}\bar{u}_{1}=const=b,$
(221)
and so
$\delta_{L}^{\prime}(r)=-\frac{(W-\bar{W}_{I})b}{\sec^{2}\delta_{L}(\bar{u}_{2}u^{\prime}/u-\bar{u}_{2}^{\prime})^{2}}.$
(222)
Further manipulations lead to
$\displaystyle\delta_{L}^{\prime}(r)$
$\displaystyle=-\frac{(W-\bar{W}_{I})(\hat{\jmath}_{L}(br)\cos\delta_{L}(r)-\hat{n}_{L}(br)\sin\delta_{L}(r))^{2}}{b}.$
Note that in case of type two reference potentials
$(\bar{W}=\bar{W}_{II}(r)=0)$ we would obtain
$\tan\gamma_{L}(r)=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))},$
(224)
and with
$\displaystyle\bar{u}_{1}(r$ $\displaystyle\rightarrow$ $\displaystyle
0)\rightarrow br,$ $\displaystyle\bar{u}_{2}(r$ $\displaystyle\rightarrow$
$\displaystyle 0)\rightarrow 1$ $\displaystyle u(r$ $\displaystyle\rightarrow$
$\displaystyle 0)=c(br)^{L+1},$ (225)
we obtain
$\tan\gamma_{L}(r\rightarrow
0)\rightarrow\frac{bc(br)^{L+1}-brcb(L+1)(br)^{L}}{cb(L+1)(br)^{L}}\rightarrow
0,$ (226)
and so we obtain the same boundary condition as with the type I reference
potentials. From Eq. (108)
$\gamma_{L}^{\prime}(r)=-\frac{W}{b}\sin^{2}(br+\gamma_{L}(r)),$ (227)
at short distances becomes
$\gamma_{L}^{\prime}(0)=-\frac{L(L+1)}{b}\sin^{2}(b+\gamma_{L}^{\prime}(0)),$
(228)
with the solution given in Eq. (109).
Next we sketch an analogous derivation for the phase shift equation which
involves long range potentials corresponding to Eq. (88) in which the Coulomb
potential appears. As discussed in the text we begin with the following two
sets of differential equations
$\displaystyle u^{\prime\prime}+(b^{2}-W)u$ $\displaystyle=0,$
$\displaystyle\bar{u}_{i}^{\prime\prime}+(b^{2}-\bar{W}_{III})\bar{u}_{i}$
$\displaystyle=0,~{}i=1,2,$ $\displaystyle\bar{W}_{III}$
$\displaystyle=-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}},$
$\displaystyle W$
$\displaystyle=\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}.$
(229)
Note that the total potential plus barrier term $W~{}$appears in the equation
for $u$. We are not including the angular momentum barrier in the definitions
of $\bar{u}_{i}(r).$
The solutions $\bar{u}_{1},\bar{u}_{2}$ to
$\bar{u}_{i}^{\prime\prime}+(b^{2}+\frac{2\varepsilon_{w}\alpha}{r}+\frac{\alpha^{2}}{r^{2}})\bar{u}_{i}=0,~{}i=1,2,$
(230)
are Coulomb wave functions
$\displaystyle\bar{u}_{1}$
$\displaystyle=aF_{\lambda_{\pm}}+cG_{\lambda_{\pm}}$
$\displaystyle\bar{u}_{2}$
$\displaystyle=dF_{\lambda_{\pm}}+fG_{\lambda_{\pm}}.$ (231)
We choose the constants so that $\bar{u}_{1}$ has the same behavior at the
origin that $u$ does.
Even though four functions are listed here, only two are linearly independent
(see Eq. (80)̇). To determine the phase shift equation let us write down first
the wave function $u(r)$ in terms of $\bar{u}_{1},\bar{u}_{2}$
$u(r)=\alpha(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}(r)).$
(232)
In that case
$\displaystyle u(r$
$\displaystyle\rightarrow\infty)\rightarrow(\cos\gamma_{\pm}(r\rightarrow\infty)\sin(br-\eta\log
2br+\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2)$
$\displaystyle+\sin\gamma_{\pm}(r$
$\displaystyle\rightarrow\infty)\cos(br-\eta\log
2br+\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2)$
$\displaystyle=\sin(br-\eta\log
2br+\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2+\gamma_{\pm}(\infty)).$ (233)
This defines the phase shift function $\gamma_{\pm}(r)$ and its relation to
the asymptotic behavior of $u(r)$. On the other hand since $u(r)$ is the wave
function for a potential that includes at $r>>2\alpha/w$ the modified angular
momentum barrier $(2-\alpha^{2})/r^{2}\equiv\kappa(\kappa+1)/r^{2}$ in
addition to the Coulomb term, we must have
$u(r\rightarrow\infty)\rightarrow\sin(br-\eta\log
2br+\sigma_{\kappa}-\kappa\pi/2+\delta_{\kappa}),$ (234)
and so comparison gives
$\sigma_{\lambda_{\pm}}-\lambda_{\pm}\pi/2+\gamma_{\pm}(\infty)=\sigma_{\kappa}-\kappa\pi/2+\delta_{\kappa}.$
(235)
Thus with
$\displaystyle\kappa(\kappa+1)$ $\displaystyle=$ $\displaystyle 2-\alpha^{2},$
(236) $\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{-1+\sqrt{9-4\alpha^{2}}}{2},$
the full phase shift is
$\displaystyle\delta_{\kappa}+\sigma_{\kappa}$
$\displaystyle=\sigma_{\lambda_{\pm}}+(\kappa-\lambda_{\pm})\pi/2+\gamma_{\pm}(\infty)$
$\displaystyle=\arg\Gamma(\lambda_{\pm}+1+i\eta)+(\kappa-\lambda_{\pm})\pi/2+\gamma_{\pm}(\infty)$
$\displaystyle\sim\arg\Gamma(\lambda_{\pm}+1+i\eta)+(1-\lambda_{\pm})\pi/2+\gamma_{\pm}(\infty).$
(237)
To find the differential equation that $\gamma_{\pm}(r)$ satisfies, define
$\displaystyle u^{\prime}(r)$
$\displaystyle=\alpha(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}^{\prime}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}^{\prime}(r)),$
$\displaystyle u(r)$
$\displaystyle=\alpha(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}(r)).$
(238)
Then following a procedure similar that given in Eqs. (218) we obtain
$\tan\gamma_{\pm}(r)=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))}.$
(239)
Also
$\gamma_{\pm}^{\prime}(r)\sec^{2}\gamma_{\pm}(r)=-\frac{Wu^{2}b}{(\bar{u}_{2}u^{\prime}-\bar{u}_{2}^{\prime}u)^{2}},$
(240)
where we have used the Wronskian relation
$\displaystyle\bar{u}_{2}\bar{u}_{1}^{\prime}-\bar{u}_{2}^{\prime}\bar{u}_{1}$
$\displaystyle=const$
$\displaystyle=\cos()\cos()(b-\frac{\eta}{r})+\sin()\sin()(b-\frac{\eta}{r})$
$\displaystyle\rightarrow b$ (241)
and so
$\gamma_{\pm}^{\prime}(r)=-\frac{(W-\bar{W}_{III})b}{\sec^{2}\gamma_{\pm}(\bar{u}_{2}u^{\prime}/u-\bar{u}_{2}^{\prime})^{2}}.$
(242)
Now use
$\frac{u^{\prime}(r)}{u(r)}=\frac{(\cos\gamma_{\pm}(r)\bar{u}_{1}^{\prime}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}^{\prime}(r))}{(\cos\gamma_{\pm}(r)\bar{u}_{1}+\sin\gamma_{\pm}(r)\bar{u}_{2})},$
(243)
and hence, with $\bar{u}_{1}=F_{\lambda_{\pm}},\bar{u}_{2}=G_{\lambda_{\pm}}$
we have
$\gamma_{\pm}^{\prime}(r)=-\frac{(W-\bar{W}_{III})(\cos\gamma_{\pm}(r)F_{\lambda_{\pm}}(r)+\sin\gamma_{\pm}(r)G_{\lambda_{\pm}}(r))^{2}}{b}.$
(244)
Because of the $2/r^{2}$ behavior of $W$ for large $r$ one will have to
integrate quite far to obtain a convergence for $\gamma_{\pm}(r)$ and after
that one must subtract the phase shift $-\pi/2$ due to the $2/r^{2}$ angular
momentum barrier. An alternative form of this equation is
$\tan^{\prime}\gamma_{\pm}(r)=-\frac{(W-\bar{W}_{III})(F_{\lambda_{\pm}}(r)+\tan\gamma_{\pm}(r)G_{\lambda_{\pm}}(r))^{2}}{b}.$
(245)
The question now arises about the boundary condition at the origin for
$\gamma_{\pm}(r).$ We focus on Eq. (239) to determine the boundary condition
at the origin for $\gamma_{\pm}(r)$,
$\tan\gamma_{\pm}(r)=\frac{\bar{u}_{1}^{\prime}(r)u(r)-\bar{u}_{1}(r)u^{\prime}(r)}{(\bar{u}_{2}(r)u^{\prime}(r)-\bar{u}_{2}^{\prime}(r)u(r))}.$
(246)
We determine the behavior at the origin by evaluating the right hand side for
very small $r$. The dominant term for the quasipotential for both case is
$-\alpha^{2}/r^{2}$. Thus it is sufficient to focus on the first case
We use Eq. (246) with
$\left\\{-\frac{d^{2}}{dr^{2}}+\frac{2}{(r+2\alpha/w)^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}u=b^{2}u,$
(247)
and
$\displaystyle\left\\{-\frac{d^{2}}{dr^{2}}-\frac{2\varepsilon_{w}\alpha}{r}-\frac{\alpha^{2}}{r^{2}}\right\\}\bar{u}_{1,2}$
$\displaystyle=$ $\displaystyle b^{2}\bar{u}_{1,2},$
$\displaystyle\bar{u}_{1}$ $\displaystyle=$ $\displaystyle F_{\lambda}(br),$
$\displaystyle\bar{u}_{2}$ $\displaystyle=$ $\displaystyle G_{\lambda}(br).$
(248)
At short distance, the potential energy for $u$ is the same as that for
$\bar{u}_{1,2}$. At very short distance, we choose
$\displaystyle u,\bar{u}_{1}$ $\displaystyle\rightarrow$ $\displaystyle
const\rightarrow r^{\lambda+1},$ $\displaystyle\bar{u}_{2}$
$\displaystyle\rightarrow$ $\displaystyle ar^{\lambda+1}+br^{-\lambda}.$
Clearly then $\tan\gamma_{\pm}(0)=0$ as the numerator vanishes in both cases
where as the denominator is proportional to the Wronskian of $\bar{u}_{1}$ and
$\bar{u}_{2}$ which is $b^{2}$. This case allows us to integrate either Eq.
(244) or (245) with the boundary condition of
$\tan\gamma_{\pm}(0)=\gamma_{\pm}(0)=0$.
To find the total wave function we need to find the additional differential
equation for the amplitude of the wave function. We use
$\displaystyle u^{\prime}(r)$
$\displaystyle=\alpha(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}^{\prime}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}^{\prime}(r))$
$\displaystyle=\alpha^{\prime}(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}(r))$
$\displaystyle+\alpha(r)(\cos\gamma_{\pm}(r)\bar{u}_{1}^{\prime}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}^{\prime}(r))$
$\displaystyle+\gamma_{\pm}^{\prime}\alpha(r)(-\sin\gamma_{\pm}(r)\bar{u}_{1}(r)+\cos\gamma_{\pm}(r)\bar{u}_{2}(r)),$
(249)
and thus, using Eq. (244)
$\displaystyle\frac{\alpha^{\prime}(r)}{\alpha(r)}$
$\displaystyle=-\frac{W(r)-W_{III}}{b}[\frac{\left(\bar{u}_{1}^{2}-\bar{u}_{2}^{2}\right)\sin
2\gamma_{\pm}(r)}{2}-\bar{u}_{1}\bar{u}_{2}\cos 2\gamma_{\pm}],$
$\displaystyle\alpha(r)$
$\displaystyle=\alpha(r_{0})\exp\\{-\int_{0}^{r}\frac{W(r)-W_{III}(r)}{b}[\frac{\left(\bar{u}_{1}^{2}-\bar{u}_{2}^{2}\right)\sin
2\gamma_{\pm}(r)}{2}-\bar{u}_{1}\bar{u}_{2}\cos 2\gamma_{\pm}(r)]\\}.$ (250)
So, the total wave function is
$\displaystyle u(r)$
$\displaystyle=\alpha(r_{0})\exp\\{-\int_{0}^{r}\frac{W(r)-W_{III}(r)}{b}[\frac{\left(\bar{u}_{1}^{2}-\bar{u}_{2}^{2}\right)\sin
2\gamma_{\pm}(r)}{2}-\bar{u}_{1}\bar{u}_{2}\cos 2\gamma_{\pm}(r)]\\}$
$\displaystyle\times(\cos\gamma_{\pm}(r)\bar{u}_{1}(r)+\sin\gamma_{\pm}(r)\bar{u}_{2}(r)).$
(251)
Acknowledgment
The authors would like to thank Profs. Jin-Hee Yoon, R. L. Becker, and L.
Hulett for helpful discussions. The research was sponsored in part by the
Office of Nuclear Physics, U.S. Department of Energy.
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|
arxiv-papers
| 2012-03-03T22:04:35 |
2024-09-04T02:49:28.218979
|
{
"license": "Public Domain",
"authors": "Horace W. Crater and Cheuk-Yin Wong",
"submitter": "Cheuk-Yin Wong",
"url": "https://arxiv.org/abs/1203.0687"
}
|
1203.0840
|
11footnotetext: E-mail: gh.dong@163.com(G. Dong); ninglw@163.com(N. Wang);
hyqq@hunnu.edu.cn(Y. Huang); hren@math.ecnu.edu.cn(H. Ren);
ypliu@bjtu.edu.cn(Y. Liu).
# Vertex Splitting and Upper Embeddable Graphs 222This work was partially
Supported by the New Century Excellent Talents in University (Grant No:
NCET-07-0276 (Y. Huang)), the National Natural Science Foundation of China
(Grant No. 11171114 (H. Ren); 10871021 (Y. Liu)), and the China Postdoctoral
Science Foundation funded project (Grant No: 20110491248 (G. Dong)).
Guanghua Dong1,2∗, Ning Wang3, Yuanqiu Huang1, Han Ren4, Yanpei Liu5
1.Department of Mathematics, Normal University of Hunan, Changsha, 410081,
China
2.Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160,
China
3.Department of Information Science and Technology, Tianjin University of
Finance
and Economics, Tianjin, 300222, China
4.Department of Mathematics, East China Normal University, Shanghai,
200062,China
5.Department of Mathematics, Beijing Jiaotong University, Beijing, 100044,
China
###### Abstract
The $weak$ $minor$ $\underline{G}$ of a graph $G$ is the graph obtained from
$G$ by a sequence of edge-contraction operations on $G$. A
$weak$-$minor$-$closed$ family of upper embeddable graphs is a set
$\mathcal{G}$ of upper embeddable graphs that for each graph $G$ in
$\mathcal{G}$, every weak minor of $G$ is also in $\mathcal{G}$. Up to now,
there are few results providing the necessary and sufficient conditions for
characterizing upper embeddability of graphs. In this paper, we studied the
relation between the vertex splitting operation and the upper embeddability of
graphs; provided not only a necessary and sufficient condition for
characterizing upper embeddability of graphs, but also a way to construct
weak-minor-closed family of upper embeddable graphs from the bouquet of
circles; extended a result in $J.$ $Graph$ $Theory$ obtained by L. Nebeský. In
addition, the algorithm complex of determining the upper embeddability of a
graph can be reduced much by the results obtained in this paper.
Key Words: maximum genus; weak minor; flexible-weak-minor; flexible-vertex;
flexible-edge
MSC(2000): 05C10
1\. Introduction
Graphs considered here are all connected, undirected, and with minimum degree
at least three. In addition, multiple edges and loops are permitted.
Terminologies and notations not defined here can be seen in [1]. The reader is
assumed to be familiar with topological graph theory, which can be find more
details in [2], [3] or [4].
A graph is denoted by $G$ = ($V(G),E(G)$), and $V(G)$, $E(G)$ denotes its
vertex set and edge set respectively. The number $|E(G)|$ $-$ $|V(G)|$ \+ 1 is
known as the _Betti number_ (or _cycle rank_) of the connected graph _G_ , and
is denoted by $\beta(\emph{G})$. A $u,v$-$path$ is a path whose vertices of
degree 1 (its endpoints) are $u$ and $v$. Let _T_ be a spanning tree of a
connected graph _G_. Define the _deficiency_ $\xi(G,T)$ of a spanning tree $T$
in a graph _G_ to be the number of components of $G-E(T)$ which have odd size.
The deficiency $\xi(G)$ of a graph _G_ is defined to be the minimum value of
$\xi(G,T)$ over all spanning tree _T_ of _G_ , $i.e.$,
$\xi(G)=min\\{\xi(G,T)\mid$ T is an spanning tree of G}. A $splitting$ $tree$
of a connected graph $G$ is a spanning tree $T$ for $G$ such that at most one
component of $G-E(T)$ has odd size. Let $v$ be a vertex of $G$, and $N_{G}(v)$
be the set of vertices in $G$ adjacent to $v$, then the subgraph induced by
$N_{G}(v)$ is referred to as the $v$-$local$ subgraph, and is denoted by
$G_{loc}(v)$. The $vertex$ $splitting$ on a vertex $v$, whose degree
deg${}_{G}(v)\geqslant 4$, is the replacement of the vertex $v$ by adjacent
vertices $v^{\prime}$ and $v^{\prime\prime}$ and the replacement of each edge
$e=vu$ incident to $v$ either by the edge $v^{\prime}u$ or by the edge
$v^{\prime\prime}u$, and the edge $v^{\prime}v^{\prime\prime}$ in the new
$G^{*}$ is called the $splitting$-$edge$. If $G^{*}$ is a graph obtained from
$G$ by a vertex splitting operation on the vertex $v\in V(G)$, then the
subgraph of $G^{*}$, which is induced by $v^{\prime}$, $v^{\prime\prime}$ and
the vertices adjacent to $v^{\prime}$ and $v^{\prime\prime}$, is refereed to
as the $v$-$spliting$ $subgraph$ and is denoted by $G^{*}_{spl}(v)$. The
$intersection$ of two graphs $G_{1}$ and $G_{2}$ is defined as $G_{1}\cap
G_{2}=(V(G_{1})\cap V(G_{2}),E(G_{1})\cap E(G_{2}))$, and the $union$ of
$G_{1}$ and $G_{2}$ is defined as $G_{1}\cup G_{2}=(V(G_{1})\cup
V(G_{2}),E(G_{1})\cup E(G_{2}))$. A $partial$ $order$ $\mathcal{R}$ on a set
$X$ is a binary relation that is reflexive, antisymmetric, and transitive. A
$poset$, which is short for $partially$ $ordered$ $set$, is a pair
($X;\mathcal{R}$) where $X$ is a set and $\mathcal{R}$ is a $partial$ $order$
$relation$ on $X$. The $weak$ $minor$ $\underline{G}$ of a graph $G$, which is
denoted by $\underline{G}\preccurlyeq G$, is the graph obtained from $G$ by a
sequence of edge-contraction operations on $G$. Furthermore, a graph $G$ is a
weak minor of itself. For example, both $G_{1}$ in Fig.2 and $G_{2}$ in Fig.3
are a weak-minor of the graph $G$ in Fig.1. A $weak$-$minor$-$closed$ family
of upper embeddable graphs is a set $\mathcal{G}$ of upper embeddable graphs
that for each graph $G$ in $\mathcal{G}$, every weak minor of $G$ is also in
$\mathcal{G}$. Obviously, the binary relation $weak$ $minor$, which is denoted
by $\preccurlyeq$, is a $partial$ $order$.
Fig.1: GFig.2: $G_{1}$Fig.3: $G_{2}$
The _maximum genus_ $\gamma_{M}(G)$ of a connected graph _G_ is the maximum
integer _k_ such that there exists an embedding of $G$ into the orientable
surface of genus $k$. A graph $G$ is said to be _upper embeddable_ if
$\gamma_{M}(\emph{G})$ = $\lfloor\frac{\beta(G)}{2}\rfloor$. Nordhaus, Stewart
and White [5] introduced the idea of the maximum genus of graphs in 1971\.
From then on, many interesting results have being made, mainly concerned with
the relation between the maximum genus and other graph parameters as diameter,
face size, connectivity, girth, etc., and the readers can find more details in
[6][7][8][9][10][11][12][13][14][15] etc.. But few papers have provided the
informations about the problems as: (I) the relation between the upper
embeddability and vertex splitting; (II) the weak-minor-closed family of upper
embeddable graphs. The following is the details for the two problems.
Problem I: Let $G$ be an upper embeddable graph, $v$ be a vertex of $G$ with
degree no less than 4, and $G^{*}$ be the graph obtained from $G$ through a
vertex splitting operation on $v$, then $G^{*}$ may be upper embeddable or
not. For example, both the graph $G_{1}$ in Fig.5 and the graph $G_{2}$ in
Fig.6 are obtained from an upper embeddable $G$ in Fig.4 through a vertex
splitting operation on $v$ in $G$. The graph $G_{1}$ is upper embeddable, but
$G_{2}$ is not upper embeddable. So, a question is naturally raised: How does
an upper embeddable graph remain the upper embeddability after the vertex
splitting operation on some vertex $v$ of this graph?
$v$Fig.4: G$v^{\prime}$$v^{\prime\prime}$Fig.6:
$G_{2}$$v^{\prime}$$v^{\prime\prime}$Fig.5: $G_{1}$
Problem II: In general, a class of upper embeddable graphs is not closed under
minors. For example, although the graph $G$ depicted in Fig.8 is upper
embeddable, the graph $G_{1}$ in Fig.7, which is a minor of $G$, is not upper
embeddable. But, if $G$ is an upper embeddable graph then every weak minor
$\underline{G}$ of $G$ is also upper embeddable. So we can easily get a poset
$\mathcal{F}$, which is a weak-minor closed family of upper embeddable graphs,
from $G$ through a sequence of edge-contraction operations on $G$. Obviously,
the bouquet of circles $B_{\beta(G)}$, which consists of a single vertex with
$\beta(G)$ loops incident to this vertex, is the smallest element of
$\mathcal{F}$, $i.e.$, every upper embeddable graph with $\beta(G)$ co-tree
edges has bouquet circles $B_{\beta(G)}$ as its weak-minor. However, from the
example in Fig.4-Fig.6 we can get that the bouquet circles $B_{\beta(G)}$ may
also be a weak-minor of a graph $G$ which is not upper embeddable. So, how to
get a poset $\mathcal{F}$, which is a weak-minor-closed family of upper
embeddable graphs, from the bouquet of circles $B_{n}$ or other upper
embeddable graph via series of vertex-splitting operations on it is the second
problem.
Fig.7: $G_{1}$Fig.8: G
In this paper, we will do some research on the above two problems. The
following is a Lemma which is obtained by Liu [4][16] and Xuong [15]
independently.
Lemma 1.1 Let _G_ be a connected graph, then
1) $\gamma_{M}(G)$ = $\frac{\beta(G)-\xi(G)}{2}$;
2) _G_ is upper embeddable if and only if $\xi(\emph{G})\leqslant 1$, or $G$
has a splitting tree.
2\. Vertex splitting and upper embeddability
As described in the introduction, an upper embeddable graph may be changed
into a non-upper embeddable graph after a vertex splitting operation. How does
a graph remain the upper embeddability after vertex splitting operations? In
this section, we provide some results on this problem.
Lemma 2.1 Let $G$ be an upper embeddable graph, $v$ be a vertex of $G$ with
deg${}_{G}(v)\geqslant$3, and $v_{1},v_{2},\dots,v_{n}$ be all the neighbors
of $v$ in $G$. If the $v$-$local$ subgraph $G_{loc}(v)$ is connected, then
there must exist a splitting tree $\mathbb{T}$ of $G$ such that all of
{$vv_{1},vv_{2},\dots,vv_{n}$} are edges of $\mathbb{T}$.
Proof Let $T$ be an arbitrary splitting tree of $G$. Since
$v_{1},v_{2},\dots,v_{n}$ are all the neighbors of $v$ in $G$, the splitting
tree $T$ must contain at least one of $\\{vv_{i}|i=1,2,\dots,n\\}$ as its
edge. Without loss of generality, it may be assumed that $vv_{1}\in E(T)$.
If each of $\\{vv_{i}|i=2,\dots,n\\}$ is an edge of $T$, then the splitting
tree $T$ is $\mathbb{T}$ itself.
If some edges of $\\{vv_{i}|i=2,\dots,n\\}$ are not in $T$, then assume,
without loss of generality, that
$vv_{i_{1}},vv_{i_{2}},\dots,vv_{i_{m}}(m\leqslant n-1)$ are all the edges of
$\\{vv_{i}|i=2,\dots,n\\}$ which are not in $T$, where the vertex set
{$v_{i_{1}},v_{i_{2}},\dots,v_{i_{m}}$}$\subseteq$ {$v_{2},\dots,v_{n}$}. Let
$v_{i_{j}}$ be an arbitrary vertex of {$v_{i_{1}},v_{i_{2}},\dots,v_{i_{m}}$}.
Because there is exactly one $u,\omega$-$path$ in $T$ for any two vertices $u$
and $\omega$ in $G$, and the edge $vv_{i_{j}}$ is not in $T$, there must be a
$vv_{i_{j}}$-$path$ in $T$, and the $vv_{i_{j}}$-$path$ in $T$ must be the
style: $v\dots v_{\alpha}v_{i_{j}}$, where $v_{\alpha}$ is a vertex of
$\\{V(G)-\\{v,v_{i_{j}}\\}$}. Let $T_{i_{j}}=\\{T-v_{\alpha}v_{i_{j}}\\}\cup
vv_{i_{j}}$. It is obvious that $T_{i_{j}}$ is a spanning tree of $G$ and the
edge $vv_{i_{j}}\in E(T_{i_{j}})$. Through series of processes similar to that
of getting $T_{i_{j}}$, a spanning tree $T^{*}$ is obtained, where all of
$\\{vv_{1},vv_{2},\dots,vv_{n}\\}$ are edges of $T^{*}$. Since all edges of
$\\{vv_{1},vv_{2},\dots,vv_{n}\\}$ are in $T^{*}$, each edge of $G_{loc}(v)$
is not in $T^{*}$, or else the spanning tree $T^{*}$ will contain cycles. So
all edges of $G_{loc}(v)$ are co-tree edges of $T^{*}$. Because the
$v$-$local$ subgraph $G_{loc}(v)$ is connected, we can get that
$\xi(G,T^{*})\leqslant\xi(G,T)=\xi(G)\leqslant 1$. So $T^{*}$ is a splitting
tree of $G$ which satisfies the Lemma.$\Box$
Lemma 2.2 Let $G$ be an upper embeddable graph with minimum degree at least 3,
$v$ be a vertex of $G$ with deg${}_{G}(v)$=4, $G^{*}$ be the graph obtained
from $G$ by splitting $v$ into two adjacent vertices $v^{\prime}$ and
$v^{\prime\prime}$. If the $splitting$-$edge$ $v^{\prime}v^{\prime\prime}$ is
not a cut-edge of the $v$-$splitting$ subgraph $G^{*}_{spl}(v)$, then $G^{*}$
is upper embeddable.
Proof Let $v_{1}$, $v_{2}$, $v_{3}$, $v_{4}$ be the four vertices adjacent to
$v$ in $G$, and $\mathbb{T}$ be a splitting tree of $G$. Since
$v^{\prime}v^{\prime\prime}$ is not a cut-edge of the $v$-$splitting$ subgraph
$G^{*}_{spl}(v)$, $G^{*}_{spl}(v)$ must contain at least one cycle which has
$v^{\prime}v^{\prime\prime}$ as one of its edges. Without loss of generality,
let $v_{i_{1}}v_{i_{2}}v^{\prime\prime}v^{\prime}$ be the 4-cycle of
$G^{*}_{spl}(v)$, which is depicted, for example, in Fig.9 or Fig.11, where
{$v_{i_{1}},v_{i_{2}}$}={$v_{1},v_{2}$}. Because $G^{*}$ is obtained from $G$
through vertex splitting operation on $v$, $v_{1}v_{2}v$ must be a 3-cycle of
$G$, which is depicted, for example, in Fig.10. In graph $G$, let
$C_{i}$$(i=1,2,3,4)$ denote the connected component which is obtained from
such connected component of $G-E(\mathbb{T})$ that contains $v_{i}$ as one of
its vertices, by deleting the edges $vv_{1},vv_{2},vv_{3},vv_{4},v_{1}v_{2}$
from it. It is possible that $C_{i}$ and $C_{j}$ may be the same connected
component of $G-E(\mathbb{T})$ $(i,j=1,2,3,4\ $and$\ i\neq j)$. If $G$ is
upper embeddable, the graph $G^{*}$ in Fig.11, which is obtained from $G$
through vertex splitting on $v$, is upper embeddable, for $G^{*}$ can also be
viewed as a subdivision of $G$. So, we should only discuss the upper
embeddability of $G^{*}$ in Fig.9. For $v_{1}$, $v_{2}$, $v_{3}$, $v_{4}$
being all the neighbors of $v$ in graph $G$, the splitting tree $\mathbb{T}$
of $G$ must contain at least one edge which belongs to the edge set
$E(v)$={$vv_{i}|i=1,2,3,4$}. It will be discussed in three cases according to
whether at least three edges of $E(v)$ are in $\mathbb{T}$, or exactly two
edges of $E(v)$ are in $\mathbb{T}$, or only one edge of $E(v)$ is in
$\mathbb{T}$. Without loss of generality, let the edges $v^{\prime}v_{i_{1}}$,
$v^{\prime\prime}v_{i_{2}}$, $v^{\prime\prime}v_{3}$, $v^{\prime}v_{4}$ in
$G^{*}$ be the replacement of $vv_{1}$, $vv_{2}$, $vv_{3}$, $vv_{4}$ in $G$
after vertex splitting on $v$, where the edge set
$\\{v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}}\\}$ may be
$\\{v^{\prime}v_{1},v^{\prime\prime}v_{2}\\}$ or
$\\{v^{\prime}v_{2},v^{\prime\prime}v_{1}\\}$.
$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.9:
$G^{*}$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.10:
$G$$v_{i_{1}}$$v_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$v_{l}$$v_{3}$$v_{p}$Fig.11:
$G^{*}$
Case 1: At least three edges of $E(v)$ are in $\mathbb{T}$.
Without loss of generality, let $vv_{1}$, $vv_{2}$, $\dots$, $vv_{n}$($n=$3 or
4) be all the edges of $E(v)$ which are in $\mathbb{T}$. Obviously, if exactly
three edges of $E(v)$, which are denoted by $E_{3}(v)$, are in $\mathbb{T}$,
and $E^{*}_{3}(v)$ denotes the replacement of $E_{3}(v)$ after vertex
splitting on $v$ in $G$, then $T^{*}=(G^{*}\cap\mathbb{T})\cup
v^{\prime}v^{\prime\prime}\cup E^{*}_{3}(v)$ is a spanning tree of $G^{*}$. If
the four edges of $E(v)$ are all in $\mathbb{T}$,
$T^{*}=(G^{*}\cap\mathbb{T})\cup
v^{\prime}v^{\prime\prime}\cup\\{v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}},v^{\prime\prime}v_{3},v^{\prime}v_{4}\\}$
is a spanning tree of $G^{*}$. Furthermore, $\xi(G^{*},T^{*})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}$ is a splitting tree of
$G^{*}$, and in Case 1 $G^{*}$ is upper embeddable.
Case 2: Exactly two edges of $E(v)$ are in $\mathbb{T}$.
The two edges of $E(v)$ in $\mathbb{T}$ may be (i) $vv_{1}$ and $vv_{2}$; or
(ii) $vv_{3}$ and $vv_{4}$; or (iii) one edge belongs to {$vv_{1},vv_{2}$} and
the other belongs to {$vv_{3},vv_{4}$}.
Subcase 2.1: The two edges of $E(v)$ in $\mathbb{T}$ are $vv_{1}$ and
$vv_{2}$.
In this case, the edge $v_{1}v_{2}$ in $G$ can not be an edge of $\mathbb{T}$,
or else $vv_{1}v_{2}$ would form a 3-cycle of $\mathbb{T}$. Let $G^{*}$, which
is depicted in Fig.9, denotes the graph obtained from $G$ through vertex
splitting on $v$, where {$C_{i_{1}},C_{i_{2}}$}={$C_{1},C_{2}$}, and
{$v_{i_{1}},v_{i_{2}}$}={$v_{1},v_{2}$}.
Subcase 2.1.1: $C_{3}$ and $C_{4}$ are the same connected component of $G$.
In this case, let $T^{*}=(G^{*}\cap\mathbb{T})\cup
v^{\prime}v^{\prime\prime}\cup\\{v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}}\\}$.
It is obvious that $T^{*}$ is a spanning tree of $G^{*}$, and
$\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}$ is a
splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable in Subcase-2.1.1.
Subcase 2.1.2: $C_{3}$ and $C_{4}$ are two different connected components of
$G$.
In graph $G^{*}$, if at least one of $C_{3}\cup v^{\prime\prime}v_{3}$ and
$C_{4}\cup v^{\prime}v_{4}$ contains an even number of edges, then let
$T^{*}=(G^{*}\cap\mathbb{T})\cup
v^{\prime}v^{\prime\prime}\cup\\{v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}}\\}$.
It is obvious that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$.
So $T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
If both $C_{3}\cup v^{\prime\prime}v_{3}$ and $C_{4}\cup v^{\prime}v_{4}$
contain an odd number of edges, then $C_{3}$ and $C_{4}$ both contain an even
number of edges. Because there is exactly one $u,\omega$-$path$ in
$\mathbb{T}$ for any two vertices $u$ and $\omega$ in $G$, and both $vv_{3}$
and $vv_{4}$ are not in $\mathbb{T}$, there must be exactly one
$v,v_{3}$-$path$ in $\mathbb{T}$, and the $v,v_{3}$-$path$ in $\mathbb{T}$
must be of the form as $vv_{1}\dots v_{p}v_{3}$ or $vv_{2}\dots v_{p}v_{3}$.
Also, there must be exactly one $v,v_{4}$-$path$ in $\mathbb{T}$, and the
$v,v_{4}$-$path$ in $\mathbb{T}$ must be of the form as $vv_{1}\dots
v_{l}v_{4}$ or $vv_{2}\dots v_{l}v_{4}$. Furthermore, the $v,v_{3}$-$path$ and
$v,v_{4}$-$path$ in $\mathbb{T}$ can not form a cycle. It is discussed in the
following three subcases.
Subcase 2.1.2-a: The $v,v_{3}$-$path$ and $v,v_{4}$-$path$ in $\mathbb{T}$ are
$vv_{1}\dots v_{p}v_{3}$ and $vv_{1}\dots v_{l}v_{4}$ respectively.
If the edges $vv_{1}$ and $vv_{2}$ in $G$ are replaced, after the vertex
splitting on $v$, by $v^{\prime}v_{i_{1}}$ and $v^{\prime\prime}v_{i_{2}}$
respectively, then
$T^{*}_{1}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime\prime}v_{3},v^{\prime\prime}v_{i_{2}}\\}$
is a spanning tree of $G^{*}$. Noticing that the size of $C_{i_{1}}\cup
v_{i_{1}}v_{i_{2}}\cup C_{i_{2}}\cup v_{i_{1}}v^{\prime}\cup
v^{\prime}v^{\prime\prime}$ and $C_{i_{1}}\cup v_{i_{1}}v_{i_{2}}\cup
C_{i_{2}}$ have the same parity, and both the size of $C_{3}$ and $C_{4}$ are
an even number, we can easily get that $\xi(G^{*},T^{*}_{1})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}_{1}$ is a splitting tree of
$G^{*}$, and $G^{*}$ is upper embeddable.
After the vertex splitting on $v$ in $G$, if the edge $vv_{1}$ is replaced by
$v^{\prime\prime}v_{i_{2}}$, and $vv_{2}$ by $v^{\prime}v_{i_{1}}$
respectively, then
$T^{*}_{2}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v_{i_{1}},v^{\prime\prime}v_{3}\\}$
is a spanning tree of $G^{*}$. It is obvious that $\xi(G^{*},T^{*}_{2})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}_{2}$ is a splitting tree of
$G^{*}$, and $G^{*}$ is upper embeddable.
Subcase 2.1.2-b: The $v,v_{3}$-$path$ and $v,v_{4}$-$path$ in $\mathbb{T}$ are
$vv_{2}\dots v_{p}v_{3}$ and $vv_{1}\dots v_{l}v_{4}$ respectively.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$
be a spanning tree of $G^{*}$. It is obvious that $\xi(G^{*},T^{*})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}$ is a splitting tree of
$G^{*}$, and $G^{*}$ is upper embeddable.
Subcase 2.1.2-c: The $v,v_{3}$-$path$ and $v,v_{4}$-$path$ in $\mathbb{T}$ are
$vv_{2}\dots v_{p}v_{3}$ and $vv_{2}\dots v_{l}v_{4}$ respectively, or
$vv_{1}\dots v_{p}v_{3}$ and $vv_{2}\dots v_{l}v_{4}$ respectively.
In this case, it is similar to that of Subcase 2.1.2-a and Subcase 2.1.2-b to
get that $G^{*}$ contains a splitting tree.
So, in Subcase-2.1.2, $G^{*}$ is upper embeddable.
Subcase 2.2: The two edges of $E(v)$ in $\mathbb{T}$ are $vv_{3}$ and
$vv_{4}$.
In this case, according to $v_{1}v_{2}$ being an edge of $\mathbb{T}$ or not,
it will be discussed in the following two subcases.
Subcase 2.2.1: The edge $v_{1}v_{2}$ of $G$ is not in $\mathbb{T}$.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$
be a spanning tree of $G^{*}$. It is obvious that $\xi(G^{*},T^{*})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}$ is a splitting tree of
$G^{*}$.
Subcase 2.2.2: The edge $v_{1}v_{2}$ of $G$ is an edge of $\mathbb{T}$.
It will be discussed in the following subcases.
Subcase 2.2.2-1: $C_{i_{1}}$ and $C_{i_{2}}$ are the same connected component
of $G$.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$
be a spanning tree of $G^{*}$. It is obvious that $\xi(G^{*},T^{*})$ =
$\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So $T^{*}$ is a splitting tree of
$G^{*}$.
Subcase 2.2.2-2: $C_{i_{1}}$ and $C_{i_{2}}$ are two different connected
components of $G$.
If at least one of $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$ and $C_{i_{2}}\cup
v^{\prime\prime}v_{i_{2}}$ contains an even number of edges, then let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$.
It is obvious that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$.
So $T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
If both $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$ and $C_{i_{2}}\cup
v^{\prime\prime}v_{i_{2}}$ contain an odd number of edges, then $C_{i_{1}}$
and $C_{i_{2}}$ both contain an even number of edges. Because there is exactly
one $u,\omega$-$path$ in $\mathbb{T}$ for any two vertices $u$ and $\omega$ in
$G$, and both $vv_{1}$ and $vv_{2}$ are not in $\mathbb{T}$, there must be
exactly one $v,v_{1}$-$path$ in $\mathbb{T}$, and this $v,v_{1}$-$path$ in
$\mathbb{T}$ may be the form as $vv_{4}\dots v_{1}v_{2}$, or $vv_{4}\dots
v_{2}v_{1}$, or $vv_{3}\dots v_{1}v_{2}$, or $vv_{3}\dots v_{2}v_{1}$. It is
discussed in the following two subcases.
Subcase 2.2.2-2a: The $v,v_{1}$-$path$ in $\mathbb{T}$ is $vv_{4}\dots
v_{1}v_{2}$ or $vv_{4}\dots v_{2}v_{1}$.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime\prime}v_{3},v^{\prime\prime}v_{i_{2}}\\}$.
Noticing that both $C_{i_{1}}\cup v_{i_{1}}v^{\prime}\cup
v^{\prime}v^{\prime\prime}$ and $C_{i_{2}}$ contain an even number of edges,
we can get that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So
$T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
Subcase 2.2.2-2b: The $v,v_{1}$-$path$ in $\mathbb{T}$ is $vv_{3}\dots
v_{1}v_{2}$ or $vv_{3}\dots v_{2}v_{1}$.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{1}},v^{\prime}v_{4},v^{\prime\prime}v_{3}\\}$.
It is obvious that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$.
So $T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
Subcase 2.3: The two edges of $E(v)$ in $\mathbb{T}$ are such two edges that
one is selected from {$vv_{1},vv_{2},$} and the other is selected from
{$vv_{3},vv_{4}$}.
Without loss of generality, let the two edges of $E(v)$ in $\mathbb{T}$ are
$vv_{1}$ and $vv_{3}$, which is illustrated in Fig.13. We will discuss in the
following two subcases.
Subcase 2.3.1: After the vertex splitting on $v$ in $G$, the replacements of
$vv_{1}$ and $vv_{3}$ are both adjacent to $v^{\prime}$ or both adjacent to
$v^{\prime\prime}$.
Without loss of generality, let the replacements of $vv_{1}$ and $vv_{3}$ are
both adjacent to $v^{\prime\prime}$, which is illustrated in Fig.12. Let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime\prime}v_{3},v^{\prime\prime}v_{i_{2}},v^{\prime}v^{\prime\prime}\\}$.
It is obvious that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$.
So $T^{*}$ is a splitting tree of $G^{*}$.
$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.12:
$G^{*}$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.13:
$G$$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.14:
$G^{*}$
Subcase 2.3.2: After the vertex splitting on $v$ in $G$, the replacements of
$vv_{1}$ and $vv_{3}$ are adjacent to $v^{\prime}$ and $v^{\prime\prime}$
respectively.
Without loss of generality, let $vv_{1}$ and $vv_{3}$ be replaced, after
vertex splitting on $v$, by $v^{\prime}v_{i_{1}}$ and $v^{\prime\prime}v_{3}$
respectively, which is illustrated in Fig.14.
Subcase 2.3.2-1: In graph $G$, the edge $v_{1}v_{2}$ is not an edge of
$\mathbb{T}$.
If $C_{4}$ and one of {$C_{i_{1}},C_{i_{2}}$} are the same connected component
of $G$, then
$T^{*}_{1}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{1}},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$
is a splitting tree of $G^{*}$.
If $C_{4}$ is a connected component of $G$ which is different from both of
{$C_{i_{1}},C_{i_{2}}$}, we will discuss in two subcases.
Subcase 2.3.2-1a: At least one of $C_{i_{1}}\cup v_{i_{1}}v_{i_{2}}\cup
C_{i_{2}}\cup v_{i_{2}}v^{\prime\prime}$ and $C_{4}\cup v^{\prime}v_{4}$
contains an even number of edges.
In this case, let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{1}},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$.
It is obvious that $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$.
So $T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
Subcase 2.3.2-1b: Both $C_{i_{1}}\cup v_{i_{1}}v_{i_{2}}\cup C_{i_{2}}\cup
v_{i_{2}}v^{\prime\prime}$ and $C_{4}\cup v^{\prime}v_{4}$ contain an odd
number of edges.
In this case, $C_{4}$ contains an even number of edges. Because there is
exactly one $u,\omega$-$path$ in $\mathbb{T}$ for any two vertices $u$ and
$\omega$ in $G$, and both $vv_{2}$ and $vv_{4}$ are not in $\mathbb{T}$, there
must be exactly one $v,v_{4}$-$path$ in $\mathbb{T}$, and the $v,v_{4}$-$path$
in $\mathbb{T}$ must be the form as $vv_{1}\dots v_{4}$ or $vv_{3}\dots
v_{4}$. If the $v,v_{4}$-$path$ in $\mathbb{T}$ is $vv_{1}\dots v_{4}$, then
$T^{*}_{1}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$
is a splitting tree of $G^{*}$. If the $v,v_{4}$-$path$ in $\mathbb{T}$ is
$vv_{3}\dots v_{4}$, then
$T^{*}_{2}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime\prime}v_{3},v^{\prime}v_{4},v^{\prime}v_{i_{1}}\\}$
is a splitting tree of $G^{*}$.
Subcase 2.3.2-2: In graph $G$, the edge $v_{1}v_{2}$ is an edge of
$\mathbb{T}$.
If at least one of $C_{i_{2}}\cup v^{\prime\prime}v_{i_{2}}$ and $C_{4}\cup
v^{\prime}v_{4}$ contains an even number of edges, then let
$T^{*}_{1}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{1}},v^{\prime}v^{\prime\prime},v^{\prime\prime}v_{3}\\}$.
It is obvious that $\xi(G^{*},T^{*}_{1})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant
1$. So $T^{*}_{1}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper
embeddable.
If both $C_{i_{2}}\cup v^{\prime\prime}v_{i_{2}}$ and $C_{4}\cup
v^{\prime}v_{4}$ contain an odd number of edges, then $C_{i_{2}}$ and $C_{4}$
both contain an even number of edges. Let
$T^{*}_{2}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}},v^{\prime\prime}v_{3}\\}$.
It is obvious that $\xi(G^{*},T^{*}_{2})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant
1$. So $T^{*}_{2}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper
embeddable.
Case 3: Only one edge of $E(v)$ is in $\mathbb{T}$.
According to this edge is selected from {$vv_{1},vv_{2}$} or
{$vv_{3},vv_{4}$}, it will be discussed in the following Subcase-3.1 and
Subcase-3.2.
Subcase 3.1: One of {$vv_{1},vv_{2}$} is the edge in $\mathbb{T}$.
Without loss of generality, let $vv_{1}$ be the edge in $\mathbb{T}$, which is
depicted in Fig.16. In addition, throughout Subcase 3.1, let $vv_{1}$ and
$vv_{2}$ be replaced by $v^{\prime}v_{1}$ and $v^{\prime\prime}v_{2}$
respectively after the vertex splitting on $v$ in $G$; and the edge set
{$vv_{3},vv_{4}$} be replaced by
{$v^{\prime\prime}v_{i_{3}},v^{\prime}v_{i_{4}}$}, where
{$v_{i_{3}},v_{i_{4}}$}={$v_{3},v_{4}$} and
{$C_{i_{3}},C_{i_{4}}$}={$C_{3},C_{4}$}, which is depicted in Fig.15.
According to the edge $v_{1}v_{2}$ of $G$ is in the splitting tree
$\mathbb{T}$ or not, it will be discussed in the following two subcases.
$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v^{\prime}$$v^{\prime\prime}$$v_{i_{4}}$$v_{l}$$C_{i_{4}}$$v_{i_{3}}$$v_{p}$$C_{i_{3}}$Fig.15:
$G^{*}$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.16:
$G$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v^{\prime}$$v^{\prime\prime}$$v_{i_{4}}$$v_{l}$$C_{i_{4}}$$v_{i_{3}}$$v_{p}$$C_{i_{3}}$Fig.17:
$G^{*}$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v^{\prime}$$v^{\prime\prime}$$v_{i_{4}}$$v_{l}$$C_{i_{4}}$$v_{i_{3}}$$v_{p}$$C_{i_{3}}$Fig.18:
$G^{*}$
Subcase 3.1.1: In graph $G$, $v_{1}v_{2}$ is not an edge of $\mathbb{T}$. It
is discussed in the following subcases.
Subcase 3.1.1-1: In graph $G^{*}$, $C_{1}\cup v_{1}v_{2}\cup C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}\cup
C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ contains an odd number of edges.
In this case,
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$. So, in Subcase 3.1.1-1, $G^{*}$ is upper
embeddable.
Subcase 3.1.1-2: In graph $G^{*}$, $C_{1}\cup v_{1}v_{2}\cup C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}\cup
C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ contains an even number of edges.
In this case, if $C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ contains an even number
of edges, then
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$.
If $C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ contains an odd number of edges, then
$C_{1}\cup v_{1}v_{2}\cup C_{2}\cup v_{2}v^{\prime\prime}\cup
v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}$ contains an odd number of edges too.
It is discussed in the following two subcases.
Subcase 3.1.1-2a: In graph $G^{*}$, the connected component $C_{i_{4}}$ is the
same with at least one of {$C_{1}$, $C_{i_{3}}$}.
In this case,
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$.
Subcase 3.1.1-2b: In graph $G^{*}$, neither of {$C_{1}$, $C_{i_{3}}$} is the
same connected component with $C_{i_{4}}$.
Because there is exactly one $u,\omega$-$path$ in $\mathbb{T}$ for any two
vertices $u$ and $\omega$ in $G$, and none of {$vv_{2},vv_{3},vv_{4}$} is an
edge of $\mathbb{T}$, there must be exactly one $v,v_{3}$-$path$ and exactly
one $v,v_{4}$-$path$ in $\mathbb{T}$, and the $v,v_{3}$-$path$,
$v,v_{4}$-$path$ in $\mathbb{T}$ must be of the form as $vv_{1}\dots v_{3}$
and $vv_{1}\dots v_{4}$ respectively. Noticing that both $C_{i_{4}}$ and
$v^{\prime}v_{1}\cup C_{1}\cup v_{1}v_{2}\cup C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}$ are
connected component of $G^{*}$ with an even number of edges, we can easily get
that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{i_{4}},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$.
Subcase 3.1.2: In graph $G$, $v_{1}v_{2}$ is an edge of $\mathbb{T}$. It is
discussed in the following subcases.
In graph $G^{*}$, if $C_{i_{4}}$ is the same connected component with at least
one of {$C_{1},C_{2},C_{i_{3}}$}, then
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$. If any pair of components, which is selected
from {$C_{1},C_{2},C_{i_{3}},C_{i_{4}}$}, is not the same connected component
of $G^{*}$, then it will be discussed in the following two subcases.
Subcase 3.1.2-1: In graph $G^{*}$, $C_{2}\cup v_{2}v^{\prime\prime}\cup
v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}\cup C_{i_{4}}\cup v^{\prime}v_{i_{4}}$
contains an odd number of edges.
Noticing that one of {$C_{i_{4}}\cup v^{\prime}v_{i_{4}},\ C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}$} is a
connected component of $G^{*}$ which contains an even number of edges, and the
other is one which contains an odd number of edges, we can easily get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$.
Subcase 3.1.2-2: In graph $G^{*}$, $C_{2}\cup v_{2}v^{\prime\prime}\cup
v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}\cup C_{i_{4}}\cup v^{\prime}v_{i_{4}}$
contains an even number of edges.
If both $C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ and $C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}$ are
connected component of $G^{*}$ which contain an even number of edges, then it
is easy to get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$.
If both $C_{i_{4}}\cup v^{\prime}v_{i_{4}}$ and $C_{2}\cup
v_{2}v^{\prime\prime}\cup v^{\prime\prime}v_{i_{3}}\cup C_{i_{3}}$ are
connected component of $G^{*}$ which contain an odd number of edges, then we
will discuss it in the following two subcases.
Subcase 3.1.2-2a: In graph $G^{*}$, $C_{2}$ is a connected component with an
even number of edges, and $C_{i_{3}}$ is one which contains an odd number of
edges.
Noticing that both $C_{2}$ and $C_{i_{3}}\cup v_{i_{3}}v^{\prime\prime}\cup
v^{\prime\prime}v^{\prime}\cup v^{\prime}v_{i_{4}}\cup C_{i_{4}}$ are
connected component of $G^{*}$ which contain an even number of edges, we can
easily get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v_{2}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$, which is depicted in Fig.17.
Subcase 3.1.2-2b: In graph $G^{*}$, $C_{2}$ is a connected component with an
odd number of edges, and $C_{i_{3}}$ is one which contains an even number of
edges.
Because there is exactly one $u,\omega$-$path$ in $\mathbb{T}$ for any two
vertices $u$ and $\omega$ in $G$, and none of {$vv_{2},vv_{3},vv_{4}$} is an
edge of $\mathbb{T}$, there must be exactly one $v,v_{3}$-$path$ and exactly
one $v,v_{4}$-$path$ in $\mathbb{T}$, and the $v,v_{3}$-$path$,
$v,v_{4}$-$path$ in $\mathbb{T}$ must be of the form as $vv_{1}\dots v_{3}$
and $vv_{1}\dots v_{4}$ respectively. Noticing that, in the graph $G^{*}$, the
connected components $C_{i_{3}}$ and $C_{2}\cup v_{2}v^{\prime\prime}\cup
v^{\prime\prime}v^{\prime}\cup v^{\prime}v_{i_{4}}\cup C_{i_{4}}$ both contain
an even number of edges, we can easily get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{1},v^{\prime\prime}v_{i_{3}}\\}$
is a splitting tree of $G^{*}$, which is depicted in Fig.18.
Subcase 3.2: One of {$vv_{3},vv_{4}$} is the edge in $\mathbb{T}$.
Without loss of generality, let $vv_{4}$ be the edge in $\mathbb{T}$, which is
depicted in Fig.20. In addition, throughout Subcase 3.2, let $vv_{3}$ and
$vv_{4}$ be replaced by $v^{\prime\prime}v_{3}$ and $v^{\prime}v_{4}$
respectively after the vertex splitting on $v$ in $G$; and the edge set
{$vv_{1},vv_{2}$} be replaced by
{$v^{\prime}v_{i_{1}},v^{\prime\prime}v_{i_{2}}$}, where
{$v_{i_{1}},v_{i_{2}}$}={$v_{1},v_{2}$} and
{$C_{i_{1}},C_{i_{2}}$}={$C_{1},C_{2}$}, which is depicted in Fig.19.
According to the edge $v_{1}v_{2}$ of $G$ is in the splitting tree
$\mathbb{T}$ or not, it will be discussed in the following two subcases.
$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.19:
$G^{*}$$v_{1}$$C_{1}$$v_{2}$$C_{2}$$v$$v_{4}$$v_{l}$$C_{4}$$v_{3}$$v_{p}$$C_{3}$Fig.20:
$G$$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$C_{4}$$v_{3}$$C_{3}$Fig.21:
$G^{*}$$v_{i_{1}}$$C_{i_{1}}$$v_{i_{2}}$$C_{i_{2}}$$v^{\prime}$$v^{\prime\prime}$$v_{4}$$C_{4}$$v_{3}$$C_{3}$Fig.22:
$G^{*}$
Subcase 3.2.1: In graph $G$, $v_{1}v_{2}$ is not an edge of $\mathbb{T}$.
In this case, it is obvious that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$, which is depicted in Fig.19. So, in Subcase
3.2.1, $G^{*}$ is upper embeddable.
Subcase 3.2.2: In graph $G$, $v_{1}v_{2}$ is an edge of $\mathbb{T}$.
In this case, if $C_{i_{1}}$ in $G^{*}$ is the same connected component with
at least one of {$C_{i_{2}},C_{3},C_{4}$}, then
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$. If any pair of components, which is selected
from {$C_{i_{1}},C_{i_{2}},C_{3},C_{4}$}, is not the same connected component
of $G^{*}$, then it will be discussed in the following two subcases.
Subcase 3.2.2-1: In graph $G^{*}$, $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$
contains an even number of edges.
In this case, it is obvious that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime}\\}$
is a splitting tree of $G^{*}$, So, in Subcase 3.2.2-1, $G^{*}$ is upper
embeddable.
Subcase 3.2.2-2: In graph $G^{*}$, $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$
contains an odd number of edges, and $C_{i_{2}}\cup
v_{i_{2}}v^{\prime\prime}\cup v^{\prime\prime}v_{3}\cup C_{3}$ contains an
even number of edges.
In this case, the connected component $C_{1}\cup v_{1}v\cup C_{2}\cup
v_{2}v\cup vv_{3}\cup C_{3}$, which contains an odd number of edges in $G$, is
replaced by $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$ and $C_{i_{2}}\cup
v_{i_{2}}v^{\prime\prime}\cup v^{\prime\prime}v_{3}\cup C_{3}$ after the
vertex splitting on $v$ in $G$. Let
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime}v^{\prime\prime}\\}$.
Because $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$ contains an odd number of edges,
and $C_{i_{2}}\cup v_{i_{2}}v^{\prime\prime}\cup v^{\prime\prime}v_{3}\cup
C_{3}$ contains an even number of edges, it is obvious that
$\xi(G^{*},T^{*})$=$\xi(G,\mathbb{T}))\leqslant 1$. So, $T^{*}$ is a splitting
tree of $G^{*}$.
Subcase 3.2.2-3: In graph $G^{*}$, both $C_{i_{1}}\cup v^{\prime}v_{i_{1}}$
and $C_{i_{2}}\cup v_{i_{2}}v^{\prime\prime}\cup v^{\prime\prime}v_{3}\cup
C_{3}$ contain an odd number of edges.
In this case, according to the parity of the number of the edges in
$C_{i_{2}}$ and $C_{3}$ respectively, it will be discussed in the following
two subcases.
Subcase 3.2.2-3a: In graph $G^{*}$, $C_{i_{2}}$ contains an odd number of
edges, and $C_{3}$ contains an even number of edges.
Because there is exactly one $u,\omega$-$path$ in $\mathbb{T}$ for any two
vertices $u$ and $\omega$ in $G$, and none of {$vv_{1},vv_{2},vv_{3}$} is an
edge of $\mathbb{T}$, there must be exactly one $v,v_{3}$-$path$ in
$\mathbb{T}$, and the $v,v_{3}$-$path$ in $\mathbb{T}$ must be of the form as
$vv_{4}\dots v_{3}$. Noticing that, in the graph $G^{*}$, the connected
components $C_{3}$ and $C_{i_{2}}\cup v_{i_{2}}v^{\prime\prime}\cup
v^{\prime\prime}v^{\prime}\cup v^{\prime}v_{i_{1}}\cup C_{i_{1}}$ both contain
an even number of edges, we can easily get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime\prime}v_{3}\\}$
is a splitting tree of $G^{*}$, which is depicted in Fig.21.
Subcase 3.2.2-3b: In graph $G^{*}$, $C_{i_{2}}$ contains an even number of
edges, and $C_{3}$ contains an odd number of edges.
In this case, noticing that in the graph $G^{*}$ the connected components
$C_{i_{2}}$ and $C_{3}\cup v_{3}v^{\prime\prime}\cup
v^{\prime\prime}v^{\prime}\cup v^{\prime}v_{i_{1}}\cup C_{i_{1}}$ both contain
an even number of edges, we can easily get that
$T^{*}=(G^{*}\cap\mathbb{T})\cup\\{v^{\prime}v_{4},v^{\prime\prime}v_{i_{2}}\\}$
is a splitting tree of $G^{*}$, which is depicted in Fig.22.
From Case 1, Case 2, and Case 3, the Lemma 2.2 is obtained. $\Box$
Theorem 2.1 Let $G$ be a graph with minimum degree at least 3, $v$ be a vertex
of $G$ with deg${}_{G}(v)$ $\geqslant$ 4, $G^{*}$ be the graph obtained from
$G$ by splitting $v$ into two adjacent vertices $v^{\prime}$ and
$v^{\prime\prime}$, furthermore, the $v$-$local$ subgraph $G_{loc}(v)$ be
connected. Then the graph $G$ is upper embeddable if and only if $G^{*}$ is
upper embeddable.
Proof ($\Longleftarrow$) Let $\textit{E}^{*}$ be an embedding of $G^{*}$ in
the orientable surfaces $S_{g}$ of genus $g$. Then we can get an embedding E
of $G$ in the surface $S_{g}$ by contracting the $splitting$-$edge$
$v^{\prime}v^{\prime\prime}$ in $\textit{E}^{*}$. So
$\lfloor\frac{\beta(G)}{2}\rfloor$=$\lfloor\frac{\beta(G^{*})}{2}\rfloor$=$\gamma_{M}(G^{*})\leqslant\gamma_{M}(G)$.
On the other hand, $\gamma_{M}(G)\leqslant\lfloor\frac{\beta(G)}{2}\rfloor$.
Therefore, $\gamma_{M}(G)=\lfloor\frac{\beta(G)}{2}\rfloor$, $i.e.$, the graph
$G$ is upper embeddable.
($\Longrightarrow$) Let $v_{1}$, $v_{2}$, $\dots$, $v_{n}(n\geqslant 4)$ be
all the vertices adjacent to $v$ in $G$, $v^{\prime}$and $v^{\prime\prime}$ be
the replacement of $v$ after the vertex splitting on $v$ in $G$, and the edge
subset $\\{vv_{i}|i=1,2,\dots n\\}$ of $E(G)$ is replaced by the subset
$\\{v^{*}v_{i}|v^{*}$ may be $v^{\prime}$ or $v^{\prime\prime}$, $i=1,2,\dots
n\\}$ of $E(G^{*})$. It can be obtained from Lemma 2.1 that there exists a
splitting tree $\mathbb{T}$ of $G$ such that all of
{$vv_{1},vv_{2},\dots,vv_{n}$} are edges of $\mathbb{T}$. Let
$T^{*}=\\{G^{*}\cap\mathbb{T}\\}\cup
v^{\prime}v^{\prime\prime}\cup\\{v^{*}v_{i}|$$v^{*}$ may be $v^{\prime}$ or
$v^{\prime\prime}$, $i=1,2,\dots n\\}$. Obviously, $T^{*}$ is a spanning tree
of $G^{*}$, and $\xi(G^{*},T^{*})$ = $\xi(G,\mathbb{T})=\xi(G)\leqslant 1$. So
$T^{*}$ is a splitting tree of $G^{*}$, and $G^{*}$ is upper embeddable.
$\Box$
Especially, for a vertex $v$ of $G$ with deg${}_{G}(v)$=4, we have the
following theorem.
Theorem 2.2 Let $G$ be a graph with minimum degree at least 3, $v$ be a vertex
of $G$ with deg${}_{G}(v)$=4, $G^{*}$ be the graph obtained from $G$ by
splitting $v$ into two adjacent vertices $v^{\prime}$ and $v^{\prime\prime}$,
where the $splitting$-$edge$ $v^{\prime}v^{\prime\prime}$ is not a cut-edge of
the $v$-$splitting$ subgraph $G^{*}_{spl}(v)$. Then the graph $G$ is upper
embeddable if and only if $G^{*}$ is upper embeddable.
Proof ($\Longleftarrow$) It is the same with that of the Theorem 2.1.
($\Longrightarrow$) It is an obvious result of the Lemma 2.2. $\Box$
3\. Weak minor and upper embeddability
In this section, we will provide a method to construct a weak-minor-closed
family of upper embeddable graphs from the bouquet of circles $B_{n}$; in
addition, we provide a corollary which extends a result obtained by L. Nebeský
[17].
Let $v$ be a vertex of the graph $G$ with deg${}_{G}(v)\geqslant 4$, $G^{*}$
be the graph obtained from $G$ by splitting $v$ into two adjacent vertices
$v^{\prime}$ and $v^{\prime\prime}$, then $v$ is referred to as a
$flexible$-$vertex$ of $G$ if it satisfies one of the following two
conditions: (I) If $v$ is a vertex of the graph $G$ with
deg${}_{G}(v)\geqslant 4$, then the $v$-$local$ subgraph $G_{loc}(v)$ is
connected (and the vertex splitting operation on this kind of vertices is
referred to as $type$-$I$ $vertex$ $splitting$); (II) If $v$ is a vertex of
the graph $G$ with deg${}_{G}(v)$=4, then the $splitting$-$edge$
$v^{\prime}v^{\prime\prime}$ is not a cut-edge of the $v$-$splitting$ subgraph
$G^{*}_{spl}(v)$ (this kind of vertex splitting operation is referred to as
$type$-$II$ $vertex$ $splitting$).
According to Theorem 2.1 and Theorem 2.2, we can get, from the bouquet of
circles $B_{n}$, a weak-minor-closed family of upper embeddable graphs through
a sequence of vertex splitting operations on the $flexible$-$vertices$.
A graph $G$ is called locally connected if for every vertex $v$ of $G$ the
$v$-$local$ subgraph $G_{loc}(v)$ is connected. In 1981, L. Nebeský [17]
obtained that every connected, locally connected graph is upper embeddable.
The following corollary extends this result.
Corollary A graph, which is obtained from a connected, locally connected
graph through a sequence of type-I or type-II vertex splitting operations on
it, is upper embeddable.
Proof According to the result obtained by L. Nebeský [17] we can get that
every connected, locally connected graph is upper embeddable. Combining with
Theorem 2.1 and Theorem 2.2 we can get the Corollary. $\Box$
4\. Conclusions
Remark 1 Let $G$ be an upper embeddable graph with minimum degree at least 3,
$v$ be a vertex of $G$ with deg${}_{G}(v)\geqslant 5$, $G^{*}$ be the graph
obtained from $G$ by splitting $v$ into two adjacent vertices $v^{\prime}$ and
$v^{\prime\prime}$. Then the condition that the $splitting$-$edge$
$v^{\prime}v^{\prime\prime}$ is not a cut-edge of the $v$-$splitting$ subgraph
$G^{*}_{spl}(v)$ can not guarantee the upper embeddability of $G^{*}$. For
example, the graph $G^{*}$ in Fig.24 is a graph obtained from the upper
embeddable graph $G$ in Fig.23 through vertex splitting on $v$ in $G$, and the
$splitting$-$edge$ $v^{\prime}v^{\prime\prime}$ is not a cut-edge of the
$v$-$splitting$ subgraph $G^{*}_{spl}(v)$. But, $G^{*}$ is not upper
embeddable.
$v$Fig.23: $G$$v^{\prime}$$v^{\prime\prime}$Fig.24: $G^{*}$
Remark 2 Let $v_{1}v_{2}$ be an edge of the graph $G$. The $edge$-$global$
$subgraph$ of $v_{1}v_{2}$, which is denoted by $G_{glo}(v_{1}v_{2})$, is the
subgraph of $G$ that is induced by the vertices of $v_{1}$, $v_{2}$ and all
the neighbors of them. The $edge$-$local$ $subgraph$ of $v_{1}v_{2}$, which is
denoted by $G_{loc}(v_{1}v_{2})$, is the subgraph of $G$ that is induced by
all the neighbors of the vertex $v_{1}$ and $v_{2}$. A $flexible$-$edge$ of
graph $G$ is such an edge $v_{1}v_{2}$ of $G$ which satisfies one of the
following two conditions: (I) $v_{1}v_{2}$ is not a cut-edge of the
$edge$-$global$ $subgraph$ of $v_{1}v_{2}$, and the adjacent vertices $v_{1}$,
$v_{2}$ are replaced by a vertex $v$ of degree 4 after contracting the edge
$v_{1}v_{2}$; (II) The $edge$-$local$ $subgraph$ $G_{loc}(v_{1}v_{2})$ of
$v_{1}v_{2}$ is connected, and the adjacent vertices $v_{1}$, $v_{2}$ are
replaced by a vertex $v$ with degree no less than 4 after contracting the edge
$v_{1}v_{2}$. A $flexible$-$weak$-$minor$ of the graph $G$ is a graph obtained
from $G$ through a sequence of edge-contraction operations on the
$flexible$-$edges$.
From Theorem 2.1 and Theorem 2.2 we can get that a graph $G$ is upper
embeddable if and only if its $flexible$-$weak$-$minor$ is upper embeddable.
So the determining of the upper embeddability of $G$ can be replaced by
determining the upper embeddability of its $flexible$-$weak$-$minor$.
Furthermore, the algorithm complexity of determining the upper embeddability
of $G$ may be reduced much by this way, because the order of the
$flexible$-$weak$-$minor$ of $G$ is less than the order of $G$.
## References
* [1] D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.
* [2] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, 2001.
* [3] J.L. Gross, and T.W. Tucker, Topological graph theory. Wiley-Interscience, New York, 1987.
* [4] Y.P. Liu, Embeddability in Graphs, Kluwer Academic, Dordrecht, Boston, London, 1995.
* [5] E.A. Nordhause, B.M. Stewart, A.T. White, On the maximum genus of a graph, J. Combin. Theory, 11 (1971) 258$-$267\.
* [6] C. Thomassen, Bidirectional retracting-free double tracings and upper embeddability of graphs, J. Combin. Theory Ser. B, 50 (1990) 198-207.
* [7] J. Chen, S.P. Kanchi, and J.L. Gross, A tight lower bound on the maximum genus of a simplicial graph, Discrete Math., 156 (1996) 83-102.
* [8] Y.Q. Huang, Y.P. Liu, Face size and the maximum genus of a graph, J. Combin. Theory Ser. B, 80 (2000) 356-370.
* [9] Z.D. Ouyang, L. Tang, and Y.Q. Huang, Upper embeddability, edge independence number and girth, Science China Math., 52(9) (2009) 1939 C1946.
* [10] H. Ren, H.T. Zhao, and H.L. Li, Fundamental cycles and graph embeddings, Science in China Ser. A, 52(9) (2009) 1920-1926.
* [11] Y.C. Chen, Y.P. Liu. Maximum genus, girth and maximum non-adjacent edge set, Ars Combin., 79 (2006) 145 C159.
* [12] D.M. Li and Y.P. Liu, Maximum genus, girth and connectivity, European. J. Combin. 21 (2000) 651-657.
* [13] L. Nebeský, A new characterization of the maximum genus of a graph, Czechoslova Math. J., 31(106) (1981) 604-613.
* [14] M. Škoviera, The maximum genus of graphs diameter two, Discrete Math., 87 (1991) 175$-$180\.
* [15] N.H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory Ser. B, 26 (1979) 217$-$225
* [16] Y.P. Liu, The maximum orientable genus of a graph, Scientia Sinical (Special Issue II) (1979) 41-55.
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|
arxiv-papers
| 2012-03-05T09:30:55 |
2024-09-04T02:49:28.242237
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guanghua Dong, Ning Wang, Yuanqiu Huang, Han Ren, and Yanpei Liu",
"submitter": "Guanghua Dong",
"url": "https://arxiv.org/abs/1203.0840"
}
|
1203.0843
|
11footnotetext: E-mail: gh.dong@163.com(G. Dong); ninglw@163.com(N. Wang);
hyqq@hunnu.edu.cn(Y. Huang); ypliu@bjtu.edu.cn(Y. Liu).
# Joint-tree model and the maximum genus of graphs 222This work was partially
Supported by the China Postdoctoral Science Foundation funded project (Grant
No: 20110491248), the New Century Excellent Talents in University (Grant No:
NCET-07-0276), and the National Natural Science Foundation of China (Grant No:
11171114).
Guanghua Dong1,2, Ning Wang3, Yuanqiu Huang1 and Yanpei Liu4
1.Department of Mathematics, Normal University of Hunan, Changsha, 410081,
China
2.Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387,
China
3.Department of Information Science and Technology, Tianjin University of
Finance
and Economics, Tianjin, 300222, China
4.Department of Mathematics, Beijing Jiaotong University, Beijing 100044,
China
###### Abstract
The vertex $v$ of a graph $G$ is called a 1-$critical$-$vertex$ for the
maximum genus of the graph, or for simplicity called 1-$critical$-$vertex$, if
$G-v$ is a connected graph and $\gamma_{M}(G-v)=\gamma_{M}(G)-1$. In this
paper, through the $joint$-$tree$ model, we obtained some types of
1-$critical$-$vertex$, and get the upper embeddability of the $Spiral$
$S_{m}^{n}$.
Key Words: joint-tree; maximum genus; graph embedding
MSC(2000): 05C10
1\. Introduction
In 1971, Nordhaus, Stewart and White [12] introduced the idea of the maximum
genus of graphs. Since then many researchers have paid attention to this
object and obtained many interesting results, such as the results in [2-8]
[13] [15][17] etc. In this paper, by means of the joint-tree model, which is
originated from the early works of Liu ([8]) and is formally established in
[10] and [11], we offer a method which is different from others to find the
maximum genus of some types of graphs.
Surfaces considered here are compact 2-dimensional manifolds without boundary.
An orientable surface $S$ can be regarded as a polygon with even number of
directed edges such that both $a$ and $a^{-1}$ occurs once on $S$ for each
$a\in S$, where the power “$-1$”means that the direction of $a^{-1}$ is
opposite to that of $a$ on the polygon. For convenience, a polygon is
represented by a linear sequence of lowercase letters. An elementary result in
algebraic topology states that each orientable surface is equivalent to one of
the following standard forms of surfaces:
$O_{p}=\left\\{\begin{array}[]{ll}a_{0}a_{0}^{-1},&\mbox{$p=0$,}\\\
\prod\limits_{i=1}^{p}a_{i}b_{i}a_{i}^{-1}b_{i}^{-1},&\mbox{$p\geq 1$
.}\end{array}\right.$
which are the sphere ($p=0$), torus ($p=1$), and the orientable surfaces of
genus $p\ (p\geq 2)$. The genus of a surface $S$ is denoted by $g(S)$. Let
$A$, $B$, $C$, $D$, and $E$ be possibly empty linear sequence of letters.
Suppose $A=a_{1}a_{2}\dots a_{r},r\geq 1$, then $A^{-1}=a_{r}^{-1}\dots
a_{2}^{-1}a_{1}^{-1}$ is called the $inverse$ of $A$. If
$\\{a,b,a^{-1},b^{-1}\\}$ appear in a sequence with the form as
$AaBbCa^{-1}Db^{-1}E$, then they are said to be an $interlaced$ $set$;
otherwise, a $parallel$ $set$. Let $\widetilde{S}$ be the set of all surfaces.
For a surface $S\in\widetilde{S}$, we obtain its genus $g(S)$ by using the
following transforms to determine its equivalence to one of the standard
forms.
Transform 1 $Aaa^{-1}\sim A$, where $A\in\widetilde{S}$ and $a\notin A$.
Transform 2 $AabBb^{-1}a^{-1}\sim AcBc^{-1}$.
Transform 3 $(Aa)(a^{-1}B)\sim(AB)$.
Transform 4 $AaBbCa^{-1}Db^{-1}E\sim ADCBEaba^{-1}b^{-1}$.
In the above transforms, the parentheses stand for cyclic order. For
convenience, the parentheses are always omitted when unnecessary to
distinguish cyclic or linear order. For more details concerning surfaces, the
reader is referred to [10], [11] and [14].
Let $T$ be a spanning tree of a graph $G=(V,E)$, then $E=E_{T}+E^{*}_{T}$,
where $E_{T}$ consists of all the tree edges, and
$E^{*}_{T}=\\{e_{1},e_{2},\dots e_{\beta}\\}$ consists of all the co-tree
edges, where $\beta=\beta(G)$ is the cycle rank of $G$. Split each co-tree
edge $e_{i}=(\mu_{e_{i}},\nu_{e_{i}})\in E^{*}_{T}$ into two semi-edges
$(\mu_{e_{i}},\omega_{e_{i}})$, $(\nu_{e_{i}},\omega^{\prime}_{e_{i}})$,
denoted by $e_{i}^{+1}$ (or simply by $e_{i}$ if no confusion) and
$e_{i}^{-1}$ respectively. Let $\widetilde{T}=(V+V_{1},E+E_{1})$, where
$V_{1}=\\{\omega_{e_{i}},\ \omega^{\prime}_{e_{i}}\ |\ 1\leqslant
i\leqslant\beta\\}$, $E_{1}=\\{(\mu_{e_{i}},\omega_{e_{i}}),\
(\nu_{e_{i}},\omega^{\prime}_{e_{i}})\ |\ 1\leqslant i\leqslant\beta\\}$.
Obviously, $\widetilde{T}$ is a tree. A $rotation$ $at$ $a$ $vertex$ $v$,
which is denoted by $\sigma_{v}$, is a cyclic permutation of edges incident on
$v$. A rotation system $\sigma=\sigma_{G}$ for a graph $G$ is a set
$\\{\sigma_{v}|\forall v\in V(G)\\}$. The tree $\widetilde{T}$ with a rotation
system of $G$ is called a $joint$-$tree$ of $G$, and is denoted by
$\widetilde{T}_{\sigma}$. Because it ia a tree, it can be embedded in the
plane. By reading the lettered semi-edges of $\widetilde{T}_{\sigma}$ in a
fixed direction (clockwise or anticlockwise), we can get an algebraic
representation of the surface which is represented by a $2\beta-$polygon. Such
a surface, which is denoted by $S_{\sigma}$, is called an associated surface
of $\widetilde{T}_{\sigma}$. A joint-tree $\widetilde{T}_{\sigma}$ of $G$ and
its associated surface is illustrated by Fig.1, where the rotation at each
vertex of $G$ complies with the clockwise rotation. From [10], there is 1-1
correspondence between associated surfaces (or joint-trees) and embeddings of
a graph.
Fig.
1.$G$$e_{1}$$e_{2}$$e_{3}$$\widetilde{T}_{\sigma}$$e_{1}$$e_{1}^{-1}$$e_{2}$$e_{2}^{-1}$$e_{3}$$e_{3}^{-1}$$\omega_{e_{1}}$$\omega^{{}^{\prime}}_{e_{3}}$$\omega^{{}^{\prime}}_{e_{1}}$$\omega_{e_{2}}$$\omega^{{}^{\prime}}_{e_{2}}$$\omega_{e_{3}}$$S_{\sigma}$$e_{1}$$e_{1}^{-1}$$e_{2}$$e_{2}^{-1}$$e_{3}$$e_{3}^{-1}$$\curvearrowright$
To $merge$ a vertex of degree two is that replace its two incident edges with
a single edge joining the other two incident vertices. $Vertex$-$splitting$ is
such an operation as follows. Let $v$ be a vertex of graph $G$. We replace $v$
by two new vertices $v_{1}$ and $v_{2}$. Each edge of $G$ joining $v$ to
another vertex $u$ is replaced by an edge joining $u$ and $v_{1}$, or by an
edge joining $u$ and $v_{2}$. A graph is called a $cactus$ if all circuits are
independent, $i.e.$, pairwise vertex-disjoint. The $maximum$ $genus$
$\gamma_{M}(G)$ of a connected graph _G_ is the maximum integer _k_ such that
there exists an embedding of $G$ into the orientable surface of genus $k$.
Since any embedding must have at least one face, the Euler characteristic for
one face leads to an upper bound on the maximum genus
$\gamma_{M}(G)\leq\lfloor\frac{|E(G)|-|V(G)|+1}{2}\rfloor.$
A graph $G$ is said to be $upper$ $embeddable$ if $\gamma_{M}(\emph{G})$ =
$\lfloor\frac{\beta(G)}{2}\rfloor$, where $\beta(\emph{G})=|E(G)|$ $-$
$|V(G)|$ \+ 1 denotes the _Betti number_ of _G_. Obviously, the maximum genus
of a cactus is zero. The vertex $v$ of a graph $G$ is called a
1-$critical$-$vertex$ for the maximum genus of the graph, or for simplicity
called 1-$critical$-$vertex$, if $G-v$ is a connected graph and
$\gamma_{M}(G-v)=\gamma_{M}(G)-1$. Graphs considered here are all connected,
undirected, and with minimum degree at least three. In addition, the surfaces
are all orientable. Notations and terminologies not defined here can be seen
in [1], [9], [10], and [11].
Lemma 1.0 If there is a joint-tree $\widetilde{T}_{\sigma}$ of $G$ such that
the genus of its associated surface equals $\lfloor\frac{\beta(G)}{2}\rfloor$
then $G$ is upper embeddable.
Proof According to the definition of joint-tree, associated surface, and
upper embeddable graph, Lemma 1.0 can be easily obtained. $\Box$
Lemma 1.1 Let $AB$ be a surface. If $x\notin A\cup B$, then
$g(AxBx^{-1})=g(AB)$ or $g(AxBx^{-1})=g(AB)+1$.
Proof First discuss the topological standard form of the surface $AB$. (I)
According to the left to right direction, let
$\\{x_{1},y_{1},x_{1}^{-1},y_{1}^{-1}\\}$ be the first interlaced set appeared
in $A$. Performing Transform 4 on $\\{x_{1},y_{1},x_{1}^{-1},y_{1}^{-1}\\}$ we
will get $A^{{}^{\prime}}Bx_{1}y_{1}x_{1}^{-1}y_{1}^{-1}$ ($\sim$ $AB$). Then
perform Transform 4 on the first interlaced set in $A^{{}^{\prime}}$. And so
on. Eventually we will get
$\widetilde{A}B\prod\limits_{i=1}^{r}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}$ ($\sim$
$AB$), where there is no interlaced set in $\widetilde{A}$. (II) For the
surface $\widetilde{A}B\prod\limits_{i=1}^{r}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}$,
from the left of $B$, successively perform Transform 4 on $B$ similar to that
on $A$ in (I). Eventually we will get
$\widetilde{A}\widetilde{B}\prod\limits_{i=1}^{r}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}\prod\limits_{j=1}^{s}a_{j}b_{j}a_{j}^{-1}b_{j}^{-1}$
($\sim$ $AB$), where there is no interlaced set in $\widetilde{B}$. (III) For
the surface
$\widetilde{A}\widetilde{B}\prod\limits_{i=1}^{r}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}\prod\limits_{j=1}^{s}a_{j}b_{j}a_{j}^{-1}b_{j}^{-1}$,
from the left of $\widetilde{A}\widetilde{B}$, successively perform Transform
4 on $\widetilde{A}\widetilde{B}$ similar to that on $A$ in (I). At last, we
will get $\prod\limits_{i=1}^{p}a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}$, which is the
topologically standard form of the surface $AB$.
As for the surface $AxBx^{-1}$, perform Transform 4 on $A$ and $B$ similar to
that on $A$ in (I) and $B$ in (II) respectively. Eventually
$\widetilde{A}x\widetilde{B}x^{-1}\prod\limits_{i=1}^{r}x_{i}y_{i}x_{i}^{-1}y_{i}^{-1}\prod\limits_{j=1}^{s}a_{j}b_{j}a_{j}^{-1}b_{j}^{-1}$
($\sim$ $AxBx^{-1}$) will be obtained. Then perform the same Transform 4 on
$\widetilde{A}x\widetilde{B}x^{-1}$ as that on $\widetilde{A}\widetilde{B}$ in
(III), and at last, one more Transform 4 than that in (III) may be needed
because of $x$ and $x^{-1}$ in $\widetilde{A}x\widetilde{B}x^{-1}$. Eventually
$\prod\limits_{i=1}^{p}a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}$ or
$\prod\limits_{i=1}^{p+1}a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}$, which is the
topologically standard form of the surface $AxBx^{-1}$, will be obtained.
From the above, Lemma 1.1 is obtained. $\Box$
Lemma 1.2 Among all orientable surfaces represented by the linear sequence
consisting of $a_{i}$ and $a_{i}^{-1}$ ($i=1,\dots,n$), the surface
$a_{1}a_{2}\dots a_{n}a_{1}^{-1}a_{2}^{-1}\dots a_{n}^{-1}$ is one whose genus
is maximum.
Proof According to Transform 4, Lemma 1.2 can be easily obtained. $\Box$
Lemma 1.3 Let $G$ be a graph with minimum degree at least three, and $\bar{G}$
be the graph obtained from $G$ by a sequence of vertex-splitting, then
$\gamma_{M}(\bar{G})\leq\gamma_{M}(G)$. Furthermore, if $\bar{G}$ is upper
embeddable then $G$ is upper embeddable as well.
Proof Let $v$ be a vertex of degree $n(\geq 4)$ in $G$, and $G^{{}^{\prime}}$
be the graph obtained from $G$ by splitting the vertex $v$ into two vertices
such that both their degrees are at least three. First of all, we prove that
the maximum genus will not increase after one vertex-splitting operation,
$i.e.$, $\gamma_{M}(G^{{}^{\prime}})\leq\gamma_{M}(G)$.
Let $e_{1}$, $e_{2}$, $\dots$ $e_{n}$ be the $n$ edges incident to $v$, and
$v$ be split into $v_{1}$ and $v_{2}$. Without loss of generality, let
$e_{i_{1}}$, $e_{i_{2}}$, $\dots$ $e_{i_{r}}$ be incident to $v_{1}$, and
$e_{i_{r+1}}$, $\dots$ $e_{i_{n}}$ be incident to $v_{2}$, where $2\leq
i_{r}\leq n-2$. Select such a spanning tree $T$ of $G$ that $e_{i_{1}}$ is a
tree edge, and $e_{i_{2}}$, $\dots$ $e_{i_{n}}$ are all co-tree edges. As for
graph $G^{{}^{\prime}}$, select $T^{*}$ be a spanning tree such that both
$e_{i_{1}}$ and $(v_{1},v_{2})$ are tree edges, and the other edges of $T^{*}$
are the same as the edges in $T$. Obviously, $e_{i_{2}}$, $\dots$ $e_{i_{n}}$
are co-tree edges of $T^{*}$. Let
$\mathcal{T}$=$\\{\hat{T}_{\sigma}|\hat{T}_{\sigma}=\overline{(T-v)}_{\sigma},$
where $\overline{(T-v)}_{\sigma}$ is a joint-tree of $G-v$},
$\mathcal{T}^{*}$=$\\{\hat{T}^{*}_{\sigma}|\hat{T}^{*}_{\sigma}=\overline{(T^{*}-\\{v_{1},v_{2}\\})}_{\sigma},$
where $\overline{(T^{*}-\\{v_{1},v_{2}\\})}_{\sigma}$ is a joint-tree of
$G^{{}^{\prime}}-\\{v_{1},v_{2}\\}$}. It is obvious that
$\mathcal{T}=\mathcal{T}^{*}$. Let $\mathcal{S}$ be the set of all the
associated surfaces of the joint-trees of $G$, and $\mathcal{S}^{*}$ be the
set of all the associated surfaces of the joint trees of $G^{{}^{\prime}}$.
Obviously, $\mathcal{S}^{*}\subseteq\mathcal{S}$. Furthermore,
$|\mathcal{S}^{*}|=r!\times(n-r)!\times|\mathcal{T}^{*}|$ $<$
$|\mathcal{S}|=(n-1)!\times|\mathcal{T}|$. So
$\mathcal{S}^{*}\subset\mathcal{S}$, and we have
$\gamma_{M}(G^{{}^{\prime}})\leq\gamma_{M}(G)$.
Reiterating this procedure, we can get that
$\gamma_{M}(\bar{G})\leq\gamma_{M}(G)$. Furthermore, because
$\beta(G)=\beta(\bar{G})$, it can be obtained that if $\bar{G}$ is upper
embeddable then $\lfloor\frac{\beta(G)}{2}\rfloor$ =
$\lfloor\frac{\beta(\bar{G})}{2}\rfloor$ = $\gamma_{M}(\bar{G})$
$\leq\gamma_{M}(G)$ $\leq\lfloor\frac{\beta(G)}{2}\rfloor$. So,
$\gamma_{M}(G)=\lfloor\frac{\beta(G)}{2}\rfloor$, and $G$ is upper embeddable.
$\Box$
2\. Results related to 1-critical-vertex
The $neckband$ $\mathcal{N}_{2n}$ is such a graph that
$\mathcal{N}_{2n}=C_{2n}+R$, where $C_{2n}$ is a 2n-cycle, and
$R=\\{a_{i}|a_{i}=(v_{2i-1},v_{2i+2}).\ (i=1,2,\dots,n,\ 2i+2\equiv r(mod\
2n),\ 1\leq r<2n)\\}$. The $m\ddot{o}bius$ $ladder$ $\mathcal{M}_{2n}$ is such
a cubic circulant graph with 2n vertices, formed from a 2n-cycle by adding
edges (called ”rungs”) connecting opposite pairs of vertices in the cycle. For
example, Fig. 2.1 and Fig. 2.5 is a graph of $\mathcal{N}_{8}$ and
$\mathcal{M}_{2n}$ respectively. A vertex like the solid vertex in Fig. 2.2,
Fig. 2.3, Fig. 2.4, Fig. 2.5, and Fig. 2.6 is called an $\alpha$-$vertex$,
$\beta$-$vertex$, $\gamma$-$vertex$, $\delta$-$vertex$, and $\eta$-$vertex$
respectively, where Fig. 2.6 is a neckband.
$v_{3}$$v_{4}$$v_{2}$$v_{5}$$v_{1}$$v_{6}$$v_{7}$$v_{8}$Fig. 2.1.$a$Fig.
2.2Fig. 2.3$a$$b$
$v_{1}$$v_{2}$$a$$b$Fig.
2.4$v_{1}$$v_{2}$$v_{n}$$v_{n+1}$$v_{n+2}$$v_{2n}$$m$$n$Fig.
2.5$v_{2}$$v_{1}$$v_{3}$$v_{2n}$$v_{4}$$v_{2n-1}$$v_{2n-2}$$v_{2n-3}$$s$$r$Fig.
2.6
Theorem 2.1 If $v$ is an $\alpha$-$vertex$ of a graph $G$, then
$\gamma_{M}(G-v)$ = $\gamma_{M}(G)$. If $v$ is a $\beta$-$vertex$, or a
$\gamma$-$vertex$, or a $\delta$-$vertex$, or an $\eta$-$vertex$ of a graph
$G$, and $G-v$ is a connected graph, then $\gamma_{M}(G-v)$ =
$\gamma_{M}(G)-1$, $i.e.$, $\beta$-$vertex$, $\gamma$-$vertex$,
$\delta$-$vertex$ and $\eta$-$vertex$ are 1-$critical$-$vertex$.
Proof If $v$ is an $\alpha$-$vertex$ of the graph $G$, then it is easy to get
that $\gamma_{M}(G-v)$ = $\gamma_{M}(G)$. In the following, we will discuss
the other cases.
Case 1: $v$ is an $\beta$-$vertex$ of $G$.
According to Fig. 2.3, select such a spanning tree $T$ of $G$ such that both
$a$ and $b$ are co-tree edges. It is obvious that the associated surface for
each joint-tree of $G$ must be one of the following four forms: (i)
$AabBa^{-1}b^{-1}$ $\sim ABaba^{-1}b^{-1}$, (ii) $AabBb^{-1}a^{-1}$ $\sim
AcBc^{-1}$, (iii) $AbaBa^{-1}b^{-1}$ $\sim AcBc^{-1}$, (iv) $AbaBb^{-1}a^{-1}$
$\sim ABbab^{-1}a^{-1}$. On the other hand, for each joint-tree
$\widetilde{T}^{*}_{\sigma}$, which is a joint-tree of $G-v$, its associated
surface must be the form as $AB$, where $A$ and $B$ are the same as that in
the above four forms. According to (i)-(iv), Lemma 1.1, and
$g(ABaba^{-1}b^{-1})$=$g(AB)+1$, we can get that $\gamma_{M}(G-v)$ =
$\gamma_{M}(G)-1$.
Case 2: $v$ is an $\gamma$-$vertex$ of $G$.
As illustrated by Fig. 2.4, both $v_{1}$ and $v_{2}$ are $\gamma$-$vertex$.
Without loss of generality, we only prove that $\gamma_{M}(G-v_{1})$ =
$\gamma_{M}(G)-1$. Select such a spanning tree $T$ of $G$ such that both $a$
and $b$ are co-tree edges. The associated surface for each joint-tree of $G$
must be one of the following 16 forms:
$\begin{array}[]{cccc}Aabb^{-1}a^{-1}B,&Aabb^{-1}Ba^{-1},&Aaba^{-1}Bb^{-1},&AabBa^{-1}b^{-1},\\\
Abab^{-1}a^{-1}B,&Abab^{-1}Ba^{-1},&Abaa^{-1}Bb^{-1},&AbaBa^{-1}b^{-1},\\\
Ab^{-1}a^{-1}Bab,&Ab^{-1}Ba^{-1}ab,&Aa^{-1}Bb^{-1}ab,&ABa^{-1}b^{-1}ab,\\\
Ab^{-1}a^{-1}Bba,&Ab^{-1}Ba^{-1}ba,&Aa^{-1}Bb^{-1}ba,&ABa^{-1}b^{-1}ba.\end{array}$
Furthermore, each of these 16 types of surfaces is topologically equivalent to
one of such surfaces as $AB$, $ABaba^{-1}b^{-1}$, and $AcBc^{-1}$. On the
other hand, for each joint-tree $\widetilde{T}^{*}_{\sigma}$, which is a
joint-tree of $G-v_{1}$, its associated surface must be the form of $AB$,
where $A$ and $B$ are the same as that in the above 16 forms. According to
Lemma 1.1 and $g(ABaba^{-1}b^{-1})$=$g(AB)+1$, we can get that
$\gamma_{M}(G-v)$ = $\gamma_{M}(G)-1$.
Case 3: $v$ is an $\delta$-$vertex$ of $G$.
In Fig. 2.5, let $a_{i}=(v_{i},v_{n+i}),i=1,2,\dots,n.$ Without loss of
generality, we only prove that $\gamma_{M}(G-v_{1})$ = $\gamma_{M}(G)-1$.
Select such a joint-tree $\widetilde{T}_{\sigma}$ of Fig. 2.5, which is
illustrated by Fig.3, where the edges of the spanning tree are represented by
solid line. It is obvious that the associated surface of
$\widetilde{T}_{\sigma}$ is $mnm^{-1}n^{-1}a_{2}a_{3}\dots
a_{n}a_{2}^{-1}a_{3}^{-1}\dots a_{n}^{-1}$. On the other hand,
$a_{2}a_{3}\dots a_{n}a_{2}^{-1}a_{3}^{-1}\dots a_{n}^{-1}$ is the associated
surface of one of the joint-trees of $G-v_{1}$. From Lemma 1.2 and
$g(mnm^{-1}n^{-1}a_{2}a_{3}\dots a_{n}a_{2}^{-1}a_{3}^{-1}\dots
a_{n}^{-1})$=$g(a_{2}a_{3}\dots a_{n}a_{2}^{-1}a_{3}^{-1}\dots a_{n}^{-1})+1$,
we can get that $\gamma_{M}(G-v)$ = $\gamma_{M}(G)-1$.
$v_{2n}$$v_{n+1}$$v_{1}$$v_{2}$$a_{2}$$n^{-1}$$a_{3}$$a_{n}$$a_{2}^{-1}$$a_{n-1}^{-1}$$a_{n}^{-1}$$m$$n$$m^{-1}$Fig.
3.$v_{1}$$v_{2}$$v_{2n-3}$$v_{2n}$$v_{2n-1}$$a_{n}^{-1}$$r$$a_{3}$$a_{2}$$a_{1}^{-1}$$a_{2}^{-1}$$a_{n-2}^{-1}$$a_{n}$$a_{1}$$s^{-1}$$r^{-1}$$s$Fig.
4.
Case 4: $v$ is an $\eta$-$vertex$ of $G$.
As illustrated by Fig. 2.6, every vertex in Fig. 2.6 is a $\eta$-$vertex$.
Without loss of generality, we only prove that $\gamma_{M}(G-v_{2n})$ =
$\gamma_{M}(G)-1$.
A joint-tree $\widetilde{T}_{\sigma}$ of Fig. 2.6 is depicted by Fig.4. It can
be read from Fig.4 that the associated surface of $\widetilde{T}_{\sigma}$ is
$S=a_{1}a_{n}(\prod\limits_{i=1}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{n}^{-1}rsr^{-1}s^{-1}$.
Performing a sequence of Transform 4 on $S$, we have
$\displaystyle S$ $\displaystyle=$ $\displaystyle
a_{1}a_{n}(\prod\limits_{i=1}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{n}^{-1}rsr^{-1}s^{-1}$
(Transform 4) $\displaystyle\sim$
$\displaystyle(\prod\limits_{i=2}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{2}rsr^{-1}s^{-1}a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}$
(Transform 4) $\displaystyle\sim$
$\displaystyle(\prod\limits_{i=4}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{4}rsr^{-1}s^{-1}a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}a_{3}a_{2}a_{3}^{-1}a_{2}^{-1}$
$\displaystyle\cdots$ $\displaystyle\ \ \cdots$ (Transform 4)
$\displaystyle\sim$
$\displaystyle\left\\{\begin{array}[]{ll}rsr^{-1}s^{-1}a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}(\prod\limits_{i=2}^{n-4}a_{i+1}a_{i}a_{i+1}^{-1}a_{i}^{-1})&\
\mbox{$n\equiv 0(mod\ 2)$;}\\\
rsr^{-1}s^{-1}a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}(\prod\limits_{i=2}^{n-3}a_{i+1}a_{i}a_{i+1}^{-1}a_{i}^{-1})&\
\mbox{$n\equiv 1(mod\ 2)$.}\\\ \end{array}\right.$ (3)
It is known from (1) that
$\displaystyle g(S)=\gamma_{M}(G)$ (4)
On the other hand,
$S^{{}^{\prime}}=a_{1}a_{n}(\prod\limits_{i=1}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{n}^{-1}$
is the associated surface of $\widetilde{T}^{*}_{\sigma}$, where
$\widetilde{T}^{*}_{\sigma}$ is a joint-tree of $G-v_{2n}$. Performing a
sequence of Transform 4 on $S^{{}^{\prime}}$, we have
$\displaystyle S^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle
a_{1}a_{n}(\prod\limits_{i=1}^{n-3}a_{i+1}a_{i}^{-1})a_{n-2}^{-1}a_{n}^{-1}$
(7) $\displaystyle\sim$
$\displaystyle\left\\{\begin{array}[]{ll}a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}(\prod\limits_{i=2}^{n-4}a_{i+1}a_{i}a_{i+1}^{-1}a_{i}^{-1})&\
\mbox{$n\equiv 0(mod\ 2)$;}\\\
a_{1}a_{n}a_{1}^{-1}a_{n}^{-1}(\prod\limits_{i=2}^{n-3}a_{i+1}a_{i}a_{i+1}^{-1}a_{i}^{-1})&\
\mbox{$n\equiv 1(mod\ 2)$.}\\\ \end{array}\right.$
It can be inferred from (3) that
$\displaystyle g(S^{{}^{\prime}})=\gamma_{M}(G-v_{2n}).$ (8)
From (1) and (3) we have
$\displaystyle g(S)=g(S^{{}^{\prime}})+1.$ (9)
From (2), (4), and (5) we have $\gamma_{M}(G-v_{2n})$ = $\gamma_{M}(G)-1$.
According to the above, we can get Theorem 2.1. $\Box$
Let $G$ be a connected graph with minimum degree at least 3. The following
algorithm can be used to get the maximum genus of $G$.
Algorithm I Step 1: Input $i=0$, $G_{0}=G$.
Step 2: If there is a 1-$critical$-$vertex$ $v$ in $G_{i}$, then delete $v$
from $G_{i}$ and go to Step 3. Else, go to Step 4.
Step 3: Deleting all the vertices of degree one and merging all the vertices
of degree two in $G_{i}-v$, we get a new graph $G_{i+1}$. Let $i=i+1$, then go
back to Step 2.
Step 4: Output $\gamma_{M}(G)=\gamma_{M}(G_{i})+i$.
Remark Using Algorithm I, the computing of the maximum genus of $G$ can be
reduced to the computing of the maximum genus of $G_{i}$, which may be much
easier than that of $G$.
3\. Upper embeddability of graphs
An $ear$ of a graph $G$, which is the same as the definition offered in [16],
is a path that is maximal with respect to internal vertices having degree 2 in
$G$ and is contained in a cycle in $G$. An $ear$ $decomposition$ of $G$ is a
decomposition $p_{0}$, …, $p_{k}$ such that $p_{0}$ is a cycle and $p_{i}$ for
$i\geqslant 1$ is an ear of $p_{0}\cup\dots\cup p_{i}$. A $spiral$
$\mathcal{S}_{m}^{n}$ is the graph which has an ear decomposition $p_{0}$, …,
$p_{n}$ such that $p_{0}$ is the m-cycle $(v_{1}v_{2}\dots v_{m})$, $p_{i}$
for $1\leqslant i\leqslant m-1$ is the 3-path
$v_{m+2i-2}v_{m+2i-1}v_{m+2i}v_{i}$ which joining $v_{m+2i-2}$ and $v_{i}$,
and $p_{i}$ for $i>m-1$ is the 3-path $v_{m+2i-2}v_{m+2i-1}v_{m+2i}v_{2i-m+1}$
which joining $v_{m+2i-2}$ and $v_{2i-m+1}$. If some edges in
$\mathcal{S}_{m}^{n}$ are replaced by the graph depicted by Fig. 6, then the
graph is called an $extended$-$spiral$, and is denoted by
$\mathcal{\textit{S}}_{m}^{n}$. Obviously, both the vertex $v_{1}$ and $v_{2}$
in Fig. 6 are $\gamma$-vertex. For convenience, a graph of
$\mathcal{S}_{5}^{6}$ is illustrated by Fig.5, and Fig. 7 is the graph which
is obtained from $\mathcal{S}_{5}^{6}$ by replacing the edge $(v_{13},v_{14})$
with the graph depicted by Fig. 6.
$v_{5}$$v_{6}$$v_{7}$$p_{1}$$v_{1}$$v_{8}$$p_{2}$$v_{9}$$v_{10}$$v_{11}$$p_{3}$$v_{12}$$p_{4}$$v_{13}$$v_{14}$$p_{5}$$v_{15}$$v_{16}$$v_{17}$$v_{4}$$p_{4}$$v_{2}$$v_{3}$Fig.
5.$v_{1}$$v_{2}$$v_{3}$$v_{4}$Fig.
6.$v_{5}$$v_{6}$$v_{7}$$p_{1}$$v_{1}$$v_{8}$$p_{2}$$v_{9}$$v_{10}$$v_{11}$$p_{3}$$v_{12}$$p_{4}$$v_{13}$$v_{14}$$p_{5}$$v_{15}$$v_{16}$$v_{17}$$v_{4}$$p_{4}$$v_{2}$$v_{3}$Fig.
7.
Theorem 3.1 The graph $\mathcal{S}_{5}^{n}$ is upper embeddable. Furthermore,
$\gamma_{M}(\mathcal{S}_{5}^{n}-v_{2n+3})$ =
$\gamma_{M}(\mathcal{S}_{5}^{n})-1$, $i.e.,$ $v_{2n+3}$ is a
1-$critical$-$vertex$ of $\mathcal{S}_{5}^{n}$.
Proof According to the definition of $\mathcal{S}_{5}^{n}$, when $n\leq 4$, it
is not a hard work to get the upper embeddability of $\mathcal{S}_{5}^{n}$. So
the following 5 cases will be considered.
Case 1: $n=5j$, where $j$ is an integer no less than 1.
Without loss of generality, a spanning tree $T$ of $\mathcal{S}_{5}^{n}$ can
be chosen as $T=T_{1}\cup T_{2}$, where $T_{1}$ is the path
$v_{2}v_{1}v_{5}v_{4}v_{3}\\{\prod\limits_{i=1}^{j-1}v_{10i+1}v_{10i}v_{10i-1}v_{10i-2}v_{10i-3}v_{10i-4}v_{10i+5}v_{10i+4}v_{10i+3}v_{10i+2}\\}v_{2n+1}$\-
$v_{2n}v_{2n-1}v_{2n-2}v_{2n-3}v_{2n-4}v_{2n+5}v_{2n+4}v_{2n+3}$,
$T_{2}=(v_{2n+1},v_{2n+2})$. Obviously, the $n+1$ co-tree edges of
$\mathcal{S}_{5}^{n}$ with respect to $T$ are $e_{1}=(v_{2},v_{3})$,
$e_{2}=(v_{2},v_{9})$, $e_{3}=(v_{1},v_{7})$,
$\prod\limits_{i=1}^{j-1}\\{e_{5i-1}=(v_{10i-5},v_{10i-4}),e_{5i}=(v_{10i-6},v_{10i+3}),e_{5i+1}=(v_{10i+1},v_{10i+2}),e_{5i+2}=(v_{10i},v_{10i+9}),e_{5i+3}=(v_{10i-2},v_{10i+7})\\}$,
$e_{n-1}=(v_{2n-5},v_{2n-4})$, $e_{n}=(v_{2n-6},v_{2n+3})$,
$e_{n+1}=(v_{2n+2},v_{2n+3})$. Select such a joint-tree
$\widetilde{T}_{\sigma}$ of $\mathcal{S}_{5}^{n}$ which is depicted by Fig.8.
After a sequence of Transform 4, the associated surface $S$ of
$\widetilde{T}_{\sigma}$ has the form as
$\displaystyle S$ $\displaystyle=$ $\displaystyle
e_{1}e_{2}e_{1}^{-1}e_{3}e_{4}e_{5}\\{\prod\limits_{i=1}^{j-2}e_{5i+1}e_{5i+2}e_{5i-3}^{-1}e_{5i+3}e_{5i-2}^{-1}e_{5i-1}^{-1}e_{5i+4}e_{5i+5}e_{5i}^{-1}e_{5i+1}^{-1}\\}$
$\displaystyle
e_{n-4}e_{n-3}e_{n-8}^{-1}e_{n-2}e_{n-7}^{-1}e_{n-6}^{-1}e_{n-1}e_{n-5}^{-1}e_{n-4}^{-1}e_{n-3}^{-1}e_{n-2}^{-1}e_{n-1}^{-1}e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}$
$\displaystyle\sim$
$\displaystyle\prod\limits_{i=1}^{\lfloor\frac{n+1}{2}\rfloor}e_{i1}e_{i2}e_{i1}^{-1}e_{i2}^{-1},$
where $e_{ij}$, $e_{ij}^{-1}$ $\in$
$\\{e_{1},\dots,e_{n+1},e_{1}^{-1},\dots,e_{n+1}^{-1}\\}$;
$i=1,\dots,\lfloor\frac{n+1}{2}\rfloor;j=1,2.$ Obviously,
$g(S)=\lfloor\frac{n+1}{2}\rfloor$. So, when $n=5j$, $\mathcal{S}_{5}^{n}$ is
upper embeddable.
$e_{2}$$e_{1}^{-1}$$v_{2}$$v_{1}$$e_{3}$$v_{5}$$e_{4}$$v_{4}$$e_{5}$$v_{3}$$e_{1}$$v_{11}$$e_{6}$$e_{n-1}$$v_{2n-5}$$e_{n}^{-1}$$v_{2n-6}$$e_{n-5}^{-1}$$v_{2n-7}$$v_{2n-8}$$e_{n-4}^{-1}$$v_{2n+1}$$v_{2n+2}$$e_{n+1}^{-1}$$v_{2n}$$e_{n-3}^{-1}$$v_{2n-1}$$v_{2n-2}$$e_{n-2}^{-1}$$v_{2n-3}$$e_{n-1}^{-1}$$v_{2n-4}$$v_{2n+5}$$v_{2n+4}$$e_{n+1}$$e_{n}$$v_{2n+3}$Fig.
8.
Case 2: $n=5j+1$, where $j$ is an integer no less than 1.
Without loss of generality, select $T=T_{1}\cup T_{2}$ to be a spanning tree
of $\mathcal{S}_{5}^{n}$, where $T_{1}$ is the path
$v_{3}v_{2}v_{1}\\{\prod\limits_{i=1}^{j}v_{10i-3}v_{10i-4}v_{10i-5}v_{10i-6}v_{10i+3}v_{10i+2}v_{10i+1}v_{10i}v_{10i-1}v_{10i-2}\\}v_{2n+5}v_{2n+4}v_{2n+3}$,
$T_{2}=(v_{2n+1},v_{2n+2})$. It is obviously that the $n+1$ co-tree edges of
$\mathcal{S}_{5}^{n}$ with respect to $T$ are $e_{1}=(v_{1},v_{5})$,
$e_{2}=(v_{3},v_{4})$, $e_{3}=(v_{3},v_{11})$, $e_{4}=(v_{2},v_{9})$,
$\prod\limits_{i=1}^{j-1}\\{e_{5i}=(v_{10i-3},v_{10i-2}),e_{5i+1}=(v_{10i-4},v_{10i+5}),e_{5i+2}=(v_{10i+3},v_{10i+4}),e_{5i+3}=(v_{10i+2},v_{10i+11}),e_{5i+4}=(v_{10i},v_{10i+9})\\}$,
$e_{n-1}=(v_{2n-5},v_{2n-4})$, $e_{n}=(v_{2n-6},v_{2n+3})$,
$e_{n+1}=(v_{2n+2},v_{2n+3})$. Similar to Case 1, select a joint tree
$\widetilde{T}_{\sigma}$ of $\mathcal{S}_{5}^{n}$. After a sequence of
Transform 4, the associated surface $S$ of $\widetilde{T}_{\sigma}$ has the
form as
$\displaystyle S$ $\displaystyle=$ $\displaystyle
e_{1}e_{2}e_{3}e_{4}e_{5}e_{6}e_{1}^{-1}e_{2}^{-1}e_{7}e_{8}e_{3}^{-1}e_{9}\\{\prod\limits_{i=1}^{j-2}e_{5i-1}^{-1}e_{5i}^{-1}e_{5i+5}e_{5i+6}e_{5i+1}^{-1}e_{5i+2}^{-1}e_{5i+7}$
$\displaystyle
e_{5i+8}e_{5i+3}^{-1}e_{5i+9}\\}e_{n-7}^{-1}e_{n-6}^{-1}e_{n-1}e_{n-5}^{-1}e_{n-4}^{-1}e_{n-3}^{-1}e_{n-2}^{-1}e_{n-1}^{-1}e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}$
$\displaystyle\sim$
$\displaystyle\prod\limits_{i=1}^{\lfloor\frac{n+1}{2}\rfloor}e_{i1}e_{i2}e_{i1}^{-1}e_{i2}^{-1},$
where $e_{ij}$, $e_{ij}^{-1}$ $\in$
$\\{e_{1},\dots,e_{n+1},e_{1}^{-1},\dots,e_{n+1}^{-1}\\}$;
$i=1,\dots,\lfloor\frac{n+1}{2}\rfloor;j=1,2.$ Obviously,
$g(S)=\lfloor\frac{n+1}{2}\rfloor$. So, when $n=5j+1$, $\mathcal{S}_{5}^{n}$
is upper embeddable.
Case 3: $n=5j+2$, where $j$ is an integer no less than 1.
Without loss of generality, select a spanning tree of $\mathcal{S}_{5}^{n}$ to
be $T=T_{1}\cup T_{2}$, where $T_{1}$ is the path
$v_{1}v_{5}v_{4}v_{3}v_{2}\\{\prod\limits_{i=1}^{j}v_{10i-1}v_{10i-2}v_{10i-3}v_{10i-4}v_{10i+5}v_{10i+4}v_{10i+3}v_{10i+2}v_{10i+1}v_{10i}\\}v_{2n+5}v_{2n+4}v_{2n+3}$,
$T_{2}=(v_{2n+1},v_{2n+2})$. It is obviously that the $n+1$ co-tree edges of
$\mathcal{S}_{5}^{n}$ with respect to $T$ are $e_{1}=(v_{1},v_{2})$,
$e_{2}=(v_{1},v_{7})$, $e_{3}=(v_{5},v_{6})$, $e_{4}=(v_{4},v_{13})$,
$e_{5}=(v_{3},v_{11})$,
$\prod\limits_{i=1}^{j-1}\\{e_{5i+1}=(v_{10i-1},v_{10i}),e_{5i+2}=(v_{10i-2},v_{10i+7}),e_{5i+3}=(v_{10i+5},v_{10i+6}),e_{5i+4}=(v_{10i+4},v_{10i+13}),e_{5i+5}=(v_{10i+2},v_{10i+11})\\}$,
$e_{n-1}=(v_{2n-5},v_{2n-4})$, $e_{n}=(v_{2n-6},v_{2n+3})$,
$e_{n+1}=(v_{2n+2},v_{2n+3})$. Similar to Case 1, select a joint-tree
$\widetilde{T}_{\sigma}$ of $\mathcal{S}_{5}^{n}$. After a sequence of
Transform 4, the associated surface $S$ of $\widetilde{T}_{\sigma}$ has the
form as
$\displaystyle S$ $\displaystyle=$ $\displaystyle
e_{1}e_{2}e_{3}e_{4}e_{5}e_{1}^{-1}e_{6}e_{7}e_{2}^{-1}e_{3}^{-1}e_{8}e_{9}e_{4}^{-1}e_{10}\\{\prod\limits_{i=1}^{j-2}e_{5i}^{-1}e_{5i+1}^{-1}e_{5i+6}e_{5i+7}e_{5i+2}^{-1}e_{5i+3}^{-1}e_{5i+8}$
$\displaystyle
e_{5i+9}e_{5i+4}^{-1}e_{5i+10}\\}e_{n-7}^{-1}e_{n-6}^{-1}e_{n-1}e_{n-5}^{-1}e_{n-4}^{-1}e_{n-3}^{-1}e_{n-2}^{-1}e_{n-1}^{-1}e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}$
$\displaystyle\sim$
$\displaystyle\prod\limits_{i=1}^{\lfloor\frac{n+1}{2}\rfloor}e_{i1}e_{i2}e_{i1}^{-1}e_{i2}^{-1},$
where $e_{ij}$, $e_{ij}^{-1}$ $\in$
$\\{e_{1},\dots,e_{n+1},e_{1}^{-1},\dots,e_{n+1}^{-1}\\}$;
$i=1,\dots,\lfloor\frac{n+1}{2}\rfloor;j=1,2.$ Obviously,
$g(S)=\lfloor\frac{n+1}{2}\rfloor$. So, when $n=5j+2$, $\mathcal{S}_{5}^{n}$
is upper embeddable.
Case 4: $n=5j+3$, where $j$ is an integer no less than 1.
Without loss of generality, a spanning tree $T$ of $\mathcal{S}_{5}^{n}$ can
be chosen as $T=T_{1}\cup T_{2}$, where $T_{1}$ is the path
$v_{2}v_{1}v_{7}v_{6}v_{5}v_{4}v_{3}\\{\prod\limits_{i=1}^{j}v_{10i+1}v_{10i}v_{10i-1}v_{10i-2}v_{10i+7}v_{10i+6}v_{10i+5}v_{10i+4}v_{10i+3}$-$v_{10i+2}\\}v_{2n+5}v_{2n+4}v_{2n+3}$,
$T_{2}=(v_{2n+1},v_{2n+2})$. It is obviously that the $n+1$ co-tree edges of
$\mathcal{S}_{5}^{n}$ with respect to $T$ are $e_{1}=(v_{1},v_{5})$,
$e_{2}=(v_{2},v_{3})$, $e_{3}=(v_{2},v_{9})$,
$\prod\limits_{i=1}^{j}\\{e_{5i-1}=(v_{10i-3},v_{10i-2}),e_{5i}=(v_{10i-4},v_{10i+5}),e_{5i+1}=(v_{10i-6},v_{10i+3}),e_{5i+2}=(v_{10i+1},v_{10i+2}),e_{5i+3}=(v_{10i},v_{10i+9})\\}$,
$e_{n+1}=(v_{2n+2},v_{2n+3})$. Similar to Case 1, select a joint-tree
$\widetilde{T}_{\sigma}$ of $\mathcal{S}_{5}^{n}$. After a sequence of
Transform 4, the associated surface $S$ of $\widetilde{T}_{\sigma}$ has the
form as
$\displaystyle S$ $\displaystyle=$ $\displaystyle
e_{1}e_{2}e_{3}e_{4}e_{5}e_{1}^{-1}e_{6}e_{2}^{-1}\\{\prod\limits_{i=1}^{j-1}e_{5i+2}e_{5i+3}e_{5i-2}^{-1}e_{5i-1}^{-1}e_{5i+4}e_{5i+5}e_{5i}^{-1}e_{5i+6}$
$\displaystyle
e_{5i+1}^{-1}e_{5i+2}^{-1}\\}e_{n-1}e_{n-5}^{-1}e_{n-4}^{-1}e_{n-3}^{-1}e_{n-2}^{-1}e_{n-1}^{-1}e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}$
$\displaystyle\sim$
$\displaystyle\prod\limits_{i=1}^{\lfloor\frac{n+1}{2}\rfloor}e_{i1}e_{i2}e_{i1}^{-1}e_{i2}^{-1},$
where $e_{ij}$, $e_{ij}^{-1}$ $\in$
$\\{e_{1},\dots,e_{n+1},e_{1}^{-1},\dots,e_{n+1}^{-1}\\}$;
$i=1,\dots,\lfloor\frac{n+1}{2}\rfloor;j=1,2.$ Obviously,
$g(S)=\lfloor\frac{n+1}{2}\rfloor$. So, when $n=5j+3$, $\mathcal{S}_{5}^{n}$
is upper embeddable.
Case 5: $n=5j+4$, where $j$ is an integer no less than 1.
Without loss of generality, a spanning tree $T$ of $\mathcal{S}_{5}^{n}$ can
be chosen as $T=T_{1}\cup T_{2}\cup T_{3}$, where $T_{1}$ is the path
$v_{1}v_{2}\\{\prod\limits_{i=1}^{j}v_{10i-1}v_{10i-2}v_{10i-3}v_{10i-4}v_{10i-5}v_{10i-6}v_{10i+3}v_{10i+2}v_{10i+1}v_{10i}\\}v_{2n+1}$-$v_{2n}v_{2n-1}v_{2n-2}v_{2n-3}v_{2n-4}v_{2n+5}v_{2n+4}v_{2n+3}$,
$T_{2}=(v_{2},v_{3})$, $T_{3}=(v_{2n+1},v_{2n+2})$. It is obviously that the
$n+1$ co-tree edges of $\mathcal{S}_{5}^{n}$ with respect to $T$ are
$e_{1}=(v_{1},v_{5})$, $e_{2}=(v_{1},v_{7})$, $e_{3}=(v_{3},v_{4})$,
$e_{4}=(v_{3},v_{11})$,
$\prod\limits_{i=1}^{j}\\{e_{5i}=(v_{10i-1},v_{10i}),e_{5i+1}=(v_{10i-2},v_{10i+7}),e_{5i+2}=(v_{10i-4},v_{10i+5}),e_{5i+3}=(v_{10i+3},v_{10i+4}),e_{5i+4}=(v_{10i+2},v_{10i+11})\\}$,
$e_{n+1}=(v_{2n+2},v_{2n+3})$. Similar to Case 1, select a joint-tree
$\widetilde{T}_{\sigma}$ of $\mathcal{S}_{5}^{n}$. After a sequence of
Transform 4, the associated surface $S$ of $\widetilde{T}_{\sigma}$ has the
form as
$\displaystyle S$ $\displaystyle=$ $\displaystyle
e_{2}e_{1}e_{3}e_{4}e_{5}e_{6}e_{2}^{-1}e_{7}e_{1}^{-1}e_{3}^{-1}\\{\prod\limits_{i=1}^{j-1}e_{5i+3}e_{5i+4}e_{5i-1}^{-1}e_{5i}^{-1}e_{5i+5}e_{5i+6}e_{5i+1}^{-1}$
$\displaystyle
e_{5i+7}e_{5i+2}^{-1}e_{5i+3}^{-1}\\}e_{n-1}e_{n-5}^{-1}e_{n-4}^{-1}e_{n-3}^{-1}e_{n-2}^{-1}e_{n-1}^{-1}e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}$
$\displaystyle\sim$
$\displaystyle\prod\limits_{i=1}^{\lfloor\frac{n+1}{2}\rfloor}e_{i1}e_{i2}e_{i1}^{-1}e_{i2}^{-1},$
where $e_{ij}$, $e_{ij}^{-1}$ $\in$
$\\{e_{1},\dots,e_{n+1},e_{1}^{-1},\dots,e_{n+1}^{-1}\\}$;
$i=1,\dots,\lfloor\frac{n+1}{2}\rfloor;j=1,2.$ Obviously,
$g(S)=\lfloor\frac{n+1}{2}\rfloor$. So, when $n=5j+4$, $\mathcal{S}_{5}^{n}$
is upper embeddable.
From the Case 1-5, the upper embeddability of $\mathcal{S}_{5}^{n}$ can be
obtained.
Similar to the Case 1-5, for each $n\geq 5$, there exists a joint-tree
$\widetilde{T}^{*}_{\sigma}$ of $\mathcal{S}_{5}^{n}-v_{2n+3}$ such that its
associated surface is
$S^{{}^{\prime}}=S-\\{e_{n+1}e_{n}e_{n+1}^{-1}e_{n}^{-1}\\}$. It is obvious
that $S^{{}^{\prime}}$ is the surface into which the embedding of
$\mathcal{S}_{5}^{n}-v_{2n+3}$ is the maximum genus embedding. Furthermore,
$g(S^{{}^{\prime}})=g(S)-1$, $i.e.,$
$\gamma_{M}(\mathcal{S}_{5}^{n}-v_{2n+3})$ =
$\gamma_{M}(\mathcal{S}_{5}^{n})-1$. So, $v_{2n+3}$ is a 1-$critical$-$vertex$
of $\mathcal{S}_{5}^{n}$. $\Box$
Similar to the proof of Theorem 3.1, we can get the following theorem.
Theorem 3.2 The graph $\mathcal{S}_{m}^{n}$ is upper embeddable. Furthermore,
$\gamma_{M}(\mathcal{S}_{m}^{n}-v_{m+2n-2})$ =
$\gamma_{M}(\mathcal{S}_{m}^{n})-1$, $i.e.,$ $v_{m+2n-2}$ is a
1-$critical$-$vertex$ of $\mathcal{S}_{m}^{n}$.
Corollary 3.1 Let $G$ be a graph with minimum degree at least three. If $G$,
through a sequence of vertex-splitting operations, can be turned into a
$spiral$ $\mathcal{S}_{m}^{n}$, then $G$ is upper embeddable.
Proof According to Lemma 1.3, Theorem 3.2, and the upper embeddability of
graphs, Corollary 3.1 can be obtained. $\Box$
In the following, we will offer an algorithm to obtain the maximum genus of
the $extended$-$spiral$ $\mathcal{\textit{S}}_{m}^{n}$.
Algorithm II Step 1: Input $i=0$ and $j=0$. Let $G_{0}$ be the
$extended$-$spiral$ $\mathcal{\textit{S}}_{m}^{n}$.
Step 2: If there is a $\gamma$-vertex $v$ in $G_{i}$, then delete $v$ from
$G_{i}$, and go to Step 3. Else, go to Step 4.
Step 3: Deleting all the vertices of degree one and merging some vertices of
degree two in $G_{i}-v$, we get a new graph $G_{i+1}$. Let $i=i+1$. If $G_{i}$
is a $spiral$ $\mathcal{S}_{m}^{n}$, then go to Step 4. Else, go back to Step
2.
Step 4: Let $G_{i+j}$ be the $spiral$ $\mathcal{S}_{m}^{n}$. Deleting
$v_{m+2n-2}$ from $\mathcal{S}_{m}^{n}$, we will get a new graph $G_{i+j+1}$,
(obviously, $G_{i+j+1}$ is either a $spiral$ $\mathcal{S}_{m}^{n-2}$ or a
$cactus$).
Step 5: If $G_{i+j+1}$ is a $cactus$, then go to Step 6. Else, Let $n=n-2$,
$j=j+1$ and go back to Step 4.
Step 6: Output $\gamma_{M}(\mathcal{\textit{S}}_{m}^{n})=i+j+1$.
Remark 1\. In the graph $G$ depicted by Fig. 6, after deleting a
$\gamma$-vertex $v_{1}$ (or $v_{2}$) from $G$, the vertex $v_{3}$ (or $v_{4}$)
is still a $\gamma$-vertex of the remaining graph.
2\. From Algorithm II we can get that the $extended$-$spiral$
$\mathcal{\textit{S}}_{m}^{n}$ is upper embeddable.
## References
* [1] Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan, London, 1976.
* [2] Cai, J., Dong, G., and Liu, Y.: A suffcient condition on upper embeddability of graphs. Science China Mathematics, 53(5), 1377-1384 (2010).
* [3] Chen, J., Kanchi, S. P., and Gross, J. L.: A tight lower bound on the maximum genus of a simplicial graph. Discrete Math., 156, 83-102 (1996)
* [4] Chen, Y., Liu, Y.: upper embeddability of a Graph by Order and Girth . Graphs and Combinatorics, 23, 521-527 (2007)
* [5] Hao, R., Xu, L., ect., Embeddable Properties of Digraphs in Orientable Surfaces, Acta Mathematicae Applicatae Sinica (Chinese Ser.), 31(4), 630 -634 (2008).
* [6] Huang, Y., Liu, Y.: Face size and the maximum genus of a graph. J Combin. Theory Ser B., 80, 356–370 (2000)
* [7] Li, Z., Ren, H., Maximum Genus Embeddings and Minimum Genus Embeddings in Non-orientable Surfaces, Acta Mathematica Sinica, Chinese Series, 54(2), 329-332, (2011).
* [8] Liu, Y.: The maximum orientable genus of a graph. _Scientia Sinical (Special Issue)_ , (II), 41-55 (1979)
* [9] Liu, Y.: Embeddability in Graphs. Kluwer Academic, Dordrecht, Boston, London, 1995.
* [10] Liu, Y.: Theory of polyhedra. Science Press, Beijing, 2008.
* [11] Liu, Y.: Topological Theory on Graphs, USTC Press, Hefei, 2008.
* [12] Nordhause, E.A., Stewart, B.M., White, A.T.: On the maximum genus of a graph. J. Combin. Theory., 11, 258-267 (1971).
* [13] Ren, H., Li, G.: Survey of maximum genus of graphs, J. East China Normal University(Natural Sc), 5: 1-13 (2010).
* [14] Ringel, G.: Map Color Theorem, Springer, 1974.
* [15] Škoviera, M.: The maximum genus of graphs diameter two. Discrete Math, 87, 175$-$180 (1991)
* [16] West, D.B.: Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 2001.
* [17] Xuong, N.H.: How to determine the maximum genus of a graph. J. Combin. Theory Ser. B., 26 217$-$225 (1979)
|
arxiv-papers
| 2012-03-05T09:46:39 |
2024-09-04T02:49:28.250012
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guanghua Dong, Ning Wang, Yuanqiu Huang, Yanpei Liu",
"submitter": "Guanghua Dong",
"url": "https://arxiv.org/abs/1203.0843"
}
|
1203.0855
|
11footnotetext: E-mail: gh.dong@163.com(G. Dong).
# Lower bound on the number of the maximum genus embedding of $K_{n,n}$
222This work was partially Supported by the China Postdoctoral Science
Foundation funded project (Grant No: 20110491248), the National Natural
Science Foundation of China (Grant No: 11171114), and the New Century
Excellent Talents in University (Grant No: NCET-07-0276).
Guanghua Dong1,2, Han Ren3, Ning Wang4, Yuanqiu Huang1
1.Department of Mathematics, Normal University of Hunan, Changsha, 410081,
China
2.Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387,
China
3.Department of Mathematics, East China Normal University, Shanghai,
200062,China
4.Department of Information Science and Technology, Tianjin University of
Finance
and Economics, Tianjin, 300222, China
###### Abstract
In this paper, we provide an method to obtain the lower bound on the number of
the distinct maximum genus embedding of the complete bipartite graph $K_{n,n}$
($n$ be an odd number), which, in some sense, improves the results of S. Stahl
and H. Ren.
Key Words: graph embedding; maximum genus; v-type-edge
MSC(2000): 05C10
1\. Introduction
Graphs considered here are all connected and finite. A $surface$ $S$ means a
compact and connected two-manifold without boundaries. A $cellular$
$embedding$ of a graph $G$ into a surface $S$ is a one-to-one mapping $\psi:$
$G\rightarrow S$ such that each component of $S-\psi(G)$ is homomorphic to an
open disc. The maximum genus $\gamma_{M}(G)$ of a connected graph _G_ is the
maximum integer _k_ such that there exists an embedding of $G$ into the
orientable surface of genus $k$. By Euler’s polyhedron formula, if a cellular
embedding of a graph $G$ with $n$ vertices, $m$ edges and $r$ faces is on an
orientable surface of genus $\gamma$, the $n-m+r=2-2\gamma$. Since
$\gamma\geqslant 1$, we have
$\gamma(G)\leqslant\frac{1}{2}\lfloor\beta(G)\rfloor$, where $\beta(G)=m-n+1$
is called the $Betti$ $number$ (or $cycle$ $rank$) of the graph $G$. It
follows that $\gamma_{M}(G)\leqslant\frac{1}{2}\lfloor\beta(G)\rfloor$. If
$\gamma_{M}(G)=\frac{1}{2}\lfloor\beta(G)\rfloor$, then the graph is called
$upper$ $embeddable$. It is not difficult to deduced that a graph is upper
embeddable if and only if its face number is not greater than two. Since the
introductory investigations on the maximum genus of graphs by Nordhaus,
Stewart, and White${}^{\cite[cite]{[\@@bibref{}{nor}{}{}]}}$, this parameter
has attracted considerable attention from mathematicians and computer
scientists. Up to now, the research about the maximum genus of graphs mainly
focus on the aspects as characterizations and complexity, the upper
embeddability, the lower bound, the enumeration of the distinct maximum genus
embedding, $etc.$. For more detailed information, the reader can be found in a
survey in [2].
It is well known that the enumeration of the distinct maximum genus embedding
plays an important role in the study of the genus distribution problem, which
may be used to decide whether two given graphs are isomorphic. It was S.
Stahl${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ who provides the first result
about the lower bound on the number of the distinct maximum genus embedding,
which is states as the following:
Lemma 1${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ A connected graph (loops and
multi-edges are allowed) of order $n$ with degree sequence $d_{1}$, $d_{2}$,
$\dots$, $d_{n}$ has at least
$(d_{1}-5)!(d_{2}-5)!(d_{3}-5)!(d_{4}-5)!\prod_{i=5}^{n}(d_{i}-2)!$
distinct orientable embeddings with at most two facial walks, where $m!=1$
whenever $m\leqslant 0$.
But up to now, except [3] and [4], there is little result concerning the
number of the maximum genus embedding of graphs. In this paper, we will
provide a method to enumerate the number of the distinct maximum genus
embedding of the complete bipartite graph $K_{n,n}$ ($n$ be an odd number),
and offer a lower bound which is better than that of S.
Stahl${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ and H.
Ren${}^{\cite[cite]{[\@@bibref{}{ren}{}{}]}}$ in some sence. Furthermore, the
enumerative method below can be used to any maximum genus embedding, other
than the method in [3] which is restricted to upper embeddable graphs.
Terminologies and notations not explained here can be seen in [5] for general
graph theory, and in [6] and [7] for topological graph theory.
2\. Main results
A simple graph $G$ is called a $complete$ $bipartite$ $graph$ if its vertex
set can be partitioned into two subsets $X$ and $Y$ so that every edge has one
end in $X$ and one end in $Y$, and every vertex in $X$ is joined to every
vertex in $Y$. We denote a $complete$ $bipartite$ $graph$ $G$ with bipartition
$X$ and $Y$ by $G_{[X][Y]}$. A 2-$path$ is called a $v$-$type$-$edge$, and is
denoted by $\mathcal{V}$. Let $\psi(G)$ be an embedding of a graph $G$. We say
that a $v$-$type$-$edge$ are inserted into $\psi(G)$ if the three endpoints of
the $v$-$type$-$edge$ are inserted into the corners of the faces in $\psi(G)$,
yielding an embedding of $G+\mathcal{V}$. The embedding $\psi(G)$ of $G$ is
called a $one$-$face$-$embedding$ (or $two$-$face$-$embedding$) if the total
face number of $\psi(G)$ is one (or two). The following observation can be
easily obtained and is essential in the proof of the Theorem A.
Observation Let $\psi(G)$ be an embedding of a graph $G$. We can insert a
$v$-$type$-$edge$ $\mathcal{V}$ to $\psi(G)$ to get an embedding
$\rho(G+\mathcal{V})$ of $G+\mathcal{V}$ so that the face number of
$\rho(G+\mathcal{V})$ is not more than that of $\psi(G)$.
Theorem A For $n\equiv 1\ (mod\ 2)$, the number of the distinct maximum genus
embedding of the complete bipartite graph $K_{n,n}$ is at least
$2^{\frac{n-1}{2}}\times\big{(}(n-2)!!\big{)}^{n}\times\big{(}(n-1)!\big{)}^{n}.$
Proof Let $n=2s+1$ and
$V(K_{n,n})=\\{x_{1},x_{2},\dots,x_{n}\\}\cup\\{y_{1},y_{2},\dots,y_{n}\\}$,
where $X=\\{x_{1},x_{2},\dots,x_{n}\\}$ and $Y=\\{y_{1},y_{2},\dots,y_{n}\\}$
are the two independent set of $K_{n,n}$. We denote the $v$-$type$-$edge$
$y_{2i}x_{j}y_{2i+1}$ by $\mathcal{V}_{ji}$, where $i\in\\{1,2,\dots,s\\}$ and
$j\in\\{1,2,\dots,n\\}$.
$y_{1}$$y_{2}$$y_{3}$$x_{1}$$x_{2}$G${}_{[x_{1},x_{2}][y_{1},y_{2},y_{3}]}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$x_{1}$$x_{2}$G${}_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$x_{1}$$x_{2}$$x_{3}$G${}_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}\cup
x_{3}y_{1}\cup\mathcal{V}_{3,1}$
Claim 1: For $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}$, the number of the
distinct $one$-$face$-$embedding$ is at least $2^{s}\times((2s-1)!!)^{2}$.
There are 2 different ways to embed $G_{[x_{1},x_{2}][y_{1},y_{2},y_{3}]}$ on
an orientable surface so that the embedding is a $one$-$face$-$embedding$.
Select any one of them and denote its face boundary by $W_{0}$. In $W_{0}$,
there are three $face$-$corner$ containing $x_{1}$ and $x_{2}$ respectively.
So, there are 3 different ways to put $\mathcal{V}_{1,2}$ in $W_{0}$, and 3
different ways to put $\mathcal{V}_{2,2}$ in $W_{0}$. Therefore, the total
number of ways to put $\mathcal{V}_{1,2}\cup\mathcal{V}_{2,2}$ in $W_{0}$ is
$3\times 3=9$. For each of the above 9 ways, there are 2 different ways to
make the embedding of $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}$ being a
$one$-$face$-$embedding$. So, for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},y_{3}]}$, there are $3\times 3\times 2$
different ways to add $\mathcal{V}_{1,2}\cup\mathcal{V}_{2,2}$ to
$G_{[x_{1},x_{2}][y_{1},y_{2},y_{3}]}$ to get a $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}$.
Similarly, we can get that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}$, there are $5\times 5\times 2$
different ways to add $\mathcal{V}_{1,3}\cup\mathcal{V}_{2,3}$ to
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{5}]}$ to get a $one$-$face$-$embedding$
of $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{7}]}$.
In general, we have that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{2k-1}]}$, there are
$(2k-1)\times(2k-1)\times 2$ different ways to add
$\mathcal{V}_{1,k}\cup\mathcal{V}_{2,k}$ to
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{2k-1}]}$ to get a
$one$-$face$-$embedding$ of $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{2k+1}]}$.
From the above we can get that the number of the distinct
$one$-$face$-$embedding$ of $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}$ is at
least
$\displaystyle 2\times(3\times 3\times 2)\times(5\times 5\times
2)\times(7\times 7\times 2)\times\dots\times((2s-1)\times(2s-1)\times 2)$
$\displaystyle=2^{s}\times((2s-1)!!)^{2}.$
Claim 2: For each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}$, there are at least
$2\times(2s-1)!!\times 2^{2s}$ different ways to make
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}$ being a
$one$-$face$-$embedding$.
Let $\mathcal{E}_{1}$ be an arbitrary $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}$. In $\mathcal{E}_{1}$, there are
two different $face$-$corner$ containing $y_{i}\ (i=1,2,3)$. So, there are
$2\times 2\times 2(=8)$ different ways to add
$y_{1}x_{3}\cup\mathcal{V}_{3,1}$ to $\mathcal{E}_{1}$ to make
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}$ being a $one$-$face$-$embedding$. For each of
the above 8 $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}$, there are 3 different $face$-$corner$
containing $x_{3}$ and 2 different $face$-$corner$ containing $y_{i}\
(i=4,5)$. So, for each of the above 8 $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}$, there are $3\times 2\times 2$ different ways
to add $\mathcal{V}_{3,2}$ to $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}$ to make
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}\cup\mathcal{V}_{3,2}$ being a
$one$-$face$-$embedding$.
In general, we have that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}\cup\mathcal{V}_{3,2}\cup\dots\cup\mathcal{V}_{3,k-1}$,
there are $(2k-1)\times 2\times 2$ different ways to add $\mathcal{V}_{3,k}$
to $G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}\cup\mathcal{V}_{3,2}\cup\dots\cup\mathcal{V}_{3,k-1}$
to get a $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{3}\cup\mathcal{V}_{3,1}\cup\mathcal{V}_{3,2}\cup\dots\cup\mathcal{V}_{3,k-1}\cup\mathcal{V}_{3,k}$.
From the above we can get that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2}][y_{1},y_{2},\dots,y_{n}]}$, there are at least
$\displaystyle(2\times 2\times 2)\times(3\times 2\times 2)\times(5\times
2\times 2)\times\dots\times((2s-1)\times 2\times 2)$
$\displaystyle=2\times(2s-1)!!\times 2^{2s}$
different ways to make $G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}$
being a $one$-$face$-$embedding$.
Claim 3: For each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}$, there are at least
$3\times(2s-1)!!\times 3^{2s}$ different ways to make
$G_{[x_{1},x_{2},x_{3},x_{4}][y_{1},y_{2},\dots,y_{n}]}$ being a
$one$-$face$-$embedding$.
Let $\mathcal{E}_{2}$ be an arbitrary $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}$. In $\mathcal{E}_{2}$,
there are three different $face$-$corner$ containing $y_{i}\ (i=1,2,3)$. So,
there are $3\times 3\times 3(=27)$ different ways to add
$y_{1}x_{4}\cup\mathcal{V}_{4,1}$ to $\mathcal{E}_{2}$ to make
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}$ being a $one$-$face$-$embedding$. For each of
the above 27 $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}$, there are 3 different $face$-$corner$
containing $x_{4}$ and 3 different $face$-$corner$ containing $y_{i}\
(i=4,5)$. So, for each of the above 27 $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}$, there are $3\times 3\times 3$ different ways
to add $\mathcal{V}_{4,2}$ to
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}$ to make
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}\cup\mathcal{V}_{4,2}$ being a
$one$-$face$-$embedding$.
In general, we have that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}\cup\mathcal{V}_{4,2}\cup\dots\cup\mathcal{V}_{4,k-1}$,
there are $(2k-1)\times 3\times 3$ different ways to add $\mathcal{V}_{4,k}$
to $G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}\cup\mathcal{V}_{4,2}\cup\dots\cup\mathcal{V}_{4,k-1}$
to get a $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}\cup
y_{1}x_{4}\cup\mathcal{V}_{4,1}\cup\mathcal{V}_{4,2}\cup\dots\cup\mathcal{V}_{4,k-1}\cup\mathcal{V}_{4,k}$.
From the above we can get that for each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},x_{3}][y_{1},y_{2},\dots,y_{n}]}$, there are at least
$\displaystyle(3\times 3\times 3)\times(3\times 3\times 3)\times(5\times
3\times 3)\times\dots\times((2s-1)\times 3\times 3)$
$\displaystyle=3\times(2s-1)!!\times 3^{2s}$
different ways to make
$G_{[x_{1},x_{2},x_{3},x_{4}][y_{1},y_{2},\dots,y_{n}]}$ being a
$one$-$face$-$embedding$.
Similarly, we can get the following general result.
Claim 4: For each of the $one$-$face$-$embedding$ of
$G_{[x_{1},x_{2},\dots,x_{k-1}][y_{1},y_{2},\dots,y_{n}]}$, there are at least
$(k-1)\times(2s-1)!!\times(k-1)^{2s}$ different ways to make
$G_{[x_{1},x_{2},\dots,x_{k-1},x_{k}][y_{1},y_{2},\dots,y_{n}]}$ being a
$one$-$face$-$embedding$.
Noticing that a $one$-$face$-$embedding$ of a graph must be its maximum genus
embedding, we can get, from Claim 1 - Claim 4, that the number of the distinct
maximum genus embedding of $K_{n,n}$ is at least
$\displaystyle\\{2^{s}\times((2s-1)!!)^{2}\\}\times\\{2\times(2s-1)!!\times
2^{2s}\\}\times\\{3\times(2s-1)!!$ $\displaystyle\times
3^{2s}\\}\times\dots\times\\{2s\times(2s-1)!!\times(2s)^{2s}\\}$
$\displaystyle=2^{s}\times((2s-1)!!)^{2s+1}\times((2s)!)^{2s+1}$
$\displaystyle=2^{\frac{n-1}{2}}\times((n-2)!!)^{n}\times((n-1)!)^{n}.\hskip
213.39566pt\Box$
Remark Through a comparison we can get that the result in Theorem A is much
better than that of Lemma 1${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ when
$n\leqslant 9$.
In [4], the second author of the present paper obtained that a connected
loopless graph of order $n$ has at least
$\frac{1}{4^{\gamma_{M}(G)}}\prod_{v\in V(G)}(d(v)-1)!$ distinct maximum genus
embedding. Let
$f_{1}(n)=2^{\frac{n-1}{2}}\times\big{(}(n-2)!!\big{)}^{n}\times\big{(}(n-1)!\big{)}^{n}$,
$f_{2}(n)=\frac{1}{4^{\gamma_{M}(G)}}\prod_{v\in
V(G)}\big{(}d(v)-1\big{)}!=\frac{1}{4^{\frac{(n-1)(n-1)}{2}}}\times\big{(}(n-1)!\big{)}^{2n}$.
Through a computation we can get $f_{1}(3)-f_{2}(3)=16$,
$f_{1}(5)-f_{2}(5)=6772211712$. So, when $n\leqslant 5$ the result obtained in
Theorem A is much better than that of [4].
## References
* [1] E. Nordhause, B. Stewart, A. White, On the maximum genus of a graph. J Combin Theory, 11(1971): 151-185.
* [2] L. Beineke, R. Wilson, Topics in topological graph theory. Cambridge University Press, Cambridge, 2009: 34-44.
* [3] S. Stahl, On the number of maximum genus embeddings of almost all graphs. Europ. J. Combinatorics. 13 (1992) 119-126.
* [4] H. Ren and Y. Gao, Lower Bound of the Number of Maximum Genus Embeddings and Genus Embeddings of $K_{12s+7}$. Graphs and Combinatorics. 27-2 (2011) 187-197.
* [5] J. Bondy, U. Murty. Graph Theory[M]. Springer, New York, 2008.
* [6] Y. Liu, Embeddability in Graphs. Dordrecht, Kluwer Academic, Boston and London, (1995).
* [7] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, 2001.
|
arxiv-papers
| 2012-03-05T10:39:20 |
2024-09-04T02:49:28.257927
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guanghua Dong, Han Ren, Ning Wang, Yuanqiu Huang",
"submitter": "Guanghua Dong",
"url": "https://arxiv.org/abs/1203.0855"
}
|
1203.0864
|
11footnotetext: E-mail: gh.dong@163.com(G. Dong); hren@math.ecnu.edu.cn(H.
Ren); ninglw@163.com(N. Wang).
# The extremal genus embedding of graphs 222This work was partially Supported
by the China Postdoctoral Science Foundation funded project (Grant No:
20110491248), the National Natural Science Foundation of China (Grant No:
11171114), and the New Century Excellent Talents in University (Grant No:
NCET-07-0276).
Guanghua Dong1,2, Han Ren3, Ning Wang4, Hao Wu3
1.Department of Mathematics, Normal University of Hunan, Changsha, 410081,
China
2.Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387,
China
3.Department of Mathematics, East China Normal University, Shanghai,
200062,China
4.Department of Information Science and Technology, Tianjin University of
Finance
and Economics, Tianjin, 300222, China
###### Abstract
Let $W_{n}$ be a wheel graph with $n$ spokes. How does the genus change if
adding a degree-3 vertex $v$, which is not in $V(W_{n})$, to the graph
$W_{n}$? In this paper, through the joint-tree model we obtain that the genus
of $W_{n}+v$ equals 0 if the three neighbors of $v$ are in the same face
boundary of $\mathbb{P}(W_{n})$; otherwise, $\gamma(W_{n}+v)=1$, where
$\mathbb{P}(W_{n})$ is the unique planar embedding of $W_{n}$. In addition,
via the independent set, we provide a lower bound on the maximum genus of
graphs, which may be better than both the result of D. Li & Y. Liu and the
result of Z. Ouyang $etc.$ in Europ. J. Combinatorics. Furthermore, we obtain
a relation between the independence number and the maximum genus of graphs,
and provide an algorithm to obtain the lower bound on the number of the
distinct maximum genus embedding of the complete graph $K_{m}$, which, in some
sense, improves the result of Y. Caro and S. Stahl respectively.
Key Words: joint-tree model; genus; maximum genus; independence number
MSC(2000): 05C10
1\. Introduction
Graph considered here are all finite and connected. If the graph $M$ can be
obtained from a graph $G$ by successively contracting edges and deleting edges
and isolated vertices, then $M$ is a $minor$ of $G$. The minimum genus
$\gamma_{min}(G)$ (or, simply, the genus $\gamma(G)$) of a graph $G$ is the
minimum integer $g$ such that there exists an embedding of $G$ into the
orientable surface $S_{g}$ of genus $g$, and the _maximum genus_
$\gamma_{M}(G)$ of a connected graph _G_ is the maximum integer _k_ such that
there exists an embedding of $G$ into the orientable surface of genus $k$. The
difference between the maximum genus and the minimum genus of a graph $G$ is
called the $genus$ $range$ of $G$. A graph $G$ is said to be _upper
embeddable_ if $\gamma_{M}(\emph{G})$ = $\lfloor\frac{\beta(G)}{2}\rfloor$,
where $\beta(G)$ is the $cycle$ $rank$ (or $Betti$ $number$) of $G$. A
$one$-$face$ embedding ($two$-$face$ embedding) $\psi(G)$ of a graph $G$ means
that the face number of $\psi(G)$ is one (two). An $odd$ $vertex$ is a vertex
whose degree is an odd number. For $n\geqslant 3$, the $wheel$ of $n$ spokes
is the graph $W_{n}$ obtained from the $n$-cycle $C_{n}$ by adding a new
vertex (called the $center$ of the $wheel$) and joint it to all vertices of
$C_{n}$. For example, $W_{3}=K_{4}$. A $subdivision$ of an edge $e\in
E(W_{n})$ means inserting a vertex of degree two to $e$, where the inserted
vertex is called a $subdividing$-$vertex$ of $W_{n}$. Let $v$ be a degree-
three vertex which is not in $V(W_{n})$, then the graph $W_{n}+v$, which is
called the $near$-$wheel$ graph, means the connected graph obtained from
$W_{n}$ by joining $v$ to $v_{i}\ (i=1,2,3)$, where $v_{i}$ may be a
$subdividing$-$vertex$ of $W_{n}$ or a vertex which belongs to $V(W_{n})$.
Furthermore, the vertices $v_{1}$, $v_{2}$, $v_{3}$ are called the
$antennal$-$vertex$ of the graph $W_{n}+v$.
Surfaces considered here are compact 2-dimensional manifold without boundary.
An orientable surface $S$ can be regarded as a polygon with even number of
directed edges such that both $a$ and $a^{-}$ occurs once on $S$ for each
$a\in S$, where the power “$-$”means that the direction of $a^{-}$ is opposite
to that of $a$ on the polygon. For convenience, a polygon is represented by a
linear sequence of lowercase letters. An elementary result in algebraic
topology states that each orientable surface is equivalent to one of the
following standard forms of surfaces:
$O_{p}=\left\\{\begin{array}[]{ll}a_{0}a_{0}^{-},&\mbox{$p=0$,}\\\
\prod\limits_{i=1}^{p}a_{i}b_{i}a_{i}^{-}b_{i}^{-},&\mbox{$p\geq 1$
.}\end{array}\right.$
which are the sphere ($p=0$), torus ($p=1$), and the orientable surfaces of
genus $p\ (p\geq 2)$. The genus of a surface $S$ is denoted by $g(S)$. Let
$A$, $B$, $C$, $D$, and $E$ be possibly empty linear sequence of letters.
Suppose $A=a_{1}a_{2}\dots a_{r},r\geq 1$, then $A^{-}=a_{r}^{-}\dots
a_{2}^{-}a_{1}^{-}$ is called the $inverse$ of $A$. If $\\{a,b,a^{-},b^{-}\\}$
appear in a sequence of the form of $AaBbCa^{-}Db^{-}E$, then they are said to
be an $interlaced$ $set$; otherwise, a $parallel$ $set$. Let $\widetilde{S}$
be the set of all surfaces. For a surface $S\in\widetilde{S}$, we obtain its
genus $g(S)$ by using the following transforms to determine its equivalence to
one of the standard forms.
Transform 1 $Aaa^{-}\sim A$, where $A\in\widetilde{S}$ and $a\notin A$.
Transform 2 $AabBb^{-}a^{-}\sim AcBc^{-}$.
Transform 3 $(Aa)(a^{-}B)\sim(AB)$.
Transform 4 $AaBbCa^{-}Db^{-}E\sim ADCBEaba^{-}b^{-}$.
In the above transforms, the parentheses stand for cyclic order. For
convenience, the parentheses are always omitted when unnecessary to
distinguish cyclic or linear order. For more details concerning surfaces, the
reader is referred to [1] and [2].
Let $T$ be a spanning tree of a graph $G=(V,E)$, then $E=E_{T}+\hat{E}_{T}$,
where $E_{T}$ consists of all the tree edges, and
$\hat{E}_{T}=\\{\hat{e}_{1},\hat{e}_{2},\dots\hat{e}_{\beta}\\}$ consists of
all the co-tree edges, where $\beta=\beta(G)$ is the cycle rank of $G$. Split
each co-tree edge $\hat{e}_{i}=(u[\hat{e}_{i}],v[\hat{e}_{i}])\in\hat{E}_{T}$
into two semi-edges $(u[\hat{e}_{i}],v_{i})$, $(v[\hat{e}_{i}],\bar{v}_{i})$,
denoted by $\hat{e}_{i}^{+}$ and $\hat{e}_{i}^{-}$ respectively. Let
$\widetilde{T}=(V+V_{1},E+E_{1})$, where $V_{1}=\\{v_{i},\bar{v}_{i}|1\leq
i\leq\beta\\}$,
$E_{1}=\\{(u[\hat{e}_{i}],v_{i}),(v[\hat{e}_{i}],\bar{v}_{i})|1\leq
i\leq\beta\\}$. Obviously, $\widetilde{T}$ is a tree. A rotation at a vertex
$v$, which is denoted by $\sigma_{v}$, is a cyclic permutation of edges
incident on $v$. A rotation system $\sigma=\sigma_{G}$ for a graph $G$ is a
set $\\{\sigma_{v}|\forall v\in V(G)\\}$. The tree $\widetilde{T}$ with a
rotation system of $G$ is called a $joint$-$tree$ of $G$, and is denoted by
$\widetilde{T}_{\sigma}$. Because $\widetilde{T}_{\sigma}$ is a tree, it can
be embedded in the plane. By reading the lettered semi-edges of
$\widetilde{T}_{\sigma}$ in a fixed direction (clockwise or anticlockwise), we
can get an algebraic representation of the surface which is represented by a
$2\beta-$polygon. Such a surface, which is denoted by $S_{\sigma}$, is called
an associated surface of $\widetilde{T}_{\sigma}$. A joint-tree
$\widetilde{T}_{\sigma}$ of $G$ and its associated surface is illustrated by
Fig.1, where the rotation at each vertex of $G$ complies with the clockwise
rotation. From [1], there is 1-1 correspondence between the associated
surfaces (or joint-trees) and the embeddings of a graph. The $joint$-$tree$ is
originated from the early works of Liu [3], and more detailed information
about the $joint$-$tree$ can be found in [1]. Terminologies and notations not
defined here can be seen in [4] for graph theory and [5] for topological graph
theory.
Fig.
1.$\hat{e}_{1}$$\hat{e}_{2}$$\hat{e}_{3}$$G$$\hat{e}_{1}$$\hat{e}_{1}^{-}$$\hat{e}_{2}$$\hat{e}_{2}^{-}$$\hat{e}_{3}^{-}$$\hat{e}_{3}$$\widetilde{T}_{\sigma}$$\hat{e}_{1}$$\hat{e}_{1}^{-}$$\hat{e}_{2}$$\hat{e}_{2}^{-}$$\hat{e}_{3}^{-}$$\hat{e}_{3}$$\curvearrowright$$S_{\sigma}$
The following lemma is essential in the whole paper.
Lemma 1.1 ${}^{\cite[cite]{[\@@bibref{}{whi}{}{}]}}$ Every simple 3-connected
planar graph has a unique planar embedding.
Lemma 1.2 The minimum genus of a minor of a graph $G$ can never be larger than
$\gamma(G)$.
Proof Let the graph $G$ be embedded in a surface $S$, then contracting an
edge $e$ of $G$ on $S$ can obtain an embedding of the contracted graph $G/e$
on $S$. Moreover, edge deletion can never increase embedding genus. Thus, the
lemma is obtained. $\Box$
Lemma 1.3 If an orientable surface $S$ has the form as $(AxByCx^{-}Dy^{-}E)$,
then $g(S)\geqslant 1$, furthermore, the genus of $S$ is $p(\geqslant 1)$ if,
and only if, $ADCBE$ is with genus $p-1$.
Proof According to the Transform 4, it is obvious. $\Box$
2\. The genus of the near-wheel graphs
It is obvious that $W_{n}$ is 3-connected and $\gamma(W_{n})=0$. So, according
to Lemma 1.1, $W_{n}$ has an unique embedding in the plane. We denote this
unique planar embedding of $W_{n}$ by $\mathbb{P}(W_{n})$.
Lemma 2.1 Let $\mathbb{P}(W_{n})$ be the planar embedding of the wheel $W_{n}$
with $n$ spokes, $v$ be a degree-three vertex which is not in $W_{n}$, then
the genus $\gamma(W_{n}+v)$ of the graph $W_{n}+v$ equals 0 if the three
$antennal$ $vertices$ of $W_{n}+v$ are in the same face boundary of
$\mathbb{P}(W_{n})$.
Proof Let $v_{1}$, $v_{2}$, $v_{3}$ be the three $antennal$ $vertices$ of
$W_{n}+v$, $f_{1}$ be the face of $\mathbb{P}(W_{n})$ with $v_{1}$, $v_{2}$,
$v_{3}$ on it, then we can get a planar embedding of $W_{n}+v$ by placing $v$
in the interior of $f_{1}$ and jointing $vv_{i}\ (i=1,2,3)$. $\Box$
Lemma 2.2 Let $\mathbb{P}(W_{n})$ be the planar embedding of the wheel $W_{n}$
with $n$ spokes, $v$ be a degree-three vertex which is not in $W_{n}$, then
the genus $\gamma(W_{n}+v)$ of the graph $W_{n}+v$ equals 1 if the following
two conditions are satisfied: (i) the three $antennal$-$vertex$ of $W_{n}+v$
are in the boundary of two different faces of $\mathbb{P}(W_{n})$; (ii) there
is no face of $\mathbb{P}(W_{n})$ whose boundary contains all the three
$antennal$-$vertex$.
Proof It is easy to find out that $K_{3,3}$ is a minor of $W_{n}+v$.
According to Lemma 1.2 we can get that $\gamma(W_{n}+v)\geqslant 1$.
Let $v_{1}$, $v_{2}$, $v_{3}$ be the three $antennal$-$vertex$ of $W_{n}+v$.
Because the three $antennal$-$vertex$ of $W_{n}+v$ are in the boundary of two
different faces of $\mathbb{P}(W_{n})$, without loss of generality, we may
assume that $v_{1}$, $v_{2}$ are in the boundary of $f_{1}$, and $v_{3}$ in
$f_{2}$, where $f_{1}$ and $f_{2}$ are two different faces of
$\mathbb{P}(W_{n})$. Putting $v$ in the interior of $f_{1}$ and joining
$vv_{i}\ (i=1,2,3)$, then we will get a torus embedding of $W_{n}+v$ if add a
handle to the plane with the edge $vv_{3}$ on it. So $\gamma(W_{n}+v)\leqslant
1$.
From the above we can get that $\gamma(W_{n}+v)\geqslant 1$ and
$\gamma(W_{n}+v)\leqslant 1$. So $\gamma(W_{n}+v)=1$. $\Box$
Lemma 2.3 Let $\mathbb{P}(W_{n})$ be the planar embedding of the wheel $W_{n}$
with $n$ spokes, $v$ be a degree-three vertex which is not in $W_{n}$, then
the genus $\gamma(W_{n}+v)$ of the graph $W_{n}+v$ equals 1 if any pair of the
three $antennal$-$vertex$ of $W_{n}+v$ are not in a same face boundary of
$\mathbb{P}(W_{n})$.
Proof It is not difficult to find out that $K_{3,3}$ is a minor of $W_{n}+v$.
According to Lemma 1.2 we can get that $\gamma(W_{n}+v)\geqslant 1$.
Case 1: The three $antennal$-$vertex$ of $W_{n}+v$ are all
$subdividing$-$vertex$ of $W_{n}$.
Let $v_{1}$, $v_{2}$, $v_{3}$ be the three $antennal$-$vertex$ of $W_{n}+v$.
For any pair of the three $antennal$-$vertex$ of $W_{n}+v$ are not in a same
face boundary of $\mathbb{P}(W_{n})$, the vertices $v_{1}$, $v_{2}$ and
$v_{3}$ must belong to one of the following two subcases: (1) $v_{1}$, $v_{2}$
and $v_{3}$ are in three different spokes of $W_{n}$, furthermore, any pair of
these three spokes are not in a same face boundary of $\mathbb{P}(W_{n})$; (2)
one of {$v_{1}$, $v_{2}$, $v_{3}$} is on the boundary of the unbounded face of
$\mathbb{P}(W_{n})$, and the other two are in two different spokes of
$\mathbb{P}(W_{n})$, where the two spokes are not on a same face boundary of
$\mathbb{P}(W_{n})$.
$a_{1}$$a_{m-1}$$a_{m}$$a_{m+p}$$a_{m+p+1}$$a_{n}$$v_{1}$$v$$v_{2}$$v_{3}$Fig.2:
$W_{n}+v$$a_{1}$$y$$x$$x^{-}$$a_{m}$$a_{m-1}^{-}$$a_{m-1}$$a_{m-2}^{-}$$a_{2}$$a_{1}^{-}$$a_{m+p}$$a_{m+p-1}^{-}$$a_{m+1}$$a_{m}^{-}$$y^{-}$$a_{m+p}^{-}$$a_{m+p+1}$$a_{m+p+1}^{-}$$a_{m+p+2}$$a_{n}^{-}$$a_{n}$$a_{n-1}^{-}$$v$$v_{1}$$v_{2}$$v_{3}$Fig.3:
$\widetilde{T}_{\sigma}$
In the first subcase, the graph $W_{n}+v$ and one of its joint-tree are shown
in Fig.2 and Fig.3 respectively, where we denoted the edge ($v$, $v_{2}$) by
$x$, and ($v$, $v_{3}$) by $y$. In Fig.2, the edges of the $n$-cycle in
$W_{n}$, according to the clockwise rotation, are denoted by $a_{1}$, $a_{2}$,
$\dots$, $a_{n}$. The surface associated with the joint-tree in Fig.3 is
$\displaystyle S$ $\displaystyle=$ $\displaystyle
a_{1}yxx^{-}a_{m}a_{m-1}^{-}a_{m-1}a_{m-2}^{-}a_{m-2}\dots
a_{2}^{-}a_{2}a_{1}^{-}a_{m+p}a_{m+p-1}^{-}a_{m+p-1}a_{m+p-2}^{-}$
$\displaystyle a_{m+p-2}\dots
a_{m+1}^{-}a_{m+1}a_{m}^{-}y^{-}a_{m+p}^{-}a_{m+p+1}a_{m+p+1}^{-}a_{m+p+2}a_{m+p+2}^{-}\dots
a_{n}a_{n}^{-}$ $\displaystyle\sim$ $\displaystyle
a_{1}ya_{m}a_{1}^{-}a_{m+p}a_{m}^{-}y^{-}a_{m+p}^{-}$ $\displaystyle\sim$
$\displaystyle a_{m+p}a_{m}^{-}a_{m}a_{m+p}^{-}a_{1}ya_{1}^{-}y^{-}$
$\displaystyle\sim$ $\displaystyle a_{1}ya_{1}^{-}y^{-}$
Obviously, $g(S)=1$. So $\gamma(W_{n}+v)\leqslant 1$. On the other hand
$\gamma(W_{n}+v)\geqslant 1$. Therefore, in the first subcase,
$\gamma(W_{n}+v)=1$.
In the second subcase, the graph $W_{n}+v$ and one of its joint-tree are shown
in Fig.4 and Fig.5 respectively, where we denoted the edge ($v$, $v_{2}$) by
$x$, and ($v$, $v_{3}$) by $y$. In Fig.4, the edges of the $n$-cycle in
$W_{n}$, according to the clockwise rotation, are denoted by $a_{1}$, $a_{2}$,
$\dots$, $a_{m-1}$, $b$, $a_{m}$, $\dots$, $a_{n}$. The surface associated
with the joint-tree in Fig.5 is
$\displaystyle S$ $\displaystyle=$ $\displaystyle
a_{1}yxx^{-}a_{m}a_{m-1}^{-}a_{m-1}a_{m-2}^{-}a_{m-2}\dots
a_{2}^{-}a_{2}a_{1}^{-}a_{m+p}a_{m+p-1}^{-}a_{m+p-1}a_{m+p-2}^{-}$
$\displaystyle a_{m+p-2}\dots
a_{m+1}^{-}a_{m+1}a_{m}^{-}y^{-}a_{m+p}^{-}a_{m+p+1}a_{m+p+1}^{-}a_{m+p+2}a_{m+p+2}^{-}\dots
a_{n}a_{n}^{-}$ $\displaystyle\sim$ $\displaystyle
a_{1}ya_{m}a_{1}^{-}a_{m+p}a_{m}^{-}y^{-}a_{m+p}^{-}$ $\displaystyle\sim$
$\displaystyle a_{m+p}a_{m}^{-}a_{m}a_{m+p}^{-}a_{1}ya_{1}^{-}y^{-}$
$\displaystyle\sim$ $\displaystyle a_{1}ya_{1}^{-}y^{-}$
Obviously, $g(S)=1$. So $\gamma(W_{n}+v)\leqslant 1$. On the other hand
$\gamma(W_{n}+v)\geqslant 1$. Therefore, in the second subcase,
$\gamma(W_{n}+v)=1$.
$a_{1}$$a_{m}$$a_{m+1}$$a_{m+p}$$a_{m+p+1}$$a_{n}$$b$$v_{1}$$v$$v_{2}$$v_{3}$Fig.4:
$W_{n}+v$$a_{1}$$y$$x$$x^{-}$$v_{2}$$b$$a_{m}$$a_{m-1}^{-}$$a_{m-1}$$a_{m-2}^{-}$$a_{2}$$a_{1}^{-}$$a_{m+p}$$a_{m+p-1}^{-}$$a_{m+1}$$a_{m}^{-}$$y^{-}$$a_{m+p}^{-}$$a_{m+p+1}$$a_{m+p+1}^{-}$$a_{m+p+2}$$a_{n}^{-}$$a_{n}$$a_{n-1}^{-}$$v$$v_{1}$$v_{3}$Fig.5:
$\widetilde{T}_{\sigma}$
According to the above, we can get that, in the Case 1, $\gamma(W_{n}+v)=1$.
Case 2: The three $antennal$-$vertex$ of $W_{n}+v$ consist of both
$subdividing$-$vertex$ of $W_{n}$ and vertices which belong to $V(W_{n})$.
Because any pair of the three $antennal$-$vertex$ of $W_{n}+v$ are not in a
same face boundary of $\mathbb{P}(W_{n})$, among these three $antennal$
$vertices$, there is one and only one vertex belongs to $V(W_{n})$, and the
other two are both $subdividing$-$vertex$ of $W_{n}$. It is not difficult to
find out that the graph $W_{n}+v$ in Case 2 is minor of the graph $W_{n}+v$ in
Case 1. So, according to Lemma 1.2 we can get that, in Case 2,
$\gamma(W_{n}+v)\leqslant 1$. On the other hand, we can get that
$\gamma(W_{n}+v)\geqslant 1$ because $K_{3,3}$ is a minor of $W_{n}+v$. So, in
the Case 2, $\gamma(W_{n}+v)=1$.
According to the Case 1 and Case 2 we can get the Lemma 2.3. $\Box$
The following theorem can be easily obtained from Lemma 2.1, Lemma 2.2 and
Lemma 2.3.
Theorem A Let $\mathbb{P}(W_{n})$ be the planar embedding of the wheel $W_{n}$
with $n$ spokes, $v$ be a degree-three vertex which is not in $W_{n}$, then
the genus $\gamma(W_{n}+v)$ of the graph $W_{n}+v$ equals 0 if the three
$antennal$-$vertex$ of $W_{n}+v$ are in the same face boundary of
$\mathbb{P}(W_{n})$, otherwise, $\gamma(W_{n}+v)=1$.
Remark (i) From theorem A we can get that there are many planar or toroidal
graphs whose genus range can be arbitrarily large; (ii) How does the genus of
a cubic planar graph $G$ change if we add a degree-three vertex $v$, which is
not in $V(G)$, to $G$? We believe its genus to be 0 or 1. So, the proof or
disproof of the result will be interesting.
3\. Lower bound on the maximum genus of graphs
A set $J\subseteq V(G)$ is called a $non$-$separating$ $independent$ $set$ of
a connected graph $G$ if $J$ is an independent set of $G$ and $G-J$ is
connected. In 1997, through the independent set of a graph, Huang and
Liu${}^{\cite[cite]{[\@@bibref{}{hua}{}{}]}}$ studied the maximum genus of
cubic graphs, and obtained the following result.
Lemma 3.1 ${}^{\cite[cite]{[\@@bibref{}{hua}{}{}]}}$ The maximum genus of a
cubic graph $G$ equals the cardinality of the maximum non-separating
independent set of $G$.
But for general graphs that is not necessary cubic, there is no result
concerning the maximum genus which is characterized by the independent set of
the graph. In the following, we will provide a lower bound of the maximum
genus, which is characterized via the independent set, for general graphs.
Furthermore, there are examples shown that the bound may be tight, and, in
some sense, may be better than the result obtained by Li and
Liu${}^{\cite[cite]{[\@@bibref{}{li}{}{}]}}$, and the result obtained by Z.
Ouyang $etc.^{\cite[cite]{[\@@bibref{}{ou}{}{}]}}$.
Theorem B Let $G$ be a connected graph whose minimum degree is at leas 3. If
$A=\\{v_{1},v_{2},\dots,v_{m}\\}$ is an independent set such that $G-A$ is
connected,then
$\gamma_{M}(G)\geqslant\frac{1}{2}\sum_{i=1}^{m}\big{(}d(v_{i})-\varepsilon_{i}\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},\dots,v_{m}\\}\big{)},$
where for each index $i(1\leqslant i\leqslant m)$, $\varepsilon_{i}=1$ if
$d(v_{i})\equiv 1(mod\ 2)$ and $\varepsilon_{i}=2$ otherwise.
Proof Without loss of generality, let $H$ be the graph obtained from $G$ by
successively deleting $v_{1},v_{2},\dots,v_{m}$ from $G$, and $\psi(H)$ be a
maximum genus embedding of $H$. We first add the vertex $v_{m}$ to $H$.
Case 1: $d_{G}(v_{m})\equiv 1\ (mod\ 2)$.
Without loss of generality, let $d_{G}(v_{m})=2i+1$, and
$x_{1},x_{2},\dots,x_{2i+1}$ be the $2i+1$ neighbors of $v_{m}$ in $G$.
According to the $2i+1$ neighbors of $v_{m}$ are in the same face boundary of
$\psi(H)$ or not, we will discuss in the following two subcases.
Subcase 1.1: All the neighbors of $v_{m}$ are in the same face boundary of
$\psi(H)$.
Let $f_{0}$, which is bounded by $B_{0}$, be the face of $\psi(H)$ that
$x_{1},x_{2},\dots,x_{2i+1}$ are on the boundary of it. Firstly, we put
$v_{m}$ in $f_{0}$ and connect each of {$x_{1},x_{2},x_{3}$} to $v_{m}$, and
denote this resulting graph by $H_{1}$. Through the manner depicted by Fig.7,
where each $vertex$-$rotation$ is the same with that of $\psi(H)$ except
$v_{m}$, we can get an embedding $\psi(H_{1})$ of $H_{1}$ such that its face
number is the same with that of $\psi(H)$. From the equation $V-E+F=2-2g$, it
can be easily deduced that the maximum genus of $H_{1}$ is at least one more
than that of $H$.
Now connect each of {$x_{4},x_{5}$} to $v_{m}$, and denote the resulting graph
by $H_{2}$. Through the manner depicted by Fig.8, we can get an embedding
$\psi(H_{2})$ of $H_{2}$, which has the same face number with that of
$\psi(H)$. From the equation $V-E+F=2-2g$, it can be easily deduced that the
maximum genus of $H_{2}$ is at least two more than that of $H$.
$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{2i+1}$$f_{0}$Fig.6:
$B_{0}$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{2i+1}$$v_{m}$Fig.7:
$\psi(H_{1})$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{2i+1}$$v_{m}$Fig.8:
$\psi(H_{2})$
Similar to the manner of connecting {$x_{4},x_{5}$} to $v_{m}$, we can connect
{$x_{6},x_{7}$}, …, {$x_{2i},x_{2i+1}$} to $v_{m}$. Eventually, we will get an
embedding of $H+v_{m}$. It can be easily deduced that the maximum genus of
$H+v_{m}$ is at least
$\frac{1}{2}\big{(}d(v_{m})-1\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},\dots,v_{m}\\}\big{)}$.
Subcase 1.2: There is no face boundary of $\psi(H)$ containing all the
neighbors of $v_{m}$.
First, add $v_{m}$ to $H$ and connect each of $\\{x_{1},x_{2},x_{3}\\}$ to
$v_{m}$. The resulting graph is denoted by $H_{1}$. If $x_{1},x_{2},x_{3}$ are
in two different face boundaries of $\psi(H)$, say $f_{1}$ and $f_{2}$, then
via the manner depicted by the left part of Fig.9, we can get an embedding
$\psi(H_{1})$ of $H_{1}$ whose face number is the same with that of $\psi(H)$.
If $x_{1},x_{2},x_{3}$ are in three different face boundaries of $\psi(H)$,
say $f_{1}$, $f_{2}$, and $f_{3}$, then through the manner depicted by the
right part of Fig.9, we can get an embedding $\psi(H_{1})$ of $H_{1}$ whose
face number is two less than that of $\psi(H)$. From the equation
$V-E+F=2-2g$, it can be easily deduced that the maximum genus of $H_{1}$ is at
least one more than that of $H$.
$x_{1}$$x_{2}$$f_{2}$$x_{3}$$v_{m}$$f_{1}$Fig.9$v_{m}$$x_{1}$$x_{2}$$x_{3}$$f_{1}$$f_{2}$$f_{3}$
Similarly, connect $\\{x_{4},x_{5}\\}$, …, $\\{x_{2i},x_{2i+1}\\}$ to $v_{m}$.
Eventually, we will get an embedding of $H+v_{m}$, and it can be easily
deduced that the maximum genus of $H+v_{m}$ is at least
$\frac{1}{2}\big{(}d(v_{m})-1\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},\dots,v_{m}\\}\big{)}$.
From Subcase 1.1 and Subcase 1.2 we can get that if $d_{G}(v_{m})=1\ (mod\
2)$, then
$\gamma_{M}(H+v_{m})\geqslant\frac{1}{2}(d(v_{m})-1)+\gamma_{M}(G-\\{v_{1},v_{2},\dots,v_{m}\\})$.
Case 2: $d_{G}(v_{m})\equiv 0\ (mod\ 2)$.
Similar to that of Case 1, we can get that if $d_{G}(v_{m})\equiv 0\ (mod\
2)$, then
$\gamma_{M}(H+v_{m})\geqslant\frac{1}{2}(d(v_{m})-2)+\gamma_{M}(G-\\{v_{1},v_{2},\dots,v_{m}\\})$.
From Case 1 and Case 2 we can get that
$\gamma_{M}(H+v_{m})\geqslant\frac{1}{2}(d(v_{m})-\varepsilon_{i})+\gamma_{M}(H)$,
where $\varepsilon_{i}=1$ if $d(v_{i})\equiv 1(mod\ 2)$ and
$\varepsilon_{i}=2$ otherwise.
Similarly to that of $v_{m}$, we can add $v_{m-1}$, $v_{m-2}$, …, $v_{1}$, one
by one, to $H+v_{m}$. Eventually we will get an embedding of $G$, and it is
not hard to obtain that the maximum genus of $G$ is at least
$\frac{1}{2}\sum_{i=1}^{m}(d(v_{i})-\varepsilon_{i})+\gamma_{M}(G-\\{v_{1},v_{2},\dots,v_{m}\\})$,
where for each index $i(1\leqslant i\leqslant m)$, $\varepsilon_{i}=1$ if
$d(v_{i})\equiv 1(mod\ 2)$ and $\varepsilon_{i}=2$ otherwise. $\Box$
Noticing that the upper embeddability of a graph would not be changed if
adding an odd vertex to it, we can get the following theorem whose proof is
similar to that of Theorem B.
Theorem C Let $G$ be a connected graph and $A_{1},A_{2},\dots A_{s}$ be a
sequence of disjoint independent vertex sets which satisfy: (i) $G_{0}=G$,
$G_{i}=G_{i-1}-A_{i}$ is connected $(i=1,2,\dots,s)$; (ii) each vertex of
$A_{i}$ $(i=1,2,\dots,s)$ is an odd vertex in $G_{i-1}$. Then for
$i=0,1,\dots,s-1$,
$\gamma_{M}(G_{i})\geqslant\frac{1}{2}\sum_{v\in
A_{i+1}}\big{(}d_{G_{i}}(v)-1\big{)}+\gamma_{M}(G_{i+1}).$
In particular, if one of the graph sequence $G_{1},G_{2},\dots,G_{s}$ is upper
embeddable, then $G$ is upper embeddable.
Remark In 2000, through the girth $g$ and connectivity of graphs, D. Li and Y.
Liu${}^{\cite[cite]{[\@@bibref{}{li}{}{}]}}$ obtained the lower bound of the
maximum genus of graphs, which is displayed by the following table, where the
first row and the first column represents the girth and connectivity
respectively.
| $g$=3 | $g$=4 | $g$=5 | $g$=6 | $g$= 7 | $g$= 8 | $g$=9 | $g$=10 | $g$=12
---|---|---|---|---|---|---|---|---|---
1 | $\frac{\beta(G)+2}{4}$ | $\frac{\beta(G)+2}{3}$ | $\frac{2\beta(G)+2}{5}$ | $\frac{3\beta(G)+2}{7}$ | $\frac{5\beta(G)+2}{11}$ | $\frac{7\beta(G)+2}{15}$ | $\frac{14\beta(G)+2}{29}$ | $\frac{17\beta(G)+2}{35}$ | $\frac{31\beta(G)+2}{63}$
2 | $\frac{\beta(G)+2}{3}$ | $\frac{\beta(G)+2}{3}$ | $\frac{2\beta(G)+3}{5}$ | $\frac{3\beta(G)+4}{7}$ | $\frac{6\beta(G)+7}{13}$ | $\frac{7\beta(G)+8}{15}$ | $\frac{14\beta(G)+15}{29}$ | $\frac{17\beta(G)+18}{35}$ | $\frac{31\beta(G)+32}{63}$
3 | $\frac{\beta(G)+2}{3}$ | $\frac{3\beta(G)+4}{7}$ | $\frac{5\beta(G)+6}{11}$ | $\frac{7\beta(G)+8}{15}$ | $\frac{11\beta(G)+12}{23}$ | $\frac{15\beta(G)+16}{31}$ | $\frac{29\beta(G)+30}{59}$ | $\frac{35\beta(G)+36}{71}$ | $\frac{63\beta(G)+64}{127}$
Ten years later, Z. Ouyang, J. Wang and Y.
Huang${}^{\cite[cite]{[\@@bibref{}{ou}{}{}]}}$ studied this parameter too, and
obtained that: Let $G$ be a $k$-$edge$-$connected$ (or $k$-$connected$) simple
graph with minimum degree $\delta$ and girth $g$. Then $\gamma_{M}(G)\geqslant
min\\{f_{k}(\delta,g)(\beta(G)+1),\lfloor\frac{\beta(G)}{2}\rfloor\\}$ for
$k=1,2,3,$ where
$\delta$ | $f_{1}(\delta,g)$ | $f_{2}(\delta,g)$ | $f_{3}(\delta,g)$
---|---|---|---
$\delta=3$ | $\frac{1}{4}$ | $\frac{1}{3}$ | $\frac{1}{2}(1-\frac{1}{4\lceil\frac{(\delta-2)(\delta+g-3)-3}{4}\rceil+3}$)
$\delta\geqslant 4$ | $\frac{1}{2}(1-\frac{3}{4\lceil\frac{(\delta-2)(\delta+g-3)-3}{4}\rceil+1}$) | $\frac{1}{2}(1-\frac{1}{2\lceil\frac{(\delta-2)(\delta+g-3)-3}{4}\rceil+1}$) | $\frac{1}{2}(1-\frac{1}{4\lceil\frac{(\delta-2)(\delta+g-3)-3}{4}\rceil+3}$)
.
There are many examples showing that the lower bound in Theorem B may be best
possible. Furthermore, it may be better than the result obtained by Li and
Liu${}^{\cite[cite]{[\@@bibref{}{li}{}{}]}}$ and the result of Z. Ouyang
$etc.^{\cite[cite]{[\@@bibref{}{ou}{}{}]}}$. The following are two examples
with girth 3 and connectivity 2, and girth 4 and connectivity 3 respectively.
$v_{2}$$v_{1}$girth 3 and
2-connectedFig.10$v_{2}$$v_{1}$$v_{3}$$v_{4}$$v_{5}$girth 4 and 3-connected
In the graph $G$ depicted in the left of Fig.10, let $A=\\{v_{1},v_{2}\\}$.
Then
$\displaystyle\frac{1}{2}\sum_{i=1}^{m}\big{(}d(v_{i})-\varepsilon_{i}\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},\dots,v_{m}\\}\big{)}$
$\displaystyle=\frac{1}{2}\big{(}(3-1)+(3-1)\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2}\\}\big{)}$
$\displaystyle=2+2=4=\gamma_{M}(G)$
Obviously, it is bigger than $\frac{\beta(G)+2}{3}\ (=\frac{10}{3}$), and is
bigger than
$min\\{f_{2}(\delta,g)(\beta(G)+1),\lfloor\frac{\beta(G)}{2}\rfloor\\}\
(=f_{2}(\delta,g)(\beta(G)+1)=3)$.
In the graph $G$ depicted in the right of Fig.10, let
$A=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$. Then
$\displaystyle\frac{1}{2}\sum_{i=1}^{m}\big{(}d(v_{i})-\varepsilon_{i}\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},\dots,v_{m}\\}\big{)}$
$\displaystyle=\frac{1}{2}\big{(}(3-1)\times
5\big{)}+\gamma_{M}\big{(}G-\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}\big{)}$
$\displaystyle=5+0=5=\gamma_{M}(G)$
Obviously, it is bigger than $\frac{3\beta(G)+4}{7}\ (=\frac{34}{7})$, and is
bigger than
$min\\{f_{3}(\delta,g)(\beta(G)+1),\lfloor\frac{\beta(G)}{2}\rfloor\\}\
(=f_{3}(\delta,g)(\beta(G)+1)=\frac{33}{7})$.
4\. Independence number and the maximum genus of graphs
Caro[10] and Wei[11] independently shown that for a graph $G$ its independence
number
$\alpha(G)\geqslant\sum\limits_{v\in V(G)}\frac{1}{d_{G}(v)+1}.$
Later, Alon and Spencer [12] gave an elegant probabilistic proof of this
bound. But, up to now, there is little result concerning the relation between
the independence number and the maximum genus of graphs. Let $N_{G}(v)$ denote
all the neighbors of the vertex $v$ in $G$, the following theorem remedies
this deficiency.
Theorem D Let $G=(V,E)$ be a connected 3-regular graph (loops and multi-edges
are permitted) with $A=\\{x_{1},x_{2},\dots,x_{\gamma_{M}(G)}\\}$ be a maximum
non-separating independent set of $G$. Then its independence number
$\alpha(G)\geqslant\gamma_{M}(G)+\alpha(G-N_{A}),$
where $\alpha(G-N_{A})$ is the independence number of the subgraph $G-N_{A}$
and $N_{A}$ is the $closed$ $closure$ of the set
$N_{G}\\{x_{1},x_{2},\dots,x_{\gamma_{M}(G)}\\}$, $i.e.$,
$N_{A}=\big{(}\bigcup_{i=1}^{\gamma_{M}(G)}N_{G}(x_{i})\big{)}\cup\\{x_{1},x_{2},\dots,x_{\gamma_{M}(G)}\\}$.
Proof From Lemma 3.1 we can get that there exists a maximum non-separating
independent set $A=\\{x_{1},x_{2},\dots,x_{\gamma_{M}(G)}\\}$ which satisfies
$G-A$ is connected. Let $\mathcal{I}$ be an arbitrary independent set of
$G-N_{A}$. It is obvious that every vertex in $A$ is not adjacent to any
vertex in $\mathcal{I}$. So, $A\cup\mathcal{I}$ is an independent set of $G$,
and the theorem is obtained. $\Box$
Remark In the graph $G$ depicted in Fig.11, we may select $A=\\{x_{1}\\}$.
Then $N_{A}=\\{x_{1},x_{2},x_{6}\\}$, and $\alpha(G-N_{A})=2$. Noticing
$\alpha(G)=3$ and $\gamma_{M}(G)=1$, we can get that
$\alpha(G)=\gamma_{M}(G)+\alpha(G-N_{A})=3>\sum_{v\in
V(G)}\frac{1}{d_{G}(v)+1}=\frac{6}{3+1}=\frac{3}{2}$. So, the lower bound in
Theorem D may be best possible, and may be better than that of Caro[10] and
Wei[11] in the case of cubic graphs.
$x_{1}$$x_{2}$$x_{6}$$x_{3}$$x_{5}$$x_{4}$Fig.11
5\. Estimating the number of the maximum genus embedding of $K_{m}$
The enumeration of the distinct maximum genus embedding plays an important
role in the study of the genus distribution problem, which may be used to
decide whether two given graphs are isomorphic. But up to now, except [13] and
[14], there is little result concerning the number of the maximum genus
embedding of graphs. In this section, we will provide an algorithm to
enumerate the number of the distinct maximum genus embedding of the complete
graph $K_{m}$, and offer a lower bound which is better than that of S.
Stahl${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ for $m\leqslant 10$.
Furthermore, the enumerative method below can be used to any maximum genus
embedding, other than the method in [13] which is restricted to upper
embeddable graphs.
A 2-$path$ is called a $\mathcal{V}$-$type$-$edge$, and is denoted by
$\mathcal{V}$. If the $\mathcal{V}$-$type$-$edge$ consists of the 2-$path$
$v_{i}v_{j}v_{k}$, then this $\mathcal{V}$-$type$-$edge$ is denoted by
$\mathcal{V}_{j}^{i,k}$ for simplicity. Let $\psi(G)$ be an embedding of a
graph $G$. We say that a $\mathcal{V}$-$type$-$edge$ are $inserted$ into
$\psi(G)$ if the three endpoints of the $\mathcal{V}$-$type$-$edge$ are
inserted into the corners of the faces in $\psi(G)$, yielding an embedding of
$G+\mathcal{V}$. The following observation can be easily obtained and is
essential in this section.
Observation Let $\psi(G)$ be an embedding of a graph $G$. We can insert a
$\mathcal{V}$-$type$-$edge$ $\mathcal{V}$ to $\psi(G)$ to get an embedding
$\rho(G+\mathcal{V})$ of $G+\mathcal{V}$ so that the face number of
$\rho(G+\mathcal{V})$ is not more than that of $\psi(G)$.
Lemma 5.1 Let $\psi(G)$ be a $one$-$face$ embedding of the graph $G$, $v_{j}$,
$v_{i}$ and $v_{k}$ be vertices of $G$. If the number of the $face$-$corner$
which containing $v_{j}$, $v_{i}$ and $v_{k}$ are $r_{1}$, $r_{2}$ and $r_{3}$
respectively, then there are $r_{1}\times r_{2}\times r_{3}$ different ways to
add the $\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{j}^{i,k}$ to $\psi(G)$ to
get a $one$-$face$ embedding of the graph $G+\mathcal{V}_{j}^{i,k}$.
Proof Let the graph depicted in the middle of Fig.12 denote a $one$-$face$
embedding $\psi(G)$ of the graph $G$. Because the number of the
$face$-$corner$ which containing $v_{j}$, $v_{i}$ and $v_{k}$ are $r_{1}$,
$r_{2}$ and $r_{3}$ respectively, we can insert the
$\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{j}^{i,k}$ into $\psi(G)$ so that
there are $r_{1}$ different ways to put the edges $v_{j}v_{k}$ and
$v_{j}v_{i}$ in the same $face$-$corner$ which containing the vertex $v_{j}$,
$r_{2}$ different ways to put the edge $v_{j}v_{i}$ in a $face$-$corner$ which
containing the vertex $v_{i}$, and $r_{3}$ different ways to put the edge
$v_{j}v_{k}$ in a $face$-$corner$ which containing the vertex $v_{k}$. For any
one of the $r_{1}\times r_{2}\times r_{3}$ different ways to insert the
$\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{j}^{i,k}$ into $\psi(G)$, we can
always get a $one$-$face$ embedding of $G+\mathcal{V}_{j}^{i,k}$ by one and
only one of the two ways which is depicted by the left and right of Fig.12. So
the lemma is obtained. $\Box$
$v_{k}$$v_{i}$$v_{j}$Fig.12$v_{k}$$v_{i}$$v_{j}$$v_{k}$$v_{i}$$v_{j}$
The following algorithm together with Lemma 5.1 provide a maximum genus
embedding of $K_{m}$ and a lower bound of the number of the maximum genus
embedding of $K_{m}$.
Algorithm
Note: Let $V=\\{v_{1},v_{2},\dots,v_{m}\\}$ be the vertex set of the complete
graph $K_{m}$. In the following algorithm, $\forall\
i\in\\{k,a,b\\}\subseteq\\{1,2,\dots,m\\}$, if $i\equiv 0\ (mod\ m)$, then let
$i=m$.
Step 1. Embed the tree $v_{2}v_{3}\dots v_{m}v_{1}$ on the plane.
Step 2. Let $k=1$, $a=2$, $b=3$.
Step 3. If the $one$-$face$ embedding of the complete graph $K_{m}$ is
obtained, then stop. Otherwise, go to Step 4.
Step 4. If there are only two vertices $v_{k}$ and $v_{a}$ that are not
adjacent, then connect them to get a $two$-$face$ embedding of the complete
graph $K_{m}$ and stop. Otherwise, go to Step 5.
Step 5. If there is no edge connecting the vertex $v_{k}$ and $v_{a}$ then go
to Step 6. Otherwise, go to Step 10.
Step 6. If any pair of $\\{v_{k},v_{a},v_{b}\\}$ are not the same, and there
is no edge connecting the vertex $v_{k}$ and $v_{b}$ then add the
$\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{k}^{a,b}$ to the graph to get a
$one$-$face$ embedding and go to Step 9. Otherwise, let $b\equiv b+1\ (mod\
m)$ and go to Step 7.
Step 7. If $b\equiv k-1\ (mod\ m)$ then go to Step 8. Otherwise, go back to
Step 6.
Step 8. Let $c=k$, $k=a$, $a=c$ ($i.e.,$ exchange $k$ and $a$). Then go back
to Step 3.
Step 9. Let $b\equiv a+3\ (mod\ m)$, $a\equiv a+2\ (mod\ m)$, and go to Step
11.
Step 10. Let $a\equiv a+1\ (mod\ m)$, and go to Step 11.
Step 11. If $a\equiv k-1\ (mod\ m)$, then go to Step 12. Otherwise, go back to
Step 3.
Step 12. Let $k=1$, and go to Step 13.
Step 13. If $d_{G}(v_{k})<m-1$, then let $a\equiv k+2\ (mod\ m)$, $b\equiv
k+3\ (mod\ m)$, and go back to Step 3. Otherwise, go to Step 14.
Step 14. If $d_{G}(v_{k})=m-1$, then let $k=k+1$, and go back to Step 13.
Using the above algorithm, we can get the maximum genus embedding of $K_{m}$
except that $m=1+8i$ or $m=6+8i$ ($i=0,1,3,\dots$). Furthermore, for
$m\leqslant 10$, our result is much better than that of
Stahl${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$. For simplicity, we give some
symbols which are used below. Let $E$ be a $one$-$face$ embedding of a graph.
Then the symbol ($\mathcal{V}_{j}^{i,k}:r_{1}\times r_{2}\times r_{3}$) means
that there are $r_{1}\times r_{2}\times r_{3}$ different ways to add the
$\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{j}^{i,k}$ to $E$ to get a
$one$-$face$ embedding of $E+\mathcal{V}_{j}^{i,k}$, and the symbol
($e_{j}^{j,k}:r_{1}\times r_{2}$) means that there are $r_{1}\times r_{2}$
different ways to add the edge $v_{j}v_{k}$ to $E$ to get a $two$-$face$
embedding of $E+v_{j}v_{k}$.
Result 1 The number of the maximum genus embedding of the complete graph
$K_{8}$ is at least $2^{26}\times 3^{11}\times 5^{5}$.
Proof Let $V=\\{v_{1},v_{2},\dots,v_{8}\\}$ be the vertex set of the complete
graph $K_{8}$. There is only one way to embed the tree $T=v_{2}v_{3}\dots
v_{8}v_{1}$ on the plane, which is a $one$-$face$ embedding, and is denoted by
$\mathcal{E}_{1}$. In $\mathcal{E}_{1}$, the number of the $face$-$corner$
which containing the vertex $v_{1}$, $v_{2}$, $v_{3}$ is 1, 1 and 2
respectively. So, according to Lemma 5.1, there are 2 different ways to add
the $\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{1}^{2,3}$ to $\mathcal{E}_{1}$
to get a $one$-$face$ embedding of $T+\mathcal{V}_{1}^{2,3}$. Let
$\mathcal{E}_{2}$ be any one of the $one$-$face$ embedding of
$T+\mathcal{V}_{1}^{2,3}$. In $\mathcal{E}_{2}$, the number of the
$face$-$corner$ which containing the vertex $v_{1}$, $v_{4}$, $v_{5}$ is 3, 2
and 2 respectively. So, according to Lemma 5.1, there are $3\times 2\times 2\
(=12)$ different ways to add the $\mathcal{V}$-$type$-$edge$
$\mathcal{V}_{1}^{4,5}$ to $\mathcal{E}_{2}$ to get a $one$-$face$ embedding
of $T+\mathcal{V}_{1}^{2,3}+\mathcal{V}_{1}^{4,5}$. Similarly, we can get that
for each of the $one$-$face$ embedding of
$T+\mathcal{V}_{1}^{2,3}+\mathcal{V}_{1}^{4,5}$, there are $5\times 2\times 2$
different ways to add the $\mathcal{V}$-$type$-$edge$ $\mathcal{V}_{1}^{6,7}$
to $T+\mathcal{V}_{1}^{2,3}+\mathcal{V}_{1}^{4,5}$ to get a $one$-$face$
embedding of
$T+\mathcal{V}_{1}^{2,3}+\mathcal{V}_{1}^{4,5}+\mathcal{V}_{1}^{6,7}$.
Similarly, we can add $\mathcal{V}$-$type$-$edges$, one by one in the
following order, to
$T+\mathcal{V}_{1}^{2,3}+\mathcal{V}_{1}^{4,5}+\mathcal{V}_{1}^{6,7}$ to get a
$two$-$face$ embedding of $K_{8}$ eventually.
($\mathcal{V}_{2}^{4,5}:2\times 3\times 3$), ($\mathcal{V}_{2}^{6,7}:4\times
3\times 3$), ($\mathcal{V}_{8}^{2,3}:2\times 6\times 3$),
($\mathcal{V}_{8}^{4,5}:4\times 4\times 4$), ($\mathcal{V}_{6}^{8,3}:4\times
6\times 4$), ($\mathcal{V}_{4}^{6,7}:5\times 6\times 4$),
($\mathcal{V}_{3}^{5,7}:5\times 5\times 5$), ($e_{5}^{5,7}:6\times 6$).
So, the number of the distinct maximum genus embedding of $K_{8}$ is at least
$\displaystyle 2\times(3\times 2\times 2)\times(5\times 2\times
2)\times(2\times 3\times 3)\times(4\times 3\times 3)\times(2\times 6\times 3)$
$\displaystyle\times(4\times 4\times 4)\times(4\times 6\times 4)\times(5\times
6\times 4)\times(5\times 5\times 5)\times(6\times 6)$
$\displaystyle=2^{26}\times 3^{11}\times 5^{5}$
Result 2 The number of the distinct maximum genus embedding of the complete
graph $K_{10}$ is at least $2^{52}\times 3^{15}\times 5^{7}\times 7^{6}$,
which is obtained from the unique $one$-$face$ embedding of the tree
$T=v_{2}v_{3}\dots v_{10}v_{1}$ by successively adding the following
$\mathcal{V}$-$type$-$edges$: ($\mathcal{V}_{1}^{2,3}:1\times 1\times 2$),
($\mathcal{V}_{1}^{4,5}:3\times 2\times 2$), ($\mathcal{V}_{1}^{6,7}:5\times
2\times 2$), ($\mathcal{V}_{1}^{8,9}:7\times 2\times 2$),
($\mathcal{V}_{2}^{4,5}:2\times 3\times 3$), ($\mathcal{V}_{2}^{6,7}:4\times
3\times 3$), ($\mathcal{V}_{2}^{8,9}:6\times 3\times 3$),
($\mathcal{V}_{10}^{2,3}:2\times 8\times 3$), ($\mathcal{V}_{10}^{4,5}:4\times
4\times 4$), ($\mathcal{V}_{10}^{6,7}:6\times 4\times 4$),
($\mathcal{V}_{8}^{10,3}:4\times 8\times 4$), ($\mathcal{V}_{8}^{4,5}:6\times
5\times 5$), ($\mathcal{V}_{6}^{8,9}:5\times 8\times 4$),
($\mathcal{V}_{6}^{3,4}:7\times 5\times 6$), ($\mathcal{V}_{3}^{5,7}:6\times
6\times 5$), ($\mathcal{V}_{9}^{3,4}:5\times 8\times 7$),
($\mathcal{V}_{9}^{5,7}:7\times 7\times 6$), ($\mathcal{V}_{7}^{4,5}:7\times
8\times 8$).
Result 3 The number of the distinct maximum genus embedding of the complete
graph $K_{7}$ is at least 49766400000, which is obtained from the unique
$one$-$face$ embedding of the tree $T=v_{2}v_{3}\dots v_{7}v_{1}$ by
successively adding the following $\mathcal{V}$-$type$-$edges$:
($\mathcal{V}_{1}^{2,3}:1\times 1\times 2$), ($\mathcal{V}_{1}^{4,5}:3\times
2\times 2$), ($\mathcal{V}_{6}^{1,2}:2\times 5\times 2$),
($\mathcal{V}_{6}^{3,4}:4\times 3\times 3$), ($\mathcal{V}_{2}^{4,5}:3\times
4\times 3$), ($\mathcal{V}_{7}^{2,3}:2\times 5\times 4$),
($\mathcal{V}_{7}^{4,5}:4\times 5\times 4$), ($e_{3}^{3,5}:5\times 5$).
Result 4 The number of the distinct maximum genus embedding of the complete
graph $K_{5}$ is at least 432, which is obtained from the unique $one$-$face$
embedding of the tree $T=v_{2}v_{3}v_{4}v_{5}v_{1}$ by successively adding the
following $\mathcal{V}$-$type$-$edges$: ($\mathcal{V}_{1}^{2,3}:1\times
1\times 2$), ($\mathcal{V}_{4}^{1,2}:2\times 3\times 2$),
($\mathcal{V}_{5}^{2,3}:2\times 3\times 3$).
The algorithm doesn’t work for $K_{6}$ and $K_{9}$. But the maximum genus
embedding of $K_{6}$ and $K_{9}$ can be obtained by the following manners.
Result 5 The number of the distinct maximum genus embedding of the complete
graph $K_{6}$ is at least 663552, which is obtained from the unique
$one$-$face$ embedding of the tree $T=v_{2}v_{3}\dots v_{6}v_{1}$ by
successively adding the following $\mathcal{V}$-$type$-$edges$:
($\mathcal{V}_{1}^{2,3}:1\times 1\times 2$), ($\mathcal{V}_{1}^{4,5}:3\times
2\times 2$), ($\mathcal{V}_{2}^{4,5}:2\times 3\times 3$),
($\mathcal{V}_{6}^{2,4}:2\times 4\times 4$), ($\mathcal{V}_{3}^{5,6}:3\times
4\times 4$).
Result 6 The number of the distinct maximum genus embedding of the complete
graph $K_{9}$ is at least $2^{27}\times 3^{12}\times 5^{7}\times 7^{6}$, which
is obtained from the unique $one$-$face$ embedding of the tree
$T=v_{2}v_{3}\dots v_{9}v_{1}$ by successively adding the following
$\mathcal{V}$-$type$-$edges$: ($\mathcal{V}_{1}^{2,3}:1\times 1\times 2$),
($\mathcal{V}_{1}^{4,5}:3\times 2\times 2$), ($\mathcal{V}_{1}^{6,7}:5\times
2\times 2$), ($\mathcal{V}_{8}^{1,2}:2\times 7\times 2$),
($\mathcal{V}_{8}^{3,4}:4\times 3\times 3$), ($\mathcal{V}_{8}^{5,6}:6\times
3\times 3$), ($\mathcal{V}_{2}^{4,5}:3\times 4\times 4$),
($\mathcal{V}_{2}^{6,7}:5\times 4\times 3$), ($\mathcal{V}_{9}^{2,3}:2\times
7\times 4$), ($\mathcal{V}_{9}^{4,5}:4\times 5\times 5$),
($\mathcal{V}_{9}^{6,7}:6\times 5\times 4$), ($\mathcal{V}_{3}^{5,6}:5\times
6\times 6$), ($\mathcal{V}_{7}^{3,5}:5\times 7\times 7$),
($\mathcal{V}_{4}^{6,7}:6\times 7\times 7$).
Remark Saul Stahl${}^{\cite[cite]{[\@@bibref{}{sta}{}{}]}}$ obtained that the
complete graph $K_{m}$ on $m$ vertices has at least
$[(m-6)!]^{4}[(m-3)!]^{m-4}$ maximum genus embeddings, and for $m\equiv 0,3\
(mod\ 4)$ $K_{m}$ has at least $(\frac{m-2}{m-1})^{2}[(m-3)!]^{m}$ maximum
genus embeddings. It is obvious that our results for $m\leqslant 10$ is much
better than that of Stahl.
$\bf{Acknowledgements}$ The authors thank the referees for their careful
reading of the paper, and for their valuable comments.
## References
* [1] Y. Liu, Theory of Polyhedra, Science Press, Beijing, 2008.
* [2] G. Ringel, Map Color Theorem, Springer, 1974.
* [3] Y. Liu, The maximum orientable genus of a graph, _Scientia Sinical (Special Issue)_ , II(1979) 41-55.
* [4] J. Bondy, U. Murty. Graph Theory[M]. Springer, New York, 2008.
* [5] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, 2001.
* [6] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933) 245-254.
* [7] Y. Huang and Y. Liu, Maximum genus and maximum nonseparating independent set of a 3-regular graph, Discrete Math. 176 (1997) 149-158.
* [8] D. Li, and Y. Liu, Maximum genus, girth and connectivity, Europ. J. Combinatorics. 21 (2000) 651-657.
* [9] Z. Ouyang, J. Wang and Y. Huang, On the lower bounds for the maximum genus for simple graphs, Europ. J. Combinatorics. 31-5 (2010) 1235-1242.
* [10] Y. Caro, New results on the independence number, Technical Report, Tel Aviv University, 1979.
* [11] V. Wei, A lower bound on the stability number of a simple graph, Bell Laboratories TM, 81-11217-9 (1981).
* [12] N. Alon and J. Spencer, The probabilistic Method, Wiley, New York, 1992\.
* [13] S. Stahl, On the number of maximum genus embeddings of almost all graphs. Europ. J. Combinatorics. 13 (1992) 119-126.
* [14] H. Ren and Y. Gao, Lower Bound of the Number of Maximum Genus Embeddings and Genus Embeddings of $K_{12s+7}$. Graphs and Combinatorics. 27-2 (2011) 187-197.
|
arxiv-papers
| 2012-03-05T11:20:00 |
2024-09-04T02:49:28.263387
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guanghua Dong, Han Ren, Ning Wang, Hao Wu",
"submitter": "Guanghua Dong",
"url": "https://arxiv.org/abs/1203.0864"
}
|
1203.0921
|
# Quark charge balance function and hadronization effects in relativistic
heavy ion collisions
Jun Song Department of Physics, Jining University, Jining, Shandong 273155,
China School of Physics, Shandong University, Jinan, Shandong 250100, China
Feng-lan Shao Department of Physics, Qufu Normal University, Shandong 273165,
China Zuo-tang Liang School of Physics, Shandong University, Jinan, Shandong
250100, China
###### Abstract
We calculate the charge balance function of the bulk quark system before
hadronization and those for the directly produced and the final hadron system
in high energy heavy ion collisions. We use the covariance coefficient to
describe the strength of the correlation between the momentum of the quark and
that of the anti-quark if they are produced in a pair and fix the parameter by
comparing the results for hadrons with the available data. We study the
hadronization effects and decay contributions by comparing the results for
hadrons with those for the bulk quark system. Our results show that while
hadronization via quark combination mechanism slightly increases the width of
the charge balance functions, it preserves the main features of these
functions such as the longitudinal boost invariance and scaling properties in
rapidity space. The influence from resonance decays on the width of the
balance function is more significant but it does not destroy its boost
invariance and scaling properties in rapidity space either. The balance
functions in azimuthal direction are also presented.
###### pacs:
25.75.Dw, 25.75.Gz, 25.75.Nq, 25.75.-q
## I introduction
The electric charge balance function for the final state hadrons has been
proposed as a probe to study the properties of the bulk matter system produced
in relativistic heavy ion collisions Bass_clockHd ; SJeon02 ; Bialas04 ;
ChengS04 ; Bozek05StatResonance ; LNbf09 ; Pratt11_bf . Measurements have
already carried out both in the rapidity space StarBF130 ; NA49SCden ;
NA49REden and in the azimuthal directionStarBF200AApp . From the data now
availableStarBF130 ; StarBF200AApp , we are already able to see clearly that
the charge balance functions for hadrons produced in high energy heavy ion
collisions are significantly narrower than those for $pp$ collisions at the
same energies and they are narrower for central collisions than those for
peripheral collisions, indicating a strong local charge compensation in the
bulk quark matter system produced in heavy ion collisions. The data
StarBF200Scaling further show that the charge balance functions have the
longitudinal boost invariance and scaling properties in the rapidity space,
and these properties hold for either transverse momentum $p_{T}$-integrated
balance functions or those for different $p_{T}$ ranges.
These features of the experimental dataStarBF130 ; StarBF200AApp ;
StarBF200Scaling are rather striking and suggest that such studies should be
able to give more insights to the understanding of the properties of the bulk
quark matter system produced in $AA$ collisions. It is thus natural to ask
whether such behavior hold also for the quark anti-quark system before
hadronization. It is also important to see how large the influence from the
hadronization and resonance decay.
In this paper, we propose a simple working model to calculate the charge
balance function for the bulk quark anti-quark system before hadronization. We
introduce the variance coefficient $\rho$ to describe the local correlation in
the momentum distribution for the quark and that for the anti-quark if they
are produced in a pair. The parameter $\rho$ measures the strength of the
quark-anti-quark momentum correlation produced in the processes. We study the
influence due to hadronization process including the contributions due to
resonance decay by simulating the hadronization process using a quark
combination model which describe the final hadron distributions.
The paper is organized as follows. In Sec. II, we study the charge balance of
the quark system before hadronization. In Sec. III, we study the charge
balance function of initial hadron system as well as final hadron system, and
compare them with that of quark system. Sec. IV gives a brief summary.
## II Charge balance function of the system of quarks and anti-quarks
We recall that , the balance function is in general defined as Bass_clockHd ,
$B(\Delta_{2}|\Delta_{1})=\frac{1}{2}\Big{\\{}\rho(b,\Delta_{2}|a,\Delta_{1})-\rho(a,\Delta_{2}|a,\Delta_{1})+\rho(a,\Delta_{2}|b,\Delta_{1})-\rho(b,\Delta_{2}|b,\Delta_{1})\Big{\\}},$
(1)
where $\rho(b,\Delta_{2}|a,\Delta_{1})$ is the conditional probability of
observing a particle of type $b$ in bin $\Delta_{2}$ given the existence of a
particle of type $a$ in bin $\Delta_{1}$. The label $a$ may e.g. refer to all
positively charged particles while $b$ refers to all negatively charged ones;
$a$ may also refer to all particles with strangeness $-1$ while $b$ refers to
those with $+1$, and so on. For a system consisting of many particles, the
conditional probability $\rho(b,\Delta_{2}|a,\Delta_{1})$ is calculated by
counting the number $N(b,\Delta_{2}|a,\Delta_{1})$ of the $ab$-pairs where $a$
is in bin $\Delta_{1}$ and $b$ is in bin $\Delta_{2}$ and the number
$N(a,\Delta_{1})$ of $a$ in bin $\Delta_{1}$, i.e.,
$\rho(b,\Delta_{2}|a,\Delta_{1})=\frac{N(b,\Delta_{2}|a,\Delta_{1})}{N(a,\Delta_{1})}.$
(2)
These numbers can be calculated using the usual two-particle joint momentum
distribution function $f_{ab}(\bm{p}_{1},\bm{p}_{2})$ and single particle
distribution function $f_{a}(\bm{p})$ or $f_{b}(\bm{p})$ respectively. They
are given by,
$N(b,\Delta_{2}|a,\Delta_{1})=\int_{\Delta_{1}}d^{3}p_{1}\int_{\Delta_{2}}d^{3}p_{2}f_{ab}(\bm{p}_{1},\bm{p}_{2}),$
(3) $N(a,\Delta_{1})=\int_{\Delta_{1}}d^{3}p_{1}f_{a}(\bm{p}_{1}).$ (4)
We see that, if $a$ is locally compensated by $b$, the balance function
$B(\Delta_{2}|\Delta_{1})$ should have a very narrower distribution. In the
opposite case, it should be flat. In the case that $a$ is globally compensated
by $b$, e.g., for electric charge balance function where $a$ and $b$ denote
positively or negatively charged particle respectively, the balance function
is normalized to unit, i.e. $\sum_{\Delta_{2}}B(\Delta_{2}|\Delta_{1})=1$.
### II.1 A working model for the two particle joint momentum distribution
functions in the bulk quark matter system
We consider the bulk quark matter system produced in heavy ion collisions at
high energies. We suppose that the system is composed of $N_{q}$ quarks and
$N_{\bar{q}}$ anti-quarks. We denote the normalized momentum distribution of
the quarks and anti-quarks by $n_{q}(\bm{p})$ and $n_{\bar{q}}(\bm{p})$
respectively. In heavy ion collisions, the bulk matter system consists of new
created quarks, anti-quarks and the quarks from the incident nuclei. Those
quarks from the incident nuclei are referred as the net quarks and the new
born quarks and anti-quarks are created in pairs.
To obtain a charge balance function that is narrower than that for the
completely uncorrelated case, we introduce a minimum correlation in the two
particle joint momentum distributions in the system. To this end, we construct
the following working model for the two particle joint momentum distribution
for the bulk quark matter system. We assume that there is no correlation
between the momentum distributions of two different quarks or two anti-quarks.
The joint distributions are simply the products of the corresponding single
particle momentum distributions, i.e.
$\displaystyle
f_{q_{1}q_{2}}(\bm{p}_{1},\bm{p}_{2})=N_{q_{1}}N_{q_{2}}n_{q_{1}}(\bm{p}_{1})n_{q_{2}}(\bm{p}_{2})(1-\delta_{q_{1},q_{2}})+N_{q_{1}}(N_{q_{2}}-1)n_{q_{1}}(\bm{p}_{1})n_{q_{2}}(\bm{p}_{2})\delta_{q_{1},q_{2}},$
(5) $\displaystyle
f_{\bar{q}_{1}\bar{q}_{2}}(\bm{p}_{1},\bm{p}_{2})=N_{\bar{q}_{1}}N_{\bar{q}_{2}}n_{\bar{q}_{1}}(\bm{p}_{1})n_{\bar{q}_{2}}(\bm{p}_{2})(1-\delta_{q_{1},q_{2}})+N_{\bar{q}_{1}}(N_{\bar{q}_{2}}-1)n_{\bar{q}_{1}}(\bm{p}_{1})n_{\bar{q}_{2}}(\bm{p}_{2})\delta_{q_{1},q_{2}},$
(6)
where $q_{1}$ and $q_{2}$ denote the flavors of the quarks. For the $q\bar{q}$
joint momentum distribution, we introduce a correlation between the moment
distribution of the quark and that of the anti-quark which are produced in the
same pair. In this case, the joint distribution for a quark $q_{1}$ and an
anti-quark $q_{2}$ is given by,
$f_{q_{1}\bar{q}_{2}}(\bm{p}_{1},\bm{p}_{2})=N_{q_{1}}N_{\bar{q_{2}}}n_{{q_{1}}}(\bm{p}_{1})n_{\bar{q_{2}}}(\bm{p}_{2})+N_{\bar{q_{1}}}\bigl{[}n^{pair}_{q\bar{q}}(\bm{p}_{1},\bm{p}_{2})-n_{{\bar{q}_{1}}}(\bm{p}_{1})n_{\bar{q_{2}}}(\bm{p}_{2})\bigr{]}\delta_{q_{1},q_{2}}.$
(7)
The single particle momentum distributions are related to
$n_{q\bar{q}}^{pair}(\bm{p}_{1},\bm{p}_{2})$ by,
$\displaystyle n_{\bar{q}}(\bm{p}_{2})=\int
d^{3}p_{1}n^{pair}_{q\bar{q}}(\bm{p}_{1},\bm{p}_{2}),$ (8) $\displaystyle
n_{q}(\bm{p})=\frac{N_{\bar{q}}}{N_{q}}n_{\bar{q}}(\bm{p})+\frac{N_{net}}{N_{q}}n_{net}(\bm{p}).$
(9)
Hence, as long as we know $n^{pair}_{q\bar{q}}(\bm{p}_{1},\bm{p}_{2})$ and
$n_{net}(\bm{p})$, we can calculate the two particle joint momentum
distributions for $qq$, $\bar{q}\bar{q}$ and $q\bar{q}$-system.
To calculate $n^{pair}_{q\bar{q}}(\bm{p}_{1},\bm{p}_{2})$, we adopt the
picture of the hydrodynamic theory. Here, we assume the local thermalization
and collectivity in the system Kolb0305084nuth ; StarRev ; PhenixRev . Hence,
in the co-moving frame of the fluid cell, due to local thermalization, we take
a Boltzmann distribution for the single quark or anti-quark distribution,
i.e.,
$n^{*}_{q}(\bm{p}^{*})=n_{th}(\bm{p}^{*})=\frac{1}{4\pi
m^{2}TK_{2}(m/T)}e^{-E^{*}/T},$ (10)
where the supscript $*$ denote that these quantities are in the co-moving
frame, $K_{2}$ is the Bessel function, $m$ is the mass of the constituent
quark (340 MeV for $u$ or $d$ quark and 500 MeV for strange quark), and
$E^{*}=\sqrt{\bm{p}^{*2}+m^{2}}$ is the energy of quark; $T$ is the
temperature of the system at hadronization (take as $T=165$ MeV Karsch2002NPA
).
For the joint momentum distribution of the quark and the anti-quark produced
in the same pair, we use the covariance coefficient $\rho$ to describe the
correlation between them. We recall that for a joint momentum distribution fof
a $q\bar{q}$-system, the covariance coefficient $\rho$ is defined as
$\rho={\rm cov}(\bm{p}_{q},\bm{p}_{\bar{q}})/{\rm var}(\bm{p}_{\bar{q}})$,
where ${\rm
cov}(\bm{p}_{q},\bm{p}_{\bar{q}})\equiv\langle{\bm{p}}_{q}\cdot{\bm{p}}_{\bar{q}}\rangle-\langle{\bm{p}}_{q}\rangle\cdot\langle{\bm{p}}_{\bar{q}}\rangle$
and ${\rm
var}(\bm{p}_{\bar{q}})\equiv\langle{\bm{p}}_{\bar{q}}^{2}\rangle-\langle{\bm{p}}_{\bar{q}}\rangle^{2}$.
We take the joint distribution for the $q\bar{q}$-system in the co-moving
frame of the $q\bar{q}$-pair in the Cholesky factorization form, i.e.,
$n^{pair*}_{q\bar{q}}(\bm{p}^{*}_{q},\,\bm{p}^{*}_{\bar{q}})=\frac{1}{2({1-\rho^{2}})^{3/2}}[n_{th}(\bm{p}^{*}_{q})n_{th}(\frac{\bm{p}^{*}_{\bar{q}}-\rho\bm{p}^{*}_{q}}{\sqrt{1-\rho^{2}}})+n_{th}(\bm{p}^{*}_{\bar{q}})n_{th}(\frac{\bm{p}^{*}_{q}-\rho\bm{p}^{*}_{\bar{q}}}{\sqrt{1-\rho^{2}}})].$
(11)
The covariance parameter $\rho$ describe the strength of the correlation. If
$\rho=0$, there is no correlation between the momentum distribution of the
quark and that of the anti-quark and we obtain the factorized form. For $\rho$
very close to unity, we get a maximum correlation between the momentum of
${\bm{p}}_{q}$ and ${\bm{p}}_{\bar{q}}$, where the probability is non-zero
only when ${\bm{p}}_{q}={\bm{p}}_{\bar{q}}$. In general $-1\leq\rho\leq 1$,
and $\rho>0$ means short range compensation of $q$ and $\bar{q}$ while
$\rho<0$ means the opposite.
The joint distribution $n^{pair}_{q\bar{q}}(\bm{p}_{q},\bm{p}_{\bar{q}})$ in
the laboratory frame is obtained from
$n^{pair*}_{q\bar{q}}(\bm{p}^{*}_{q},\,\bm{p}^{*}_{\bar{q}})$. Here, we first
make the Lorentz transformation ($\bm{\beta}$) from the co-moving frame of the
fluid cell to the laboratory frame to obtain
$n^{pair}_{q\bar{q}}(\bm{p}_{q},\,\bm{p}_{\bar{q}},\bm{\beta})$, then sum up
the contributions from different fluid cells in the system with different
collective velocities, i.e.,
$n^{pair}_{q\bar{q}}(\bm{p}_{q},\bm{p}_{\bar{q}})=\int
h(\bm{\beta})\,n^{pair}_{q\bar{q}}(\bm{p}_{q},\,\bm{p}_{\bar{q}},\bm{\beta})\,d^{3}\beta,$
(12)
where $h(\bm{\beta})$ is the so-called velocity function which corresponds to
the velocity distribution of the fluid cell in the system.
The velocity function $h(\bm{\beta})$ is normalized to unity and can be
decomposed into the longitudinal part $h_{L}$ and the transverse part
$h_{\perp}$. The longitudinal velocity $\beta_{z}$ is usually replaced by the
rapidity $y$. The azimuthal dependence is isotropic, we integrate it out and
obtain, $\int h(\bm{\beta})\,d^{3}\beta=\int
h_{L}(y)h_{\perp}(\beta_{\perp})\,dyd\beta_{\perp}$. This velocity function
$h(\bm{\beta})$ determines, together with the momentum distribution of the
quark and anti-quark in the fluid cell, the single quark spectrum thus the
inclusive momentum distribution of the hadrons after hadronization. In
practice, it is parameterized by fitting the data for the hadron momentum
distributions with the aid of hadronization models. According to the
transparency observed in experimentsbearden04stop , and because of that the
observed rapidity spectra of hadrons show a roughly Gaussian shape in the full
rapidity rangeBeardMeson04 , we take the transverse part as a uniform
distribution between $[0,\beta_{\perp}^{max}]$ and parameterize the
longitudinal part in a Gaussian-like form,
$h_{L}(y)=\frac{1}{2\sigma^{\frac{2}{a}}\Gamma(1+\frac{1}{a})}e^{-|y|^{a}/\sigma^{2}}.$
(13)
The free parameters $a$, $\sigma$ and $\beta_{\perp}^{max}$ are fixed using
the data for the rapidity and the $p_{T}$ spectra of hadrons. For example, in
the following of this paper, we just use the results obtained by fitting the
data of rapidity and $p_{T}$ spectra of final hadrons in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 200 GeV BeardMeson04 ; ptPhenix with the aid
of the combination model for hadronizationQBXie1988PRD ; FLShao2005PRC . The
results are $a=2.40$, $\sigma=2.54$ and $\beta_{\perp}^{max}=0.30$ for $u$ and
$d$ newborn quarks and $a=2.36$, $\sigma=2.73$ and $\beta_{\perp}^{max}=0.34$
for strange quarks. The numbers of light and strange (anti-)quarks and
momentum distribution of net-quarks from the colliding nuclei have been fixed
in Ref. JSong2009MPA .
### II.2 Charge balance function of the bulk quark anti-quark system
Having the joint momentum distribution functions, we can calculate the charge
balance function in a straight forward way. In the following, we present the
results in rapidity space for different transverse momentum intervals. In
practice, the balance function in rapidity space is often rewritten as a
function of the rapidity difference $\delta y=y_{a}-y_{b}$ between two
particles in a limited window $y_{w}$, i.e.,
$B_{ab}(\delta y|y_{w})=\frac{1}{2}\Bigl{\\{}\frac{N_{ba}(\delta
y,y_{w})-N_{aa}(\delta y,y_{w})}{N_{a}(y_{w})}+\frac{N_{ab}(\delta
y,y_{w})-N_{bb}(\delta y,y_{w})}{N_{b}(y_{w})}\Bigr{\\}}.$ (14)
Since quarks of different flavors posses different electric charges, it is not
straight forward to extend the definition of the the electric charge balance
function given by Eq.(1) or (14) to the quark anti-quark system. There is no
direct extension of Eq.(14) to such cases. We have many different
possibilities at the quark level, e.g.,
$B_{q}^{(c1)}(\delta
y|y_{w})=-\frac{1}{2N_{f}}\sum_{a,b}\frac{e_{a}e_{b}N_{ab}(\delta
y,y_{w})}{e_{a}^{2}N_{a}(y_{w})},$ (15) $B_{q}^{(c2)}(\delta
y|y_{w})=-\frac{1}{2N_{f}}\sum_{a,b}\frac{{\rm sgn}(e_{a}e_{b})N_{ab}(\delta
y,y_{w})}{N_{a}(y_{w})},$ (16)
where both $a$ and $b$ run over all the quarks and the anti-quarks, $N_{f}$ is
the number of flavor involved. We can also defined it as,
$B_{q}^{(c3)}(\delta y|y_{w})=-\frac{1}{2}\Bigl{\\{}\frac{\sum_{a,b}{\rm
sgn}(e_{a}e_{b})N_{ba}(\delta
y,y_{w})}{\sum_{a}N_{a}(y_{w})}+\frac{\sum_{a,b}{\rm
sgn}(e_{a}e_{b})N_{ab}(\delta y,y_{w})}{\sum_{b}N_{b}(y_{w})}\Bigr{\\}},$ (17)
where $a=u$, $\bar{d}$ or $\bar{s}$ while $b=\bar{u}$, $d$ or $s$ represent
the positively and negatively charged particles respectively. We may also
define the baryon number balance function $B_{q}(\delta y|y_{w})$ for the
quark anti-quark system instead, which is given by,
$B_{q}^{(b1)}(\delta
y|y_{w})=-\frac{1}{2N_{f}}\sum_{a,b}\frac{B_{a}B_{b}N_{ab}(\delta
y,y_{w})}{B_{a}^{2}N_{a}(y_{w})},$ (18)
where the summations over $a$ and $b$ run over all different flavors of quarks
and those of anti-quarks, and $B_{a}$ and $B_{b}$ stand for the baryon
numbers. We can also defined it as,
$B_{q}^{(b2)}(\delta
y|y_{w})=-\frac{1}{2}\Bigl{\\{}\frac{\sum_{a,b}B_{a}B_{b}N_{ba}(\delta
y,y_{w})}{\sum_{a}B_{a}^{2}N_{a}(y_{w})}+\frac{\sum_{a,b}B_{a}B_{b}N_{ab}(\delta
y,y_{w})}{\sum_{b}B_{b}^{2}N_{b}(y_{w})}\Bigr{\\}},$ (19)
where $a$ denotes all the quarks of different flavors and $b$ all the anti-
quarks of different flavors respectively. All these definitions satisfy $\int
d\delta yB_{q}(\delta y|y_{w})=1$.
We note that, so far as the kind of correlations between the momentum
distributions of the quarks and that of the anti-quarks described in the
working model presented in Sec.A are concerned, all these definitions do not
make much differences. More precisely, in the working model presented in Sec.
A, only a correlation between the momentum of the quark and that of the anti-
quark from the same $q\bar{q}$ pair is introduced as given by Eq.(11). There
is no correlation between the quarks and anti-quarks from different pairs and
there is no difference between different flavors. In this case, all the
definitions given by Eqs.(15-19) are equivalent in the sense that they are all
different suppositions of the correlations given by Eq.(11) for different
flavors and Eq.(11) does not distinguish between different flavors. The only
differences come from the net quark contributions where no strange quark
exists.
For comparison, we made the calculations using the different definitions
Eqs.(15-19) and the results are indeed similar. In the following part of this
section, we show the results obtained by using Eq. (17).
We first study the case where $\rho=0$. In this case, there is no correlation
between the momentum distribution of the quarks and anti-quarks. The balance
is obtained only from the global flavor compensation of the new created quarks
and anti-quarks. This is also the minimum compensation in the produced system.
In Fig. 1 (a), we show the results of $y_{w}$=1 for different rapidity
positions with transverse momentum $p_{\perp}$ integrated. In Fig. 1 (b), we
show the results for $\rho=0.5$ and a comparison of the results for different
values of $\rho$ is given in Fig. 1 (c).
Figure 1: The electric charge balance function $B_{q}(\delta y|y_{w})$ for the
bulk quark system for same window size as a function of $\delta y$ at the
variance coefficient $\rho=0$ in panel (a) and $\rho=0.5$ in panel (b),
respectively; A comparison of the results at different values of $\rho$ is
shown in panel (c).
From the results, we see that in all cases, also for $\rho=0$, the balance
function $B_{q}(\delta y|y_{w})$ decrease with increasing $\delta y$ showing a
local compensation of the electric charge in the rapidity space for the bulk
quark system. It is also clear that $B_{q}(\delta y|y_{w})$ decreases faster
with increasing $\delta y$ for larger value of $\rho$ indicating stronger
local charge compensation. We also see that $B_{q}(\delta y|y_{w})$ does not
change much for different rapidity window with the same window size showing
the longitudinal boost invariance. This is hold for different values of the
variance coefficient $\rho$.
The existence of the approximate boost invariance for the charge balance
function for the bulk quark system can easily be understood. We note that by
looking at the different rapidity window in the case that the window size is
much smaller than the total rapidity range of the bulk quark system, we are in
fact looking at different fluid cells. Since we do not differentiate these
fluid cells in any significant way, the results should be similar. This
results in similar charge balance function as indicated by the calculated
results shown in Fig. 1. In other words, the boost invariance of the charge
balance function just reflects the homogeneity of the fluid cell at
hadronization in different rapidity windows.
We continue to study the dependence of the balance function $B_{q}(\delta
y|y_{w})$ on the window size and/or transverse momentum. In Fig. 2 (a), we
show $B_{q}(\delta y|y_{w})$ in the different rapidity positions with the same
window size $y_{w}=1$ and in Fig. 2 (b), we show $B_{q}(\delta y|y_{w})$ at
different window sizes $y_{w}=1,2,3,4$. We see that $B_{q}(\delta y|y_{w})$
varies with window size and becomes flatter with increasing window size. This
qualitative feature is naturally expected from the definition since the
balance function is normalized to unity but the range of the allowed values of
$\delta y$ becomes larger for the larger window size. This effect can be
eliminated by scaling the balance function $B_{q}(\delta y|y_{w})$ with the
factor $1-\delta y/|y_{w}|$ as suggested in Ref SJeon02 , i.e. we study the
scaled balance function,
$B_{s}(\delta y)=\frac{B_{q}(\delta y|y_{w})}{1-\delta y/y_{w}}.$ (20)
In Fig. 2(c), we show the results obtained for the scaled $B_{s}(\delta y)$ of
the bulk quark system. We see clearly that the scaled balance functions fall
on one curve showing that they are independent of the size and position of
rapidity window. For comparison, we also present the balance function in the
full rapidity region (open cross) for the case that the net charge of the
system is taken to be zero. We see that the result is also consistent with
those for the limited rapidity windows so far as the scaled balance function
is studied. This is very nice feature since it suggests that the scaled
balance function for particles in the limited rapidity window can indeed be
regarded as an example for the charge balance function of the system.
Figure 2: The $p_{T}$-integrated $B_{q}(\delta y|y_{w})$ of the constituent
quark system at different rapidity positions with same (panel a) and different
(panel b) window sizes, as well as the $B_{s}(\delta y)$ (panel c).
Correlation coefficient $\rho$ is taken to be 0.3.
We emphasize that these properties of the balance functions of the bulk quark
system are results of the momentum distributions of the quarks and anti-quarks
in the system . These distributions including the correlations given by Eq.
(11) are results of the local thermalization and collectivity for the system
produced in relativistic heavy ion collisions in the hydrodynamic theory.
These qualitative features for the charge balance functions for the bulk quark
system before hadronization is consistent with those for the final hadrons as
observed by STAR Collaboration at RHIC StarBF200Scaling .
In Fig.3, we show $B_{q}(\delta y|y_{w})$ and $B_{s}(\delta y)$ in different
rapidity windows and in the different $p_{T}$ ranges. We clearly see that the
scaling properties of balance function still hold in the different $p_{T}$
ranges. We can also see that the width of the scaled balance function
decreases with increasing $p_{T}$. This is because, in general, the quarks and
anti-quarks with larger $p_{T}$ come from the fluid cell with larger
transverse flow, which results in a smaller longitudinal rapidity interval and
hence smaller width for balance function. Such a feature was expected earlier
at the hadron levelBialas04 and observed in central Au+Au collisions at
$\sqrt{s_{NN}}=200$ GeV StarBF200Scaling .
Figure 3: The $B_{q}(\delta y|y_{w})$ of quark system (top panels) at
different rapidity positions with different window sizes as well as the
$B_{s}(\delta y)$ (below panels) in the different $p_{T}$ (GeV/c) ranges.
Correlation coefficient $\rho$ is taken to be 0.3.
## III charge balance functions of the hadron system
With the momentum distribution functions of the bulk quark system discussed in
last section, we study the charge balance functions of hadrons produced in the
hadronization of this system. We compare the results obtained for the directly
produced hadrons and those of the final state hadrons with those for the
quarks and anti-quarks to study the influence of the hadronization and
resonance decay on the balance functions.
We describe the hadronization of the bulk quark system with the
(re-)combination or coalescence mechanism. Such a hadronization mechanism is
tested by various data and is implemented in different forms such as the quark
recombination model Fries2003PRL ; RCHwa04 , the parton coalescence model
Greco2003PRL ; Molnar2003PRL , and the quark combination mode (SDQCM)
QBXie1988PRD ; FLShao2005PRC . All these models are tested against the various
features of the hadrons produced in heavy ion collisions at high energies.
Here, in this paper, we use SDQCM QBXie1988PRD ; FLShao2005PRC for our
calculations since this model takes the exclusive description and is
implemented by a Monte-Carlo program so that can be apply to calculate the
balance functions for the directly produced hadrons as well as the final
hadrons after the resonance decays in a very convenient way. Also, this model
guarantees that mesons and baryons exhaust all the quarks and anti-quarks in
the deconfined color-neutral system at hadronization.
### III.1 Charge balance functions in rapidity space
We insert the momentum distributions including the correlations given by Eq.
(11) to determine the momenta of the quarks and anti-quarks before
hadronization. We then apply the quark combination rules as implemented in the
Monte-Carlo program of SDQCMFLShao2005PRC to calculate the momentum
distribution of the directly produced hadrons. Those resonances will decay
accordingly and the momentum distributions are simulated also in the program
by using the material from the particle data grouppdg08p355 .
In Fig. 4, we show the results for the $p_{T}$-integrated balance functions
for the directly produced hadrons. Here, in Fig. 4(a), we see the results in
different rapidity windows with the same width $y_{w}=1$, while in Fig. 4(b)
and (c), we see the results at different window sizes as well as the scaled
function $B_{s}(\delta y)$. In Fig.5, we show the corresponding results in
different $p_{T}$ ranges.
Figure 4: The $p_{T}$-integrated $B(\delta y|y_{w})$ of initial hadron system
at different rapidity positions with same (panel a) and different (panel b)
window sizes, as well as the $B_{s}(\delta y)$ (panel c). Correlation
coefficient $\rho$ is taken to be 0.3. Figure 5: The $B(\delta y|y_{w})$ of
initial hadron system (top panels) at different rapidity positions with
different window sizes as well as the $B_{s}(\delta y)$ (below panels) in the
different $p_{T}$ (GeV/c) ranges. Correlation coefficient $\rho$ is taken to
be 0.3.
From these results, we see that both the longitudinal boost invariance and
rapidity scaling for the balance functions are hold for the hadrons directly
produced in the quark combination mechanism, either for the $p_{T}$-integrated
quantities or those for different $p_{T}$ ranges. This is in fact not
surprising because the formation of hadrons in this hadronization mechanism is
realized by the combination of two or three nearest quarks/antiquarks in
momentum space. This means that the combination happens locally and does not
destroy the locality nature of charge balance of the system.
We further study the resonance decay contributions by calculating the balance
functions for the final hadrons where decays of the resonances are taken into
account. We show the corresponding results in Figs. 6 and 7.
Figure 6: The $p_{T}$-integrated $B(\delta y|y_{w})$ of final hadron system at
different rapidity positions with same (panel a) and different (panel b)
window sizes, as well as the $B_{s}(\delta y)$ (panel c). Correlation
coefficient $\rho$ is taken to be 0.3. Figure 7: The $B(\delta y|y_{w})$ of
final hadron system (top panels) at different rapidity positions with
different window sizes as well as the $B_{s}(\delta y)$ (below panels) in the
different $p_{T}$ (GeV/c) ranges. Correlation coefficient $\rho$ is taken to
be 0.3.
From Figs. 6 and 7, we see that both the boost invariance in rapidity space
and the scaling property still preserved after the contributions from the
resonance decays are taken into account. Together with those results given in
Figs. 4 and 5, these results show clearly, although there are definitely
influences from hadronization and resonance decay on the form of the charge
balance functions, these effects do not significantly influence the boost
invariance and the scaling in rapidity space.
The influences from hadronization and resonance decay to the balance function
can be studied more quantitatively by calculating the averaged width of the
balance function, which is defined as,
$\langle\delta y\rangle=\frac{\int\nolimits_{0}^{y_{w}}B(\delta y|y_{w})\
\delta y\ d\delta y}{\int\nolimits_{0}^{y_{w}}B(\delta y|y_{w})d\delta y}.$
(21)
We note that the averaged width $\langle\delta y\rangle$ is in general a
charactering quantity describing the radius of charge balance of the system.
For the final hadron system in heavy ion collisions, it can be sensitive to
different effects such as delayed hadronization or hadron freeze-out
Bass_clockHd ; FuQHbf11 , possibly highly localized charge balance at freeze-
out Schlichting11_bf , transverse flow Bialas04 ; bozek05 , multiplicity
effect NA22BFkp ; DuJX07 and hadronic weak decay. Here, by comparing the
results for the balance functions of the quark anti-quark system with those
for the initial hadrons and those for the final hadrons, we can study the
magnitudes of the influences of hadronization and those from resonance decay.
Figure 8: In (a), we see the averaged widths $\langle\delta\eta\rangle$ of the
balance functions for the quark and anti-quark, the directly produced and the
final hadron systems as functions of $\rho$. In (b), we see the averaged
widths $\langle\delta\eta\rangle$ of the balance functions for the daughter
hadrons from all the resonance decays, those for the daughter hadrons from the
meson decays and those from the baryon decays, respectively. The band area
represents the experimental of $\langle\delta\eta\rangle$ for charged
particles in central Au+Au $\sqrt{s_{NN}}=$ 200 GeV StarBF200Scaling .
In Fig. 8(a), we show the results for the averaged width
$\langle\delta\eta\rangle$ of the balance function for the directly produced
hadrons as the function of $\rho$, compared with that for the quarks and anti-
quarks and that for the final hadrons. Here, we show the results for pseudo-
rapidity $\eta$ in order to compare with the experimental data now available
StarBF200AApp . We choose also the same pseudo-rapidity and $p_{T}$ regions as
the experiments StarBF200AApp , i.e. $|\eta|<1$, $0.1\leq|\delta\eta|\leq 2.0$
and $0.2<p_{T}<2.0$ GeV/c. In Fig. 8(a), the data of
$\langle\delta\eta\rangle$ in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200
GeV StarBF200AApp is shown as a band area. We see clearly that, in all the
three cases, the averaged width $\langle\delta\eta\rangle$ decreases with the
increasing $\rho$. We see also that, the $\langle\delta\eta\rangle$ for the
directly produced hadron system decreases with the increasing $\rho$ in
exactly the same way as that for the bulk quark system does. The difference
between $\langle\delta\eta\rangle$ for the directly produced hadron system and
that for the bulk quark system is almost a constant 0.04 for all the different
values of $\rho$. This is because, as mentioned above, the combination of
quark(s) and/or anti-quark(s) in neighbor does not change the electric charge
balance in any essential way. However, the formation of electrically neutral
hadrons may delay the charge balance in momentum space. We take a quark and an
anti-quark from a given $q\bar{q}$-pair as an example. As given by Eq.(11),
their momentum distributions posses a correlation measured by the covariance
coefficient $\rho$. If both of them enter into the respectively charged
hadrons in the combination process, the correlation will pass to the hadronic
level. However, if one of them enters into an electrically neutral hadron, the
correlation will be lost in the charged hadrons. This will decrease the local
charge balance at the hadron level.
From Fig. 8(a), we also see that the resonance decay contributions change
$\langle\delta\eta\rangle$ significantly. It is also interesting to see that
these contributions strengthen the local charge balance for relatively small
values of $\rho$ but weaken the balance for larger values of $\rho$. This
indicates that the decay contributions dilute the balance functions quite
significantly. To see where these different behaviors come from, we calculate
the averaged width $\langle\delta y\rangle$ for those hadrons from resonance
decay separately. We note that the influence of resonance decay to the charge
balance function is in general different for hyperon decay from that for
vector meson decay. The decay of the hyperons such as $\Lambda\to p\pi$ and
$\Xi^{0}\to\Lambda\pi$ produces a pair of charged daughter particles with
quite narrow rapidity interval, e.g. about one third for $\Lambda\to p\pi$ and
$\Xi^{0}\to\Lambda\pi$, due to the small amount of energy released in the
decay process. This leads also to a smaller $\langle\delta\eta\rangle$ for the
charge balance function. However, in vector meson decay such as
$\rho^{0}\to\pi^{+}\pi^{-}$ and $K^{*0}\to K^{+}\pi^{-}$, the energy released
is much larger leading to a much larger rapidity difference between the
daughter particles, e.g. up to 1.7 for $\rho^{0}\to\pi^{+}\pi^{-}$ and 1.3 for
$K^{*0}\to K^{+}\pi^{-}$ in the rest frame of parent particle. To study this
effect in a more quantitative manner, we calculate the averaged width
$\langle\delta\eta\rangle$ of the balance function only for the daughter
particles from baryon or meson decay respectively. The results are shown in
Fig. 8 (b). Here, we see clearly that, the averaged width
$\langle\delta\eta\rangle$ for the daughter particles from baryon decay is
indeed much smaller than those from meson decay. We see also that the charge
balance for the daughter particles from baryon decays is dominated by the
decay effect which leads to an averaged width of about one third. However, for
those from meson decays, the charge balance is dominated by the effect from
the mother particles.
### III.2 Charge balance in the azimuthal direction
The charge balance in the azimuthal direction for hadrons in high energy heavy
ion collisions can be sensitive to jet production. Experimental studies have
already been carried out by STAR Collaboration for hadrons of different
$p_{T}$ regions StarBF200AApp . It is thus also interesting to see how the
charge balance function behaves for the bulk quark matter system and the
resulting hadrons.
The balance function of hadrons in the azimuthal direction is defined
similarly to that in rapidity,
$\displaystyle
B_{ba,azi}(\delta\phi,\phi)=\frac{1}{2}\left\\{\frac{N_{ba}(\delta\phi,\phi)-N_{aa}(\delta\phi,\phi)}{N_{a}(\phi)}+{N_{ab}(\delta\phi,\phi)-N_{bb}(\delta\phi,\phi)\over
N_{b}(\phi)}\right\\},$ (22)
where, e.g. the quantity $N_{ba}(\delta\phi,\phi)$ is defined as the number of
pairs of particles with the particle $a$ flying at an angle $\phi$ (measured
with respect to the reaction plane) and the particle ${b}$ at an angle between
$\phi$ and $\phi+\delta\phi$. In the following we study the azimuthally
averaged balance function
$B_{ab,azi}(\delta\phi)=\int_{0}^{2\pi}d\phi B_{ab}(\delta\phi,\phi).$ (23)
Having the Monte-Carlo program at hand, the extension of the calculations
mentioned above to azimuthal direction is straight forward. We show the
results obtained for the quark, the directly produced hadron and the final
hadron system at different $\rho$ values in Fig. 9. Comparison with the
available dataStarBF200AApp is also given in the figure.
Figure 9: Balance function $B_{azi}(\delta\phi)$ of the quark anti-quark
system (a), the directly produced hadron system (b) and the final hadron
system (c) as functions of $\delta\phi$ in the pseudo-rapidity region
$-1<\eta<1$ and $0.2<p_{T}<2.0$GeV/c. A comparison of the results for the
final hadron system at $\rho=0.5,0.6$ and 0.7 and the experimental data for
all charged particles with $0.2<p_{T}<2.0$GeV/c in central Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV in Ref.StarBF200AApp is given in (d).
From the results for the bulk quark system Fig. 9(a), we see clearly that the
dependence of the quark charge balance function on the variance parameter
$\rho$ is quite obvious. For $\rho$ close to unity, the momentum of the quark
and that of the anti-quark produced in pair are closely correlated, and we see
a sharp peak at $\delta\phi=0$. For $\rho=0$, there is no correlation and the
balance function is almost a flat function showing only the influence from the
global charge compensation.
Comparing the results in Fig. 9(b) with those in Fig. 9(a), we see that the
charge balance functions in the azimuthal direction for the directly produced
hadron system are slightly broader than the corresponding results for the
quark system, showing a slightly loose correlation. This is similar to the
case in rapidity space studied in last subsection. However, the influences
from the resonance decays are quite significant in the azimuthal direction. We
see quite significant differences between the results for the final hadrons
and the corresponding results for the hadron system before resonance decay. We
see in particular that the very much pronounced peak at $\delta\phi=0$ is
smoothed by the decay influences. This is also obvious since such strong
correlation can be destroyed by the resonance decay because of the transverse
momentum conservation in the decay processes. From Fig. 9(c), we see that the
dataStarBF200AApp is well be described except for the peak at $\delta\phi=0$.
This peak could be an indication of jet contribution which is not included in
our calculations.
## IV summary
In summary, we have calculated the charge balance functions of the bulk quark
system before hadronization, those for the directly produced and the final
hadron system in relativistic heavy ion collisions. The momentum distributions
for the quarks and the anti-quarks in the bulk quark system are taken as
determined in the hydrodynamic picture with local thermalization and
collectivity. A correlation between the momentum distribution of the quark and
that of the anti-quark is introduced if they are from the same new produced
$q\bar{q}$ pair and the correlation strength is described by the variance
coefficient $\rho$. Our results show that the charge balance functions for the
bulk quark system have the longitudinal boost invariance and the scaling
behavior in rapidity space. Such properties are preserved by the subsequent
hadronization via combination mechanism and the resonance decay, although both
hadronization and resonance decay can influence the width of the balance
function. With the same inputs, we also studied the balance function in the
azimuthal direction.
## ACKNOWLEDGMENTS
The authors thank Q. B. Xie, Q. Wang and G. Li for helpful discussions. The
work is supported in part by the National Natural Science Foundation of China
under grant 11175104, 10947007, 10975092, and by the Natural Science
Foundation of Shandong Province, China under grant ZR2011AM006.
## References
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|
arxiv-papers
| 2012-03-05T14:00:13 |
2024-09-04T02:49:28.272629
|
{
"license": "Public Domain",
"authors": "Jun Song, Feng-lan Shao, and Zuo-tang Liang",
"submitter": "Jun Song",
"url": "https://arxiv.org/abs/1203.0921"
}
|
1203.0987
|
# ADDITIVE RELATION AND ALGEBRAIC SYSTEM OF EQUATIONS
Wu Zi qian Xi’an Shiyou University.Address:Xi’an city,China.
woodschain$@$sohu.com
###### Abstract.
Additive relations are defined over additive monoids and additive operation is
introduced over these new relations then we build algebraic system of
equations. We can generate profuse equations by additive relations of two
variables. To give an equation with several known parameters is to give an
additive relation taking these known parameters as its variables or value and
the solution of the equation is just the reverse of this relation which always
exists. We show a core result in this paper that any additive relation of many
variables and their inverse can be expressed in the form of the superposition
of additive relations of one variable in an algebraic system of equations if
the system satisfies some conditions. This result means that there is always a
formula solution expressed in the superposition of additive relations of one
variable for any equation in this system. We get algebraic equations if
elements of the additive monoid are numbers and get operator equations if they
are functions.
###### Key words and phrases:
Additive relation of many variables, algebraic system of equations,
decomposition of additive relations
## 1\. introduction
To give explicit solutions are always difficult for not only general algebraic
equations but also for general operator equations. May be one consider that no
one try to find formula solution for a general polynomial equation and further
for general algebraic equation after Abel proof that there isn’t an algebraic
solution for it. But actually the most outstanding mathematicians devote
themselves to this problem in each times.
Camille.Jordan (1838-1922) shows that algebraic equations of any degree can be
solved in terms of modular functions in 1870. Ferdinand.von.Lindemann
(1852-1939) expresses the roots of an arbitrary polynomial in terms of theta
functions in 1892. In 1895 Emory.McClintock (1840-1916) gives series solutions
for all the roots of a polynomial. Robert Hjalmal.Mellin (1854-1933) solves an
arbitrary polynomial equation with Mellin integrals in 1915. In 1925
R.Birkeland shows that the roots of an algebraic equation can be expressed
using hypergeometric functions in several variables. Hiroshi.Umemura expresses
the roots of an arbitrary polynomial through elliptic Siegel functions
in1984[3].All of solutions mentioned above are not ones expressed in binary
function. David.Hilbert presumed that there is no solution expressed in binary
function for polynomial equations of n when n$\geq$7 and wrote his doubt into
his famous 23 problems as the 13th one[2].
In 1957 V.I.Arnol’d proved that every continuous function can be represented
as a superposition of functions of two variables and refuted Hilbert
conjecture[4][5]. Furthermore, A.N.Kolmogorov proved that every continuous
function can be represented as a superposition of continuous functions of one
variable and additive operation[1].Thus we can’t say we refuse to find formula
solutions for general transcendental equations because they don’t exist.
On the hand modern algebra is running in its own direction but not in the
direction of classical mathematics which takes solving equations including to
give formula solutions as one of its main tasks. We can clear this point if we
note that so many results gotten by modern algebra but there are so many
algebraic equations with no explicit solution. We will never give explicit
solution to most of algebraic equations by so few operations which meet axioms
of arithmetic and actually many algebraic equations can’t be generated only by
them. But modern algebra limits itself in these operations.
In this paper we limit the problem in finite sets when we construct algebraic
equations and operator equations thus the problem becomes simpler and clearer.
We get perfect results then we develop a new algebraic system called algebraic
system of equations by these results.
This is a three grade algebraic system which defines relations on a finite set
and defines operations on these relations whereas modern algebraic structures
such as group,ring and field which can be called two grade algebraic systems
are built by set and operations defined on it. One shall feel rich and
colorful of this new algebraic system when he read this paper.
## 2\. Basic Definitions
We shall introduce a new type of relations named additive relation and define
additive operation on these new relations. We build equations by these
relations and try to solve these equations but we have to face the problem of
polykeys in our theory of equation. Function should be the most suit concept
to express polykeys if function can be many-valued but it is defined to be
single valued in modern algebra strictly. We don’t use function but use
relation to express polykeys then we can avoid conflict with modern algebra.
Relations which can be many-valued shall be taken not as operations but as
elements and on these relations we define operations which are single valued
thus the new algebraic system will never be inconsistent with modern algebra.
Definition 2.1 A M+1-ary relation R over non-empty sets $B_{i}(1\leq i\leq M)$
and $B_{0}$ is a subset of their Cartesian product written as $R\subset
B_{1}\times B_{2}\ldots\times B_{M}\times B_{0}=\\{\langle
b_{1},b_{2},\ldots,b_{M},b_{0}\rangle|b_{i}\in B_{i}(1\leq i\leq M),b_{0}\in
B_{0}\\}$. Specially R is called M+1-ary relation over B if
$R\subset\underbrace{B\times B\ldots\times B\times B}_{M+1}=B^{M+1}$ and all
M+1-ary relations over B form the power set of $B^{M+1}$ so we denote them as
$P(B^{M+1})$.
Definition 2.2 Let B is a finite set and (B,+) is a monoid,R is a M+1-ary
relation over B, we shall call it an additive relation of M variables if we
take ith element of its ordered M+1-tuple as its ith variable and the last
element as its value. We denote it by $R^{M}$ and all $R^{M}$ as $P(B^{M+1})$.
B is called the basic set of $P(B^{M+1})$ and the number of elements in set B
will be taken as the order of $R^{M}$.
The number of all ordered M+1-tuples gotten by B with N elements will be
$N^{M+1}$ and the number of $P(B^{M+1})$ will be $2^{(N^{M+1})}$.
For example let $B=\\{0,1\\}$,then all additive relations of one variables
shall be shown below:
$R^{1}_{1}=\emptyset$,$R^{1}_{2}=\\{\langle
0,0\rangle\\}$,$R^{1}_{3}=\\{\langle 0,1\rangle\\}$,$R^{1}_{4}=\\{\langle
1,0\rangle\\}$,$R^{1}_{5}=\\{\langle 1,1\rangle\\}$, $R^{1}_{6}=\\{\langle 0,$
$0\rangle,\langle 0,1\rangle\\}$,$R^{1}_{7}$ $=\\{\langle 0,0\rangle,\langle
1,0\rangle\\}$,$R^{1}_{8}=\\{\langle 0,0\rangle,\langle 1,1\rangle\\}$,
$R^{1}_{9}=\\{\langle 0,1\rangle,\langle 1,0\rangle\\}$,
$R^{1}_{10}=\\{\langle 0,1\rangle,\langle
1,1\rangle\\}$,$R^{1}_{11}=\\{\langle 1,0\rangle,\langle 1,1\rangle\\}$,
$R^{1}_{12}=\\{\langle 0,0\rangle,\langle 0,1\rangle\,\langle
1,0\rangle\\}$,$R^{1}_{13}=\\{\langle 0,0\rangle,\langle 0,1\rangle,$ $\langle
1,1\rangle\\}$, $R^{1}_{14}=\\{\langle 0,0\rangle,\langle 1,0\rangle,\langle
1,$$1\rangle\\}$,$R^{1}_{15}=\\{\langle 0,1\rangle,\langle 1,0\rangle,\langle
1,1\rangle\\}$, $R^{1}_{16}=\\{\langle 0,0\rangle,\langle 0,1\rangle,$
$\langle 1,0\rangle,\langle 1,1\rangle\\}$.
$\emptyset$ and $\\{\langle 0,0\rangle,\langle 0,1\rangle,\langle
1,0\rangle,\langle 1,$ $1\rangle\\}$ is also called empty relation and
universal relation respectively.
Definition 2.3 R will be single valued in point $(b_{1},b_{2},\ldots,b_{M})$
if $\exists 1\langle b_{1},b_{2},$ $\ldots,b_{M},y\rangle\in R,y\in B$.In this
case R will be called function too. R will be many-valued in the same point if
$\exists 1\langle b_{1},b_{2},\ldots,$ $b_{M},y1\rangle\in R,y1\in
B\wedge\exists 1\langle b_{1},b_{2},\ldots,b_{M},y2\rangle\in R,y2\in B$. R
will be undefined in this point if $\exists 0\langle b_{1},b_{2},$
$\ldots,b_{M},y\rangle\in R,y\in B$.
For example,$\\{\langle 0,0\rangle\\}$ and $\\{\langle 0,1\rangle,\langle
1,0\rangle,\langle 1,1\rangle\\}$ are single valued in point ‘0’. $\\{\langle
0,0\rangle,$ $\langle 0,1\rangle$ and $\\{\langle 0,0\rangle,\langle
0,1\rangle,\langle 1,0\rangle,\langle 1,1\rangle\\}$are many-valued in point
‘0’. $\\{\langle 1,0\rangle\\}$ and $\\{\langle 1,0\rangle,\langle
1,1\rangle\\}$are undefined in point ‘0’. $\emptyset$ is undefined in each
point.
It’s so wonderful to describe the single solution and polykeys and no solution
for equations by the single valued and many-valued and undefined of additive
relations respectively.
N-th-order additive relations of one variable can be expressed by a table
$2\times N$ and elements in the first line are variable and ones in the the
second line are value. Many-valued numbers will be partitioned by the symbol
‘*’ and N means undefined.For example $R^{1}_{6}=\\{\langle 0,0\rangle,\langle
0,1\rangle\\}$ will be denoted as below. N-th-order additive relations of two
variables can be expressed by table(N+1)x(N+1) and we give an example of
3-order one as below too:
0 | 1
---|---
0*1 | N
| 0 | 1 | 2
---|---|---|---
0 | 0 | 1 | 2
1 | 0*1 | 0*2 | 1*2
2 | $\phi$ | 0*1*2 | $\phi$
For convenience we remaind only the second line and denote
$R^{1}_{6}=\\{\langle 0,0\rangle,\langle 0,1\rangle\\}$ as (0*1,N).
We use the expression form for function to denote additive relation of M
variables:
$b_{0}=R^{M}(b_{1},b_{2},\ldots,b_{M})$
But to additive relations of two variables which are used so frequently, we
sturdy use the form of binary operation $b_{1}R^{2}b_{2}$ to denote it. It’s
too clear and we are so accustomed to this form.
Definition 2.4 Inverse of an additive relation is defined as
$R^{-1}=\\{\langle x,y\rangle|$ $\langle y,x\rangle\in R\\}$.
We denote inverse of R by T(R). Actually inverse means the transformation to
elements of additive relation so we extend it to all transformations of
(1,2,$\ldots$,M,0) and denote transformations as T with superscript
$(i_{1},i_{2},\ldots,i_{M},i_{0})$.
$T^{i_{1},i_{2},\ldots,i_{M},i_{0}}(R)=\\{\langle
b_{i_{1}},b_{i_{2}},\ldots,b_{i_{M}},b_{i_{0}}\rangle|\langle
b_{1},b_{2},\ldots,b_{i},\ldots,b_{M},b_{0}\rangle\in R\\}$
There are 6 transformations for additive relations of two variables and ’M!’
ones for additive relations of M variables.
Transformations of R can be expressed by the form of function:
$b_{i_{0}}=T^{i_{1},i_{2},\ldots,i_{M},i_{0}}(R)(b_{i_{1}},b_{i_{2}},\ldots,b_{i_{M}})$
Definition 2.5 Composition of additive relations
Let $R^{M}$ is an additive relation of M variables and $\beta$ one of one
variable, here we define the ith parameter composition of $R^{M}$ and $\beta$.
For $1\leq i\leq M$
$R^{M}\times_{i}\beta=\\{\langle
b_{1},b_{2},\ldots,b_{i-1},b_{i},b_{i+1},\ldots,b_{M},b_{0}\rangle|y(\langle
b_{1},b_{2},$ $\ldots,b_{i-1},y,b_{i+1},b_{M},$ $b_{0}\rangle\in R^{M}\langle
b_{i},y\rangle\in\beta)\\}(1\leq i\leq M)$
$R^{M}\times_{i}\beta$ is written in the form of functions:
$[R^{M}\times_{i}\beta](b_{1},b_{2},\ldots,b_{i-1},b_{i},b_{i+1},\ldots,b_{M1})=R^{M}[b_{1},b_{2},\ldots,b_{i-1},\beta(b_{i}),b_{i+1},$
$\ldots,b_{M}]$
For i=0
$\beta(b_{0})=R^{M}(b_{1},b_{2},\ldots,b_{i},\ldots,b_{M})$
$[R^{M}\times_{0}\beta](b_{1},b_{2},\ldots,b_{i},\ldots,b_{M})=b_{0}=\beta^{-1}[R^{M}(b_{1},b_{2},\ldots,b_{i},\ldots,b_{M})]$
If both of $R^{M}$ and $\beta$ is an additive relation of one variable
$\beta_{1}$ and $\beta_{2}$ respectively then:
$b_{0}=[\beta_{1}\times_{0}\beta_{2}](b)=\beta_{2}^{-1}[\beta_{1}(b)]$
Definition 2.6 Let both of $R^{M}_{1}$ and $R^{M}_{2}$ is additive relation
and the sum $R^{M}_{1}+R^{M}_{2}$ is defined as:
$R=(R^{M}_{1}+R^{M}_{2})=\\{\langle b_{1},b_{2},\ldots
b_{M},b_{01}+b_{02}\rangle|(b_{1},b_{2},\ldots b_{M},b_{01})\in
R^{M}_{1}\wedge(b_{1},b_{2},\ldots$ $b_{M},b_{02})\in R^{M}_{2}\\}$
write it in the form of function:
$R=(R^{M}_{1}+R^{M}_{2})(b_{1},b_{2},\ldots,b_{M})=R^{M}_{1}(b_{1},b_{2},\cdots,b_{M})+R^{M}_{2}(b_{1},b_{2},\cdots,b_{M})$
$b_{01}+b_{02}$ must exist and be unique because (B,+) is a monoid. The sum R
includes only ordered M+1-tuple so R will exist and be unique despite it may
contains undefined points or many-valued points.
Note,in $P(B^{M+1})$ there is ‘o’ additive relation being ‘0’ value in its any
point and o+R=R $(R\in P(B^{M+1}))$. There is domination additive relation
being undefined in its any point and $\emptyset+R=\emptyset(R\in P(B^{M+1}))$.
There is local domination additive relation being undefined in some but not
all points of it and the sum will be undefined in these points when it adds an
other additive relation.
($P(B^{M+1})$,+) is a monoid because there is no an inverse for some additive
relations. We can take $P(B^{M+1})$ as a basic set to define additive
relations so we can understand why we use not group but monoid in the
definition of additive relations.
The sum of two additive relations will be undefined or single valued or many-
valued in a point upon the below rule which can be gotten by definition of
addition operation.
1 Sum will be undefined in an point if any of additive relations is undefined
in this point.
2 Sum will be single valued in an point if both of additive relations is
single valued in this point.
3 Sum will be many-valued in an point if one additive relation is many-valued
and the other is defined in this point.
The below example includes all situations:
| 0 | 1 | 2
---|---|---|---
0 | $\phi$ | $\phi$ | $\phi$
1 | 0 | 1 | 2
2 | 0*1 | 1*2 | 0*1*2
\+ 0 1 2 0 $\phi$ 1 0*1*2 1 $\phi$ 0 0*2 2 $\phi$ 0 1*2 = 0 1 2 0 $\phi$
$\phi$ $\phi$ 1 $\phi$ 1 1*2 2 $\phi$ 1*2 0*1*2
Definition 2.7 Let $R^{M1}$ is an additive relations of M variables and
$R^{M2}$ is a false additive relations of M2 variables defined by $R^{M1}$.
$R^{M2}=F^{M2}_{i_{1},i_{2},\cdots,i_{M1}}(R^{M1})=\\{\langle
y,\cdots,y,b_{i_{1}},y,\cdots,y,b_{i_{2}},y,\cdots,y,b_{{}_{iM1}},$
$y,\cdots,y,$ $b_{0}\rangle|\langle b_{1},b_{2},\cdots,b_{M1},b_{0}\rangle\in
R^{M1}\wedge y\in B,(b_{i_{j}}=b_{j})\\}$
Here B is the basic set. $R^{M2}$ will take jth variable of $R^{M1}$ as its
$i_{j}$th variable. The value of $R^{M2}$ will change with only M1 variables
and other variables are false ones.
For example,$R^{1}=(0,1*2,\emptyset)$, $F^{2}_{1}(R^{1})$ taking variable of
$R^{1}$ as its first variable and $F^{2}_{2}(R^{1})$ taking it as its second
variable as below respectively:
| 0 | 1 | 2
---|---|---|---
0 | 0 | 0 | 0
1 | 1*2 | 1*2 | 1*2
2 | $\emptyset$ | $\emptyset$ | $\emptyset$
| 0 | 1 | 2
---|---|---|---
0 | 0 | 1*2 | $\emptyset$
1 | 0 | 1*2 | $\emptyset$
2 | 0 | 1*2 | $\emptyset$
In the expression $w(x,y)=f(x)+g(y)=[F^{2}_{1}(f)+F^{2}_{2}(g)](x,y)$, we
change additive relations of one variable ‘f’ and ‘g’ to false one of two
variables and get ‘w’ by adding them then we express ‘w’ by
$[F^{2}_{1}(f)+F^{2}_{2}(g)]$ clearly. Certainly we will never get ‘w’ by
‘f+g’. This denotation is useful when we express the explicit solution of an
equation.
Definition 2.8 Let B is a finite set and (B,+) is a monoid. We define additive
operation ‘+’ over set of additive relations $P(B^{i+1})(1\leq i\leq M)$
defined over (B,+).[B,$P(B^{i+1})(1\leq i\leq M)$,+] will be called an
algebraic system of equations over B.
## 3\. Basic theorems
We take additive relations of two variables as examples to show some law below
and they are easy to be extended to ones of many variables.
Here $R^{2}_{i}$ is an additive relation of two variables and $\beta_{i}$ one
of one variable.
Theorem 3.1 Addition satisfies commutative law and associative law.
$R^{2}_{1}+R^{2}_{2}=R_{2}^{2}+R^{2}_{1}$
$(R^{2}_{1}+R^{2}_{2})+R^{2}_{3}=R^{2}_{1}+(R^{2}_{2}+R^{2}_{3})$
Theorem 3.2 Composition satisfies the transposal law.
$R^{2}_{1}\times_{i}\beta_{1}\times_{j}\beta_{2}=R^{2}_{1}\times_{j}\beta_{2}\times_{i}\beta_{1}\qquad(i\neq
j)$
Theorem 3.3 Addition and composition satisfy left distributive law.
$(R^{2}_{1}+R^{2}_{2})\times_{i}\beta=R^{2}_{1}\times_{i}\beta+R^{2}_{2}\times_{i}\beta\qquad(i\neq
0)$
Theorem 3.4 Transformation satisfies associative law.
$[(c_{1}c_{2})c_{3}](R^{2})=[c_{1}(c_{2}c_{3})](R^{2})$
Transformations make a group with identity element $T^{1,2,0}$ but not an
abelian group because it doesn’t satisfy the commutative law.
Theorem 3.5 Transformation $T^{2,1,0}$ and composition is equal a new
composition and the same transformation.
$[T^{2,1,0}(R^{2})]\times_{1}\beta=T^{2,1,0}(R^{2}\times_{2}\beta)$
$[T^{2,1,0}(R^{2})]\times_{2}\beta=T^{2,1,0}(R^{2}\times_{1}\beta)$
Theorem 3.6 Addition and transformation $T^{2,1,0}$ satisfy left distributive
law.
$T^{2,1,0}(R^{2}_{1}+R^{2}_{2})=T^{2,1,0}(R^{2}_{1})+T^{2,1,0}(R^{2}_{2})$
## 4\. Solvability of an algebraic system of equations
If an additive relation of M variables can be written in the expression
consisting of additive relations of one variable $f_{i},g_{ij}$ like this:
(4.1)
$R(x_{1},x_{2},\cdots,x_{M})=\sum_{i=1}^{L}f_{i}\big{[}g_{i1}(x_{1})+g_{i2}(x_{2})+\cdots+g_{in}(x_{M})\big{]}$
then we say that it can be represented as a superposition of additive
relations of one variable or decomposing it to form in additive relations of
one variable. We have our core theorem below,a very very important theorem!
Theorem 4.1 $B=\\{0,1,\cdots,N\\}(N\geq 3)$ then $R^{i}(B)(2\leq i\leq M)$can
be represented as a superposition of additive relations of one variable.
Definition 4.1 A singular additive relation of M variables is one with only
one non-zero point.
Proof step1: It holds for the below singular additive relation of two
variables.
| 0 | 1 | 2
---|---|---|---
0 | 1 | 0 | 0
1 | 0 | 0 | 0
2 | 0 | 0 | 0
It can be expressed as:
$R^{2}_{1}(x_{1},x_{2})=(0,0,1)\Big{[}(1,0,0)(x_{1})+(1,0,0)(x_{2})\Big{]}$
Step 2: It holds for a general singular additive relation of two variables.
(1,0,0) in (1,0,0)$(x_{1})$ in the expression of $R^{2}_{1}$ is called row
additive relation of one variable. Row including the none-zero point will
change if we adjust the location of ‘1’ in (1,0,0)$(x_{1})$. (1,0,0) in
(1,0,0)$(x_{2})$ in it is called column additive relation of one variable.
Column including the none-zero point will change if we adjust the location of
‘1’in (1,0,0)$(x_{2})$. Both row additive relation of one variable or column
additive relation of one variable is called location additive relation of one
variable.(0,0,1) is called value additive relation of one variable. Value
which may be single-valued or many-valued or no-valued will change if we
replace ‘1’ with other numbers. Thus we know that it can be represented as a
superposition of additive relations of one variable wherever the none-zero
point is.
Step 3: It holds for a general 2-th-order additive relation of two variables.
Because every additive relation of two variables can be transformed to sum of
9 singular additive relation of two variables then we get our conclusion.
Step 4: We can extend the result to general N-th-order additive relations of M
variables.
In the expression of $R^{2}_{1}$ if we replace row additive relation of one
variable (1,0,0)$(x_{1})$ or column additive relation of one variable
(1,0,0)$(x_{2})$ by (1,0,$\cdots$,0)$(x_{1})$ or(1,0,$\cdots$,0)$(x_{2})$ and
value function (0,0,1) by (0,0,$\cdots$,0,v) respectively then we can extend
this expression to order N additive relations of two variables. There will be
more location additive relations of one variable (1,0,0, $\cdots$,0) when we
extend the result to general N-th-order additive relations of M variables.
Here v in (0,0,$\cdots$,0,v)can be single valued or many-valued or undefined
so theorem 4.1 will be hold in any case.
Step 5: If N$<$M+1, the number of location additive relation of one variable
in the expression of singular N-th-order additive relation of two variables
will be bigger than M+1. For example a 3-th-order singular additive relation
of three variables $R^{3}(x_{1},x_{2},x_{3})$ can be represented as:
$R^{3}(x_{1},x_{2},x_{3})=(0,0,1)\Big{\\{}(0,0,1)\big{[}(1,0,0)(x_{1})+(1,0,0)(x_{2})\big{]}+(1,0,0)(x_{3})\Big{\\}}$
Note there is an additional location additive relation of one variable (0,0,1)
between symbols ‘$\Big{\\{}$’ and ‘[’.
It’s easy to check that 2-th-order singular additive relation of two variables
can’t be represented as a superposition of additive relations of one variable.
The location additive relation of one variable $g_{ij}$ will not change with W
but the value additive relation of one variable $f_{i}$ is not the same for
different W so we express $f_{i}$ as below:
(4.2) $f_{i}=V_{i}(W)\qquad\qquad\qquad\qquad(1\leq i\leq K)$
The decomposing method shown above is called trivial method and the number of
terms of expression gotten by it is equal to $N^{M}$. The method will be
called non-trivial one if the number of terms of expression gotten by it is
less than $N^{M}$.
Let B=$\\{0,1,2,\cdots N-1\\}$, here N is a odd number, additive relations of
two variables defined over it can be decomposed to N+1 terms.
$\sum_{i=0}^{N}h_{i}[f_{i}(x_{1})+g_{i}(x_{2})]=$
($a_{11}$,$a_{12}$,$a_{13}$,$\cdots$,$a_{1N-1}$,$a_{1N}$)[(1,0,$\cdots$,0,0,0)(x)+(N-2,$\cdots$,3,2,1,0,0)(y)]
+($a_{21}$,$a_{22}$,$\cdots$,$a_{2N-1}$,0)[(0,0,1,2,3,$\cdots$,N-2)(x)+(0,0,$\cdots$,0,0,0,1)(y)]
+($a_{31}$,$a_{32}$,$\cdots$,$a_{3N-1}$,0)[(N-2,$\cdots$,3,2,1,0,0)(x)+(0,0,$\cdots$,0,0,0,1)(y)]
+($a_{41}$,$a_{42}$,$\cdots$,$a_{4N-1}$,0)[(N-1,$\cdots$,4,3,2,1,0)(x)+(1,0,$\cdots$,0,0,0,0)(y)]
+($a_{51}$,$a_{52}$,$\cdots$,$a_{5N-1}$,0)[(N-1,$\cdots$,4,3,2,1,0)(x)+(0,1,$\cdots$,0,0,0,0)(y)]
$\cdots$
+($a_{N1}$,$a_{N2}$,$\cdots$,$a_{NN-1}$,0)[(N-1,$\cdots$,4,3,2,1,0)(x)+(0,0,$\cdots$1,0,0,0)(y)]
+($a_{N+11}$,$a_{N+12}$,$\cdots$,$a_{N+1N-1}$,0)[(N-1,$\cdots$,4,3,2,1,0)(x)+(0,0,$\cdots$,0,1,0,0)(y)]
(3)
The fourth term and ones before it are called original items and others
plagiarized items. The correctness of the decomposition is easy to be validate
by building and solving a set of equations. Expressions with only N terms may
be gotten. It’s need to give the procedure to decompose any additive relation
of many variables.
Let $B=\\{0,1,\cdots,N\\}(N\geq 3)$,$A=R^{1}(B)$, [A,$P(A^{i+1})(1\leq i\leq
M)$,+] will be an algebraic system of equations over A and there will be many
operator equations.
An algebraic system of equations is solvable if any equation in it can be
given a formula solution. Actually theorem 4.1 give us a conclusion that the
system is solvable. First we judge a new system if it’s solvable when we meet
it. We shall downplay it if it is unsolvable because there is few elements in
the basic set and is very poor content needs to be studied. There is plenty to
be researched in both algebraic system of equations $[B,P(B^{i+1})(1\leq i\leq
M),+]$ and $[A,P(A^{i+1})(1\leq i\leq M),+]$. One of both includes so many
algebraic equations and another includes so many operator equations. Theorem
4.1 give us a constructive method to give a formula solution for any equation
in that system actually. But the number of terms is too big so it will be a
core task for us to get the shortest expression then we can get the perfect
formula solution.
We can proof the solvability of $[A,P(A^{i+1})(1\leq i\leq M),+]$ and study
equations in it then actually we break a new path for functional analysis.
## 5\. Formula solution of the double branches equation
Let $B=\\{0,1,2\\}$and equation$(xR^{2}_{1}a)R^{2}_{3}$ $(xR^{2}_{2}b)=c$ is
called the double branches equation in which $R^{2}_{j}\in P(B^{3})(1\leq
j\leq 3)$. We solve it as the following procedure:
Step 1: Decompose $R^{2}_{3}$ to:
$R^{2}_{3}(u,v)=\sum_{i=1}^{4}f_{i}[g_{i1}(u)+g_{i2}(v)]$
$\sum_{i=1}^{4}f_{i}\Big{[}g_{i1}(xR^{2}_{1}a)+g_{i2}(xR^{2}_{2}b)\Big{]}=c$
Step 2: By composition change $g_{i1}(xR^{2}_{1}a)$ to
$x(R^{2}_{1}\times_{0}g_{i1}^{-1})a$ and change $g_{i2}(xR^{2}_{2}b)$ to
$x(R^{2}_{2}\times_{0}g_{i2}^{-1})b$:
$\sum_{i=1}^{4}f_{i}\Big{[}x(R^{2}_{1}\times_{0}g_{i1}^{-1})a+x(R^{2}_{2}\times_{0}g_{i2}^{-1})b\Big{]}=c$
Step 3: Change $R^{2}_{1}\times_{0}g_{i1}^{-1}$ and
$R^{2}_{2}\times_{0}g_{i2}^{-1}$ to false ones of three variables and add
them:
$R^{3}_{3i}(x,a,b)=[F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})](x,a,b)\qquad(1\leq
i\leq 4)$
Step 4: By composition of $R^{3}_{3i}$ we get:
$R^{3}_{4i}(x,a,b)=R^{3}_{3i}\times_{0}f_{i}^{-1}(x,a,b)\qquad(1\leq i\leq 4)$
Step 5: To sum $R^{3}_{4i}$ we get:
$R^{3}_{5}(x,a,b)=\sum_{i=1}^{4}R^{2}_{4i}(x,a,b)$
The equation will become to:
$R^{3}_{5}(x,a,b)=c$
Step 6: By inverse we get:
$x=\Big{[}T^{2,3,0,1}(R^{3}_{5})\Big{]}(a,b,c)=W(a,b,c)$
Step 7: By decomposing the additive relation of three variables we get:
$x=\sum_{k=1}^{27}(V_{k}W)\Bigg{\\{}(g_{k4})\Big{[}(g_{k1})(a)+(g_{k2})(b)\Big{]}+(g_{k3})(c)\Bigg{\\}}$
We replace logogram symbols by complete ones.
$x=\sum_{k=1}^{27}V_{k}\Big{[}T^{2,3,0,1}\Big{(}\sum_{i=1}^{4}\big{\\{}\big{[}F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})]\times_{0}(V_{i}R^{2}_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{\\{}g_{k4}\Big{[}T^{2,3,0,1}\Big{(}\sum_{i=1}^{4}\big{\\{}\big{[}F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})]\times_{0}(V_{i}R^{2}_{3})^{-1}\big{\\}}\Big{)}\Big{]}$
$\Bigg{[}g_{k1}\Big{[}T^{2,3,0,1}\Big{(}\sum_{i=1}^{4}\big{\\{}\big{[}F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})]\times_{0}(V_{i}R^{2}_{3})^{-1}\big{\\}}\Big{)}\Big{]}(a)$
$+g_{k2}\Big{[}T^{2,3,0,1}\Big{(}\sum_{i=1}^{4}\big{\\{}\big{[}F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})]\times_{0}(V_{i}R^{2}_{3})^{-1}\big{\\}}\Big{)}\Big{]}(b)\Bigg{]}$
$+g_{k3}\Big{[}T^{2,3,0,1}\Big{(}\sum_{i=1}^{4}\big{\\{}\big{[}F^{3}_{1,2}(R^{2}_{1}\times_{0}g_{i1}^{-1})+F^{3}_{1,3}(R^{2}_{2}\times_{0}g_{i2}^{-1})]\times_{0}(V_{i}R^{2}_{3})^{-1}\big{\\}}\Big{)}\Big{]}(c)\Bigg{\\}}$
$(xR^{2}_{1}a)R^{2}_{3}$$(xR^{2}_{2}b)=c$ will be an operator equation if we
replace the elements of B=$\\{0,1,2\\}$ by functions of one variable then
B=$\\{(0,0,0),(2,0,1),$ $(1,0,2)\\}$ and we can also give its explicit
solution!
We can see that we give the formula solution to this equation not by axioms of
arithmetic but by decomposing additive relations of many variables.
## 6\. Conclusion and expectation
We build the algebraic system of equations which is a new three grade
algebraic system then we can research general additive relations which include
all operations.We show that there is always formula solution for any equation
in this system and developed a constructive method for it. Importance of the
results shown in this paper is twofold.In the first place,the new system study
all relations which include all operations and very few of them satisfy axioms
of arithmetic and are studied by algebra. So the new system is a great break
to algebra. In the second place, we shall find formula solutions of equations
not by axioms of arithmetic but by expressing additive relations of many
variables in the superposition of ones of one variable.
reference
[1] A.N.Kolmogorov, On the representation of continuous functions of several
variables by superpositions of continuous functions of one variable and
addition, Dokl.Akad.Nauk SSSR 114 (1957), 953-956; English transl., Amer.Math.
Soc.Transl. (2) 28 (1963), 55-59.
[2] D.Hilbert, Mathematical Problems, Bull.Amer.Math. Soc.8(1902),461-462.
[3]H.Umemura. Solution of algebraic equations in terms of theta constants. In
D.Mumford, Tata.Lectures on Theta II, Progress in Mathematics 43, Birkh user,
Boston, 1984.
[4] V.I.Arnol d, On functions of three variables, Dokl.Akad. Nauk SSSR 114
(1957), 679 C681; English transl., Amer. Math. Soc. Transl.(2) 28 (1963), 51
C54.
[5] V.I.Arnol d, On the representation of continuous functions of three
variables by superpositions of continuous functions of two variables, Mat.Sb.
48 (1959), 3 C74; English transl., Amer.Math. Soc.Transl.(2) 28 (1963), 61
C147.
|
arxiv-papers
| 2012-03-01T02:44:40 |
2024-09-04T02:49:28.282566
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ziqian Wu",
"submitter": "Ziqian Wu Professor",
"url": "https://arxiv.org/abs/1203.0987"
}
|
1203.1031
|
# Are megaquakes clustered?
###### Abstract
We study statistical properties of the number of large earthquakes over the
past century. We analyze the cumulative distribution of the number of
earthquakes with magnitude larger than threshold $M$ in time interval $T$, and
quantify the statistical significance of these results by simulating a large
number of synthetic random catalogs. We find that in general, the earthquake
record cannot be distinguished from a process that is random in time. This
conclusion holds whether aftershocks are removed or not, except at magnitudes
below $M=7.3$. At long time intervals ($T$ = 2-5 years), we find that
statistically significant clustering is present in the catalog for lower
magnitude thresholds ($M$ = 7-7.2). However, this clustering is due to a large
number of earthquakes on record in the early part of the 20th century, when
magnitudes are less certain.
Gindraft=false DAUB ET AL. ARE MEGAQUAKES CLUSTERED? E. Ben-Naim, Theoretical
Division, Los Alamos National Laboratory, MS B213, Los Alamos, NM 87545, USA.
(ebn@lanl.gov) E. G. Daub, R. A. Guyer, P. A. Johnson, Geophysics Group, Los
Alamos National Laboratory, MS D443, Los Alamos, NM 87545, USA.
(edaub@lanl.gov)
11affiliationtext: Earth and Environmental Sciences Division, Los Alamos
National Laboratory, Los Alamos, New Mexico, USA.22affiliationtext: Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico,
USA.33affiliationtext: Theoretical Division, Los Alamos National Laboratory,
Los Alamos, New Mexico, USA.44affiliationtext: Physics Department, University
of Nevada, Reno, Nevada, USA.
## 1 Introduction
The number of powerful earthquakes worldwide has increased over the past
decade (Fig. 1 (left)). This increase has prompted debate whether large
earthquakes cluster in time [Kerr, 2011]. If so, this would have an impact on
how seismic hazard is assessed worldwide. Multiple studies have investigated
this question [Bufe and Perkins, 2005; Brodsky, 2009; Michael, 2011; Shearer
and Stark, 2012; Ammon et al., 2011; Bufe and Perkins, 2011]. Conclusions have
been mixed, with some studies finding evidence of clustering [Bufe and
Perkins, 2005; 2011], while others have concluded that earthquakes cannot be
distinguished from a process that is random in time [Michael, 2011; Shearer
and Stark, 2012].
Figure 1: (left) The number of large earthquakes, $n$, in a calendar year over
the past century (1900-2011). Here, large earthquakes are defined as events
with magnitude greater than or equal to $M$. Three thresholds were used:
$M=7.0$ (top), $M=7.5$ (middle), and $M=8.0$ (bottom). (right) The cumulative
number $N_{M}$ of large earthquakes with magnitude of at least $M$ during the
time period $1900-2011$.
In parallel, recent studies show that earthquakes can be dynamically triggered
by seismic waves [Hill et al., 1993; Gomberg et al., 2004; Freed, 2005]. It is
not clear if large earthquakes can trigger other large earthquakes; one recent
study did not find evidence of such triggering [Parsons and Velasco, 2011],
although this remains an open question in seismology. If large earthquakes do
cluster in time, this might suggest that large earthquakes can be dynamically
triggered.
We study the statistics of large ($M\geq 7$) earthquakes from 1900-2011 to
assess whether earthquakes deviate from random occurrence. We examine the
catalog both with and without removal of aftershocks, and use transparent
statistical measures to quantify the likelihood that a random process could
produce the earthquake record.
## 2 Data and Aftershock Removal
Our statistical analysis uses the USGS PAGER catalog of large earthquakes
[Allen et al., 2009], supplemented with the Global CMT catalog through the end
of 2011. The catalog consists of 1761 events with magnitude $M>7.0$. As can be
seen from the magnitude-frequency plot in Fig. 1 (right), this catalog adheres
to the ubiquitous Gutenberg-Richter law [Gutenberg and Richter, 1954], and is
complete for magnitude $M>7.0$. The magnitudes in the PAGER catalog are a mix
of magnitude types – the majority of events are given in moment magnitude, but
events early in the century often use a different magnitude measure, such as
surface wave magnitude. Because very large earthquakes are rare, any study of
the statistics of this dataset is inherently limited by the small number of
extremely powerful earthquakes on record.
We have studied two additional catalogs, one compiled by Pacheco and Sykes
[1992], and one based on the NOAA Significant Earthquake Database (National
Geophysical Data Center/World Data Center (NGDC/WDC) Significant Earthquake
Database, Boulder, CO, USA, available at
http://www.ngdc.noaa.gov/hazard/earthqk.shtml). We find that the results
depend on the catalog choice due to discrepancies in magnitude between the
catalogs. Because PAGER contains more events, and the magnitudes in PAGER are
the most consistent with the Gutenberg-Richter Law, we focus on PAGER in our
analysis. A comprehensive study of the discrepancies between catalogs will be
the subject of future work.
While the PAGER catalog is the most complete record of large earthquakes, the
data has limitations. First, because seismic instruments were relatively
sparse in the first half of the 20th century, data for these events have
larger uncertainties. Additionally, the data includes aftershocks. Aftershock
removal is not trivial, and it requires assumptions that cannot be tested
rigorously due to limited data.
We remove aftershocks by flagging any event within a specified time and
distance window of a larger magnitude main shock [Gardner and Knopoff, 1974].
We use the time window from the original Gardner and Knopoff study. The
distance window should be similar to the rupture length of the main shock.
However, rupture length data does not exist for the entire catalog. Therefore,
we must estimate the rupture length based on magnitude. This is problematic
because the catalog contains multiple types of faulting (i.e. subduction
megathrust, crustal strike-slip, etc.), each with a different typical rupture
length for a given magnitude. For example, the 2002 $M=7.9$ Denali earthquake
and the 2011 $M=9.0$ Tohoku Earthquake did not have substantially different
rupture lengths [Eberhart-Philips et al., 2003; Simons et al., 2011] despite a
large difference in seismic moment. We use an empirical rupture length formula
[Wells and Coppersmith, 1994], and choose to be conservative by doubling the
Wells and Coppersmith subsurface rupture length estimate for reverse faulting.
We have studied various choices for this rupture length multiplicative factor,
and find that doubling the rupture length estimate makes the rupture lengths
large enough to be fairly conservative, but not so large as to excessively
remove events from the catalog. This may remove some events from the catalog
that are not aftershocks, but it will not bias our results by leaving many
aftershocks in the catalog. After removal of aftershocks, the PAGER catalog is
reduced to 1253 events. In this investigation, we first examine the entire
catalog to draw as much information from the raw data as possible before
introducing assumptions about aftershocks.
## 3 Statistical Analysis
Our study utilizes the cumulative probability distribution of the number of
large earthquakes in a fixed time interval $Q_{n}$. The cumulative
distribution gives the probability that there are at least $n$ earthquakes
with magnitude of at least $M$ in a given time interval $T$, measured in
months. We compare the observed frequency distribution $Q_{n}$ with the
frequency distribution for a random Poisson process. Let the average number of
large earthquakes in a time interval be $\alpha$. If large earthquakes are not
correlated in time, then the probability $P^{{\rm rand}}_{n}$ that there are
$n$ events during a time interval is
$P^{{\rm rand}}_{n}=\frac{\alpha^{n}}{n!}e^{-\alpha}.$ (1)
The Poisson distribution is characterized by a single parameter, the average.
We also note that the average and the variance are identical, $\langle
n\rangle=\langle n^{2}\rangle-\langle n\rangle^{2}=\alpha$. The cumulative
distribution for a Poissonian catalog $Q^{{\rm rand}}_{n}$ is given by the
following sum:
$Q^{{\rm rand}}_{n}=\sum_{m=n}^{\infty}P^{\rm
rand}_{m}=\sum_{m=n}^{\infty}\frac{\alpha^{m}}{m!}e^{-\alpha}.$ (2)
Note that $Q^{\rm rand}_{n}$ depends on the choice of $M$ and $T$, as these
determine the average event rate $\alpha$. We calculate $Q_{n}$ for the
earthquake data, and compare the data with the expected distribution for a
Poissonian catalog $Q^{{\rm rand}}_{n}$. Note that the cumulative distribution
forms the basis of one of the statistical tests used in Shearer and Stark
[2012], but here we explore many time bin sizes to see if the results depend
on the choice of the time window.
Figure 2 (left) shows an example of the cumulative distribution plot for the
raw PAGER catalog for $M=7$ and $T=12$ months. The cumulative distribution
$Q_{n}$ quantifies the probability that a time window contains at least $n$
events. Thus, the curves always begin at $Q_{0}=1$, and decrease as $n$
increases. The final point on each plot corresponds to the maximum number of
events observed in the chosen time window.
Figure 2: The cumulative frequency distribution at different threshold
magnitudes and time intervals. (left) $Q_{n}$ versus $n$ for $M=7.0$ and
$T=12$ months, compared to the distribution expected for a random catalog.
(right) $Q_{n}$ versus $n$, obtained using magnitude thresholds $M=7.0$ (top),
$M=7.5$ (middle), and $M=8.0$ (bottom) and time intervals $T=1$ month (left),
$T=12$ months (middle), and $T=60$ months (right). The solid lines indicate
the expected distribution for a Poissonian catalog.
Figure 2 (left) shows that the frequency of large earthquakes with $M\geq 7.0$
is roughly Poissonian below the average $\alpha=15.7$ events/year. However,
the tail of the cumulative distribution is overpopulated with respect to the
Poisson distribution. An overpopulated tail indicates that events are
clustered in time. We perform this analysis for higher magnitude thresholds
($M=7.5$, $M=8$) and both longer and shorter time window sizes ($T=1$ month,
$T=60$ months), and the results are shown in Fig. 2 (right). The bins evenly
divide the catalog into an integer number of fixed time windows: $T=1$ month
corresponds to $112\times 12=1344$ bins, and $T=12$ months corresponds to 112
bins. For $T=60$ months, the catalog cannot be evenly divided into 5 year
bins. Therefore, it is instead divided into the closest integer number of bins
(22), which means that the bin size is actually slightly larger than 60
months.
We find that the catalog exhibits an overpopulated tail only for $M=7$. Within
the $M=7$ data, the overpopulation is found for all $T$. The strength of this
overpopulation is significant because it can be a few orders of magnitude.
However, the catalog at $M=7.5$ and $M=8$ agrees very well with the prediction
for a Poissonian catalog. This is remarkable, as even with a relatively small
number of earthquakes, the data is in agreement with a random distribution.
To quantify the statistical significance of the overpopulation, we utilize the
normalized variance:
$V=\frac{\langle n^{2}\rangle-\langle n\rangle^{2}}{\langle n\rangle}.$ (3)
An observed distribution with a strongly overpopulated tail necessarily has a
large variance. Moreover, a value close to unity is expected for a catalog
that is random in time, while a value larger than unity indicates clustering.
Hence, the normalized variance $V$ is a convenient, scalar, measure of
clustering. The normalized variance is shown as a function of $M$ and $T$ in
Fig. 3 (left), and confirms that at $M=7$ the catalog is clustered. In this
analysis, we compute $V$ with many different bin sizes, ranging from 1 month
up to approximately 5 years. In each case, the number of bins is chosen to be
an integer so that we always utilize the entire catalog (i.e. the time bin
size is not always an integer number of months).
Figure 3: (left) The normalized variance $V$ versus magnitude threshold $M$
and the time interval $T$ (in months). The normalized variance is color coded
with red indicating strong overpopulation and blue indicating a random
distribution. (right) Standard deviations above the mean variance $\sigma$ as
a function of $M$ and $T$, determined from $10^{6}$ Poissonian synthetic
catalogs. Again, statistically significant overpopulation is indicated in red,
while blue indicates a random distribution.
To test whether the clustering observed in the data is statistically
significant, we generate $10^{6}$ synthetic Poissonian catalogs with an
average event rate given by $\alpha=1761/112$ events/year, the same as in the
PAGER catalog. Each event is assigned a magnitude, drawn randomly from the
actual catalog magnitudes with replacement. Using the $10^{6}$ Poissonian
realizations of the earthquake catalog, we compute the average normalized
variance $\bar{V}$ and the standard deviation of the normalized variance
$\delta V$ as a function of $M$ and $T$. Conveniently, the normalized variance
for an ensemble of synthetic random catalogs is approximately described by a
normal distribution. This makes this quantity useful for determining the
statistical significance of the observed clustering. The normalized variance
determined from the earthquake data $V$ can then be expressed as a certain
number of standard deviations above the mean $\sigma$,
$\sigma=\frac{V-\bar{V}}{\delta V}.$ (4)
Since $V$ is normally distributed for an ensemble of random catalogs, we know
that if the value of $V$ determined from the data is larger than $\bar{V}$ by
several standard deviations, this indicates that the catalog contains
statistically significant clustering.
The number of standard deviations above the mean $\sigma$ is shown as a
function of $M$ and $T$ in Fig. 3 (right). In the plot, red indicates
statistically significant clustering, and blue indicates a variance consistent
with a random catalog. This analysis verifies the results from the cumulative
distribution: clustering is observed at low magnitudes ($M<7.3$), while no
significant clustering is observed at higher magnitudes ($M\geq 7.3$). This
observation is robust over time bin sizes ranging from 1 month to 5 years.
Note that while the normalized variance is much larger for $M=7$ and $T=60$
months than for $M=7$ and $T=1$ month, in both cases the normalized variance
is several standard deviations above the mean. This is because there is more
variability in the normalized variance for longer time bins – we find that
$\delta V\sim T^{1/2}$, independent of the magnitude threshold. We stress that
our analysis thus far relies on the complete earthquake record which
necessarily includes aftershocks. Hence, aftershock removal is not even
necessary to demonstrate that the statistics of large earthquakes with
magnitude $M>7.3$ show no significant clustering.
We repeat the above analysis, with aftershocks removed, to test if the
clustering observed for $M<7.3$ is due to aftershocks. The results of the
cumulative distribution analysis with aftershocks removed is shown in Fig. 4.
The catalog now closely follows the cumulative distribution for a Poissonian
catalog for $M=7$, $T=1$ month, demonstrating that the clustering at short
times and lower magnitudes is due to aftershocks. There is still
overpopulation for $M=7$ at longer times. At higher magnitudes, many of the
curves appear slightly underpopulated for large numbers of events. This could
be due to our conservative aftershock removal procedure, which may have
removed some independent events.
Figure 4: The cumulative frequency distribution for the catalog with
aftershocks removed at different threshold magnitudes and time intervals.
Shown is $Q_{n}$ versus $n$, obtained using magnitude thresholds $M=7.0$
(top), $M=7.5$ (middle), and $M=8.0$ (bottom) and time intervals $T=1$ month
(left), $T=12$ months (middle), and $T=60$ months (right). The solid lines
indicate the expected distribution for a Poissonian catalog.
Calculations using synthetic catalogs and the variance measure $V$ confirm
these results. Figure 5 shows that the clustering observed for small
magnitudes ($M<7.3$) and short times ($T<12$ months) no longer occurs once
aftershocks are removed from the catalog. Interestingly, the clustering at
longer time intervals ($T>24$ months) persists. Most likely, this clustering
is due to the fact that there is a mismatch between the event rates in the
first and the second halves of the century, the former being larger by about
$20\%$. This can be seen in Fig. 1 (left, top), which shows several spikes in
the number of $M\geq 7$ events during the first half of the century. If we
divide the catalog into two time periods (1900-1955 and 1956-2011), we find
that each half of the data is consistent with random earthquake occurrence,
with a different rate for each half. Because magnitude estimates early in the
century are subject to larger uncertainties and may be systematically
overestimated [Engdahl and Villaseñor, 2002], it is not clear if this
clustering is real or due to less reliable data.
Figure 5: (left) The normalized variance $V$ versus magnitude threshold $M$
and the time interval $T$ (in months) for the catalog with aftershocks
removed. The normalized variance is color coded with red indicating strong
overpopulation and blue indicating a random distribution. (right) Standard
deviations above the mean variance $\sigma$ for the catalog with aftershocks
removed as a function of $M$ and $T$, determined from $10^{6}$ Poissonian
synthetic catalogs. Again, statistically significant overpopulation is
indicated in red, while blue indicates a random distribution.
## 4 Conclusions
Our studies using the PAGER earthquake catalog demonstrate that the catalog
cannot be distinguished from random earthquake occurrence. This is in
agreement with several other recent studies [Michael, 2011; Shearer and Stark,
2012]. We do find evidence of clustering for $M=7$ and $T=2$-5 years, which
was not identified in the other studies. However, we note that this clustering
is due to a large number of events on record early in the 20th Century.
For large events ($M>7.3$), the catalog with aftershocks is well described by
a process that is random in time. This is because large aftershocks are rare,
and there are relatively few large events in the catalog to begin with.
Because clustering due to aftershocks, which is known to be present in the
data, is not detectable by our statistical tests, it is possible that there is
clustering in the catalog at large magnitudes that is obscured by the small
amount of data. Future studies will examine the likelihood of identifying
clustering in synthetic clustered catalogs given the small amount of data in
the earthquake catalog.
These findings underscore that we have very little megaquake data, due to
limited instrumentation. Increases in the number of seismic and geodetic
instruments in recent years has led not only to the improved identification
and characterization of large earthquakes, but also to the discovery of novel
slip behaviors such as low frequency earthquakes [Katsumata and Kamaya 2003],
very low frequency earthquakes [Ito et al., 2006], slow slip events [Dragert
et al., 2001], and silent earthquakes [Kawasaki et al., 1995]. Integrating
observations of other types of events with earthquake data may prove to be the
key to identifying causal links between events, providing a comprehensive
picture of the interactions that may underlie the physics of great
earthquakes.
###### Acknowledgements.
The USGS PAGER catalog is available on the web at
http://earthquake.usgs.gov/earthquakes/pager/, and the Global CMT catalog is
available at http://www.globalcmt.org/. We thank Terry Wallace, Thorne Lay,
Charles Ammon, and Joan Gomberg for useful comments. This research has been
supported by DOE grant DE-AC52-06NA25396 and institutional (LDRD) funding at
Los Alamos.
## References
* [1] Allen, T.I., K. Marano, P. S. Earle, and D. J. Wald (2009), PAGER-CAT: A composite earthquake catalog for calibrating global fatality models, Seism. Res. Lett., 80, 50-56.
* [2] Ammon, C. J., R. C. Aster, T. Lay, and D. W. Simpson (2011), The Tohoku Earthquake and a 110-year Spatiotemporal Record of Global Seismic Strain Release, Seismol. Res. Lett., 82, 455.
* [3] Brodsky, E. E. (2009), The 2004-2008 Worldwide Superswarm, Eos. Trans. AGU, Fall Meet. Suppl., 90, S53B.
* [4] Bufe, C. G., and D. M. Perkins (2005), Evidence for a Global Seismic-Moment Release Sequence, Bull. Seismol. Soc. Am., 95, 833-843.
* [5] Bufe, C. G., and D. M. Perkins (2011), The 2011 Tohoku Earthquake: Resumption of Temporal Clustering of Earth’s Megaquakes, Seismol. Res. Lett., 82, 455.
* [6] Dragert, H., K. Wang, and T. S. James (2001), A Silent Slip Event on the Deeper Cascadia Subduction Interface, Science 292, 5521, 1525-1528.
* [7] Eberhart-Philips, D., et al. (2003), The 2002 Denali fault earthquake, Alaska: A large-magnitude, slip-partitioned event, Science, 300, 1113-1118.
* [8] Engdahl, E. R., and A. Villaseñor (2002), Global seismicity: 1900-1999, International Handbook of Earthquake and Engineering Seismology, Volume 81A, ISBN:0-12-440652-1, 665-690.
* [9] Freed, A. M. (2005), Earthquake triggering by static dynamic, and postseismic stress transfer, Ann. Rev. Earth Plant. Sci. 33, 335-367, doi:10.1146/annurev.earth.33.092203.122505.
* [10] Gardner, J. K., Knopoff, L. (1974), Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonian? Bull. Seismol. Soc. Am., 64, 1363-1367.
* [11] Gomberg, J., P. Bodin, K. Larson, and H. Dragert (2004), Earthquake nucleation by transient deformations caused by the M = 7.9 Denali, Alaska, earthquake, Nature, 427, 621-624.
* [12] Gutenberg, B., and C. F. Richter (1954), Seismicity of the Earth and Associated Phenomena, 2nd ed., Princeton University Press, Princeton.
* [13] Hill, D. P., et al. (1993), Remote seismicity triggered by the M7.5 Landers, California earthquake of June 28, 1992, Science, 260, 1617-1623.
* [14] Ito, Y., K. Obara, K. Shiomi, S. Sekine, and H. Hirose (2006), Slow earthquakes coincident with episodic tremors and slow slip events, Science, 26, 503 506.
* [15] Katsumata, A., and N. Kamaya (2003), Low-frequency continuous tremor around the Moho discontinuity away from volcanoes in the southwest Japan, Geophys. Res. Lett., 30, 1020, doi:10.1029/2002GL015981.
* [16] Kawasaki, I. et al. (1995), The 1992 Sanriku-oki, Japan, ultra-slow earthquake, J. Phys. Earth, 43, 105 116.
* [17] Kerr, R. A. (2011), More Megaquakes on the Way? That Depends on Your Statistics, Science, 332, 411.
* [18] Michael, A. J. (2011), Random Variability Explains Apparent Global Clustering of Large Earthquakes, Geophys. Res. Lett., 38, L21301, doi:10.1029/2011GL049443.
* [19] Pacheco, J. F., and L. R. Sykes (1992), Seismic moment catalog of large shallow earthquakes, 1900 to 1989, Bull. Seismol. Soc. Am., 82, 1306-1349.
* [20] Peng., Z., and J. Gomberg (2010), An integrated perspective of the continuum between earthquakes and slow-slip phenomena, Nat. Geosci., 3, 599-607.
* [21] Shearer, P. M., and P. B. Stark, (2012), The global risk of big earthquakes has not recently increased, Proc. Nat. Acad. Sci., 109(3), 717-721.
* [22] Simons, M., et al. (2011), The 2011 Magnitude 9.0 Tohoku-Oki Earthquake: Mosaicking the Megathrust from Seconds to Centuries, Science, 332, 1421-1425.
* [23] Parsons, T., and A. A. Velasco (2011), Absence of remotely triggered large earthquakes beyond the mainshock region, Nat. Geosci., 4, 312-316.
* [24] Wells, D. L., and K. J. Coppersmith (1994), New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement, Bull. Seismol. Soc. Am., 84, 1053-1069.
|
arxiv-papers
| 2012-03-05T20:51:34 |
2024-09-04T02:49:28.289923
|
{
"license": "Public Domain",
"authors": "Eric G. Daub, Eli Ben-Naim, Robert A. Guyer, and Paul A. Johnson",
"submitter": "Eric Daub",
"url": "https://arxiv.org/abs/1203.1031"
}
|
1203.1146
|
# Slant helices in three dimensional Lie groups
O. Zeki Okuyucu1∗, İ.Gök2, Y. Yaylı2 and N. Ekmekci2 1 Bilecik Şeyh Edeabali
University, Faculty of Sciences and Arts, Department of Mathematics, 11210,
Bilecik, Turkey. osman.okuyucu@bilecik.edu.tr 2 Ankara University, Faculty of
Science, Department of Mathematics, 06100, Tandog̃an, Ankara, Turkey.
igok@science.ankara.edu.tr yayli@science.ankara.edu.tr
nekmekci@science.ankara.edu.tr
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ; Date: ∗
Corresponding author; Date: Received: xxxxxx; Revised: yyyyyy; Accepted:
zzzzzz. ; Date: ∗ Corresponding author)
###### Abstract.
In this paper, we define slant helices in three dimensional Lie Groups with a
bi-invariant metric and obtain a characterization of slant helices. Moreover,
we give some relations between slant helices and their involutes, spherical
images.
###### Key words and phrases:
Slant helices, curves in a Lie groups.
###### Key words and phrases:
Slant helices, curves in a Lie groups.
###### Key words and phrases:
Slant helices, curves in a Lie groups.
###### Key words and phrases:
Slant helices, curves in a Lie groups.
###### 2010 Mathematics Subject Classification:
Primary 53A04; Secondary 22E15.
## 1\. Introduction
In differential geometry, we think that curves are geometric set of points of
loci. Curves theory is important workframe in the differential geometry
studies and we have a lot of special curves such as geodesics, circles,
Bertrand curves, circular helices, general helices, slant helices etc.
Characterizations of these special curves are heavily studied for a long time
and are still studied. We can see helical structures in nature and mechanic
tools. In the field of computer aided design and computer graphics, helices
can be used for the tool path description, the simulation of kinematic motion
or design of highways. Also we can see the helix curve or helical structure in
fractal geometry, for instance hyperhelices. In differential geometry; a curve
of constant slope or general helix in Euclidean 3-space $\mathbb{E}^{3}$, is
defined by the property that its tangent vector field makes a constant angle
with a fixed straight line (the axis of the general helix). A classical result
stated by M. A. Lancret in 1802 and first proved by B. de Saint Venant in 1845
(see [1, 2] for details) is: A necessary and sufficient condition that a curve
be a general helix is that the ratio of curvature to torsion is constant. If
both of $\varkappa$ and $\tau$ are non-zero constants then the curve is called
as a circular helix. It is known that a straight line and a circle are
degenerate-helix examples ($\varkappa=0$, if the curve is straight line and
$\tau=0$, if the curve is a circle).
The Lancret theorem was revisited and solved by Barros [3] in $3$-dimensional
real space forms by using killing vector fields along curves. Also in the same
spaceforms, a characterization of helices and Cornu spirals is given by
Arroyo, Barros and Garay in [4].
The degenarete semi-Riemannian geometry of Lie group is studied by Çöken and
Çiftçi [5]. Moreover, they obtanied a naturally reductive homogeneous semi-
Riemannian space using the Lie group. Then Çiftçi [6] defined general helices
in three dimensional Lie groups with a bi-invariant metric and obtained a
generalization of Lancret’s theorem and gave a relation between the geodesics
of the so-called cylinders and general helices.
Recently, Izumiya and Takeuchi, in [7], have introduced the concept of slant
helix in Euclidean $3$-space. A slant helix in Euclidean space
$\mathbb{E}^{3}$ was defined by the property that its principal normal vector
field makes a constant angle with a fixed direction. Moreover, Izumiya and
Takeuchi showed that $\alpha$ is a slant helix if and only if the geodesic
curvature of spherical image of principal normal indicatrix $\left(N\right)$
of a space curve $\alpha$
$\sigma_{N}\left(s\right)=\left(\frac{\varkappa^{2}}{\left(\varkappa^{2}+\tau^{2}\right)^{3/2}}\left(\frac{\tau}{\varkappa}\right)^{\prime}\right)\left(s\right)$
is a constant function. In [8]; Kula and Yayli have studied spherical images
of a slant helix and showed that the spherical images of a slant helix are
spherical helices. In [9], the authors characterize slant helices by certain
differential equations verified for each one of spherical indicatrix in
Euclidean $3$-space. Ali and Lopez, in [10], have studied slant helix in
Minkowski $3$-space. They showed that the spherical indicatrix of a slant
helix are helices in $\mathbb{E}_{1}^{3}$. Then Ali and Turgut studied
position vector of a time-like slant helix with respect to standard frame of
Minkowski space $\mathbb{E}_{1}^{3}$ in terms of Frenet equations (see [11]
for details). Also slant helices are used in some applications in quaternion
algebra (see [12, 13] for details).
In this paper, first of all, we define slant helices in a three dimensional
Lie group $G$ with a bi-invariant metric as a curve
$\alpha:I\subset\mathbb{R\rightarrow}G$ whose normal vector field makes a
constant angle with a left invariant vector field (Definition 3.1). And then
the main result to this paper is given as (Theorem 3.6): A curve
$\alpha:I\subset\mathbb{R\rightarrow}G$ with the Frenet apparatus
$\left\\{T,N,B,\varkappa,\tau\right\\}$ is a slant helix if and only if
$\frac{\varkappa(H^{2}+1)^{\frac{3}{2}}}{H^{\shortmid}}$
is a constant function where $H$ is a harmonic curvature function of the curve
$\alpha$ (Definition 3.2).
Then we define the involutes and spherical image of a curve in three
dimensional Lie group $G$. Also we show that the spherical image of a slant
helix and the involutes of a slant helix are general helices. Finally, we give
characterization of a slant helix if $G$ are Abellian, $SO^{3}$ and $S^{3}$.
Note that three dimensional Lie groups admitting bi-invariant metrics are
$SO\left(3\right),SU^{2}$ and Abellian Lie groups. So we believe that
characterizations of slant curves in this study will be useful for curves
theory in Lie groups.
## 2\. Preliminaries
Let $G$ be a Lie group with a bi-invariant metric $\left\langle\text{
},\right\rangle$ and $D$ be the Levi-Civita connection of Lie group $G.$ If
$\mathfrak{g}$ denotes the Lie algebra of $G$ then we know that $\mathfrak{g}$
is issomorphic to $T_{e}G$ where $e$ is neutral element of $G.$ If
$\left\langle\text{ },\right\rangle$ is a bi-invariant metric on $G$ then we
have
$\left\langle
X,\left[Y,Z\right]\right\rangle=\left\langle\left[X,Y\right],Z\right\rangle$
(2.1)
and
$D_{X}Y=\frac{1}{2}\left[X,Y\right]$ (2.2)
for all $X,Y$ and $Z\in\mathfrak{g}.$
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted curve and
$\left\\{X_{1},X_{2,}...,X_{n}\right\\}$ be an orthonormal basis of
$\mathfrak{g}.$ In this case, we write that any two vector fields $W$ and $Z$
along the curve $\alpha\ $as $W=\sum_{i=1}^{n}w_{i}X_{i}$ and
$Z=\sum_{i=1}^{n}z_{i}X_{i}$ where $w_{i}:I\rightarrow\mathbb{R}$ and
$z_{i}:I\rightarrow\mathbb{R}$ are smooth functions. Also the Lie bracket of
two vector fields $W$ and $Z$ is given
$\left[W,Z\right]=\sum_{i=1}^{n}w_{i}z_{i}\left[X_{i},X_{j}\right]$
and the covariant derivative of $W$ along the curve $\alpha$ with the notation
$D_{\alpha^{\shortmid}}W$ is given as follows
$D_{\alpha^{\shortmid}}W=\overset{\cdot}{W}+\frac{1}{2}\left[T,W\right]$ (2.3)
where $T=\alpha^{\prime}$ and
$\overset{\cdot}{W}=\sum_{i=1}^{n}\overset{\cdot}{w_{i}}X_{i}$ or
$\overset{\cdot}{W}=\sum_{i=1}^{n}\frac{dw}{dt}X_{i}.$ Note that if $W$ is the
left-invariant vector field to the curve $\alpha$ then $\overset{\cdot}{W}=0$
(see [14] for details).
Let $G$ be a three dimensional Lie group and
$\left(T,N,B,\varkappa,\tau\right)$ denote the Frenet apparatus of the curve
$\alpha$, and calculate $\varkappa=\overset{\cdot}{\left\|T\right\|}.$
###### Definition 2.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve. Then
$\alpha$ is called a general helix if it makes a constant angle with a left-
invariant vector field $X$. That is,
$\left\langle T(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
for the left-invariant vector field $X\in g$ is unit length and $\theta$ is a
constant angle between $X$ and $T$ which is the tangent vector field of the
curve $\alpha$ (see [6]).
###### Definition 2.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$ then
$\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle$ (2.4)
or
$\tau_{G}=\frac{1}{2\varkappa^{2}\tau}\overset{\cdot\cdot\text{ \ \ \ \ \ \ \
\ }\cdot}{\left\langle
T,\left[T,T\right]\right\rangle}+\frac{1}{4\varkappa^{2}\tau}\overset{\text{ \
\ }\cdot}{\left\|\left[T,T\right]\right\|^{2}}$
(see [6]).
###### Theorem 2.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. If the curve $\alpha$ is
a general helix, if and only if,
$\tau=c\varkappa+\tau_{G}$
where c is a constant (see [6]).
## 3\. Slant helices in a three dimensional Lie group
In this section we define slant helix and its axis in a three dimensional Lie
group $G$ with a bi-invariant metric $\left\langle\text{ },\right\rangle$.
Also we give a characterization and some characterizations of the slant
helices in the special cases of $G$.
###### Definition 3.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve. Then $\alpha$ is called a slant helix if its principal normal vector
makes a constant angle with a left-invariant vector field $X$ which is unit
length. That is,
$\left\langle N(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$
where $\theta\neq\frac{\pi}{2}$ is a constant angle between $X$ and $N$ which
is the principal normal vector field of the curve $\alpha$.
###### Definition 3.2.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}.$ Then
the harmonic curvature function of the curve $\alpha$ is defined by
$H=\dfrac{\tau-\tau_{G}}{\varkappa}$
where $\tau_{G}=\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle.$
###### Definition 3.3.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B,\varkappa,\tau\right\\}$. Then
the geodesic curvature of the spherical image of the principal normal
indicatrix $\left(N\right)$ of the curve $\alpha$ is defined by a constant
$\sigma_{N}$ given by
$\sigma_{N}=\frac{\varkappa(1+H^{2})^{\frac{3}{2}}}{H^{\shortmid}}$
where $H$ is harmonic curvature function of the curve $\alpha.$
###### Proposition 3.4.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B\right\\}$. Then the following
equalities
$\displaystyle\left[T,N\right]$
$\displaystyle=\left\langle\left[T,N\right],B\right\rangle B=2\tau_{G}B$
$\displaystyle\left[T,B\right]$
$\displaystyle=\left\langle\left[T,B\right],N\right\rangle N=-2\tau_{G}B$
hold.
###### Proof.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc length parametrized
curve with the Frenet apparatus $\left\\{T,N,B\right\\}$. Since
$\left[T,N\right]\in Sp\left\\{T,N,B\right\\},$ we can write
$\left[T,N\right]=\lambda_{1}T+\lambda_{2}N+\lambda_{3}B.$ (3.1)
If we multiply the two sides of the Eq. (3.1) with $T,$ $N$ and $B,$
respectively
$\displaystyle\left\langle\left[T,N\right],T\right\rangle$
$\displaystyle=\lambda_{1}=0,$
$\displaystyle\left\langle\left[T,N\right],N\right\rangle$
$\displaystyle=\lambda_{2}=0,$
$\displaystyle\left\langle\left[T,N\right],B\right\rangle$
$\displaystyle=\lambda_{3}.$
Thus we can write
$\left[T,N\right]=\left\langle\left[T,N\right],B\right\rangle B,$
or using the Eq. (2.4) and the last equation, we get
$\left[T,N\right]=2\tau_{G}B.$
On the other hand, using a similar method we can easily show that
$\left[T,B\right]=-2\tau_{G}N.$
Which complete the proof. ∎
###### Proposition 3.5.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a parametrized curve with arc
length parameter s and $\left\\{T,N,B\right\\}$ denote the Frenet frame of the
curve $\alpha$. If the curve $\alpha$ is a slant helix in $G$, then the axis
of $\alpha$ is
$X=\left\\{\frac{\varkappa
H\left(1+H^{2}\right)}{H^{\shortmid}}T+N+\frac{\varkappa\left(1+H^{2}\right)}{H^{\shortmid}}B\right\\}\cos\theta$
where $H=\dfrac{\tau-\tau_{G}}{\varkappa}$ is harmonic curvature function of
the curve $\alpha$ and $\theta\neq\frac{\pi}{2}$ is a constant angle.
###### Proof.
If the axis of slant helix $\alpha$ is $X$, then we can write
$X=\lambda_{1}T+\lambda_{2}N+\lambda_{3}B$
where $\lambda_{1}=\left\langle T,X\right\rangle,$ $\lambda_{2}=\left\langle
N,X\right\rangle$ and $\lambda_{3}=\left\langle B,X\right\rangle.$
And we know from the Definition 3.1 that
$\left\langle N(s),X\right\rangle=\cos\theta\text{ for all }s\in I,$ (3.2)
where the left-invariant vector field $X\in\mathfrak{g}$ is unit length and
$\theta$ is a constant angle between $X$ and $N$ which is the principal normal
vector field of the curve $\alpha$. By differentiating $\left\langle
N(s),X\right\rangle=\cos\theta,$ we get
$\left\langle D_{T}N,X\right\rangle+\left\langle N,D_{T}X\right\rangle=0,$
or using the Eq. (2.3) and the Frenet formulas
$-\kappa\left\langle T,X\right\rangle+\tau\left\langle
B,X\right\rangle-\dfrac{1}{2}\left\langle\left[T,N\right],X\right\rangle=0,$
and with the help of the Proposition 3.4, we get
$\left\langle T,X\right\rangle=H\left\langle B,X\right\rangle,$ (3.3)
where $H=\dfrac{\tau-\tau_{G}}{\varkappa}$ is harmonic curvature function of
the curve $\alpha$.
Again differentiating the Eq. (3.3), we have
$\left\langle D_{T}T,X\right\rangle+\left\langle
T,D_{T}X\right\rangle=H^{\shortmid}\left\langle
B,X\right\rangle+H\left\\{\left\langle D_{T}B,X\right\rangle+\left\langle
B,D_{T}X\right\rangle\right\\}$
then by using the Eq. (2.3) and the Proposition 3.4 we obtain
$\left\langle
B,X\right\rangle=\frac{\varkappa\left(1+H^{2}\right)}{H^{\shortmid}}\left\langle
N,X\right\rangle.$ (3.4)
Then if we write the Eq. (3.4) in the Eq. (3.3), we get
$\left\langle T,X\right\rangle=\frac{\varkappa
H}{H^{\shortmid}}\left(1+H^{2}\right)\left\langle N,X\right\rangle.$ (3.5)
Consequently, using the equations (3.2), (3.4) and (3.5) the axis of slant
helix $\alpha$ is given by
$X=\left\\{\frac{\varkappa
H\left(1+H^{2}\right)}{H^{\shortmid}}T+N+\frac{\varkappa\left(1+H^{2}\right)}{H^{\shortmid}}B\right\\}\cos\theta,$
which completes the proof. ∎
###### Theorem 3.6.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be a unit speed curve with the
Frenet apparatus $\left(T,N,B,\varkappa,\tau\right)$. Then $\alpha$ is a slant
helix if and only if
$\sigma_{N}=\frac{\varkappa(1+H^{2})^{\frac{3}{2}}}{H^{\shortmid}}=\tan\theta$
is a constant where $H$ is a harmonic curvature function of the curve $\alpha$
and $\theta\neq\frac{\pi}{2}$ is a constant.
###### Proof.
If the axis of slant helix $\alpha$ is $X$, then using the Proposition 3.5 we
have
$X=\left\\{\frac{\varkappa
H\left(1+H^{2}\right)}{H^{\shortmid}}T+N+\frac{\varkappa\left(1+H^{2}\right)}{H^{\shortmid}}B\right\\}\cos\theta.$
Since $X$ is unit lenght vector field then we can easily see that
$\frac{\varkappa(H^{2}+1)^{\frac{3}{2}}}{H^{\shortmid}}=\tan\theta$
is a constant.
Conversely, if $\sigma_{N}\left(s\right)$ is constant then the result is
obvious. This complete the proof. ∎
In the following remark, we note that three dimensional Lie groups admitting
bi-invariant metrics are $S^{3},$ $SO^{3}$ and Abelian Lie groups using the
same notation as in [6] and [15] as follows:
###### Remark 3.7.
Let $G$ be a Lie group with a bi-invariant metric $\left\langle\text{
},\right\rangle$. Then the following equalities can be given in different Lie
groups.
$i$ ) If $G$ is abelian group then $\tau_{G}=0.$
$ii)$ If $G$ is $SO^{3}$ then $\tau_{G}=\frac{1}{2}$.
$iii)$ If $G$ is $SU^{2}$ then $\tau_{G}=1$
(see for details [6] and [15]).
###### Corollary 3.8.
Let $\alpha$ be a unit speed curve with the Frenet apparatus
$\left\\{T,N,B\right\\}$ in the Abellian Lie group $G$. Then $\alpha$ is a
slant helix if and only if
$\sigma_{N}=\frac{\left(\varkappa^{2}+\tau^{2}\right)^{3/2}}{\varkappa^{2}\left(\dfrac{\tau}{\varkappa}\right)^{\shortmid}}$
is a constant function.
###### Proof.
If $G$ is Abellian Lie group then using the above Remark and the Theorem 3.6
we have the result. ∎
So, the above Corollary shows that the study is a generalization of slant
helices defined by Izimuya [7] in Euclidean 3-space. Moreover, with a similar
proof, we have the following two corollaries.
###### Corollary 3.9.
Let $\alpha$ be unit speed curve with the Frenet apparatus
$\left\\{T,N,B\right\\}$ in the Lie group $SU^{2}$. Then $\alpha$ is a slant
helix if and only if
$\sigma_{N}=\frac{\left(\varkappa^{2}+\left(\tau-1\right)^{2}\right)^{3/2}}{\varkappa^{2}\left(\dfrac{\tau-1}{\varkappa}\right)^{\shortmid}}$
is a constant function.
###### Corollary 3.10.
Let $\alpha$ be unit speed curve with the Frenet apparatus
$\left\\{T,N,B\right\\}$ in the Lie group $SO^{3}$. Then $\alpha$ is a slant
helix if and only if
$\sigma_{N}=\frac{\left(\varkappa^{2}+\left(\tau-\frac{1}{2}\right)^{2}\right)^{3/2}}{\varkappa^{2}\left(\dfrac{\tau-\frac{1}{2}}{\varkappa}\right)^{\shortmid}}$
is a constant function.
## 4\. Spherical Images of Slant Helices in the three dimensional Lie group
In Euclidean geometry, the spherical indicatrix of a space curve is defined as
follows: Let $\alpha$ be a unit speed regular curve in Euclidean $3$-space
with Frenet vectors $t$ , $n$ and $b$. The unit tangent vectors along the
curve $\alpha$ generate a curve $\alpha_{T}$ on the sphere of radius 1 about
the origin. The curve $\alpha_{T}$ is called the spherical indicatrix of $t$
or more commonly, $\alpha_{T}$ is called tangent indicatrix of the curve
$\alpha$. If $\alpha=\alpha(s)$ is a natural representation of $\alpha$, then
$\alpha_{T}=T(s)$ will be a representation of $\alpha_{T}$. Similarly one
considers the principal normal indicatrix $\alpha_{N}=N(s)$ and binormal
indicatrix $\alpha_{B}=B(s)$. It is clear that, this definition is related
with the spherical curve [2].
In this section, firstly we define spherical indicatrices of slant helices
with the help of the studies [16, 17] and then investigate the relation
between slant helices and their spherical indicatrices in 3-dimensional Lie
group. Morever, we give some theorems with their proofs and some examples in
special Lie groups.
### 4.1. Tangent indicatrices of slant helices:
###### Definition 4.1.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve.
Its tangent indicatrix is the parametrized curve
$\beta:I\subset\mathbb{R\rightarrow}S^{2}\subset\mathfrak{g}$ defined by
$\beta\left(s^{\ast}\right)=T(s)=\sum_{i=1}^{3}\text{ }t_{i}X_{i}\text{ for
all }s\in I$
where $\left\\{X_{1},X_{2},X_{3}\right\\}$ is an orthonormal basis of
$\mathfrak{g}$ and $s^{\ast}$ is the arc length parameter of $\beta.$
###### Theorem 4.2.
Let $\alpha$ be an arc-lenghted regular curve and $\beta$ be the tangent
indicatrix of the curve $\alpha.$ Then the curve $\alpha$ is a slant helix in
three dimensional Lie group $G$ if and only if the curve $\beta$ is a general
helix on $S^{2}$.
###### Proof.
We assume that the curve $\alpha$ is a slant helix in a three dimensional Lie
group and $\alpha_{T}$ is the tangent indicatrix of the curve $\alpha.$ From
the Definition 4.1 we get
$\beta\left(s^{\ast}\right)=T(s)$
then differentiating the last equation and using the Eq. (2.3), we have
$\displaystyle\frac{d\beta}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=\overset{\cdot}{T}=D_{T}T-\dfrac{1}{2}\left[T,T\right]$
$\displaystyle\frac{d\beta}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=\varkappa N.$
Then assuming that $\varkappa\rangle 0$ we obtain
$\frac{ds^{\ast}}{ds}=\varkappa$ (4.1)
and
$T_{\beta}\left(s^{\ast}\right)=N(s).$ (4.2)
If we differentiate the last equation and use Frenet formulas then we obtain
$\displaystyle\varkappa_{\beta}N_{\beta}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}$
$\displaystyle=\overset{\cdot}{N}=D_{T}N-\dfrac{1}{2}\left[T,N\right]$
$\displaystyle\varkappa_{\beta}N_{\beta}\left(s^{\ast}\right)\varkappa$
$\displaystyle=-\kappa T+\tau
B-\dfrac{1}{2}\left\langle\left[T,N\right],B\right\rangle B$
or with the help of the Proposition 3.4, we get
$\varkappa_{\beta}N_{\beta}\left(s^{\ast}\right)=-T+HB$
where $\varkappa_{\beta}$ is the curvature of $\beta.$ Hence
$\varkappa_{\beta}=\sqrt{1+H^{2}}$
and
$N_{\beta}\left(s^{\ast}\right)=-\tfrac{1}{\sqrt{1+H^{2}}}T+\tfrac{H}{\sqrt{1+H^{2}}}B$
(4.3)
Then using the Eq.(4.2) and the Eq.(4.3) we have
$\displaystyle B_{\beta}\left(s^{\ast}\right)$
$\displaystyle=T_{\beta}\left(s^{\ast}\right)\times
N_{\beta}\left(s^{\ast}\right)$
$\displaystyle=\tfrac{H}{\sqrt{1+H^{2}}}T+\tfrac{1}{\sqrt{1+H^{2}}}B.$ (4.4)
Using the differentiation of the last equation and the Proposition 3.4, this
implies
$\left(\tau_{\beta}-\tau_{G_{\beta}}\right)N_{\beta}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}=-\tfrac{H^{\prime}}{\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\left(1+H^{2}\right)^{3/2}}B$
or using the Eq.(4.1), we have
$\left(\tau_{\beta}-\tau_{G_{\beta}}\right)N_{\beta}\left(s^{\ast}\right)=-\tfrac{H^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}B$
where
$\tau_{G_{\beta}}=\frac{1}{2}\left\langle\left[T_{\beta},N_{\beta}\right],B_{\beta}\right\rangle.$
Thus we compute
$\tau_{\beta}=\frac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)}+\tau_{G_{\beta}}$
where $\tau_{\beta}$ is the torsion of $\beta.$ The we can easily see that
$\tfrac{\tau_{\beta}-\tau_{G_{\beta}}}{\varkappa_{\beta}}=\frac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)^{3/2}}$
is a constant function. In other words, using the Theorem 2.3 we can easily
obtain that $\beta$ is a general helix.
Conversely, we assume that $\beta$ is a general helix then we can easily see
that $\alpha$ is a slant helix. These complete the proof. ∎
###### Corollary 4.3.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $G$ and $\beta$ be the tangent
indicatrix of the curve $\alpha.$ Then $\tau_{G_{\beta}}=\tau_{G}$ for the
curves $\alpha$ and $\beta.$
###### Proof.
It is obvious using the equations (4.2), (4.3) and (4.4). ∎
### 4.2. Normal indicatrices of slant helices:
###### Definition 4.4.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve.
Its normal indicatrix is the parametrized curve
$\gamma:I\subset\mathbb{R\rightarrow}S^{2}\subset\mathfrak{g}$ defined by
$\gamma\left(s^{\ast}\right)=N(s)=\sum_{i=1}^{3}\text{ }n_{i}X_{i}\text{ for
all }s\in I$
where $\left\\{X_{1},X_{2},X_{3}\right\\}$ is an orthonormal basis of
$\mathfrak{g}$ and $s^{\ast}$ is the arc length parameter of $\gamma.$
###### Theorem 4.5.
Let $\alpha$ be an arc-lenghted slant helix in three dimensional Lie Group $G$
and $\gamma$ be the normal indicatrix of the curve $\alpha.$ Then the curve
$\gamma$ is a plane curve on $S^{2}$.
###### Proof.
We assume that the curve $\alpha$ is a slant helix in a three dimensional Lie
group and $\gamma$ is the normal indicatrix of the curve $\alpha.$ From the
Definition 4.4 we get
$\gamma\left(s^{\ast}\right)=N(s).$ (4.5)
Then differentiating the Eq. (4.5) and using the Eq. (2.3) we have
$\displaystyle\frac{d\gamma}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=\overset{\cdot}{N}=D_{T}N-\dfrac{1}{2}\left[T,N\right]$
$\displaystyle=-\varkappa T+\tau
B-\frac{1}{2}\left\langle\left[T,N\right],B\right\rangle B$
$\displaystyle=-\varkappa T+\left(\tau-\tau_{G}\right)B$
$\displaystyle=-\varkappa T+\varkappa HB$
Then assuming that $\varkappa\rangle 0$ we obtain
$\frac{ds^{\ast}}{ds}=\varkappa\sqrt{1+H^{2}}$ (4.6)
and
$\frac{d\gamma}{ds^{\ast}}=\frac{1}{\sqrt{1+H^{2}}}\left(-T+HB\right).$
If we differentiate the last equation, then we obtain
$\displaystyle\frac{d^{2}\gamma}{ds^{\ast^{2}}}\frac{ds^{\ast}}{ds}$
$\displaystyle=-\dfrac{HH^{\shortmid}}{\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\frac{1}{\sqrt{1+H^{2}}}\left(-\overset{\cdot}{T}+H^{\shortmid}B+H\overset{\cdot}{B}\right)$
$\displaystyle=-\dfrac{HH^{\shortmid}}{\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\frac{1}{\sqrt{1+H^{2}}}\left\\{-\varkappa
N+H^{\shortmid}B+H\left(-\tau N-\dfrac{1}{2}\left[T,B\right]\right)\right\\}$
and by using the Eq. (4.6) with together Proposition 3.4 we obtain
$\displaystyle\frac{d^{2}\gamma}{ds^{\ast^{2}}}$
$\displaystyle=-\frac{H}{\sqrt{1+H^{2}}}\dfrac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\dfrac{1}{\varkappa\left(1+H^{2}\right)}\left\\{\left(-\varkappa-H\left(\tau-\tau_{G}\right)\right)N+H^{\shortmid}B\right\\}$
$\displaystyle=-\frac{H}{\sqrt{1+H^{2}}}\dfrac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\dfrac{1}{\varkappa\left(1+H^{2}\right)}\left\\{-\varkappa\left(1+H^{2}\right)N+H^{\shortmid}B\right\\}.$
Since $\alpha$ is a slant helix, $\sigma_{N}(s)$ is a constant function. So,
we can obtain
$\frac{d^{2}\gamma}{ds^{\ast^{2}}}=\frac{1}{\sigma_{N}(s)}\frac{H}{\sqrt{1+H^{2}}}T-N+\frac{1}{\sigma_{N}(s)}\frac{1}{\sqrt{1+H^{2}}}B$
(4.7)
Hence
$\varkappa_{\gamma}=\left\|\frac{d^{2}\gamma}{ds^{\ast^{2}}}\right\|=\frac{1}{\left|\sigma_{N}\right|}\sqrt{1+\sigma_{N}^{2}}$
where $\varkappa_{\gamma}$ is the curvature of $\gamma$ . Then differentiating
the Eq. (4.7) and using the Definition 3.3 we have
$\displaystyle\frac{d^{3}\gamma}{ds^{\ast^{3}}}\varkappa\sqrt{1+H^{2}}$
$\displaystyle=-\frac{1}{\sigma_{N}}\left\\{\dfrac{H^{\shortmid}}{\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\frac{H}{\sqrt{1+H^{2}}}\left(-\overset{\cdot}{T}+H^{\shortmid}B+H\overset{\cdot}{B}\right)\right\\}-\overset{\cdot}{N}$
$\displaystyle+\frac{1}{\sigma_{N}}\left(\frac{HH^{\shortmid}}{\sqrt{1+H^{2}}}B+\sqrt{1+H^{2}}\overset{\cdot}{B}\right)$
$\displaystyle=-\frac{1}{\sigma_{N}}\left\\{\dfrac{H^{\shortmid}}{\left(1+H^{2}\right)^{3/2}}\left(-T+HB\right)+\frac{H}{\sqrt{1+H^{2}}}\left(-\varkappa\left(1+H^{2}\right)N+H^{\shortmid}B\right)\right\\}$
$\displaystyle-
D_{T}N+\dfrac{1}{2}\left[T,N\right]+\frac{1}{\sigma_{N}}\left(\frac{HH^{\shortmid}}{\sqrt{1+H^{2}}}B+\sqrt{1+H^{2}}\left(D_{T}B-\dfrac{1}{2}\left[T,B\right]\right)\right)$
then by using the Proposition 3.4, we obtain
$\frac{d^{3}\gamma}{ds^{\ast^{3}}}=\varkappa\frac{\sigma_{N}^{2}+1}{\sigma_{N}^{2}}T-\varkappa
H\frac{\sigma_{N}^{2}+1}{\sigma_{N}^{2}}B$ (4.8)
Thus we compute
$\tau_{\gamma}=\frac{\det\left(\gamma^{\shortmid},\gamma^{\shortparallel},\gamma^{\shortmid\shortmid\shortmid}\right)}{\left\|\gamma^{\shortmid}\times\gamma^{\shortparallel}\right\|^{2}}=0$
where $\tau_{\gamma}$ is the torsion of $\gamma.$ Hence $\gamma$ is a plane
curve. This complete the proof. ∎
### 4.3. Binormal indicatrices of slant helices:
###### Definition 4.6.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve.
Its binormal indicatrix is the parametrized curve
$\delta:I\subset\mathbb{R\rightarrow}S^{2}\subset\mathfrak{g}$ defined by as
$\delta\left(s^{\ast}\right)=B(s)=\sum_{i=1}^{3}\text{ }b_{i}X_{i}\text{ for
all }s\in I$
where $\left\\{X_{1},X_{2},X_{3}\right\\}$ is an orthonormal basis of
$\mathfrak{g}$ and $s^{\ast}$ is the arc length parameter of $\delta.$
###### Theorem 4.7.
Let $\alpha$ be an arc-lenghted regular curve and $\gamma$ be the binormal
indicatrix of the curve $\alpha.$ Then the curve $\alpha$ is a slant helix in
three dimensional Lie group $G$ if and only if the curve $\delta$ is a general
helix on $S^{2}$.
###### Proof.
We assume that $\alpha$ be a slant helix in a three dimensional Lie group and
$\alpha_{B}$ be the tangent indicatrix of the curve $\alpha.$ From the
Definition 4.6 we get
$\delta\left(s^{\ast}\right)=B(s)$ (4.9)
then differentiating the Eq.(4.9) and using the Eq.(2.3), we have
$\displaystyle\frac{d\delta}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=\overset{\cdot}{B}=D_{T}B-\dfrac{1}{2}\left[T,B\right]$
$\displaystyle\frac{d\delta}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=-\varkappa HN.$
Then assuming that $\varepsilon=\left\\{\begin{array}[c]{cc}1&\text{ },\text{
if }\varkappa H\rangle 0\\\ -1&\text{ },\text{ if }\varkappa H\langle
0\end{array}\right\\}$ we have
$\frac{ds^{\ast}}{ds}=\varepsilon\varkappa H$
and
$T_{\delta}\left(s^{\ast}\right)=-\varepsilon N(s).$ (4.10)
If we differentiate the last equation then we obtain
$\displaystyle\varkappa_{\delta}N_{\delta}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}$
$\displaystyle=-\varepsilon\overset{\cdot}{N}=-\varepsilon
D_{T}N+\varepsilon\dfrac{1}{2}\left[T,N\right]$
$\displaystyle\varkappa_{\delta}N_{\delta}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}$
$\displaystyle=\varepsilon\varkappa T-\varepsilon\tau
B+\varepsilon\dfrac{1}{2}\left\langle\left[T,N\right],B\right\rangle B$
$\displaystyle\varkappa_{\delta}N_{\delta}\left(s^{\ast}\right)\varepsilon\varkappa
H$ $\displaystyle=\varepsilon\varkappa T-\varepsilon(\tau-\tau_{G})B$
$\displaystyle\varkappa_{\delta}N_{\delta}\left(s^{\ast}\right)$
$\displaystyle=\frac{1}{H}T-B$
where $\varkappa_{\delta}$ is the curvature of $\delta.$ Hence
$\varkappa_{\delta}=\frac{1}{\left|H\right|}\sqrt{1+H^{2}}$
and assuming that $\varkappa\rangle 0$ we have
$N_{\delta}\left(s^{\ast}\right)=\tfrac{\varepsilon}{\sqrt{1+H^{2}}}T-\tfrac{\varepsilon
H}{\sqrt{1+H^{2}}}B$ (4.11)
Then using the Eq.(4.10) and the Eq.(4.11) we have
$\displaystyle B_{\delta}\left(s^{\ast}\right)$
$\displaystyle=T_{\delta}\left(s^{\ast}\right)\times
N_{\delta}\left(s^{\ast}\right)$
$\displaystyle=-\tfrac{H}{\sqrt{1+H^{2}}}T+\tfrac{1}{\sqrt{1+H^{2}}}B.$ (4.12)
Using the differentiation of the last equation and the Proposition 3.4, this
implies
$\left(\tau_{\delta}-\tau_{G_{\delta}}\right)N_{\delta}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}=\tfrac{H^{\prime}}{\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\left(1+H^{2}\right)^{3/2}}B$
or using the equality $\frac{ds^{\ast}}{ds}=\varepsilon\varkappa H$, we have
$\left(\tau_{\delta}-\tau_{G_{\delta}}\right)N_{\delta}\left(s^{\ast}\right)=\tfrac{H^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}B$
where
$\tau_{G_{\delta}}=\frac{1}{2}\left\langle\left[T_{\delta},N_{\delta}\right],B_{\delta}\right\rangle.$
Thus we have
$\tau_{\delta}=\frac{H^{\shortmid}}{\varkappa
H\left(1+H^{2}\right)}+\tau_{G_{\delta}}$
where $\tau_{\delta}$ is the torsion of $\delta$ and so
$\dfrac{\tau_{\delta}-\tau_{G_{\delta}}}{\varkappa_{\delta}}=\dfrac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)^{3/2}}$
is a constant function, that is $\delta$ is a general helix.
Conversely, we assume that $\delta$ is a general helix then we can see easily
that $\alpha$ is a slant helix. These complete the proof. ∎
###### Corollary 4.8.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $G$ and $\delta$ be the binormal
indicatrix of the curve $\alpha.$ Then $\tau_{G_{\delta}}=\tau_{G}$ for the
curves $\alpha$ and $\delta.$
###### Proof.
It is obvious using the equations (4.10), (4.11) and (4.12). ∎
### 4.4. Involutes of slant helices:
###### Definition 4.9.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve.
Then the curve $x:I^{\ast}\subset\mathbb{R\rightarrow}G$ is called the
involute of the curve $\alpha$ if the tangent vector field of the curve
$\alpha$ is perpendicular to the tangent vector field of the curve $x.$ That
is,
$\left\langle T(s),T_{x}(s^{\ast})\right\rangle=0$
where $T$ and $T_{x}$ are the tangent vector fields of the curves $\alpha$ and
$x,$ respectively. Moreover $\left(x,\alpha\right)$ is called the involute-
evolute curve couple which are given by $\left(I,\alpha\right)$ and
$\left(I^{\ast},x\right)$ coordinate neighbourhoods, respectively. Then the
distance between the curves $x$ and $\alpha$ are given by
$d_{L}\left(\alpha\left(s\right),x\left(s\right)\right)=\left|c-s\right|\text{,
}c=\text{constant }\forall s\in I,$
[2]. We should remark that the parameter $s$ generally is not an arc-length
parameter of $x.$ So, we define the arc-length parameter of the curve $x$ by
$s^{\ast}=\psi\left(s\right)=\int\limits_{0}^{s}\left\|\frac{dx\left(s\right)}{ds}\right\|ds$
where $\psi:I\longrightarrow I^{\ast}$ is a smooth function and holds the
following equality
$\psi^{\prime}\left(s\right)=\left(c-s\right)\varkappa$ (4.13)
for $s\in I.$
###### Theorem 4.10.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve
and $x$ be an involute of $\alpha$. Then $\alpha$ is a slant helix in a three
dimensional Lie group if and only if $x$ is a general helix.
###### Proof.
Let $x$ be the involute of $\alpha$, then we have
$x(s)=\alpha(s)+\left(c-s\right)T\left(s\right),\text{ }c=\text{constant.}$
Let us derive both side with respect to $s$
$\displaystyle\frac{d\beta}{ds^{\ast}}\frac{ds^{\ast}}{ds}$
$\displaystyle=\left(c-s\right)\overset{\cdot}{T}(s),$ $\displaystyle
T_{x}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}$
$\displaystyle=\left(c-s\right)\varkappa N,$
where $s$ and $s^{\ast}$ are arc-parameters of $\alpha$ and $x$, respectively.
Then we calculate as
$\frac{ds^{\ast}}{ds}=\psi^{\prime}\left(s\right)=\left(c-s\right)\varkappa.$
and using this fact we can write
$T_{x}\left(s^{\ast}\right)=N.$ (4.14)
If we differentiate the last equation and use Frenet formulas then we obtain
$\displaystyle\varkappa_{x}N_{x}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}$
$\displaystyle=\overset{\cdot}{N}=D_{T}N-\dfrac{1}{2}\left[T,N\right]$
$\displaystyle\varkappa_{x}N_{x}\left(s^{\ast}\right)\varkappa$
$\displaystyle=-\kappa T+\tau
B-\dfrac{1}{2}\left\langle\left[T,N\right],B\right\rangle B$
or with the help of the Proposition 3.4, we get
$\varkappa_{x}N_{x}\left(s^{\ast}\right)=-T+HB$
where $\varkappa_{x}$ is the curvature of $x.$ Hence
$\varkappa_{x}=\sqrt{1+H^{2}}$
and
$N_{x}\left(s^{\ast}\right)=-\tfrac{1}{\sqrt{1+H^{2}}}T+\tfrac{H}{\sqrt{1+H^{2}}}B$
(4.15)
Then using the Eq.(4.14) and the Eq.(4.15) we have
$\displaystyle B_{x}\left(s^{\ast}\right)$
$\displaystyle=T_{x}\left(s^{\ast}\right)\times N_{x}\left(s^{\ast}\right)$
$\displaystyle=\tfrac{H}{\sqrt{1+H^{2}}}T+\tfrac{1}{\sqrt{1+H^{2}}}B.$ (4.16)
Using the differentiation of the last equation and the Proposition 3.4, this
implies
$\left(\tau_{x}-\tau_{G_{x}}\right)N_{x}\left(s^{\ast}\right)\frac{ds^{\ast}}{ds}=-\tfrac{H^{\prime}}{\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\left(1+H^{2}\right)^{3/2}}B$
or using the Eq.(4.13), we have
$\left(\tau_{x}-\tau_{G_{x}}\right)N_{\beta}\left(s^{\ast}\right)=-\tfrac{H^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}T+\tfrac{HH^{\prime}}{\varkappa\left(1+H^{2}\right)^{3/2}}B$
where
$\tau_{G_{x}}=\frac{1}{2}\left\langle\left[T_{x},N_{x}\right],B_{x}\right\rangle.$
Thus we compute
$\tau_{x}=\frac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)}+\tau_{G_{x}}$
where $\tau_{x}$ is the torsion of $x.$ The we can easily see that
$\tfrac{\tau_{x}-\tau_{G_{x}}}{\varkappa_{x}}=\frac{H^{\shortmid}}{\varkappa\left(1+H^{2}\right)^{3/2}}$
is a constant function. In other words, using the Theorem 2.3 $x$ is a general
helix.
Conversely, we assume that $x$ is a general helix then we can easily see that
$\alpha$ is a slant helix. These complete the proof. ∎
###### Corollary 4.11.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve
and $\beta:I\subset\mathbb{R\rightarrow}S^{2}\subset\mathfrak{g}$ be the
tangent indicatrix of the curve $\alpha.$ If $\alpha$ is a slant helix, then
$\beta$ is one of the involutes of the curve $\alpha.$
###### Proof.
It is obvious from the Theorem 4.2 and the Theorem 4.10. ∎
###### Corollary 4.12.
Let $\alpha:I\subset\mathbb{R\rightarrow}G$ be an arc-lenghted regular curve
and $\delta:I\subset\mathbb{R\rightarrow}S^{2}\subset\mathfrak{g}$ be the
binormal indicatrix of the curve $\alpha.$ If $\alpha$ is a slant helix, then
$\delta$ is one of the involutes of the curve $\alpha.$
###### Proof.
It is obvious from the Theorem 4.7 and the Theorem 4.10. ∎
###### Corollary 4.13.
Let $\alpha$ be an arc-lenghted regular curve with the Frenet vector fields
$\left\\{T,N,B\right\\}$ in the Lie group $G$ and $x$ be the involute of the
curve $\alpha.$ Then $\tau_{G_{x}}=\tau_{G}$ for the curves $\alpha$ and $x.$
###### Proof.
It is obvious using the equations (4.14), (4.15) and (4.16). ∎
## References
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* [2] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New-York, 1988.
* [3] M. Barros, General Helices and a theorem of Lancert, Proc. Amer. Math. Soc. 125 (5) (1997) 1503-1509.
* [4] J. Arroyo, M. Barros and J. O. Garay, A characterization of helices and Cornu spirals in real space forms, Bull. Austral. Math. Soc. 56 (1) (1997) 37-49.
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* [6] Ü. Çiftçi, A generalization of Lancert’s theorem, J. Geom. Phys. 59 (2009) 1597-1603.
* [7] S. Izumiya and N. Tkeuchi, New special curves and developable surfaces, Turk. J. Math 28 (2004), 153-163.
* [8] L. Kula and Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput. 169 (1) (2005) 600-607.
* [9] L. Kula, N. Ekmekci, Y. Yaylı and K. İlarslan, Characterizations of slant helices in Euclidean 3-space, Turk. J. Math. 34 (2) (2010) 261–273.
* [10] A. T. Ali and R. López, Slant helices in Minkowski space $\mathbb{E}_{1}^{3},$ J. Korean Math. Soc. 48 (1) (2011) 159-167.
* [11] A. T. Ali, M. Turgut, Position vector of time-like slant helix in Minkowski 3-space, J. Math. Anal. Appl. 365 (2010) 559-569.
* [12] İ. Gök, O. Zeki Okuyucu, F. Kahraman and H. H. Hacısalihoğlu, On the Quaternionic $B_{2}$-Slant Helices in the Euclidean Space $E^{4}$, Adv. Appl. Clifford Al. 21 (2011) 707–719.
* [13] F. Kahraman, İ. Gök and H. H. Hacısalihoğlu, On the quaternionic $B_{2}$ slant helices in the semi-Euclidean space $E_{2}^{4}$, App. Math. and Comp. 218 (2012) 6391-6400.
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|
arxiv-papers
| 2012-03-06T09:35:45 |
2024-09-04T02:49:28.298755
|
{
"license": "Public Domain",
"authors": "O. Zek\\.i Okuyucu, I.G\\\"ok, Y. Yayli and N. Ekmekc\\.i",
"submitter": "Osman Zeki Okuyucu",
"url": "https://arxiv.org/abs/1203.1146"
}
|
1203.1230
|
# Zero dissipation limit of full compressible Navier-Stokes equations with
Riemann initial data
Feimin Huang, Song Jiang and Yi Wang F. Huang is supported was supported in
part by NSFC Grant No. 10825102 for distinguished youth scholar, and National
Basic Research Program of China (973 Program) under Grant No. 2011CB808002.
E-mail: fhuang@amt.ac.cn.S. Jiang is supported by NSFC Grant No. 40890154 and
the National Basic Research Program under the Grant 2011CB309705. E-mail:
jiang@iapcm.ac.cn.Corresponding author. Y. Wang is supported by NSFC grant No.
10801128 and No. 11171326\. E-mail: wangyi@amss.ac.cn.
∗ ‡ Institute of Applied Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of
Mathematics, CAS, Beijing 100190, China
† Institute of Applied Physics and Computational Mathematics, Beijing 100088,
China
Abstract: We consider the zero dissipation limit of the full compressible
Navier-Stokes equations with Riemann initial data in the case of superposition
of two rarefaction waves and a contact discontinuity. It is proved that for
any suitably small viscosity $\varepsilon$ and heat conductivity $\kappa$
satisfying the relation (1.3), there exists a unique global piecewise smooth
solution to the compressible Navier-Stokes equations. Moreover, as the
viscosity $\varepsilon$ tends to zero, the Navier-Stokes solution converges
uniformly to the Riemann solution of superposition of two rarefaction waves
and a contact discontinuity to the corresponding Euler equations with the same
Riemann initial data away from the initial line $t=0$ and the contact
discontinuity located at $x=0$.
## 1 Introduction
We study the zero dissipation limit of the solution to the Navier-Stokes
equations of a compressible heat-conducting gas in Lagrangian coordinate:
$\left\\{\begin{array}[]{l}\displaystyle v_{t}-u_{x}=0,\\\ \displaystyle
u_{t}+p_{x}=\varepsilon(\frac{u_{x}}{v})_{x},\\\
\displaystyle(e+\frac{u^{2}}{2})_{t}+(pu)_{x}=\kappa(\frac{\theta_{x}}{v})_{x}+\varepsilon(\frac{uu_{x}}{v})_{x}\end{array}\right.$
(1.1)
with Riemann initial data
$(v,u,\theta)(0,x)=\left\\{\begin{array}[]{ll}(v_{-},u_{-},\theta_{-}),&x<0,\\\
(v_{+},u_{+},\theta_{+}),&x>0,\end{array}\right.$ (1.2)
where the functions $v(x,t)>0,u(x,t),\theta(x,t)>0$ represent the specific
volume, velocity and the absolute temperature of the gas, respectively. And
$p=p(v,\theta)$ is the pressure, $e=e(v,\theta)$ is the internal energy,
$\varepsilon>0$ is the viscosity constant and $\kappa>0$ is the coefficient of
heat conduction. Here we consider an ideal and polytropic gas, that is
$p=\frac{R\theta}{v},\qquad e=\frac{R\theta}{\gamma-1},$
with $\gamma>1,R>0$ being gas constants.
The study of the asymptotic behavior of viscous flows, as the viscosity tends
to zero, is one of the important problems in the theory of compressible fluid
flows. When the solution of the inviscid flow is smooth, the zero dissipation
limit problem can be solved by classical scaling method. However, the inviscid
compressible flow contains discontinuities, such as shock waves, in general.
In this case, it is also conjectured that a general weak entropy solution to
the inviscid flow should be the strong limit of the solution to the
corresponding viscous flows with the same initial data as the viscosity
vanishes.
It is well known that the solution to the Riemann problem for the Euler
equations consists of three basic wave patterns, that is, shock, rarefaction
wave and contact discontinuity. Moreover, the Riemann solution is essential in
the theory for the Euler equations as it captures both local and global
behavior of general solutions.
For hyperbolic conservation laws with the uniform viscosity
$u_{t}+f(u)_{x}=\varepsilon u_{xx},$
where $f(u)$ satisfies some assumptions to ensure the hyperbolic nature of the
corresponding inviscid system, Goodman-Xin [4] verified the limit for
piecewise smooth solutions separated by non-interacting shock waves using a
matched asymptotic expansion method. Later, Yu [33] proved it for hyperbolic
conservation laws with both shock and initial layers. In 2005, important
progress made by Bianchini-Bressan[1] justifies the vanishing viscosity limit
in BV-space even though the problem is still unsolved for the physical system
such as the compressible Navier-Stokes equations.
For the compressible isentropic Navier-Stokes equations where the conservation
of energy in (1.1) is neglected in the isentropic regime, Hoff-Liu [11]
firstly proved the vanishing viscosity limit for a piecewise constant shock
with initial layer. Later, Xin [31] justified the limit for rarefaction waves.
Then, Wang [29] generalized the result of Goodman-Xin [4] to the isentropic
Navier-Stokes equations.
Recently, Chen-Perepelitsa [2] proved the convergence of the isentropic
compressible Navier-Stokes equations to the compressible Euler equations as
the viscosity vanishes in Eulerian coordinates for general initial data by
using compensated compactness method if the far field does not contain vacuum.
Note that this result allows the initial data containing vacuum in the
interior domain. However, the framework of compensated compactness is
basically limited to $2\times 2$ systems so far, so that this result could not
be applied to the full compressible Navier-Stokes equations (1.1).
For the full compressible Navier-Stokes equations, there are investigations on
the limits to the Euler system for the basic wave patterns in the literature.
We refer to Jiang-Ni-Sun [17] and Xin-Zeng [32] for the rarefaction wave, Wang
[30] for the shock wave, Ma [21] for the contact discontinuity and Huang-Wang-
Yang [14, 15] for the superposition of two rarefaction waves and a contact
discontinuity and the superposition of rarefaction and shock waves,
respectively. We should point out that the limit shown in [17] was for the
discontinuous initial data while the other results mentioned were for (well-
prepared) smooth data.
In this paper, we shall investigate the zero dissipation limit of the full
Navier-Stokes equations (1.1) with Riemann initial data (1.2) in the case of
the superposition of two rarefaction waves and a contact discontinuity. The
local and global well-posedness of the full system (1.1) or the corresponding
isentropic system with discontinuous initial data is systematically studied by
Hoff, etc., see [5, 6, 7, 8, 9, 10, 3]. In order to get the zero dissipation
limit to the Riemann solution of the Euler system, we shall combine the local
existence of solutions with discontinuous data from [7] and the time-
asymptotic stability analysis to the compressible Navier-Stokes equations
(2.2). Compared with the previous result [14] where the same limit process is
studied for (well-prepared) smooth initial data, the main difficulty in the
proof here lies in the discontinuity of the initial data. The discontinuity of
the initial data for the volume $v(t,x)$ will propagate for all the time along
the particle path due to the hyperbolic regime while the smoothing effects
will also be performed on the velocity $u(t,x)$ and the temperature
$\theta(t,x)$ by the parabolic structure, and this interaction of the
discontinuity and smoothing effects brings technical difficulties. To
circumvent such difficulties, we shall choose suitable weight functions to
carry out the weighted energy estimates in terms of the superposition wave
structure (see Remark 3.7), and use the energy method of Huang-Li-Matsumura
[12] for the stability of two rarefaction waves with a contact discontinuity
in the middle, where the authors obtained a new estimate on the heat kernel
which can be applied to the study of the stability of the viscous contact wave
in the framework of the rarefaction wave (see Lemma 3.6). Namely, the anti-
derivative variable of the perturbation is not necessary and the estimates to
the perturbation itself are also available to get the stability of the viscous
contact wave.
Without loss of generality, we assume the following relation between the
viscosity constant $\varepsilon$ and the heat-conducing coefficient $\kappa$
of system (1.1) as in [17]:
$\left\\{\begin{array}[]{l}\displaystyle\kappa=O(\varepsilon),\qquad\qquad\rm
as\qquad\varepsilon\rightarrow 0;\\\
\displaystyle\nu\doteq\frac{\kappa(\varepsilon)}{\varepsilon}\geq
c>0\qquad{\rm for~{}some~{}positive~{}constant}~{}c,\quad\rm
as\quad\varepsilon\rightarrow 0.\end{array}\right.$ (1.3)
If $\kappa=\varepsilon=0$ in (1.1), then the corresponding Euler system reads
as
$\left\\{\begin{array}[]{l}v_{t}-u_{x}=0,\\\ u_{t}+p_{x}=0,\\\\[5.69054pt]
\displaystyle{\Big{(}e+\frac{u^{2}}{2}\Big{)}_{t}+(pu)_{x}=0.}\end{array}\right.$
(1.4)
It can be easily computed that the eigenvalues of the Jacobi matrix of the
flux function to (1.4) are
$\lambda_{1}=-\sqrt{\frac{\gamma
p}{v}},\quad\lambda_{2}=0,\quad\lambda_{3}=\sqrt{\frac{\gamma p}{v}}.$ (1.5)
It is well known that the first and third characteristic fields of (1.4) are
genuinely nonlinear and the second one is linearly degenerate (see[28]).
For the Euler equations, we know that there are three basic wave patterns,
shock, rarefaction wave and contact discontinuity. And the Riemann solution to
the Euler equations has a basic wave pattern consisting the superposition of
these three waves with the contact discontinuity in the middle. For later use,
let us firstly recall the wave curves for the two types of basic waves studied
in this paper.
Given the right end state $(v_{+},u_{+},\theta_{+})$ with
$v_{+},\theta_{+}>0$, the following wave curves in the phase space
$\\{(v,u,\theta)|v>0,\theta>0\\}$ are defined for the Euler equations.
$\bullet$ Contact discontinuity curve:
$CD(v_{+},u_{+},\theta_{+})=\\{(v,u,\theta)|u=u_{+},p=p_{+},v\not\equiv
v_{+}\\}.$ (1.6)
$\bullet$ $i$-Rarefaction wave curve $(i=1,3)$:
$R_{i}(v_{+},u_{+},\theta_{+}):=\Bigg{\\{}(v,u,\theta)\Bigg{|}u<u_{+},~{}u=u_{+}-\int^{v}_{v_{+}}\lambda_{i}(\eta,s_{+})\,d\eta,~{}s(v,\theta)=s_{+}\Bigg{\\}},$
(1.7)
where $s_{+}=s(v_{+},\theta_{+})$ and $\lambda_{i}=\lambda_{i}(v,s)$ defined
in (1.5) is the $i$-th characteristic speed of the Euler system (1.4).
Now, we define the solution profile that consists of the superposition of two
rarefaction waves and a contact discontinuity. Let
$(v_{-},u_{-},\theta_{-})\in$ $R_{1}$-$CD$-$R_{3}(v_{+},u_{+},\theta_{+})$.
Then, there exist uniquely two intermediate states $(v_{*},u_{*},\theta_{*})$
and $(v^{*},u^{*},\theta^{*})$, such that $(v_{*},u_{*},\theta_{*})\in
R_{1}(v_{-},u_{-},\theta_{-})$, $(v_{*},u_{*},\theta_{*})\in
CD(v^{*},u^{*},\theta^{*})$ and $(v^{*},u^{*},\theta^{*})\in
R_{3}(v_{+},u_{+},\theta_{+})$.
Thus, the wave pattern $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ consisting of
1-rarefaction wave, 2-contact discontinuity and 3-rarefaction wave that solves
the corresponding Riemann problem of the Euler system (1.4) can be defined by
$\displaystyle\left(\begin{array}[]{cc}\bar{V}\\\ \bar{U}\\\
\bar{\Theta}\end{array}\right)(t,x)=\left(\begin{array}[]{cc}v^{r_{1}}+v^{cd}+v^{r_{3}}\\\
u^{r_{1}}+u^{cd}+u^{r_{3}}\\\
\theta^{r_{1}}+\theta^{cd}+\theta^{r_{3}}\end{array}\right)(t,x)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{*}+u^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),$ (1.17)
where $(v^{r_{1}},u^{r_{1}},\theta^{r_{1}})(t,x)$ is the 1-rarefaction wave
defined in (1.7) with the right state $(v_{+},u_{+},\theta_{+})$ replaced by
$(v_{*},u_{*},\theta_{*})$, $(v^{cd},u^{cd},\theta^{cd})(t,x)$ is the contact
discontinuity defined in (1.6) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(v^{r_{3}},u^{r_{3}},\theta^{r_{3}})(t,x)$ is the 3-rarefaction wave defined
in (1.7) with the left state $(v_{-},u_{-},\theta_{-})$ replaced by
$(v^{*},u^{*},\theta^{*})$.
Now we state the main result as follows.
###### Theorem 1.1.
Given a Riemann solution $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ defined in
(1.17), which is superposition of two rarefaction waves and a contact
discontinuity for the Euler system (1.4), there exist small positive constants
$\delta_{0}$ and $\varepsilon_{0}$, such that if
$\varepsilon\leq\varepsilon_{0}$ and the wave strength
$\delta\doteq|(v_{+}-v_{-},u_{+}-u_{-},\theta_{+}-\theta_{-})|\leq\delta_{0}$,
then the compressible Navier-Stokes equations (1.1) with (1.2) and (1.3)
admits a unique global piece-wise smooth solution
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)$ satisfying that
* •
The quantities $u^{\varepsilon},\theta^{\varepsilon}$,
$p(v^{\varepsilon},\theta^{\varepsilon})-\varepsilon\frac{u^{\varepsilon}_{x}}{v^{\varepsilon}}$
and $\frac{\theta^{\varepsilon}_{x}}{v^{\varepsilon}}$ are continuous for
$t>0$, and the jumps in
$v^{\varepsilon},u^{\varepsilon}_{x},\theta^{\varepsilon}_{x}$ at $x=0$
satisfies
$|([v^{\varepsilon}(t,0)],[u^{\varepsilon}_{x}(t,0)],[\theta^{\varepsilon}_{x}(t,0)])|\leq
Ce^{-\frac{ct}{\varepsilon}},$
where the constants $C$ and $c$ are independent of $t$ and $\varepsilon$.
* •
Moreover, under the condition (1.3), it holds that
$\lim_{\varepsilon\rightarrow
0}\sup_{(t,x)\in\Sigma_{h}}|(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)-(\bar{V},\bar{U},\bar{\Theta})(t,x)|=0,\quad\forall
h>0,$ (1.18)
where $\Sigma_{h}=\big{\\{}(t,x)|t\geq h,\frac{x}{\sqrt{\varepsilon+t}}\geq
h\varepsilon^{\alpha},0\leq\alpha<\frac{1}{2}\big{\\}}$.
###### Remark 1.2.
Theorem 1.1 shows that, away from the initial time $t=0$ and the contact
discontinuity located at $x=0$, there exists a unique global solution
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)$ of the
compressible Navier-Stokes equations (1.1) which converges to the Riemann
solution $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ consisting of two rarefaction
waves and a contact discontinuity when $\varepsilon$ and $\kappa$ satisfy the
relation (1.3) and $\varepsilon$ tends to zero. Moreover, the convergence is
uniform on the set $\Sigma_{h}$ for any $h>0$.
Notations. In the paper, we always use the notation
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}=\int_{\mathbf{R}^{+}}+\int_{\mathbf{R}^{-}}$,
$\|\cdot\|$ to denote the usual $L^{2}(\mathbf{R})$ norm,
$\|\cdot\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel$
to denote the piecewise $L^{2}$ norm, that is,
$\displaystyle\|f\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}f^{2}dy$.
$\|\cdot\|_{1}$ and
$\|\cdot\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel_{1}$
represent the $H^{1}(\mathbf{R})$ norm and piece-wise
$H^{1}(\mathbf{R}^{\pm})$ norm, respectively. And the notation $[\cdot]$
represents the jump of the function $\cdot$ at $x=0$ or $y=0$ if without
confusion.
## 2 Approximate profiles
Introduce the following scaled variables
$y=\frac{x}{\varepsilon},\quad\tau=\frac{t}{\varepsilon},$ (2.1)
and set
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)=(v,u,\theta)(\tau,y).$
Then the new unknown functions $(v,u,\theta)(\tau,y)$ satisfies the system
$\left\\{\begin{array}[]{l}\displaystyle v_{\tau}-u_{y}=0,\\\ \displaystyle
u_{\tau}+p_{y}=(\frac{u_{y}}{v})_{y},\\\
\displaystyle\frac{R}{\gamma-1}\theta_{\tau}+pu_{y}=\nu(\frac{\theta_{y}}{v})_{y}+\frac{u^{2}_{y}}{v},\end{array}\right.$
(2.2)
with the scaled heat conductivity $\nu=\frac{\kappa}{\varepsilon}$ in (1.3)
satisfying
$\nu_{0}\leq\nu\leq\nu_{1},~{}~{}{\rm uniformly}~{}{\rm
in}~{}\varepsilon~{}{\rm as}~{}\varepsilon\rightarrow 0+,~{}{\rm for}~{}{\rm
some}~{}{\rm positive}~{}{\rm constants}~{}\nu_{0}~{}{\rm and}~{}\nu_{1}.$
Note that the Riemann solution $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ in (1.17)
is invariant under the scaling transformation (2.1), thus to prove the limit
(1.18) in Theorem 1.1, it is sufficient to show the following limit
$\lim_{\varepsilon\rightarrow
0}\sup_{(\tau,y)\in\Sigma^{1}_{h}}|(v,u,\theta)(\tau,y)-(\bar{V},\bar{U},\bar{\Theta})(\tau,y)|=0,\quad\forall
h>0,$ (2.3)
where $\Sigma_{h}^{1}$ is the corresponding region of $\Sigma_{h}$ in the new
coordinates $(\tau,y)$ defined by
$\Sigma^{1}_{h}=\Big{\\{}(\tau,y)|\tau\geq\frac{h}{\varepsilon},\frac{y}{\sqrt{1+\tau}}\geq\frac{h}{\varepsilon^{\frac{1}{2}-\alpha}},0\leq\alpha<\frac{1}{2}\Big{\\}}.$
(2.4)
Now we study the Navier-Stokes equations (2.2). The corresponding wave
profiles to (1.6) and (1.7) can be defined approximately as follows. We start
from the viscous contact wave to (1.6).
### 2.1 Viscous contact wave
If $(v_{-},u_{-},\theta_{-})\in CD(v_{+},u_{+},\theta_{+})$, i.e.,
$u_{-}=u_{+},~{}p_{-}=p_{+},~{}v_{-}\neq v_{+},$
then the Riemann problem, that is, the Euler system (1.4) with Riemann initial
data
$(v,u,\theta)(\tau=0,y)=\left\\{\begin{array}[]{ll}(v_{-},u_{-},\theta_{-}),&y<0,\\\
(v_{+},u_{+},\theta_{+}),&y>0,\end{array}\right.$
admits a single contact discontinuity solution
$(v^{cd},u^{cd},\theta^{cd})(\tau,y)=\left\\{\begin{array}[]{ll}(v_{-},u_{+},\theta_{-}),&y<u_{+}\tau,~{}\tau>0,\\\
(v_{+},u_{+},\theta_{+}),&y>u_{+}\tau,~{}\tau>0.\end{array}\right.$
As in [13], the viscous version of the above contact discontinuity, called
viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(\tau,y)$, can be defined as
follows. Since it is expected that
$P^{CD}\approx p_{+}=p_{-},\quad{\rm and}\quad|U^{CD}-u_{+}|\ll 1,$
the leading order of the energy equation $\eqref{NS1}_{3}$ is
$\frac{R}{\gamma-1}\Theta_{\tau}+p_{+}U_{y}=\nu(\frac{\Theta_{y}}{V})_{y}.$
Then, similar to [12] or [14], one can get the following nonlinear diffusion
equation
$\Theta_{\tau}=a\Big{(}\frac{\Theta_{y}}{\Theta}\Big{)}_{y},\quad\Theta(\tau,\pm)=\theta_{\pm},\quad
a=\frac{\nu p_{+}(\gamma-1)}{R^{2}\gamma}.$
The above diffusion equation has a unique self-similar solution
$\hat{\Theta}(\tau,y)=\hat{\Theta}(\frac{y}{\sqrt{1+\tau}})$.
Thus, the viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(\tau,y)$ can be
defined by
$\begin{array}[]{ll}\displaystyle
V^{CD}(\tau,y)=\frac{R\hat{\Theta}(\tau,y)}{p_{+}},\\\\[11.38109pt]
\displaystyle
U^{CD}(\tau,y)=u_{+}+\frac{\nu(\gamma-1)}{R\gamma}\frac{\hat{\Theta}_{y}(\tau,y)}{\hat{\Theta}(\tau,y)},\\\\[14.22636pt]
\displaystyle\Theta^{CD}(\tau,y)=\hat{\Theta}(\tau,y)+\frac{R\gamma-\nu(\gamma-1)}{\gamma
p_{+}}\hat{\Theta}_{\tau}.\end{array}$ (2.5)
Here, it is straightforward to check that the viscous contact wave defined in
(2.5) satisfies
$|\hat{\Theta}-\theta_{\pm}|+(1+\tau)^{\frac{1}{2}}|\hat{\Theta}_{y}|+(1+\tau)|\hat{\Theta}_{yy}|=O(1)\delta^{CD}e^{-\frac{c_{0}y^{2}}{1+\tau}},\quad\mbox{as
}|y|\rightarrow+\infty,$ (2.6)
where $\delta^{CD}=|\theta_{+}-\theta_{-}|$ represents the strength of the
viscous contact wave and $c_{0}$ is a positive constant. Note that in (2.5),
the higher order term $\frac{R\gamma-\nu(\gamma-1)}{\gamma
p_{+}}\hat{\Theta}_{\tau}$ is introduced in $\Theta^{CD}(\tau,y)$ to make the
viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(\tau,y)$ satisfy the
momentum equation exactly. Correspondingly,
$(V^{CD},U^{CD},\Theta^{CD})(\tau,y)$ satisfies the system
$\left\\{\begin{array}[]{l}\displaystyle V^{\scriptscriptstyle
CD}_{\tau}-U^{CD}_{y}=0,\\\\[5.69054pt] \displaystyle
U^{CD}_{\tau}+P^{CD}_{y}=\Big{(}\frac{U^{CD}_{y}}{V^{CD}}\Big{)}_{y},\\\\[11.38109pt]
\displaystyle\frac{R}{\gamma-1}\Theta^{CD}_{\tau}+P^{CD}U^{CD}_{y}=\nu\Big{(}\frac{\Theta^{CD}_{y}}{V^{CD}}\Big{)}_{y}+\frac{(U^{CD}_{y})^{2}}{V^{CD}}+Q^{CD},\end{array}\right.$
(2.7)
where $\displaystyle P^{CD}=\frac{R\Theta^{CD}}{V^{CD}}$ and the error term
$Q^{CD}$ satisfies
$\displaystyle
Q^{CD}=O(1)\delta^{CD}(1+\tau)^{-2}e^{-\frac{c_{0}y^{2}}{1+\tau}},\qquad{\rm
as}~{}~{}|y|\rightarrow+\infty,$ (2.8)
for some positive constant $c_{0}$.
### 2.2 Approximate rarefaction waves
We now turn to the approximate rarefaction waves to (1.7). Since there is no
exact rarefaction wave profile for the Navier-Stokes equations, the following
approximate rarefaction wave profile, which satisfies the Euler equations, is
motivated by [31]. For the completeness of presentation, we include its
definition and the properties in this subsection.
If $(v_{-},u_{-},\theta_{-})\in R_{i}(v_{+},u_{+},\theta_{+}),(i=1,3)$, then
there exists an $i$-rarefaction wave
$(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})(y/\tau)$ which is a global solution of
the following Riemann problem:
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
v_{\tau}-u_{y}=0,\\\\[2.84526pt] \displaystyle
u_{\tau}+p_{y}(v,\theta)=0,\\\\[5.69054pt]
\displaystyle\frac{R}{\gamma-1}\theta_{\tau}+p(v,\theta)u_{y}=0,\\\\[2.84526pt]
\displaystyle(v,u,\theta)(0,y)=\left\\{\begin{array}[]{l}\displaystyle(v_{-},u_{-},\theta_{-}),~{}~{}~{}~{}y<0,\\\
\displaystyle(v_{+},u_{+},\theta_{+}),~{}~{}~{}~{}y>0.\end{array}\right.\end{array}\right.$
(2.15)
Consider the following inviscid Burgers equation with Riemann data:
$\left\\{\begin{array}[]{l}w_{\tau}+ww_{y}=0,\\\\[5.69054pt]
w(\tau=0,y)=\left\\{\begin{array}[]{ll}w_{-},&y<0,\\\
w_{+},&y>0.\end{array}\right.\end{array}\right.$ (2.16)
If $w_{-}<w_{+}$, then the Riemann problem (2.16) admits a rarefaction wave
solution
$w^{r}(\tau,y)=w^{r}(\frac{y}{\tau})=\left\\{\begin{array}[]{ll}w_{-},&\frac{y}{\tau}\leq
w_{-},\\\\[2.84526pt] \frac{y}{\tau},&w_{-}\leq\frac{y}{\tau}\leq
w_{+},\\\\[2.84526pt] w_{+},&\frac{y}{\tau}\geq w_{+}.\end{array}\right.$
(2.17)
Thus, the Riemann solution in (2.15) can be expressed explicitly through the
above rarefaction wave (2.17) to the Burgers equation, that is,
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
s^{r_{i}}(\tau,y)=s(v^{r_{i}}(\tau,y),\theta^{r_{i}}(\tau,y))=s_{+},\\\\[5.69054pt]
\displaystyle
w_{\pm}=\lambda_{i\pm}:=\lambda_{i}(v_{\pm},\theta_{\pm}),\\\\[2.84526pt]
\displaystyle
w^{r}(\frac{y}{\tau})=\lambda_{i}(v^{r_{i}}(\tau,y),s_{+}),\\\\[2.84526pt]
\displaystyle
u^{r_{i}}(\tau,y)=u_{+}-\int^{v^{r_{i}}(\tau,y)}_{v_{+}}\lambda_{i}(v,s_{+})dv.\end{array}\right.$
(2.22)
In order to construct the approximate rarefaction wave
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)$ corresponding to (1.7), we
first consider the following approximate rarefaction wave to the Burgers
equation:
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle w_{\tau}+ww_{y}=0,\\\
\displaystyle w(0,y)=w_{0}(y)=\frac{w_{+}+w_{-}}{2}+\frac{w_{+}-w_{-}}{2}\tanh
y.\end{array}\right.$ (2.25)
Note that the solution $w^{R}(\tau,y)$ of the problem (2.25) is given by
$w^{R}(\tau,y)=w_{0}(x_{0}(\tau,y)),\qquad
x=x_{0}(\tau,y)+w_{0}(x_{0}(\tau,y))\tau.$
And $w^{R}(\tau,y)$ has the following properties, the proof of which can be
found in [22, 31]:
###### Lemma 2.1.
Let $w_{-}<w_{+}$, then $\eqref{AB}$ has a unique smooth solution
$w^{R}(\tau,y)$ satisfying
1. (1)
$w_{-}<w^{R}(\tau,y)<w_{+},~{}(w^{R})_{y}(\tau,y)>0$;
2. (2)
For any $1\leq p\leq+\infty$, there exists a constant $C$ such that
$\begin{array}[]{ll}\|\frac{\partial}{\partial
y}w^{R}(\tau,\cdot)\|_{L^{p}(\mathbf{R})}\leq
C\min\big{\\{}(w_{+}-w_{-}),~{}(w_{+}-w_{-})^{1/p}\tau^{-1+1/p}\big{\\}},\\\\[5.69054pt]
\|\frac{\partial^{2}}{\partial
y^{2}}w^{R}(\tau,\cdot)\|_{L^{p}(\mathbf{R})}\leq
C\min\big{\\{}(w_{+}-w_{-}),~{}\tau^{-1}\big{\\}};\end{array}$
3. (3)
If $y-w_{-}\tau<0$, then
$\begin{array}[]{l}|w^{R}(\tau,y)-w_{-}|\leq(w_{+}-w_{-})e^{-2|y-w_{-}\tau|},\\\\[5.69054pt]
|\frac{\partial}{\partial y}w^{R}(\tau,y)|\leq
2(w_{+}-w_{-})e^{-2|y-w_{-}\tau|};\end{array}$
If $y-w_{+}\tau>0$, then
$\begin{array}[]{l}|w^{R}(\tau,y)-w_{+}|\leq(w_{+}-w_{-})e^{-2|y-w_{+}\tau|},\\\\[5.69054pt]
|\frac{\partial}{\partial x}w^{R}(\tau,y)|\leq
2(w_{+}-w_{-})e^{-2|y-w_{+}\tau|};\end{array}$
4. (4)
$\sup\limits_{y\in\mathbf{R}}|w^{R}(\tau,y)-w^{r}(\frac{y}{\tau})|\leq\min\big{\\{}w_{+}-w_{-},\frac{1}{\tau}\ln(1+\tau)\big{\\}}$.
Then, corresponding to (2.22), the approximate rarefaction wave profile
denoted by $(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)~{}(i=1,3)$ to (1.7)
can be defined by
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
S^{R_{i}}(\tau,y)=s(V^{R_{i}}(\tau,y),\Theta^{R_{i}}(\tau,y))=s_{+},\\\\[2.84526pt]
\displaystyle
w_{\pm}=\lambda_{i\pm}:=\lambda_{i}(v_{\pm},\theta_{\pm}),\\\\[5.69054pt]
\displaystyle
w^{R}(1+\tau,y)=\lambda_{i}(V^{R_{i}}(\tau,y),s_{+}),\\\\[2.84526pt]
\displaystyle
U^{R_{i}}(\tau,y)=u_{+}-\int^{V^{R_{i}}(\tau,y)}_{v_{+}}\lambda_{i}(v,s_{+})dv.\end{array}\right.$
(2.30)
Note that $(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)$ defined above
satisfies
$\left\\{\begin{array}[]{ll}\displaystyle
V^{R_{i}}_{\tau}-U^{R_{i}}_{y}=0,\\\\[2.84526pt] \displaystyle
U^{R_{i}}_{\tau}+P^{R_{i}}_{y}=0,\\\\[5.69054pt]
\displaystyle\frac{R}{\gamma-1}\Theta^{R_{i}}_{\tau}+P^{R_{i}}U^{R_{i}}_{y}=0,\end{array}\right.$
(2.31)
where $P^{R_{i}}=p(V^{R_{i}},\Theta^{R_{i}})$.
By virtue of Lemmas 2.1, the properties on the approximate rarefaction waves
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)$ can be summarized as follows.
###### Lemma 2.2.
The approximate rarefaction waves
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)~{}(i=1,3)$ constructed in (2.30)
have the following properties:
1. (1)
$U^{R_{i}}_{x}(\tau,y)>0$ for $y\in\mathbf{R}$, $\tau>0$;
2. (2)
For any $1\leq p\leq+\infty,$ the following estimates holds,
$\begin{array}[]{ll}\|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})_{y}\|_{L^{p}(dy)}\leq
C\min\big{\\{}\delta^{R_{i}},~{}(\delta^{R_{i}})^{1/p}(1+\tau)^{-1+1/p}\big{\\}},\\\\[5.69054pt]
\|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})_{yy}\|_{L^{p}(dy)}\leq
C\min\big{\\{}\delta^{R_{i}},~{}(1+\tau)^{-1}\big{\\}},\\\ \end{array}$
where $\delta^{R_{i}}=|(v_{+},v_{-},u_{+},u_{-},\theta_{+},\theta_{-})|$ is
the $i$-rarefaction wave strength and the positive constant $C$ is independent
of $\tau$, but may only depend on $p$ and the wave strength;
3. (3)
If $y\geq\lambda_{1+}(1+\tau)$, then
$\begin{array}[]{l}|(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})(\tau,y)-(v_{-},u_{-},\theta_{-})|\leq
C\delta^{R_{1}}e^{-2|y-\lambda_{1+}(1+\tau)|},\\\\[5.69054pt]
|(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})_{y}(\tau,y)|\leq
C\delta^{R_{1}}e^{-2|y-\lambda_{1+}(1+\tau)|};\end{array}$
If $y\leq\lambda_{3-}(1+\tau)$, then
$\begin{array}[]{l}|(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})(\tau,y)-(v_{+},u_{+},\theta_{+})|\leq
C\delta^{R_{3}}e^{-2|y-\lambda_{3-}(1+\tau)|},\\\\[5.69054pt]
|(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})_{y}(\tau,y)|\leq
C\delta^{R_{3}}e^{-2|y-\lambda_{3-}(1+\tau)|};\end{array}$
4. (4)
There exists a positive constant $C$, such that for all $\tau>0,$
$\sup_{y\in\mathbf{R}}|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(\tau,y)-(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})(\frac{y}{\tau})|\leq\frac{C}{1+\tau}\ln(1+\tau).$
### 2.3 Superposition of rarefaction waves and contact discontinuity
Corresponding to (1.17), the approximate wave pattern $(V,U,\Theta)(\tau,y)$
of the compressible Navier-Stokes equations (2.2) can be defined by
$\displaystyle\left(\begin{array}[]{cc}V\\\ U\\\
\Theta\end{array}\right)(\tau,y)=\left(\begin{array}[]{cc}V^{R_{1}}+V^{CD}+V^{R_{3}}\\\
U^{R_{1}}+U^{CD}+U^{R_{3}}\\\
\Theta^{R_{1}}+\Theta^{CD}+\Theta^{R_{3}}\end{array}\right)(\tau,y)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{*}+u^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),$ (2.41)
where $(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})(\tau,y)$ is the approximate
1-rarefaction wave defined in (2.30) with the right state
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$,
$(V^{CD},U^{CD},\Theta^{CD})(\tau,y)$ is the viscous contact wave defined in
(2.5) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})(\tau,y)$ is the approximate
3-rarefaction wave defined in (2.30) with the left state
$(v_{-},u_{-},\theta_{-})$ replaced by $(v^{*},u^{*},\theta^{*})$.
Thus, from the properties of the viscous contact wave in (2.6) and the
approximate rarefaction wave in Lemma 2.3, we have the following relation
between the approximate wave pattern $(V,U,\Theta)(\tau,y)$ and the exact
inviscid wave pattern $(\bar{V},\bar{U},\bar{\Theta})(\tau,y)$ of the Euler
equations
$\displaystyle|(V,U,\Theta)(\tau,y)-(\bar{V},\bar{U},\bar{\Theta})(\tau,y)|\displaystyle\leq\frac{C}{1+\tau}\ln(1+\tau)+C\delta^{CD}e^{-\frac{cy^{2}}{1+\tau}}.$
(2.42)
Hence, to prove the zero dissipation limit (2.3) on the set $\Sigma_{h}^{1}$
defined in (2.4), it is sufficient to show the following time-asymptotic
behavior of the solution to (2.2) around the approximate wave profile (2.41),
i.e.,
$\lim_{\tau\rightarrow+\infty}\|(v,u,\theta)(\tau,\cdot)-(V,U,\Theta)(\tau,\cdot)\|_{L^{\infty}}=0.$
(2.43)
First, by (2.7) and (2.31), the superposition wave profile
$(V,U,\Theta)(\tau,y)$ defined in (2.41) satisfies the following system
$\left\\{\begin{array}[]{ll}\displaystyle V_{\tau}-U_{y}=0,\\\ \displaystyle
U_{\tau}+P_{y}=(\frac{U_{y}}{V})_{y}+Q_{1},\\\
\displaystyle\frac{R}{\gamma-1}\Theta_{\tau}+PU_{y}=\nu(\frac{\Theta_{y}}{V})_{y}+\frac{U_{y}^{2}}{V}+Q_{2},\end{array}\right.$
where $P=p(V,\Theta)$ and
$\begin{array}[]{ll}\displaystyle
Q_{1}&\displaystyle=(P-P^{R_{1}}-P^{CD}-P^{R_{3}})_{y}-\left(\frac{U_{y}}{V}-\frac{U^{CD}_{y}}{V^{CD}}\right)_{y},\\\
\displaystyle
Q_{2}&\displaystyle=(PU_{y}-P^{R_{1}}U^{R_{1}}_{y}-P^{CD}U^{CD}_{y}-P^{R_{3}}U^{R_{3}}_{y})-\nu\left(\frac{\Theta_{y}}{V}-\frac{\Theta^{CD}_{y}}{V^{CD}}\right)_{y}\\\
&\displaystyle-\left(\frac{U_{y}^{2}}{V}-\frac{(U^{CD}_{y})^{2}}{V^{CD}}\right)-Q^{CD}.\end{array}$
A direct calculation shows that
$\begin{array}[]{lll}\displaystyle Q_{1}&=&\displaystyle
O(1)\Big{\\{}|(V^{R_{1}}_{y},\Theta^{R_{1}}_{y})||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|(V^{R_{3}}_{y},\Theta^{R_{3}}_{y})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|(V^{CD}_{y},\Theta^{CD}_{y},U^{CD}_{y})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|(U^{CD}_{y},V^{CD}_{y})||(U^{R_{1}}_{y},V^{R_{1}}_{y},U^{R_{3}}_{y},V^{R_{3}}_{y})|+|(U^{R_{1}}_{y},V^{R_{1}}_{y})||(U^{R_{3}}_{y},V^{R_{3}}_{y})|\Big{\\}}\\\\[5.69054pt]
&&\displaystyle+O(1)\Big{\\{}|U^{R_{1}}_{yy}|+|U^{R_{3}}_{yy}|+|U^{R_{1}}_{y}||V^{R_{1}}_{y}|+|U^{R_{3}}_{y}||V^{R_{3}}_{y}|\Big{\\}}\\\\[5.69054pt]
&:=&\displaystyle Q_{11}+Q_{12}.\end{array}$ (2.44)
Similarly, we have
$\begin{array}[]{lll}\displaystyle Q_{2}&=&\displaystyle
O(1)\Big{\\{}|U^{R_{1}}_{y}||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|U^{R_{3}}_{y}||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|(U^{CD}_{y},V^{CD}_{y},\Theta^{CD}_{y})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\\[5.69054pt]
&&\displaystyle+|(U^{CD}_{y},V^{CD}_{y},\Theta^{CD}_{y})||(U^{R_{1}}_{y},V^{R_{1}}_{y},\Theta^{R_{1}}_{y},U^{R_{3}}_{y},V^{R_{3}}_{y},\Theta^{R_{1}}_{y})|\\\\[5.69054pt]
&&\displaystyle+|(U^{R_{1}}_{y},V^{R_{1}}_{y},\Theta^{R_{1}}_{y})||(U^{R_{3}}_{y},V^{R_{3}}_{y},\Theta^{R_{3}}_{y})|\Big{\\}}\\\\[5.69054pt]
&&\displaystyle+O(1)\Big{\\{}|\Theta^{R_{1}}_{yy}|+|\Theta^{R_{3}}_{yy}|+|(U^{R_{1}}_{y},V^{R_{1}}_{y},\Theta^{R_{1}}_{y},U^{R_{3}}_{y},V^{R_{3}}_{y},\Theta^{R_{3}}_{y})|^{2}\Big{\\}}+|Q^{CD}|\\\\[5.69054pt]
&:=&\displaystyle Q_{21}+Q_{22}+|Q^{CD}|.\end{array}$ (2.45)
Here $Q_{11}$ and $Q_{21}$ represent the wave interaction terms coming from
the wave patterns in the different family, $Q_{12}$ and $Q_{22}$ stand for the
error terms due to the inviscid approximate rarefaction wave profiles, and
$Q^{CD}$ is the error term defined in (2.8) due to the viscous contact wave.
In fact, one can estimate the interaction terms $Q_{11}$ and $Q_{21}$ by
dividing the whole domain
$\Omega=\\{(\tau,y)|(\tau,y)\in\mathbf{R}\times\mathbf{R}\\}$ into three
regions:
$\displaystyle\Omega_{-}=\\{(\tau,y)\;|\;2y\leq\lambda_{1*}(1+\tau)\\},$
$\displaystyle\Omega_{CD}=\\{(\tau,y)\;|\;\lambda_{1*}(1+\tau)<2y<\lambda_{3}^{*}(1+\tau)\\},$
$\displaystyle\Omega_{+}=\\{(\tau,y)\;|\;2y\geq\lambda_{3}^{*}(1+\tau)\\},$
where $\lambda_{1*}=\lambda_{1}(v_{*},\theta_{*})$ and
$\lambda_{3}^{*}=\lambda_{3}(v^{*},\theta^{*})$. Then, in each section the
following estimates follow from (2.6) and Lemma 2.2.
* •
In $\Omega_{-}$,
$\displaystyle|(V^{R_{3}}-v^{*},V^{R_{3}}_{y})|=O(1)\delta^{R_{3}}e^{-2\\{|y|+|\lambda_{3}^{*}|(1+\tau)\\}},$
$\displaystyle|(V^{CD}-v_{*},V^{CD}-v^{*},V^{CD}_{y})|=O(1)\delta^{CD}e^{-\frac{C\\{|\lambda_{1*}|(1+\tau)\\}^{2}}{1+\tau}}=O(1)\delta^{CD}e^{-C(1+\tau)};$
* •
In $\Omega_{CD}$,
$\displaystyle|(V^{R_{1}}-v_{*},V^{R_{1}}_{y})|=O(1)\delta^{R_{1}}e^{-2\\{|y|+|\lambda_{1*}|(1+\tau)\\}},$
$\displaystyle|(V^{R_{3}}-v^{*},V^{R_{3}}_{y})|=O(1)\delta^{R_{3}}e^{-2\\{|x|+|\lambda_{3}^{*}|(1+\tau)\\}};$
* •
In $\Omega_{+}$,
$\displaystyle|(V^{R_{1}}-v_{*},V^{R_{1}}_{y})|=O(1)\delta^{R_{1}}e^{-2\\{|x|+|\lambda_{1*}|(1+\tau)\\}},$
$\displaystyle|(V^{CD}-v_{*},V^{CD}-v^{*},V^{CD}_{y})|=O(1)\delta^{CD}e^{-\frac{C\\{|\lambda_{3}^{*}|(1+\tau)\\}^{2}}{1+\tau}}=O(1)\delta^{CD}e^{-C(1+\tau)}.$
Keep in mind that each individual wave strength is controlled by the total
wave strength by (1.6) and (1.7), that is,
$\delta^{R_{1}}+\delta^{R_{3}}+\delta^{CD}\leq C\delta.$
Hence, in summary, it follows from (2.44), (2.45) and the above arguments that
$|(Q_{11},Q_{21})|=O(1)\delta e^{-C\\{|y|+(1+\tau)\\}},$
for some positive constant $C$ independent of $\tau$ and $y$.
## 3 Proof of the main result
In this section, we shall prove the main result Theorem 1.1. By virtue of the
arguments in Section 2.3, it is sufficient to show (2.43) besides the
regularity of the solution. To this end, we first reformulate the problem.
### 3.1 Reformulation of the problem
Set the perturbation around the wave profile $(V,U,\Theta)(\tau,y)$ by
$(\phi,\psi,\zeta)(\tau,y)=(v,u,\theta)(\tau,y)-(V,U,\Theta)(\tau,y).$
Then, after a straightforward calculation, the perturbation
$(\phi,\psi,\zeta)(\tau,y)$ satisfies the system
$\left\\{\begin{array}[]{ll}\displaystyle\phi_{\tau}-\psi_{y}=0,\\\
\displaystyle\psi_{\tau}+(p-P)_{y}=(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-Q_{1},\\\\[5.69054pt]
\displaystyle\frac{R}{\gamma-1}\zeta_{\tau}+(pu_{y}-PU_{y})=\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}+(\frac{u_{y}^{2}}{v}-\frac{U^{2}_{y}}{V})-Q_{2},\\\\[11.38109pt]
\displaystyle(\phi,\psi,\zeta)(\tau=0,y)=(\phi_{0},\psi_{0},\zeta_{0})(y),\end{array}\right.$
(3.1)
where the initial data $(\phi_{0},\psi_{0},\zeta_{0})(y)$ and its derivatives
are sufficiently smooth away from but up to $y=0$, and
$(\phi_{0},\psi_{0},\zeta_{0})(y)\in L^{2}(\mathbf{R}),\phi_{0y}\in L^{2}(\bf
R^{\pm}).$
For simplicity, denote
$\mathcal{N}_{0}:=\|(\phi_{0},\psi_{0},\zeta_{0})\|^{2}+\|\phi_{0y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}.$
In order to prove (2.43), we easily see that it suffices to show
###### Proposition 3.1.
There exists a positive constant $\delta_{0}$, such that if the wave strength
$\delta$ and the initial data satisfy
$\delta+\mathcal{N}_{0}\leq\delta_{0},$
then the problem (3.1) admits a unique global solution
$(\phi,\psi,\zeta)(t,y)$ satisfying
* (i)
There exists a positive constant $C$ independent of $t$, such that
$\sup_{\tau\geq
0}\Big{(}\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\|\phi_{y}(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\Big{)}+\int_{0}^{+\infty}\|(\phi_{y},\psi_{y},\zeta_{y})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C(\mathcal{N}_{0}+\delta^{\frac{1}{4}}).$
* (ii)
For any $\tau_{0}>0$, there exists a positive constant $C=C(\tau_{0})$, such
that
$\sup_{\tau\geq\tau_{0}}\|(\psi_{y},\zeta_{y},\psi_{\tau},\zeta_{\tau})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\int_{\tau_{0}}^{+\infty}\|(\psi_{yy},\zeta_{yy},\psi_{y\tau},\zeta_{y\tau})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C(\tau_{0})(\mathcal{N}_{0}+\delta^{\frac{1}{4}}).$
* (iii)
The jump condition of $\phi(\tau,y)$ at $y=0$ admits the bound
$|[\phi](\tau)|\leq Ce^{-c\tau}$ (3.2)
where the positive constants $C$ and $c$ are independent of
$\tau\in(0,+\infty)$.
Assume that Proposition 3.1 holds, then for any $\tau_{0}>0$, one has
$\int_{\tau_{0}}^{+\infty}\Big{(}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+|\frac{d}{d\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}|\Big{)}d\tau<+\infty,$
whence,
$\lim_{\tau\rightarrow\infty}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}=0,$
which, together with Proposition 3.1 and Sobolev’s inequality, implies that
$\lim_{\tau\rightarrow\infty}\sup_{y\neq
0}\|(\phi,\psi,\zeta)\|_{L^{\infty}}^{2}\leq
C\lim_{\tau\rightarrow\infty}\|(\phi,\psi,\zeta)\|\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel\
\leq
C\lim_{\tau\rightarrow\infty}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel\
=0.$
The above inequality combined with (3.2) gives (2.43). Thus, the main result
Theorem 1.1 follows from (2.43) and (2.42).
Denote
$\begin{array}[]{l}\displaystyle
N(\tau_{*},\tau^{*})=\sup_{\tau\in[\tau_{*},\tau^{*}]}\Big{\\{}\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\|(\phi_{y},\psi_{y},\zeta_{y})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\|(\psi_{\tau},\zeta_{\tau})(\tau,\cdot)\|^{2}\Big{\\}},\\\\[11.38109pt]
\displaystyle N(\tau_{*})=N(\tau_{*},\tau_{*}),\end{array}$
and define the solution space by
$X[\tau_{*},\tau^{*}]=\left\\{(\phi,\psi,\zeta)\left|\begin{array}[]{l}\displaystyle(\phi,\psi,\zeta)(\tau,y)\in
C([\tau_{*},\tau^{*}];H^{1}({\bf R}^{\pm})),\\\\[2.84526pt]
\displaystyle(\psi_{y},\zeta_{y})\in L^{2}(\tau_{*},\tau^{*};H^{1}({\bf
R}^{\pm})),~{}\phi_{y}\in L^{2}(\tau_{*},\tau^{*};L^{2}({\bf R}^{\pm})),\\\
\displaystyle(\psi_{\tau},\zeta_{\tau})\in
L^{\infty}(\tau_{*},\tau^{*};L^{2}({\bf R}^{\pm}))\cap
L^{2}(\tau_{*},\tau^{*};H^{1}({\bf R}^{\pm})).\end{array}\right.\right\\}$
Since the local existence of solutions to (3.1) is proved in [7], we just
state it and omit its proof for brevity.
###### Proposition 3.2.
(Local existence) Suppose that $\mathcal{N}_{0}$ and the wave strength
$\delta$ are suitably small such that $\inf v_{0}$ and $\inf\theta_{0}$ are
positive. Then there exists a positive time
$\tau_{0}=\tau_{0}(N(0),\delta)>0$, such that the Cauchy problem (3.1) admits
a unique solution $(\phi,\psi,\zeta)(\tau,y)\in X[0,\tau_{0}]$ satisfying
$A(\tau_{0})+B(\tau_{0})+F(\tau_{0})\leq C(\mathcal{N}_{0}+\delta),$
where
$\displaystyle
A(\tau_{0})=\displaystyle\sup_{0\leq\tau\leq\tau_{0}}\Big{\\{}\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\|\phi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\Big{\\}}+\int_{0}^{\tau_{0}}\|(\psi_{y},\zeta_{y})\|^{2}d\tau,$
$\displaystyle
B(\tau_{0})=\displaystyle\sup_{0\leq\tau\leq\tau_{0}}\Big{\\{}g(\tau)^{\frac{1}{2}}\|\psi_{y}\|^{2}+g(\tau)\|\phi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\Big{\\}}+\int_{0}^{\tau_{0}}g(\tau)^{\frac{1}{2}+\vartheta}(\|\psi_{\tau}\|^{2}+\|(\frac{u_{y}}{v})_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2})d\tau$
$\displaystyle\displaystyle\qquad+\int_{0}^{\tau_{0}}g(\tau)(\|\psi_{y}^{2}\|^{2}+\|\theta_{\tau}\|^{2}+\|(\frac{\theta_{y}}{v})_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2})d\tau,$
$\displaystyle
F(\tau_{0})=\displaystyle\sup_{0\leq\tau\leq\tau_{0}}\Big{\\{}g(\tau)^{\frac{3}{2}+\vartheta}(\|\psi_{\tau}\|^{2}+\|(\frac{u_{y}}{v})_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2})+g(\tau)^{3}(\|\zeta_{\tau}\|^{2}+\|(\frac{\theta_{y}}{v})_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2})\Big{\\}}$
$\displaystyle\qquad\displaystyle+\int_{0}^{\tau_{0}}g(\tau)^{\frac{3}{2}+\vartheta}\|\psi_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+g(\tau)^{3}\|\zeta_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2})d\tau,$
with $g(\tau)=\tau\wedge 1=\min\\{\tau,1\\}$ and $\vartheta\in(0,1)$.
Moreover, $v,u,\theta$ have the same regularity as in Theorem 1.1. Thus,
$v,u_{x},\theta_{x}$ have one-side limit at $y=0$ and satisfy the jump
conditions
$\Big{[}p-\frac{u_{y}}{v}\Big{]}=\Big{[}\frac{\theta_{y}}{v}\Big{]}=0.$
Finally, one has the following estimate on the jump at $y=0$,
$|[v](\tau)|\leq C\delta e^{-c\tau},\qquad\tau>0$
for some positive constants $C$ and $c$ independent of $\tau$.
Hence, in view of the local existence and the standard continuation process,
we see that to prove Proposition 3.1, it suffices to show the following
(uniform) a priori estimate.
###### Proposition 3.3.
(A priori estimate) Suppose that the Cauchy problem (3.1) has a solution
$(\phi,\psi,\zeta)(\tau,y)\in X[\tau_{1},\tau_{2}]$. There exists a positive
constant $\eta_{1}$, such that if
$N(\tau_{1},\tau_{2})+\delta\leq\eta_{1},$ (3.3)
then,
$N(\tau_{1},\tau_{2})+\int_{\tau_{1}}^{\tau_{2}}\Big{\\{}\|\phi_{y}(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\|(\psi_{y},\zeta_{y})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel_{1}^{2}+\|(\psi_{y\tau},\zeta_{y\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(\tau,\cdot)\Big{\\}}d\tau\leq
C(N(\tau_{1})+\delta^{\frac{1}{4}}),$ (3.4)
where the positive constant $C$ is independent of $\tau$.
### 3.2 Energy estimates
In this section we will derive the a priori estimate given in Proposition 3.3.
Note that under the a priori assumption (3.3), if $\eta\ll 1,$ then if holds
that
$\inf_{[\tau_{1},\tau_{2}]\times\mathbf{R}}\\{(V+\phi,\Theta+\zeta)(\tau,y)\\}\geq
C_{0}$
for some positive constant $C_{0}$. First, one has the following Lemma:
###### Lemma 3.4.
Under the assumptions of Proposition 3.3, there exists a constant $C>0$, such
that for any $\tau\in[\tau_{1},\tau_{2}]$,
$\begin{array}[]{ll}\displaystyle\|(\phi,\psi,\zeta,\phi_{y})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\int_{\tau_{1}}^{\tau}\Big{\\{}\|\sqrt{(U^{R_{1}}_{y},U^{R_{3}}_{y})}(\phi,\zeta)\|^{2}+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\Big{\\}}d\tau\\\\[8.53581pt]
\leq\displaystyle
C\|(\phi,\psi,\zeta,\phi_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(\tau_{1})\displaystyle+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)(\cdot,\tau)\|^{2}d\tau+C\delta^{\frac{1}{4}}\\\\[11.38109pt]
\displaystyle~{}~{}+C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
Proof: Let
$\Phi(z)=z-1-\ln z.$
Arguing similarly to that in [12] or [14], one can get the following equality
$\begin{array}[]{ll}&\displaystyle
I_{1\tau}(\tau,y)+H_{1y}(\tau,y)+\frac{\Theta\psi_{y}^{2}}{v\theta}+\nu\frac{\Theta\zeta_{y}^{2}}{v\theta^{2}}+P(U^{R_{1}}_{y}+U^{R_{3}}_{y})\left(\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})\right)\\\\[8.53581pt]
&\displaystyle=Q_{3}-Q_{1}\psi-Q_{2}\frac{\zeta}{\theta},\end{array}$ (3.5)
where
$I_{1}(\tau,y)=R\Theta\Phi(\frac{v}{V})+\frac{\psi^{2}}{2}+\frac{R\Theta}{\gamma-1}\Phi(\frac{\theta}{\Theta}),$
$H_{1}(\tau,y)=(p-P)\psi-(\frac{u_{y}}{v}-\frac{U_{y}}{V})\psi-\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\frac{\zeta}{\theta},$
(3.6)
and
$\begin{array}[]{ll}Q_{3}=&\displaystyle-PU^{CD}_{y}\left(\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})\right)+\left(\nu(\frac{\Theta_{y}}{V})_{y}+\frac{U_{y}^{2}}{V}+Q_{2}\right)\Big{\\{}(\gamma-1)\Phi(\frac{v}{V})\\\\[11.38109pt]
&\displaystyle+\Phi(\frac{\theta}{\Theta})-\frac{\zeta^{2}}{\theta\Theta}\Big{\\}}-(\frac{1}{v}-\frac{1}{V})U_{y}\psi_{y}+(\frac{1}{v}-\frac{1}{V})U_{y}^{2}\frac{\zeta}{\theta}+2\frac{\zeta\psi_{y}U_{y}}{v\theta}+\nu\frac{\Theta_{y}\zeta_{y}\zeta}{v\theta^{2}}\\\\[11.38109pt]
&\displaystyle-\nu(\frac{1}{v}-\frac{1}{V})\frac{\Theta\Theta_{y}\zeta_{y}}{\theta^{2}}+\nu(\frac{1}{v}-\frac{1}{V})\frac{\zeta\Theta_{y}^{2}}{\theta^{2}}.\end{array}$
(3.7)
Integration of the equality (3.5) with respect to $y$ and $\tau$ over
${\mathbf{R}}^{\pm}\times[\tau_{1},\tau]$ yields that
$\begin{array}[]{ll}\displaystyle\int
I_{1}(\tau,y)dy+\int_{\tau_{1}}^{\tau}\big{[}H_{1}\big{]}(\tau)d\tau+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\bigg{(}\frac{\Theta\psi_{y}^{2}}{v\theta}+\nu\frac{\Theta\zeta_{y}^{2}}{v\theta^{2}}\bigg{)}dyd\tau\\\\[11.38109pt]
\qquad\displaystyle+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}P(U^{R_{1}}_{y}+U^{R_{3}}_{y})\left(\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})\right)dyd\tau\\\\[11.38109pt]
\displaystyle=\int
I_{1}(\tau_{1},y)dy+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\big{(}Q_{3}-Q_{1}\psi-Q_{2}\frac{\zeta}{\theta}\big{)}dyd\tau.\end{array}$
(3.8)
It is easy to observe that the jump of $H_{1}$ in (3.6) across $y=0$ vanishes,
i.e.,
$\begin{array}[]{ll}\displaystyle\big{[}H_{1}\big{]}(\tau)&\displaystyle=\big{[}(p-\frac{u_{y}}{v})\psi\big{]}-\big{[}(P-\frac{U_{y}}{V})\psi\big{]}-\nu\big{[}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\frac{\zeta}{\theta}\big{]}\\\
&\displaystyle=\big{[}p-\frac{u_{y}}{v}\big{]}\psi(\tau,0)-\big{[}P-\frac{U_{y}}{V}\big{]}\psi(\tau,0)-\nu\Big{(}\big{[}\frac{\theta_{y}}{v}\big{]}-\big{[}\frac{\Theta_{y}}{V}\big{]}\Big{)}\frac{\zeta(\tau,0)}{\theta(\tau,0)}=0.\end{array}$
Recalling that
$\Phi(1)=\Phi^{\prime}(1)=0,\qquad\Phi^{\prime\prime}(z)=z^{-2}>0,$
there exists a positive constant $C$, such that if $z$ is near 1, then
$C^{-1}(z-1)^{2}\leq\Phi(z)\leq C(z-1)^{2}.$
Thus under the a priori assumptions (3.3), one gets
$C^{-1}|\phi|^{2}\leq\Phi(\frac{v}{V})\leq C|\phi|^{2},\qquad
C^{-1}|\zeta|^{2}\leq\Phi(\frac{\theta}{\Theta})\leq C|\zeta|^{2}$ (3.9)
and
$C^{-1}|(\phi,\zeta)|^{2}\leq\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})\leq C|(\phi,\zeta)|^{2}.$ (3.10)
Now it follows from (3.7), (3.9), (3.10) and Cauchy-Schwarz’s inequality that
$\begin{array}[]{ll}\displaystyle|Q_{3}|\leq&\displaystyle\frac{\Theta\psi_{y}^{2}}{4v\theta}+\frac{\nu\Theta\zeta_{y}^{2}}{4v\theta^{2}}+C\Big{\\{}(|\Theta^{CD}_{y}|^{2},|\Theta^{CD}_{yy}|)+(|(V^{R_{1}}_{y},U^{R_{1}}_{y},\Theta^{R_{1}}_{y})|^{2},|\Theta^{R_{1}}_{yy}|)\\\\[8.53581pt]
&\displaystyle+(|(V^{R_{3}}_{y},U^{R_{3}}_{y},\Theta^{R_{3}}_{y})|^{2},|\Theta^{R_{3}}_{yy}|)+|Q_{2}|\Big{\\}}(\phi^{2}+\zeta^{2}).\end{array}$
(3.11)
By the properties of the viscous contact wave, one can obtain
$\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(|\Theta^{CD}_{y}|^{2},|\Theta^{CD}_{yy}|)(\phi^{2}+\zeta^{2})dyd\tau\leq
C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau,$
while by the properties of the approximate rarefaction wave in Lemma 2.2, we
have that for $i=1,3,$
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(|(V^{R_{i}}_{y},U^{R_{i}}_{y},\Theta^{R_{i}}_{y})|^{2},|\Theta^{R_{i}}_{yy}|)(\phi^{2}+\zeta^{2})dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\int_{\tau_{1}}^{\tau}(\|(V^{R_{i}}_{y},U^{R_{i}}_{y},\Theta^{R_{i}}_{y})\|^{2}+\|\Theta^{R_{i}}_{yy}\|_{L^{1}})\|(\phi,\zeta)\|^{2}_{L^{\infty}}d\tau\\\\[11.38109pt]
&\displaystyle\leq
C\int_{\tau_{1}}^{\tau}(1+\tau)^{-1}\|(\phi,\zeta)\|\|(\phi_{y},\zeta_{y})\|d\tau\\\\[11.38109pt]
&\displaystyle\leq\mu\int_{\tau_{1}}^{\tau}\|(\phi_{y},\zeta_{y})\|^{2}d\tau+C_{\mu}\int_{\tau_{1}}^{\tau}(1+\tau)^{-2}\|(\phi,\zeta)\|^{2}d\tau,\end{array}$
where and in the sequel $\mu$ is a small positive constant to be determined
and $C_{\mu}$ is some positive constant depending on $\mu$.
Now, it remains to estimate the terms $Q_{1}\psi$, $Q_{2}\frac{\zeta}{\theta}$
on the right-hand side of (3.8) and the term $|Q_{2}|(\phi^{2}+\zeta^{2})$ on
the right-hand side of (3.11). For simplicity, we only estimate
$Q_{2}\frac{\zeta}{\theta}$. By (2.45), we find that
$\begin{array}[]{ll}\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}|Q_{2}\frac{\zeta}{\theta}|dyd\tau\leq
C\int_{\tau_{1}}^{\tau}\|\zeta\|_{L^{\infty}_{y}}\|Q_{2}\|_{L^{1}_{y}}d\tau\\\\[11.38109pt]
\quad\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\|\zeta\|^{\frac{1}{2}}\|\zeta_{y}\|^{\frac{1}{2}}\Big{(}\|Q_{21}\|_{L^{1}_{y}}+\|Q_{22}\|_{L^{1}_{y}}+\|Q^{CD}\|_{L^{1}_{y}}\Big{)}d\tau\\\\[11.38109pt]
\quad\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\|\zeta\|^{\frac{1}{2}}\|\zeta_{y}\|^{\frac{1}{2}}\Big{(}\delta
e^{-C(1+\tau)}+(\delta^{r_{1}}+\delta^{r_{3}})^{\frac{1}{8}}(1+\tau)^{-\frac{7}{8}}+\delta(1+\tau)^{-\frac{3}{2}}\Big{)}d\tau\\\\[11.38109pt]
\quad\displaystyle\leq\mu\int_{\tau_{1}}^{\tau}\|\zeta_{y}\|^{2}d\tau+C_{\mu}~{}\delta^{\frac{1}{6}}\int_{\tau_{1}}^{\tau}\|\zeta\|^{\frac{2}{3}}(1+\tau)^{-\frac{7}{6}}d\tau\\\\[11.38109pt]
\quad\displaystyle\leq\mu\int_{\tau_{1}}^{\tau}\|\zeta_{y}\|^{2}d\tau+C_{\mu}\int_{\tau_{1}}^{\tau}\|\zeta\|^{2}(1+\tau)^{-\frac{7}{6}}d\tau+C_{\mu}~{}\delta^{\frac{1}{4}}.\end{array}$
Similarly, one can control the term $Q_{1}\psi$ and
$|Q_{2}|(\phi^{2}+\zeta^{2})$.
Thus, substituting all the above estimates into (3.8) and choosing $\mu$ in
the front of the integral
$\displaystyle\int_{\tau_{1}}^{\tau}\|(\psi_{y},\zeta_{y})\|^{2}d\tau$ small
enough, so that the integral can be absorbed by the left-hand side of (3.8),
one concludes
$\begin{array}[]{ll}\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\int_{\tau_{1}}^{\tau}\big{\\{}\|(\psi_{y},\zeta_{y})(\tau,\cdot)\|^{2}+\|\sqrt{(U^{R_{1}}_{y},U_{y}^{R_{3}})}(\phi,\zeta)(\tau,\cdot)\|^{2}\big{\\}}d\tau\\\
\displaystyle\leq
C\|(\phi,\psi,\zeta)(\tau_{1},\cdot)\|^{2}+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\delta^{\frac{1}{4}}\\\
\displaystyle+C\mu\int_{\tau_{1}}^{\tau}\|\phi_{y}(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
(3.12)
Next, we estimate $\|\phi_{y}\|^{2}$. Denote $\tilde{v}=\frac{v}{V}.$ From the
system $\eqref{P}_{2}$, one has
$(\frac{\tilde{v}_{y}}{\tilde{v}})_{\tau}-\psi_{\tau}-(p-P)_{y}-Q_{1}=0.$
Multiplying the above equation by $\frac{\tilde{v}_{y}}{\tilde{v}}$ and
noticing that
$-(p-P)_{y}=\frac{R\theta}{v}\frac{\tilde{v}_{y}}{\tilde{v}}-\frac{R\zeta_{y}}{v}+(p-P)\frac{V_{y}}{V}-R\Theta_{y}(\frac{1}{V}-\frac{1}{v}),$
one obtains
$\displaystyle\displaystyle\left(\frac{1}{2}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}-\psi\frac{\tilde{v}_{y}}{\tilde{v}}\right)_{\tau}+\left(\psi\frac{\tilde{v}_{\tau}}{\tilde{v}}\right)_{y}+\frac{R\theta}{v}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}$
$\displaystyle\quad=\displaystyle\psi_{y}(\frac{u_{y}}{v}-\frac{U_{y}}{V})+\left(\frac{R\zeta_{y}}{v}-(p-P)\frac{V_{y}}{V}+R\Theta_{y}(\frac{1}{V}-\frac{1}{v})-Q_{1}\right)\frac{\tilde{v}_{y}}{\tilde{v}}.$
Integrating the above equality with respect to $y$ and $\tau$ over ${\bf
R}^{\pm}\times[\tau_{1},\tau]$ and using Cauchy-Schwarz’s inequality, we infer
that
$\begin{array}[]{ll}&\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\left(\frac{1}{2}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}-\psi\frac{\tilde{v}_{y}}{\tilde{v}}\right)(\tau,y)dy+\int_{\tau_{1}}^{\tau}\left[\psi\frac{\tilde{v}_{\tau}}{\tilde{v}}\right](\tau)d\tau+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{R\theta}{2v}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\left(\frac{1}{2}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}-\psi\frac{\tilde{v}_{y}}{\tilde{v}}\right)(\tau_{1},y)dy+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}|\psi_{y}(\frac{u_{y}}{v}-\frac{U_{y}}{V})|dyd\tau\\\
&\displaystyle+C\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\left|\frac{R\zeta_{y}}{v}-(p-P)\frac{V_{y}}{V}+R\Theta_{y}(\frac{1}{V}-\frac{1}{v})-Q_{1}\right|^{2}dyd\tau,\end{array}$
(3.13)
where the jump across $y=0$ can be bounded as follows.
$\begin{array}[]{ll}\displaystyle\int_{\tau_{1}}^{\tau}\left[\psi\frac{\tilde{v}_{\tau}}{\tilde{v}}\right](\tau)d\tau=\int_{\tau_{1}}^{\tau}\psi(\tau,0)\left[\frac{u_{y}}{v}-\frac{U_{y}}{V}\right](\tau)d\tau=\int_{\tau_{1}}^{\tau}\psi(\tau,0)\left[p\right](\tau)d\tau\\\\[11.38109pt]
\qquad\displaystyle=R\int_{\tau_{1}}^{\tau}\psi(\tau,0)\theta(\tau,0)\left[\frac{1}{v}\right](\tau)d\tau=-R\int_{\tau_{1}}^{\tau}\frac{\psi(\tau,0)\theta(\tau,0)}{v(\tau,0+)v(\tau,0-)}\left[v\right](\tau)d\tau\\\\[11.38109pt]
\qquad\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\|\psi\|_{L^{\infty}}(\tau)|[v]|(\tau_{1})e^{-C(\tau-\tau_{1})}d\tau\leq
C\delta\int_{\tau_{1}}^{\tau}\|\psi\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{1}{2}}e^{-C(\tau-\tau_{1})}d\tau\\\\[11.38109pt]
\qquad\displaystyle\leq\delta\int_{\tau_{1}}^{\tau}\|\psi_{y}\|^{2}d\tau+\delta\sup_{\tau\in[\tau_{1},\tau_{2}]}\|\psi\|^{2}(\tau)+C\delta.\end{array}$
Using the equality
$\frac{\tilde{v}_{y}}{\tilde{v}}=\frac{v_{y}}{v}-\frac{V_{y}}{V}=\frac{\phi_{y}}{v}-\frac{V_{y}\phi}{vV},$
we see that
$C^{-1}(|\phi_{y}|^{2}-|V_{y}\phi|^{2})\leq(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}\leq
C(|\phi_{y}|^{2}+|V_{y}\phi|^{2}).$
From the definition of $Q_{1}$ in (2.44) it follows that
$\begin{array}[]{ll}\displaystyle\int_{\tau_{1}}^{\tau}\|Q_{1}\|^{2}d\tau\leq
C\int_{\tau_{1}}^{\tau}\Big{(}\|Q_{11}\|^{2}+\|Q_{12}\|^{2}\Big{)}d\tau\\\
\displaystyle\qquad\leq
C\int_{\tau_{1}}^{\tau}\Big{(}\|Q_{11}\|^{2}+\|(U^{R_{1}}_{yy},U^{R_{3}}_{yy},U^{R_{1}}_{y}V^{R_{1}}_{y},U^{R_{3}}_{y}V^{R_{3}}_{y})\|^{2}\Big{)}d\tau\leq
C\delta^{\frac{1}{4}}.\end{array}$
Therefore, substituting all the above estimates into (3.13), we conclude that
$\begin{array}[]{ll}\displaystyle\quad\|\phi_{y}(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\int_{\tau_{1}}^{\tau}\|\phi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C\|(\phi,\psi,\phi_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(\tau_{1})+C\|(\phi,\psi)(\tau,\cdot)\|^{2}\\\
\displaystyle+C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau+C\int_{\tau_{1}}^{\tau}\|(\psi_{y},\zeta_{y})\|^{2}d\tau\\\\[11.38109pt]
\displaystyle+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\delta^{\frac{1}{4}}.\end{array}$
(3.14)
Multiplying the inequality (3.12) by a large constant $C_{1}>0$, and summing
the resulting inequality with (3.14), we obtain Lemma 3.4. This completes the
proof. $\hfill\Box$
Next, we derive the higher order estimates, which are summarized in the
following Lemma:
###### Lemma 3.5.
Under the assumptions of Proposition 3.3, it holds that
$\begin{array}[]{ll}\displaystyle
N(\tau_{1},\tau_{2})+\int_{\tau_{1}}^{\tau_{2}}\big{\\{}\|\sqrt{(U^{R_{1}}_{y},U^{R_{3}}_{y})}(\phi,\zeta)\|^{2}+\|\phi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\|(\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel_{1}^{2}+\|(\psi_{y\tau},\zeta_{y\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\big{\\}}d\tau\\\
\displaystyle\leq
CN(\tau_{1})+C\int_{\tau_{1}}^{\tau_{2}}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\delta^{\frac{1}{4}}+C\delta\int_{\tau_{1}}^{\tau_{2}}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
Proof: Multiplying the equation $\eqref{P}_{2}$ by $\displaystyle-\psi_{yy}$,
one gets
$\begin{array}[]{ll}\displaystyle\left(\frac{\psi_{y}^{2}}{2}\right)_{\tau}-\left(\psi_{\tau}\psi_{y}\right)_{y}+\frac{\psi_{yy}^{2}}{v}=\Big{\\{}(p-P)_{y}+\frac{v_{y}}{v^{2}}\psi_{y}-\big{(}U_{y}(\frac{1}{v}-\frac{1}{V})\big{)}_{y}+Q_{1}\Big{\\}}\psi_{yy}.\end{array}$
Integration of the above equation with respect to $y$ and $\tau$ over ${\bf
R}^{\pm}\times[\tau_{1},\tau]$ gives
$\begin{array}[]{ll}\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{y}^{2}}{2}(\tau,y)dy+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{yy}^{2}}{v}dyd\tau=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{y}^{2}}{2}(\tau_{1},y)dy-\int_{\tau_{1}}^{\tau}\left[\psi_{\tau}\psi_{y}\right](\tau)d\tau\\\
\displaystyle~{}~{}+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\Big{\\{}(p-P)_{y}+\frac{v_{y}}{v^{2}}\psi_{y}-\big{(}U_{y}(\frac{1}{v}-\frac{1}{V})\big{)}_{y}+Q_{1}\Big{\\}}\psi_{yy}dyd\tau=:\sum_{i=1}^{3}J_{i}.\end{array}$
(3.15)
We have to estimate $J_{i}$. First, the jump $J_{2}$ can be bounded as
follows.
$\begin{array}[]{ll}J_{2}&\displaystyle=-\int_{\tau_{1}}^{\tau}\left[\psi_{\tau}\psi_{y}\right](\tau)d\tau=-\int_{\tau_{1}}^{\tau}\psi_{\tau}(\tau,0)\left[\psi_{y}\right](\tau)d\tau\\\\[8.53581pt]
&\displaystyle=-\int_{\tau_{1}}^{\tau}\psi_{\tau}(\tau,0)\left[u_{y}\right](\tau)d\tau=-\int_{\tau_{1}}^{\tau}\psi_{\tau}(\tau,0)\left[(\frac{u_{y}}{v}-p)v\right](\tau)d\tau\\\\[8.53581pt]
&\displaystyle=-\int_{\tau_{1}}^{\tau}\psi_{\tau}(\tau,0)(\frac{u_{y}}{v}-p)(\tau,0)\left[v\right](\tau)d\tau\\\\[8.53581pt]
&\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\|\psi_{\tau}\|_{L^{\infty}}\big{(}\|\psi_{y}\|_{L^{\infty}}+1\big{)}\left[v\right](\tau_{1})e^{-C(\tau-\tau_{1})}d\tau\\\\[8.53581pt]
&\displaystyle\leq
C\delta\int_{\tau_{1}}^{\tau}\|\psi_{\tau}\|^{\frac{1}{2}}\|\psi_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}\big{(}\|\psi_{y}\|^{\frac{1}{2}}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}+1\big{)}e^{-C(\tau-\tau_{1})}d\tau.\end{array}$
(3.16)
In view of $\eqref{P}_{2}$ and (3.3), one has
$\begin{array}[]{ll}\|\psi_{\tau}\|&\displaystyle\leq
C\Big{(}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(U_{yy},V_{y},U_{y},\Theta_{y})\phi\|+\|Q_{1}\|\Big{)}\\\\[8.53581pt]
&\displaystyle\leq
C\Big{(}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\delta\Big{)}.\end{array}$
(3.17)
Substituting (3.17) into (3.16), we obtain
$\begin{array}[]{ll}\displaystyle|J_{2}|\leq
C\delta\int_{\tau_{1}}^{\tau}\Big{(}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\delta\Big{)}^{\frac{1}{2}}\|\psi_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}\Big{(}\|\psi_{y}\|^{\frac{1}{2}}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}+1\Big{)}e^{-C(\tau-\tau_{1})}d\tau\\\
\displaystyle\leq\mu\int_{\tau_{1}}^{\tau}\|(\psi_{yy},\psi_{y\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C_{\mu}~{}\delta\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C_{\mu}\delta.\end{array}$
(3.18)
On the other hand, $J_{3}$ can be estimates as follows.
$\begin{array}[]{ll}J_{3}&\displaystyle=\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\bigg{\\{}(p-P)_{y}+\frac{v_{y}}{v^{2}}\psi_{y}-\big{(}U_{y}(\frac{1}{v}-\frac{1}{V})\big{)}_{y}+Q_{1}\bigg{\\}}\psi_{yy}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\Big{\\{}|(\phi_{y},\zeta_{y})|+|(\phi,\zeta)||(\phi_{y},V_{y},\Theta_{y},U_{yy})|\\\
&\displaystyle\qquad\qquad\qquad~{}~{}+|(\phi_{y},V_{y})||(\psi_{y},U_{y},U_{y}\phi)|+|Q_{1}|\Big{\\}}|\psi_{yy}|dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\mu\int_{\tau_{1}}^{\tau}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C_{\mu}\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C_{\mu}~{}\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)\|^{2}d\tau\\\\[8.53581pt]
&\displaystyle~{}~{}~{}+C_{\mu}~{}\delta+C_{\mu}~{}\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
(3.19)
Substituting (3.18) and (3.19) into (3.15) and choosing $\mu$ suitably small
in the front of the integral
$\int_{\tau_{1}}^{\tau}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau$,
we deduce that
$\begin{array}[]{ll}\displaystyle\|\psi_{y}\|^{2}(\tau)+\int_{\tau_{1}}^{\tau}\|\psi_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C\|\psi_{y}\|^{2}(\tau_{1})+C\mu\int_{\tau_{1}}^{\tau}\|\psi_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\\\\[11.38109pt]
\displaystyle\quad+C_{\mu}\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\zeta)\|^{2}d\tau+C_{\mu}~{}\delta+C_{\mu}\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\\\\[11.38109pt]
\displaystyle\quad+C_{\mu}~{}\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
(3.20)
Multiplication of the equation $\eqref{P}_{3}$ with $-\zeta_{yy}$ yields that
$\begin{array}[]{ll}\displaystyle\frac{R}{\gamma-1}\left(\frac{\zeta_{y}^{2}}{2}\right)_{\tau}-\frac{R}{\gamma-1}\left(\zeta_{\tau}\zeta_{y}\right)_{y}+\nu\frac{\zeta_{yy}^{2}}{v}\\\
\displaystyle=\bigg{\\{}(pu_{y}-PU_{y})+\nu\frac{\zeta_{y}v_{y}}{v^{2}}-\nu\big{(}\Theta_{y}(\frac{1}{v}-\frac{1}{V})\big{)}_{y}-(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+Q_{2}\bigg{\\}}\zeta_{yy}.\end{array}$
Integrating the above equality with respect to $y$ and $\tau$ over ${\bf
R}^{\pm}\times[\tau_{1},\tau]$, and employing almost the same arguments as
those used for
$\|\psi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(\tau)$
in (3.20), we obtain
$\begin{array}[]{ll}&\displaystyle\|\zeta_{y}\|^{2}(\tau)+\int_{\tau_{1}}^{\tau}\|\zeta_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C\|\zeta_{y}\|^{2}(\tau_{1})+C\delta^{\frac{1}{4}}+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\zeta)\|^{2}d\tau\\\
&\displaystyle\quad+C\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C(\delta)^{2}\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau,\end{array}$
(3.21)
where we have used the following jump estimate across $y=0$
$\begin{array}[]{ll}\displaystyle-\frac{R}{\gamma-1}\int_{\tau_{1}}^{\tau}\left[\zeta_{\tau}\zeta_{y}\right](\tau)d\tau=-\frac{R}{\gamma-1}\int_{\tau_{1}}^{\tau}\zeta_{\tau}(\tau,0)\left[\zeta_{y}\right](\tau)d\tau\\\\[11.38109pt]
\displaystyle=-\frac{R}{\gamma-1}\int_{\tau_{1}}^{\tau}\zeta_{\tau}(\tau,0)\left[\theta_{y}\right](\tau)d\tau=-\frac{R}{\gamma-1}\int_{\tau_{1}}^{\tau}\zeta_{\tau}(\tau,0)\frac{\theta_{y}}{v}(\tau,0)\left[v\right](\tau)d\tau\\\
\displaystyle\leq
C\int_{\tau_{1}}^{\tau}\|\zeta_{\tau}\|_{L^{\infty}}\big{(}1+\|\zeta_{y}\|_{L^{\infty}}\big{)}[v](\tau_{1})e^{-C(\tau-\tau_{1})}d\tau\\\
\displaystyle\leq
C\delta\int_{\tau_{1}}^{\tau}\|\zeta_{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}\|\zeta_{y\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}\big{(}1+\|\zeta_{y}\|^{\frac{1}{2}}\|\zeta_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{\frac{1}{2}}\big{)}e^{-C(\tau-\tau_{1})}d\tau\end{array}$
and the estimate
$\begin{array}[]{ll}\displaystyle\|\zeta_{\tau}\|\leq
C\Big{(}\|\zeta_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(U_{y},\Theta_{yy},\Theta_{y}V_{y},U_{y}^{2})(\phi,\zeta)\|+\|Q_{2}\|\Big{)}\\\
\displaystyle\qquad\leq
C\Big{(}\|\zeta_{yy}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel+\delta\Big{)}.\end{array}$
(3.22)
It follows from (3.17) and (3.22) that
$\begin{array}[]{ll}\displaystyle\int_{\tau_{1}}^{\tau_{2}}\|(\psi_{\tau},\zeta_{\tau})(\tau,\cdot)\|^{2}d\tau\\\
\displaystyle\leq
C\Big{(}\int_{\tau_{1}}^{\tau_{2}}\|(\psi_{yy},\zeta_{yy})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+\int_{\tau_{1}}^{\tau_{2}}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\\\\[11.38109pt]
\displaystyle\qquad\qquad+\int_{\tau_{1}}^{\tau_{2}}\|(U_{y},\Theta_{yy},\Theta_{y}V_{y},U_{y}^{2})(\phi,\zeta)\|^{2}d\tau+\int_{\tau_{1}}^{\tau_{2}}\|Q_{2}\|^{2}d\tau\Big{)}\\\\[11.38109pt]
\displaystyle\leq
C\int_{\tau_{1}}^{\tau_{2}}\|(\psi_{yy},\zeta_{yy})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C\int_{\tau_{1}}^{\tau_{2}}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+\int_{\tau_{1}}^{\tau_{2}}(1+\tau)^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\delta^{\frac{1}{4}}.\end{array}$
(3.23)
Now we turn to control
$\displaystyle\sup_{\tau\in[\tau_{1},\tau_{2}]}\|(\psi_{\tau},\zeta_{\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}$.
First, applying the operator $\partial_{\tau}$ to the equation
$(\ref{P})_{2}$, we get
$\psi_{\tau\tau}=\big{(}\frac{u_{y}}{v}-p\big{)}_{y\tau}-\big{(}\frac{U_{y}}{V}-P\big{)}_{y\tau}-Q_{1\tau}.$
Multiplication of the above equation by $\psi_{\tau}$ gives
$\begin{array}[]{ll}\displaystyle\left(\frac{\psi_{\tau}^{2}}{2}\right)_{\tau}+\frac{\psi_{y\tau}^{2}}{v}=\Big{\\{}\psi_{\tau}\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}-\psi_{\tau}\big{(}\frac{U_{y}}{V}-P\big{)}_{\tau}\Big{\\}}_{y}\\\\[11.38109pt]
\displaystyle\qquad-\psi_{y\tau}\frac{U_{y\tau}}{v}+\psi_{y\tau}\frac{u_{y}}{v^{2}}v_{\tau}+\psi_{y\tau}(\frac{U_{y}}{V})_{\tau}+\psi_{y\tau}(p-P)_{\tau}-\psi_{\tau}Q_{1\tau}.\end{array}$
If we integrate the above equality with respect to $y$ and $\tau$ over ${\bf
R}^{\pm}\times[\tau_{1},\tau]$, we find that
$\begin{array}[]{ll}\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{\tau}^{2}}{2}(\tau,y)dy+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{y\tau}^{2}}{v}dyd\tau\\\\[11.38109pt]
\displaystyle~{}~{}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{\psi_{\tau}^{2}}{2}(\tau_{1},y)dy-\int_{\tau_{1}}^{\tau}\Big{[}\psi_{\tau}\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}-\psi_{\tau}\big{(}\frac{U_{y}}{V}-P\big{)}_{\tau}\Big{]}(\tau)d\tau\\\\[11.38109pt]
\displaystyle+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\Big{\\{}-\psi_{y\tau}\frac{U_{y\tau}}{v}+\psi_{y\tau}\frac{u_{y}}{v^{2}}v_{\tau}+\psi_{y\tau}(\frac{U_{y}}{V})_{\tau}+\psi_{y\tau}(p-P)_{\tau}-\psi_{\tau}Q_{1\tau}\Big{\\}}dyd\tau,\end{array}$
(3.24)
where the jump across $y=0$ in fact vanishes, i.e.,
$\begin{array}[]{ll}\displaystyle\Big{[}\psi_{\tau}\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}-\psi_{\tau}\big{(}\frac{U_{y}}{V}-P\big{)}_{\tau}\Big{]}(\tau)\\\\[8.53581pt]
\displaystyle=[\psi_{\tau}](\tau)\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}(\tau,0-)+\psi_{\tau}(\tau,0+)\Big{[}\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}\Big{]}(\tau)-[\psi_{\tau}](\tau)\big{(}\frac{U_{y}}{V}-P\big{)}_{\tau}(\tau,0)\\\\[8.53581pt]
\displaystyle=[\psi]_{\tau}(\tau)\big{(}\frac{u_{y}}{v}-p\big{)}_{\tau}(\tau,0-)+\psi_{\tau}(\tau,0+)\Big{[}\frac{u_{y}}{v}-p\Big{]}_{\tau}(\tau)-[\psi]_{\tau}(\tau)\big{(}\frac{U_{y}}{V}-P\big{)}_{\tau}(\tau,0)\\\\[5.69054pt]
\displaystyle=0.\end{array}$ (3.25)
Now we apply $\partial_{\tau}$ to the equation $(\ref{P})_{3}$ to deduce that
$\frac{R}{\gamma-1}\zeta_{\tau\tau}=\nu\big{(}\frac{\theta_{y}}{v}\big{)}_{y\tau}-\nu\big{(}\frac{\Theta_{y}}{V}\big{)}_{y\tau}+\Big{\\{}u_{y}\big{(}\frac{u_{y}}{v}-p\big{)}\Big{\\}}_{\tau}-\Big{\\{}u_{y}\big{(}\frac{U_{y}}{V}-P\big{)}\Big{\\}}_{\tau}-Q_{2\tau}.$
Multiplying the above equation by $\zeta_{\tau}$, one has
$\begin{array}[]{ll}\displaystyle\frac{R}{\gamma-1}(\frac{\zeta_{\tau}^{2}}{2})_{\tau}+\nu\frac{\zeta_{y\tau}^{2}}{v}=\Big{\\{}\nu\zeta_{\tau}\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}-\nu\zeta_{\tau}\big{(}\frac{\Theta_{y}}{V}\big{)}_{\tau}\Big{\\}}_{y}\\\\[8.53581pt]
\displaystyle\qquad+\nu\zeta_{y\tau}\frac{\Theta_{y\tau}}{v}+\nu\zeta_{y\tau}\frac{\theta_{y}}{v^{2}}v_{\tau}+\nu\zeta_{y\tau}(\frac{\Theta_{y}}{V})_{\tau}+\zeta_{\tau}u_{y\tau}(\frac{u_{y}}{v}-p)\\\\[8.53581pt]
\displaystyle\qquad+\zeta_{\tau}u_{y}(\frac{u_{y}}{v}-p)_{\tau}-\zeta_{\tau}U_{y\tau}(\frac{U_{y}}{V}-P)-\zeta_{\tau}U_{y}(\frac{U_{y}}{V}-P)_{\tau}-\zeta_{\tau}Q_{2\tau}.\end{array}$
Integrating the above equality with respect to $y$ and $\tau$ over ${\bf
R}^{\pm}\times[\tau_{1},\tau]$, we deduce that
$\begin{array}[]{ll}\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{R\zeta_{\tau}^{2}}{2(\gamma-1)}(\tau,y)dy+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\nu\frac{\zeta_{y\tau}^{2}}{v}dyd\tau=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{R\zeta_{\tau}^{2}}{2(\gamma-1)}(\tau_{1},y)dy\\\\[14.22636pt]
\displaystyle-\int_{\tau_{1}}^{\tau}\Big{[}\nu\zeta_{\tau}\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}-\nu\zeta_{\tau}\big{(}\frac{\Theta_{y}}{V}\big{)}_{\tau}\Big{]}(\tau)d\tau+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\Big{\\{}\nu\zeta_{y\tau}\frac{\Theta_{y\tau}}{v}+\nu\zeta_{y\tau}\frac{\theta_{y}}{v^{2}}v_{\tau}+\nu\zeta_{y\tau}(\frac{\Theta_{y}}{V})_{\tau}\\\\[14.22636pt]
\displaystyle+\zeta_{\tau}u_{y\tau}(\frac{u_{y}}{v}-p)+\zeta_{\tau}u_{y}(\frac{u_{y}}{v}-p)_{\tau}-\zeta_{\tau}U_{y\tau}(\frac{U_{y}}{V}-P)-\zeta_{\tau}U_{y}(\frac{U_{y}}{V}-P)_{\tau}-\zeta_{\tau}Q_{2\tau}\Big{\\}}dyd\tau,\end{array}$
(3.26)
where the jump in fact vanishes.
$\begin{array}[]{ll}\displaystyle\Big{[}\nu\zeta_{\tau}\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}-\nu\zeta_{\tau}\big{(}\frac{\Theta_{y}}{V}\big{)}_{\tau}\Big{]}(\tau)\\\\[5.69054pt]
\displaystyle=\nu[\zeta_{\tau}](\tau)\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}(\tau,0-)+\nu\zeta_{\tau}(\tau,0+)\Big{[}\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}\Big{]}(\tau)-\nu[\zeta_{\tau}](\tau)\big{(}\frac{\Theta_{y}}{V}\big{)}_{\tau}(\tau,0)\\\\[8.53581pt]
\displaystyle=\nu[\zeta]_{\tau}(\tau)\big{(}\frac{\theta_{y}}{v}\big{)}_{\tau}(\tau,0-)+\nu\zeta_{\tau}(\tau,0+)\Big{[}\frac{\theta_{y}}{v}\Big{]}_{\tau}(\tau)-\nu[\zeta]_{\tau}(\tau)\big{(}\frac{\Theta_{y}}{V}\big{)}_{\tau}(\tau,0)\\\\[5.69054pt]
\displaystyle=0.\end{array}$ (3.27)
Hence, taking into account (3.25) and (3.27), we get from (3.24) and (3.26)
that
$\begin{array}[]{ll}\displaystyle\|(\psi_{\tau},\zeta_{\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(t)+\int_{\tau_{1}}^{\tau}\|(\psi_{y\tau},\zeta_{y\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\leq
C\|(\psi_{\tau},\zeta_{\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}(\tau_{1})+C\int_{\tau_{1}}^{\tau}\|(\psi_{\tau},\zeta_{\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau\\\\[11.38109pt]
\displaystyle\quad+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\zeta)\|^{2}d\tau+C~{}\delta+C\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau\\\\[11.38109pt]
\displaystyle\quad+C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
(3.28)
Combing the estimates (3.20), (3.21), (3.23), (3.28) and Lemma 3.4 together,
we obtain Lemma 3.5, and the proof is completed. $\hfill\Box$
It remains to control the term
$\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau,$
which comes from the viscous contact wave. We shall use the estimate on the
heat kernel in [12] to get the desired estimates.
###### Lemma 3.6.
Suppose that $Z(t,y)$ satisfies
$Z\in L^{\infty}(0,T;L^{2}(\mathbf{R}^{\pm})),~{}~{}Z_{y}\in
L^{2}(0,T;L^{2}(\mathbf{R}^{\pm})),~{}~{}Z_{\tau}\in
L^{2}(0,T;H^{-1}(\mathbf{R}^{\pm})),$
then
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}(1+\tau)^{-1}Z^{2}e^{-\frac{2\beta
y^{2}}{1+\tau}}dyd\tau\\\\[11.38109pt] &\displaystyle\leq
C_{\beta}\bigg{\\{}\|Z(\tau_{1},y)\|^{2}+\int_{\tau_{1}}^{\tau}\|h_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+\int_{\tau_{1}}^{\tau}\langle
Z_{\tau},Zg_{\beta}^{2}\rangle_{H^{1}\times
H^{-1}(\mathbf{R}^{\pm})}d\tau\bigg{\\}}\end{array}$ (3.29)
where
$g_{\beta}(\tau,y)=\displaystyle(1+\tau)^{-\frac{1}{2}}\int_{0}^{y}e^{-\frac{\beta\eta^{2}}{1+\tau}}d\eta$
(3.30)
and $\beta>0$ is the constant to be determined.
###### Remark 3.7.
Lemma 3.6 can be shown using arguments similar to those in [12], and hence its
proof will be omitted here for simplicity. Note that the domain considered
here consists of two half lines $\mathbf{R}^{\pm}$, and hence the jump across
$y=0$ should be treated. In view of this, the functional $g_{\beta}$ should be
chosen in (3.30), so that $g_{\beta}$ is continuous at $y=0$. Furthermore, it
holds that $g_{\beta}(\tau,0)\equiv 0.$
###### Lemma 3.8.
Under the assumptions of Proposition 3.3, it holds that
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{e^{-\frac{c_{0}y^{2}}{1+\tau}}}{1+\tau}|(\phi,\psi,\zeta)|^{2}dyd\tau\leq
C\delta+C\|(\phi,\psi,\zeta)(\tau_{1},\cdot)\|^{2}+C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}\\\\[5.69054pt]
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\displaystyle+C\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi)\|^{2}d\tau.\end{array}$
Proof: From the equation $\eqref{P}_{2}$ and the fact
$p-P=\frac{R\zeta-P\phi}{v}$ one gets
$\psi_{\tau}+(\frac{R\zeta-P\phi}{v})_{y}=(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-Q_{1}.$
(3.31)
Let
$G_{\alpha}(\tau,y)=(1+\tau)^{-1}\int_{0}^{y}e^{-\frac{\alpha\eta^{2}}{1+\tau}}d\eta,$
where $\alpha$ is a positive constant to be determined. Multiplying the
equation (3.31) by $G_{\alpha}(R\zeta-P\phi)$, we find that
$\begin{array}[]{ll}&\displaystyle\left(\frac{G_{\alpha}(R\zeta-P\phi)^{2}}{2v}\right)_{y}-(G_{\alpha})_{y}\frac{(R\zeta-P\phi)^{2}}{2v}+\frac{G_{\alpha}(R\zeta-P\phi)^{2}}{2v^{2}}(V_{y}+\phi_{y})\\\\[5.69054pt]
&\displaystyle=-G_{\alpha}(R\zeta-P\phi)\psi_{\tau}+G_{\alpha}(R\zeta-P\phi)(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-G_{\alpha}(R\zeta-P\phi)Q_{1}.\end{array}$
(3.32)
Noticing that
$-G_{\alpha}(R\zeta-P\phi)\psi_{\tau}=-\big{(}G_{\alpha}(R\zeta-P\phi)\psi\big{)}_{\tau}+(G_{\alpha})_{\tau}(R\zeta-P\phi)\psi+G_{\alpha}\psi(R\zeta-P\phi)_{\tau}$
(3.33)
and
$\begin{array}[]{ll}\displaystyle(R\zeta-P\phi)_{\tau}=R\zeta_{\tau}-P_{\tau}\phi-P\phi_{\tau}\\\\[5.69054pt]
\displaystyle=-\gamma
P\psi_{y}+(\gamma-1)\Big{\\{}-(p-P)(U_{y}+\psi_{y})+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}-Q_{2}\Big{\\}}-P_{\tau}\phi,\end{array}$
(3.34)
if we insert (3.34) into (3.33) and use the equality
$\displaystyle-G_{\alpha}\gamma P\psi_{y}\psi=-\Big{(}\gamma
G_{\alpha}P\frac{\psi^{2}}{2}\Big{)}_{y}+\gamma
P(G_{\alpha})_{y}\frac{\psi^{2}}{2}+\gamma P_{y}\frac{\psi^{2}}{2},$
we get from (3.32) that
$\frac{e^{-\frac{\alpha
y^{2}}{1+\tau}}}{2(1+\tau)}\Big{\\{}(R\zeta-P\phi)^{2}+\gamma
P\psi^{2}\Big{\\}}=\big{\\{}G_{\alpha}v(R\zeta-P\phi)\psi\big{\\}}_{\tau}+H_{2y}+Q_{4},$
(3.35)
where
$\displaystyle H_{2}=\frac{G_{\alpha}(R\zeta-P\phi)^{2}}{2v}+\gamma
G_{\alpha}P\frac{\psi^{2}}{2}-\nu(\gamma-1)G_{\alpha}\psi(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})-G_{\alpha}(R\zeta-P\phi)(\frac{u_{y}}{v}-\frac{U_{y}}{V})$
and
$\begin{array}[]{ll}\displaystyle
Q_{4}=\frac{G_{\alpha}(R\zeta-P\phi)^{2}}{2v^{2}}(V_{y}+\phi_{y})-(G_{\alpha})_{\tau}(R\zeta-P\phi)\psi+\big{(}G_{\alpha}(R\zeta-P\phi)\big{)}_{y}(\frac{u_{y}}{v}-\frac{U_{y}}{V})\\\\[5.69054pt]
\displaystyle\qquad+(\gamma-1)G_{\alpha}\psi\left\\{(p-P)(U_{y}+\psi_{y})-(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+Q_{2}\right\\}\\\\[5.69054pt]
\displaystyle\qquad+(\gamma-1)\nu(G_{\alpha}\psi)_{y}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})+G_{\alpha}(R\zeta-P\phi)Q_{1}+G_{\alpha}\psi
P_{\tau}\phi-\gamma P_{y}\frac{\psi^{2}}{2}.\end{array}$
Integrating (3.35) over $\mathbf{R}^{\pm}\times[\tau_{1},\tau]$, one infers
that
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{e^{-\frac{\alpha
y^{2}}{1+\tau}}}{1+\tau}\big{\\{}(R\zeta-P\phi)^{2}+\psi^{2}\big{\\}}dyd\tau=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\big{\\{}G_{\alpha}v(R\zeta-P\phi)\psi\big{\\}}(\tau,y)dy\\\\[11.38109pt]
&\displaystyle\qquad-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\big{\\{}G_{\alpha}v(R\zeta-P\phi)\psi\big{\\}}(\tau_{1},y)dy+\int_{\tau_{1}}^{\tau}\left[H_{2}\right](\tau)d\tau+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}Q_{4}dyd\tau.\end{array}$ (3.36)
Here we only analyze the jump term $[H_{2}]$ across $y=0$, the other terms in
(3.36) can be estimated similarly to those in [12] or [14]. Recalling that
$G_{\alpha}(\tau,y)$ is continuous at $y=0$ and $G_{\alpha}(\tau,0)\equiv 0$,
we easily see that
$[H_{2}](\tau)=0.$
Thus, from (3.36) one gets
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{e^{-\frac{\alpha
y^{2}}{1+\tau}}}{1+\tau}\big{\\{}(R\zeta-P\phi)^{2}+\psi^{2}\big{\\}}dyd\tau\leq
C\delta+C\|(\phi,\psi,\zeta)(\tau_{1},\cdot)\|^{2}\\\\[8.53581pt]
&\displaystyle\qquad\qquad+C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}d\tau\\\\[5.69054pt]
&\displaystyle\qquad\qquad+C\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})(\tau,\cdot)\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C\delta\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{e^{-\frac{\alpha
y^{2}}{1+\tau}}}{1+\tau}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.37)
In order to get the desired estimate in Lemma 3.8, we will use Lemma 3.6 to
derive another similar estimate from the energy equation $\eqref{P}_{3}$. To
this end, we set
$Z=\frac{R}{\gamma-1}\zeta+P\phi$
in Lemma 3.6. Thus we only need to compute the last term in (3.29). From the
energy equation $\eqref{P}_{3}$, we have
$Z_{\tau}=P_{\tau}\phi-(p-P)u_{y}+\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})-Q_{2},$
whence
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\langle
Z_{\tau},Zg_{\beta}^{2}\rangle_{H^{1}\times H^{-1}({\bf
R}^{\pm})}d\tau\\\\[14.22636pt]
=&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}\big{(}P_{\tau}\phi-(p-P)U_{y}\big{)}Zg_{\beta}^{2}dyd\tau-\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}(p-P)\psi_{y}Zg_{\beta}^{2}dyd\tau\\\\[11.38109pt]
&\displaystyle+\int_{\tau_{1}}^{\tau}\Big{[}\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})Zg_{\beta}^{2}\Big{]}(\tau)d\tau-\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})(Zg_{\beta}^{2})_{y}dyd\tau\\\\[8.53581pt]
&\displaystyle+\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}(\frac{u^{2}_{y}}{v}-\frac{U^{2}_{y}}{V})Zg_{\beta}^{2}dyd\tau-\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\mathbf{R}}Q_{2}Zg_{\beta}^{2}dyd\tau=:\displaystyle\sum_{i=1}^{6}K_{i}.\end{array}$
Here the jump term $K_{3}$ can be estimated as follows, recalling
$g_{\beta}(\tau,0)\equiv 0$.
$K_{3}=\int_{\tau_{1}}^{\tau}\Big{[}\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})Zg_{\beta}^{2}\Big{]}(\tau)d\tau=\nu\int_{\tau_{1}}^{\tau}g_{\beta}^{2}(\tau,0)(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})(\tau,0)\big{[}Z\big{]}(\tau)d\tau\equiv
0,$
while the terms $K_{i}$ ($i=1,4,5,6$) can be directly dealt with in the same
manner as in [12] or [14]. To bound the term $K_{2}$, we make use of the mass
equation $\eqref{P}_{1}$ to write $K_{2}$ in the form
$\begin{array}[]{lll}\displaystyle~{}~{}~{}-(p-P)\psi_{y}Zg_{\beta}^{2}=\frac{\gamma
P\phi-(\gamma-1)Z}{v}Zg_{\beta}^{2}\phi_{\tau}=\displaystyle\frac{\gamma
PZg_{\beta}^{2}}{2v}(\phi^{2})_{\tau}-\frac{(\gamma-1)Z^{2}g_{\beta}^{2}}{v}\phi_{\tau}\\\\[11.38109pt]
=\displaystyle\Big{(}\frac{\gamma PZ\phi^{2}g_{\beta}^{2}-2(\gamma-1)\phi
Z^{2}g_{\beta}^{2}}{2v}\Big{)}_{\tau}-\frac{\gamma
PZ\phi^{2}-2(\gamma-1)Z^{2}\phi}{v}g_{\beta}(g_{\beta})_{\tau}\\\\[8.53581pt]
~{}~{}~{}\displaystyle+\frac{\gamma
PZ\phi^{2}-2(\gamma-1)Z^{2}\phi}{2v^{2}}g_{\beta}^{2}v_{\tau}-\Big{(}\frac{2(\gamma-1)g_{\beta}^{2}\phi
Z}{v}+\frac{\gamma Pg_{\beta}^{2}\phi^{2}}{2v}\Big{)}Z_{\tau}-\frac{\gamma
g_{\beta}^{2}\phi^{2}Z}{2v}P_{\tau},\end{array}$
where all terms on the right-side hand of the above identity can be directly
bounded in the same way as in [12] or [14]. Therefore, we have bounded
$K_{2}$.
Taking $\beta=\frac{c_{0}}{2}$, one can get from Lemma 3.6 that
$\begin{array}[]{ll}&\displaystyle\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}\frac{e^{-\frac{c_{0}y^{2}}{1+\tau}}}{1+\tau}Z^{2}dyd\tau\leq
C\delta+C\|(\phi,\psi,\zeta)(\tau_{1},\cdot)\|^{2}+C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}\\\\[11.38109pt]
&\quad\displaystyle+C\int_{\tau_{1}}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}d\tau+C\int_{\tau_{1}}^{\tau}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi)\|^{2}d\tau\\\\[11.38109pt]
&\quad\displaystyle+C(\delta+\eta_{1})\int_{\tau_{1}}^{\tau}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.375pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-2.925pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.0625pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.6875pt}}\\!\int_{\bf
R}(1+\tau)^{-1}e^{-\frac{c_{0}y^{2}}{1+\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$
(3.38)
Now, taking $\alpha=c_{0}$ in (3.37) and choosing $\delta$ and $\eta_{1}$
suitably small, we combine (3.37) with (3.38) to obtain the desired estimate
in Lemma 3.8. $\hfill\Box$
By Lemmas 3.5 and 3.8, we conclude
$\begin{array}[]{ll}\displaystyle
N(\tau_{1},\tau_{2})+\int_{\tau_{1}}^{\tau_{2}}\Big{\\{}\|\phi_{y}\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}+\|(\psi_{y},\zeta_{y})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel_{1}^{2}+\|(\psi_{y\tau},\zeta_{y\tau})\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-5.70006pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.96004pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.79002pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.25002pt}}\\!\parallel^{2}\Big{\\}}d\tau\\\
\qquad\displaystyle\leq
CN(\tau_{1})+C\int_{0}^{t}(1+\tau)^{-\frac{7}{6}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\delta^{\frac{1}{4}}.\end{array}$
An application of Gronwall’s inequality to the above inequality gives the
estimate (3.4) in Proposition 3.3. This completes the proof of Proposition
3.1.
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|
arxiv-papers
| 2012-03-06T15:36:41 |
2024-09-04T02:49:28.330672
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feimin Huang, Song Jiang and Yi Wang",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/1203.1230"
}
|
1203.1249
|
# Magnetic-non-magnetic superlattice chain with external electric field: Spin
transport and the selective switching effect
Moumita Dey Theoretical Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti santanu.maiti@isical.ac.in Physics and Applied Mathematics
Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700
108, India S. N. Karmakar Theoretical Condensed Matter Physics Division,
Saha Institute of Nuclear Physics, Sector-I, Block-AF, Bidhannagar,
Kolkata-700 064, India
###### Abstract
Based on Green’s function formalism, the existence of multiple mobility edges
in a one-dimensional magnetic-non-magnetic superlattice geometry in presence
of external electric field is predicted, and, it leads to the possibility of
getting a metal-insulator transition at multiple values of Fermi energy. The
role of electric field on electron localization is discussed for different
arrangements of magnetic and non-magnetic atomic sites in the chain. We also
analyze that the model quantum system can be used as a perfect spin filter for
a wide range of energy.
###### pacs:
73.63.Nm, 72.20.Ee, 73.21.-b, 73.63.Rt
## I Introduction
Quantum transport in low-dimensional systems has been a topic of interest
within the past few decades due to its potential applicability in the field of
nanoscience and nanotechnology. Exploitation of the spin degree of freedom
adds a possibility of integrating memory and logic into a single device,
leading to remarkable development in the fields on magnetic data storage
application, device processing technique, quantum computation wolf , etc.
Naturally a lot of attention has been paid to study spin transport in low-
dimensional systems both from experimental expt1 ; expt2 ; expt3 and
theoretical theo1 ; theo2 ; theo3 ; theo4 ; shokri2 ; shokri4 ; shokri5 ;
shokri6 ; san3 ; san4 ; san5 ; sannew1 ; bellucci1 ; bellucci2 points of
view.
The understanding of electronic localization in low-dimensional model quantum
systems is always an interesting issue. Whereas, it is a well established fact
that in an infinite one-dimensional ($1$D) system with random site potentials
all energy eigenstates are exponentially localized irrespective of the
strength of randomness due to Anderson localization anderson , there exists
another kind of localization, known as Wannier-Stark localization, which
results from a static bias applied to a regular $1$D lattice, even in absence
of any disorder wannier . Till date a large number of works have been done to
explore the understanding of Anderson localization and scaling hypothesis in
one- and two-dimensional systems tvr . Similarly, Wannier-Stark localization
has also drawn the attention of many theorists starktheo1 ; starktheo2 ;
starktheo3 ; starktheo4 ; starktheo5 as well as experimentalists starkexp .
For both these two cases, viz, infinite $1$D materials with random site
energies and $1$D systems subjected to an external electric field, one never
encounters any mobility edge i.e., energy eigenvalues separating localized
states from the extended ones, since all the eigenstates are localized. But
there exist some special types of $1$D materials, like quasi-periodic Aubry-
Andre model and correlated disordered systems where mobility edge phenomenon
at some particular energies is obtained dun ; sanch ; fa ; fm ; dom ; aubry ;
san6 ; eco ; das ; rolf . Although the studies involving mobility edge
phenomenon in low-dimensional systems have already generated a wealth of
literature eco ; das ; rolf ; sch ; san1 ; san2 ; sannew there is still need
to look deeper into the problem to address several interesting issues those
have not yet been explored. For example, whether the mobility edges can be
observed in some other simple $1$D materials or the number of mobility edges
separating the extended and localized regions in the full energy band of an
$1$D material can be regulated, are still to be investigated.
To address these issues in the present article we investigate two-terminal
spin dependent transport in a $1$D mesoscopic chain composed of magnetic and
non-magnetic atomic sites in presence of external electric field. To the best
of our knowledge, no rigorous effort has been made so far to explore the
effect of an external electric field on electron transport in such a $1$D
magnetic-non-magnetic superlattice geometry. Here we show that, depending on
the unit cell configuration, a $1$D superlattice structure subjected to an
external electric field exhibits multiple mobility edges at different values
of the carrier energy. We use a simple tight-binding (TB) framework to
illustrate the model quantum system and numerically evaluate two-terminal spin
dependent transmission probabilities through the superlattice geometry based
on the Green’s function formalism. From our exact numerical analysis we
establish that a sharp crossover from a completely opaque to a fully or partly
transmitting zone takes place which leads to a possibility of tuning the
electron transport by gating the transmission zone. In addition to this
behavior we also show that the magnetic-non-magnetic superlattice structure
can be used as a pure spin filter for a wide range of energy. These phenomena
enhance the prospect of such simple superlattice structures as switching
devices at multiple energies as well as spin filter devices, the design of
which has significant impact in the present age of nanotechnology.
With an introduction in Section I, we organize the paper as follows. In
Section II, first we present the model, then describe the theoretical
formulation which include the Hamiltonian and the formulation for transmission
probabilities through the model quantum system. The numerical results are
illustrated in Section III and finally, in Section IV, we draw our
conclusions.
## II Theoretical Framework
Let us start with Fig. 1 where a $1$D mesoscopic chain composed of magnetic
and non-magnetic atomic sites is attached to two semi-infinite $1$D non-
magnetic electrodes, namely, source and drain. The chain consists
Figure 1: (Color online). A $1$D mesoscopic chain composed of magnetic (filled
magenta circle) and non-magnetic (filled green circle) atomic sites is
attached to two semi-infinite $1$D non-magnetic metallic electrodes, namely,
source and drain.
of $p$ ($p$ being an integer) number of unit cells in which each unit cell
contains $n$ and $m$ numbers of magnetic and non-magnetic atoms, respectively.
Both the chain and side-attached electrodes are described by simple TB
framework within nearest-neighbor hopping approximation.
The Hamiltonian for the entire system can be written as a sum of three terms
as,
$H=H_{c}+H_{l}+H_{tun}.$ (1)
The first term represents the Hamiltonian for the chain and it reads
$H_{c}=\sum_{i}\mbox{\boldmath$c$}_{i}^{{\dagger}}(\mbox{\boldmath$\epsilon$}_{i}+\mbox{\boldmath$\vec{h}_{i}.\vec{\sigma}$})\mbox{\boldmath$c$}_{i}+\sum_{i}\left[\mbox{\boldmath$c$}_{i}^{{\dagger}}\mbox{\boldmath$t$}\mbox{\boldmath$c$}_{i+1}+h.c.\right]$
(2)
where,
$\mbox{\boldmath$c$}^{\dagger}_{i}=\left(\begin{array}[]{cc}c_{i\uparrow}^{\dagger}&c_{i\downarrow}^{\dagger}\end{array}\right);$
$\mbox{\boldmath$c$}_{i}=\left(\begin{array}[]{c}c_{i\uparrow}\\\
c_{i\downarrow}\end{array}\right);$
$\mbox{\boldmath$\epsilon$}_{i}=\left(\begin{array}[]{cc}\epsilon_{i}&0\\\
0&\epsilon_{i}\end{array}\right)$;
$\mbox{\boldmath$t$}=t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right);$
and
$\vec{h_{i}}.\vec{\sigma}$ =
$h_{i}\left(\begin{array}[]{cc}\cos\theta_{i}&\sin\theta_{i}e^{-j\phi_{i}}\\\
\sin\theta_{i}e^{j\phi_{i}}&-\cos\theta_{i}\end{array}\right)$.
Here, $\epsilon_{i}$ refers to the on-site energy of an electron at the site
$i$ with spin $\sigma$ ($\uparrow,\downarrow$), $t$ is the nearest-neighbor
hopping strength, $c_{i\sigma^{\dagger}}$ ($c_{i\sigma}$) is the creation
(annihilation) operator of an electron at the $i$th site with spin $\sigma$
and $h_{i}$ is the strength of local magnetic moment where $h_{i}=0$ for non-
magnetic sites. The term $\vec{h_{i}}.\vec{\sigma}$ corresponds to the
interaction of the spin of the injected electron with the local magnetic
moment placed at the site $i$. The direction of magnetization in each magnetic
site is chosen to be arbitrary and specified by angles $\theta_{i}$ and
$\phi_{i}$ in spherical polar co-ordinate system for the $i$th atomic site.
Here, $\theta_{i}$ represents the angle between the direction of magnetization
and the chosen $Z$ axis, and $\phi_{i}$ represents the azimuthal angle made by
the projection of the local moment on $X$-$Y$ plane with the $X$ axis. In
presence of bias voltage $V$ between the source and drain an electric field is
developed, and therefore, the site energies of the chain becomes voltage
dependent. Mathematically we can express it as
$\epsilon_{i}=\epsilon_{i}^{0}+\epsilon_{i}(V)$, where $\epsilon_{i}^{0}$ is
the voltage independent term. The voltage dependence of $\epsilon_{i}(V)$
reflects the bare electric field in the bias junction as well as screening due
to longer range electron-electron interaction. In the absence of such
screening the electric field varies uniformly along the chain and it reads
$\epsilon_{i}(V)=V/2-iV/(N+1)$, where $N$ corresponds to the total number of
atomic sites in the chain. In our present work, we consider both the linear
Figure 2: (Color online). Voltage dependent site energies in a $1$D chain
considering $100$ atomic sites for three different electrostatic potential
profiles when the bias voltage $V$ is set equal to $1$.
and screened electric field profiles. As illustrative example, in Fig. 2 we
show the variation of voltage dependent site energies for three different
electrostatic potential profiles for a chain considering $100$ atomic sites
and describe the nature of electronic localization for these profiles in the
forthcoming section.
The second and third terms of Eq. 1 describe the TB Hamiltonians for the $1$D
semi-infinite non-magnetic electrodes and the chain-to-electrode coupling.
These Hamiltonians are written as follows.
$H_{l}=\sum\limits_{\alpha=S,D}\left[\sum_{n}\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$\epsilon_{l}$}\mbox{\boldmath$c$}_{n}+\sum_{n}\left[\mbox{\boldmath$c$}_{n}^{{\dagger}}\mbox{\boldmath$t_{l}$}\mbox{\boldmath$c$}_{n+1}+h.c.\right]\right]$
(3)
and,
$\displaystyle H_{tun}$ $\displaystyle=$ $\displaystyle H_{tun,S}+H_{tun,D}$
(4) $\displaystyle=$
$\displaystyle\tau_{s}[\mbox{\boldmath$c$}_{1}^{{\dagger}}\mbox{\boldmath$c$}_{0}+h.c.]+\tau_{d}[\mbox{\boldmath$c$}_{N}^{{\dagger}}\mbox{\boldmath$c$}_{N+1}+h.c.].$
The summation over S and D in Eq. 3 implies the incorporation of both the two
electrodes, viz, source and drain. $\epsilon_{l}$ and $t_{l}$ stand for the
site energy and nearest-neighbor coupling, respectively. The electrodes are
directly coupled to the chain through the lattice sites $1$ and $N$, and the
coupling strengths between these electrodes with the chain are described by
$\tau_{s}$ and $\tau_{d}$, respectively.
To obtain spin resolved transmission probabilities of an electron through the
source-chain-drain bridge system, we use Green’s function formalism. The
single particle Green’s function operator representing the entire system for
an electron with energy $E$ is defined as,
$G=\left(E-H+i\eta\right)^{-1}$ (5)
where, $\eta\rightarrow 0^{+}$.
Following the matrix form of $H$ and $G$ the problem of finding $G$ in the
full Hilbert space $H$ can be mapped exactly to a Green’s function
$G$${}_{c}^{eff}$ corresponding to an effective Hamiltonian in the reduced
Hilbert space of the chain itself and we have,
$\mbox{\boldmath${\mathcal{G}}$}=\mbox{\boldmath$G$}_{c}^{eff}=\sum\limits_{\sigma}\left(\mbox{\boldmath$E$}-\mbox{\boldmath$H$}_{c}-\mbox{\boldmath$\Sigma$}_{S}^{\sigma}-\mbox{\boldmath$\Sigma$}_{D}^{\sigma}\right)^{-1},$
(6)
where,
$\displaystyle\mbox{\boldmath$\Sigma$}_{S(D)}^{\sigma}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$H$}_{tun,S(D)}^{{\dagger}}\mbox{\boldmath$G$}_{S(D)}\mbox{\boldmath$H$}_{tun,S(D)}.$
(7)
These $\Sigma_{S}$ and $\Sigma_{D}$ are the self-energies introduced to
incorporate the effect of coupling of the chain to the source and drain,
respectively. Using Dyson equation the analytic form of the self energies can
be evaluated as follows,
$\Sigma_{S(D)}^{\sigma}=\frac{\tau_{s(d)}^{2}}{E-\epsilon_{l}-\xi_{l}}$ (8)
where, $\xi_{l}=(E-\epsilon_{l})/2-i\sqrt{t_{l}^{2}-(E-\epsilon_{l})^{2}/4}$.
Following Fisher-Lee relation, the transmission probability of an electron
from the source to drain is given by the expression,
$T_{\sigma\sigma^{\prime}}=\mbox{Tr}[\mbox{\boldmath$\Gamma$}_{S}^{\sigma}\mbox{\boldmath$\mathcal{G}$}^{r}\mbox{\boldmath$\Gamma$}_{D}^{\sigma^{\prime}}\mbox{\boldmath$\mathcal{G}$}^{a}].$
(9)
where, $\Gamma$${}_{S(D)}^{\sigma}$’s are the coupling matrices representing
the coupling between the chain and the electrodes and they are defined as,
$\mbox{\boldmath$\Gamma$}_{S(D)}^{\sigma}=i\left[\mbox{\boldmath$\Sigma$}^{\sigma}_{S(D)}-\mbox{\boldmath$\Sigma$}^{\sigma{\dagger}}_{S(D)}\right].$
(10)
Here, $\Sigma$${}_{k}^{\sigma}$ and $\Sigma$${}_{k}^{\sigma{\dagger}}$ are the
retarded and advanced self-energies associated with the $k$-th ($k=S,D$)
electrode, respectively.
Finally, we determine the average density of states (ADOS), $\rho(E)$, from
the following relation,
$\rho(E)=-\frac{1}{N\pi}{\mbox{Im}}\left[{\mbox{Tr}}[\mbox{\boldmath${\mathcal{G}}$}]\right].$
(11)
In what follows we limit ourselves to absolute zero temperature and use the
units where $c=e=h=1$. For the numerical calculations we set $t=1$,
$\epsilon_{i}^{0}=0\,\forall\,i$, $h_{i}=1$ for the magnetic sites,
$\theta_{i}=\phi_{i}=0$, $\epsilon_{l}=0$, $t_{l}=1$ and
$\tau_{s}=\tau_{d}=0.8$. The energy scale is measured in unit of $t$.
## III Numerical Results and Discussion
Throughout our numerical calculations we assume that the magnetic moments are
aligned along $+Z$ direction ($\theta_{i}=\phi_{i}=0$), which yields vanishing
spin flip transmission probability, viz,
$T_{\uparrow\downarrow}=T_{\downarrow\uparrow}=0$, across the bridge system.
The net transmission probability is therefore a sum
$T(E)=T_{\uparrow\uparrow}(E)+T_{\downarrow\downarrow}(E)$, and the origin of
this zero spin flipping can be explained from the following arguments. The
operators $\sigma_{+}$ $(=\sigma_{x}+i\sigma_{y})$ and $\sigma_{-}$
$(=\sigma_{x}-i\sigma_{y})$ associated with the term
$\vec{h}_{i}.\vec{\sigma}$ in the TB Hamiltonian Eq. 2 are responsible for the
spin flipping, where $\vec{\sigma}$ being the Pauli spin vector with
components $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ for the injecting
electron. In our present model since we consider that all the magnetic moments
are aligned along $+Z$ direction, the term $\vec{h}_{i}.\vec{\sigma}$
$(=h_{ix}\sigma_{x}+h_{iy}\sigma_{y}+h_{iz}\sigma_{z})$ gets the form
$h_{iz}\sigma_{z}$, and accordingly, the Hamiltonian does not contain
$\sigma_{x}$
Figure 3: (Color online). Transmission probability $T$ and ADOS as a function
of energy for a $1$D magnetic-non-magnetic superlattice geometry considering a
linear bias drop along the chain, as shown by the pink curve in Fig. 2, where
(a)-(c) correspond to the results for three different values of bias voltage
$V$.
and $\sigma_{y}$ and so $\sigma_{+}$ and $\sigma_{-}$ do not appear, which
leads to the vanishing spin flip transmission probability across the $1$D
chain. Below, we address the central results of our study i.e, the possibility
of getting multiple mobility edges in $1$D magnetic-non-magnetic superlattice
geometries and how such a simple model quantum system can be used as a perfect
spin filter for a wide range of energy.
In Fig. 3 we show the variation of total transmission probability $T$ along
with the average density of states for a $1$D magnetic-non-magnetic
superlattice geometry considering a linear bias drop. Here we consider a
$400$-site chain in which each unit cell contains one magnetic and four non-
magnetic sites and the results are shown for three different bias voltages.
For the particular case when the chain is free from external electric field
i.e., $V=0$ electronic conduction through the bridge takes place for the
entire energy band as shown in Fig. 3(a) which predicts that all the energy
eigenstates are extended in nature. The situation becomes really very
interesting when the superlattice geometry is subjected to an external
electric field.
Figure 4: (Color online). Transmission probability $T$ and ADOS as a function
of energy for a $1$D magnetic-non-magnetic superlattice geometry when the
electrostatic potential profile varies following the green curve shown in Fig.
2, where (a)-(c) represent the identical meaning as in Fig. 3.
It is illustrated in Figs. 3(b) and 3(c). From these spectra we notice that
there are some energy regions for which the transmission probability
completely drops to zero which reveals that the eigenstates associated with
these energies are localized, and they are separated from the extended energy
eigenstates. Thus, sharp mobility edges are obtained in the spectrum, and, the
total number of such mobility edges separating the extended and localized
regions in a superlattice geometry in presence electric field strongly depends
on the unit cell configuration and it can be regulated by adjusting the number
of magnetic and non-magnetic sites. This phenomenon describes the existence of
multiple mobility edges in a superlattice geometry under finite bias
condition. Now if the Fermi energy is fixed at a suitable energy zone where
$T$ drops to zero an insulating phase will appear, while for the other case,
where $T$ is finite, a metallic phase is observed and it leads to the
possibility of controlling the electronic transmission by gating the
transmission zone. The width of the localized regions between the band of
extended regions increases with the strength of the electric
Figure 5: (Color online). Transmission probability $T$ and ADOS as a function
of energy for a $1$D magnetic-non-magnetic superlattice geometry when the
electrostatic potential profile varies following the blue curve shown in Fig.
2, where (a)-(c) represent the identical meaning as in Fig. 3.
field as clearly shown by comparing the spectra given in Figs. 3(b) and 3(c),
and, for strong enough field strength almost all energy eigenstates are
localized. In that particular limit metal-to-insulator transition will no
longer be observed.
The above results are analyzed for a particular (linear) variation of electric
field along the chain. To explore the sensitivity of getting metal-to-
insulator transition on the distribution of electric field, in Figs. 4 and 5
we present the results for two different screened electric field profiles
taking the identical chain length. From the spectra we clearly observe that
the width of the localized region gradually disappears with the flatness of
the electric field profile in the interior of the bridge system. If the
potential drop takes place only at the chain-to-electrode interfaces, i.e.,
when the potential profile becomes almost flat along the chain the width of
the localized region almost vanishes and the metal-to-insulator transition is
not observed, as is the case for the zero bias limit.
Finally, we illustrate how such a simple magnetic-non-magnetic superlattice
geometry can be utilized as a perfect spin filter for
Figure 6: (Color online). $T_{\uparrow\uparrow}$, $T_{\downarrow\downarrow}$
and ADOS as a function of energy for a $1$D magnetic-non-magnetic superlattice
geometry in absence of external electric field.
a wide range of energy in absence of any external electric field. As
illustrative example, in Fig. 6 we present the transmission probabilities for
up and down spin electrons together with the average density of states as a
function of energy for a $1$D magnetic-non-magnetic superlattice geometry.
From the spectra we observe that the up and down spin electrons follow two
different channels while traversing through the superlattice geometry, since
the spin flipping is completely blocked for this configuration. This splitting
of up and down spin conduction channels is responsible for spin filtering
action and the total number of these channels strongly depends on the unit
cell configuration. From Figs. 6(a) and (b) we clearly see that for a wide
range of energy for which the transmission probability of up spin electrons
drops to zero value, shows non-zero transmission probability of down spin
electrons. Therefore, setting the Fermi energy to a suitable energy region we
can control the transmission characteristics of up and down spin electrons,
and, a spin selective transmission is thus obtained through the bridge system.
Before we end, we would like to point out that since the overlap between the
up and down spin conduction channels depends on the magnitudes of the local
magnetic moments, we can regulate the spin degree of polarization (DOP) simply
by tuning the strength of these magnetic moments and for a wide range of
energies it (DOP) almost reaches to $100\%$. Thus, our proposed magnetic-non-
magnetic superlattice geometry is a very good example for designing a spin
filter.
## IV Conclusion
To conclude, in the present work we investigate in detail the spin dependent
transport under finite bias condition through a $1$D magnetic-non-magnetic
superlattice geometry using Green’s function formalism. We use a simple TB
framework to describe the model quantum system where all the calculations are
done numerically. From our exact numerical analysis we predict that in such a
simple $1$D magnetic-non-magnetic superlattice geometry multiple mobility
edges separating the localized and extended regions are obtained in presence
of external electric field and the total number of mobility edges in the full
energy spectrum can be controlled by arranging the unit cell configuration.
This phenomenon reveals that the superlattice geometry can be used as a
switching device for multiple values of Fermi energy. The sensitivity of
metal-to-insulator transition and vice versa on the electrostatic potential
profile is thoroughly discussed. Finally, we analyze how such a superlattice
geometry can be utilized in designing a tailor made spin filter device for
wide range of energies. Setting the Fermi energy at a suitable energy zone, a
spin selective transmission is obtained through the bridge system. All these
predicted results may be utilized in fabricating spin based nano electronic
devices.
The results presented in this communication are worked out for absolute zero
temperature. However, they should remain valid even in a certain range of
finite temperatures ($\sim 300$ K). This is because the broadening of the
energy levels of the chain due to the chain-to-electrode coupling is, in
general, much larger than that of the thermal broadening datta1 .
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moumita Dey, Santanu K. Maiti and S. N. Karmakar",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1203.1249"
}
|
1203.1684
|
2011 Vol. 9 No. 00, 000–000
11institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China;
zfan@bao.ac.cn 22institutetext: Graduate University of Chinese Academy of
Sciences, Beijing 100049, China
Received [year] [month] [day]; accepted [year] [month] [day]
# Spectroscopic Study of Globular Clusters in the Halo of M31 with Xinglong
2.16m Telescope II: Dynamics, Metallicity and Age ∗ 00footnotetext: $*$
Supported by the National Natural Science Foundation of China.
Zhou Fan 11 Ya-Fang Huang 1122 Jin-Zeng Li 11 Xu Zhou 11 Jun Ma 11 Yong-Heng
Zhao 11
###### Abstract
In our Paper I, we performed the spectroscopic observations of 11 confirmed
GCs in M31 with the Xinglong 2.16m telescope and we mainly focus on the fits
method and the metallicity gradient for the M31 GC sample. In this paper, we
analyzed and discussed more about the dynamics, metallicity and age, and their
distributions as well as the relationships between these parameters. In our
work, eight more confirmed GCs in the halo of M31 were observed, most of which
lack the spectroscopic information before. These star clusters are located far
from the galactic center at a projected radius of $\sim 14$ to $\sim 117$ kpc,
which are more spatially extended than that in the previous work. The Lick
absorption-line indices and the radial velocities have been measured
primarily. Then the ages, metallicities $\rm[Fe/H]$ and $\rm[\alpha/Fe]$ have
been fitted by comparing the observed spectral feature indices and the SSP
model of Thomas et al. in the Cassisi and Padova stellar evolutionary tracks,
respectively. Our results show that most of the star clusters of our sample
are older than 10 Gyr except B290$\sim 5.5$ Gyr, and most of them are metal-
poor with the metallicity $\rm[Fe/H]<-1$, suggesting that these clusters were
born at the early stage of the galaxy’s formation. We find that the
metallicity gradient for the outer halo clusters with $r_{p}>25$ kpc may not
exist with a slope of $-0.005\pm 0.005$ dex kpc-1 and if the outliers G001 and
H11 are excluded, the slope dose not change significantly with a value of
$-0.002\pm 0.003$ dex kpc-1. We also find that the metallicity is not a
function of age for the GCs with age $<7$ Gyr while for the old GCs with age
$>7$ Gyr there seems to be a trend that the older ones have lower metallicity.
Besides, We plot metallicity distributions with the largest sample of M31 GCs
so far and it shows the bimodality is not significant and the number of the
metal-poor and metal-rich groups becomes comparable. The spatial distributions
shows that the metal-rich group is more centrally concentrated while the
metal-poor group is occupy a more extended halo and the young population is
centrally concentrated while the old populaiton is more extended spatially to
the outer halo.
###### keywords:
galaxies: individual (M31) — galaxies: star clusters — globular clusters:
general — star clusters: general
## 1 Introduction
One way to better understand the formation and evolution of the galaxies is
through detailed studies of globular clusters (GCs), which are often
considered to be the fossils of galactic formation and evolution processes,
since they formed at the early stages of their host galaxies’ life cycles
(Barmby et al., 2000). GCs are densely packed, very luminous, which usually
contains several thousands to approximately one million stars. Therefore, they
can be detected from great distances and are suitable as probes for studying
the properties of extragalactic systems. Since the halo globular clusters
(HGCs) are located far away from the galaxy center, they are very important
and useful to study the dark matter distribution of the galaxy. Besides, since
the HGCs are far from the galaxy center, the background of galaxy becomes much
lower, which makes the observations much easier, compared to the disc GCs in
the projected direction of galaxies.
As the nearest ($\sim 780$ kpc) and large spiral galaxy in our Local Group,
M31 (Andromeda) contains a great number of GCs from $460\pm 70$ (Barmby &
Huchra, 2001) to $\sim$530 (Perina et al., 2010), which is an ideal laboratory
for us to study the nature of the HGCs. A great many of new M31 HGCs have been
discovered in the recent years, which are important to study the formation
history of M31 and its dark matter content. Huxor et al. (2004) discovered
nine previously unknown HGCs of M31 using the INT survey. Subsequently, Huxor
et al. (2005) found three new, extended GCs in the halo of M31, which have
characteristics between typical GCs and dwarf galaxies. Mackey et al. (2006)
reported four extended, low-surface-brightness clusters in the halo of M31
based on Hubble Space Telescope/Advanced Camera for Surveys (ACS) imaging.
These star clusters are structurally very different from typical M31 GCs. On
the other side, since they are old and metal-poor, they look like the typical
Milky Way GCs. Huxor (2007) found 40 new extended GCs in the halo of M31 (out
to $\sim 100$ kpc from the galactic center) based on INT and CFHT imaging.
These extended star clusters in the M31 halo are very similar to the diffuse
star clusters (DSCs) associated with early-type galaxies in the Virgo Cluster
reported by Peng et al. (2006) based on the ACS Virgo Cluster survey. Indeed,
the evidence shows that DSCs are usually fainter than typical GCs. Later,
Mackey et al. (2007) reported 10 outer-halo GCs in M31, at $\sim$15 kpc to 100
kpc from the galactic center, eight of which were newly discovered based on
deep ACS imaging. The HGCs in their sample are very luminous, compact with low
metallicity, which are quite different from their counterparts in our Galaxy.
More recently, Ma et al. (2010) constrained the age, metallicity, reddening
and distance modulus of B379, which also is an HGCs of M31, based on the
multicolor photometry.
In Fan et al. (2011) (hereafter Paper I) we observed 11 confirmed star
clusters, most of which are located in the halo of M31, with the OMR
spectrograph on 2.16m telescope at Xinglong site of National Astronomical
Observatories, Chinese Academy of Sciences, in fall of 2010\. We estimated the
ages, metallicities, $\alpha$-elements with the SSP models as well as the the
radial velocities and they found that most of the halo clusters are old and
metal-poor, which were supposed to be born at the early stage of the galaxy
formation history. In this paper, we will continue the study of the HGCs of
M31 with the same instruments and a larger sample. This allows us to be able
to better understand the properties of the M31 outer halo. This paper is
organized as follows. In §2 we describe how we selected our sample of M31 GCs
and their spatial distribution. In §3, we reported the spectroscopic
observations with 2.16 m telescope and how the data was reduced and the radial
velocities and Lick indices were measured and calibrated. Subsequently, in §4,
we derive the ages, metallicities and $\alpha$-element with
$\chi^{2}-$minimization fitting. We also discuss our final results on the
metallicity distribution in the M31 halo. Finally, we summarized our work and
give our conclusions in §5.
## 2 Sample selection
The sources were selected from the updated Revised Bologna Catalogue of M31
globular clusters and candidates (RBC v.4, available from
http://www.bo.astro.it/M31; Galleti et al., 2004, 2006, 2007, 2009), which is
the latest and most comprehensive M31 GC catalogue so far. The catalogue
contains 2045 objects, including 663 confirmed star clusters, 604 cluster
candidates, and 778 other objects that were previously thought to be GCs but
later proved to be stars, asterisms, galaxies, or Hii regions. Indeed, many of
the halo clusters were from Mackey et al. (2007), who reported 10 GCs in the
outer halo of M31 from their deep ACS images, of which eight were detected for
the first time (see for details in §1). In our work, our sample clusters are
completely selected from RBC v.4. We selected the confirmed and luminous
clusters as well as being located as far as they could from the galaxy center,
where the local background is too luminous for our observations. Finally,
there are eight bright confirmed clusters in our sample, all of which are
located in the halo of the galaxy. These clusters lack spectroscopic
observational data, especially for the metallicity measurements. Thus it is
necessary to observe the spectra of our sample clusters systematically and
constrain the spectroscopic metallicities and ages in detail.
The observational information of our sample GCs are listed in Table 1, which
includes the names, coordinates, projected radii in kpc, exposures and
observation dates. All the coordinates (R.A. and Dec. in Cols. 2 and 3) and
projected radii from the galaxy center $r_{\rm p}$ (Col. 4) are all from RBC
v.4, which were calculated with M31 center coordinate $00:42:44.31$,
$+41:16:09.4$ (Perrett et al., 2002), $PA=38^{\circ}$ and distance $d=785$ kpc
(McConnachie et al., 2005).
Table 1: The observations of our sample GCs.
ID | R.A. | Dec. | $r_{p}$ | Exposure | Date
---|---|---|---|---|---
| (J2000) | (J2000) | (kpc) | (second) |
B289 | 00:34:20.882 | +41:47:51.14 | 22.65 | 6000 | 08/28/2011
B290 | 00:34:20.947 | +41:28:18.18 | 21.69 | 7200 | 09/01/2011
H11 | 00:37:28.028 | +44:11:26.41 | 42.10 | 5400 | 09/01/2011
H18 | 00:43:36.030 | +44:58:59.30 | 50.87 | 5400 | 08/29/2011
SK108A | 00:47:14.240 | +40:38:12.30 | 14.47 | 3600 | 08/28/2011
SK112A | 00:48:15.870 | +41:23:31.20 | 14.28 | 5400 | 08/29/2011
MGC1 | 00:50:42.459 | +32:54:58.78 | 117.05 | 3600 | 08/28/2011
H25 | 00:59:34.560 | +44:05:39.10 | 57.35 | 5400 | 09/01/2011
We show the spatial distribution of our sample eight halo GCs and all the
confirmed GCs from RBC v.4 in Figure 1. The large ellipse is the M31 disk/halo
boundary as defined by Racine (1991). Note that all of our sample are located
in the halo of M31, which can help us to access the nature of galaxy halo with
an enlarged cluster sample, compared to Fan et al. (2011).
Figure 1: Spatial distribution of M31 GCs. Our sample halo GCs are shown with
filled circles and the confirmed GCs from RBC v.4 are marked with points. The
large ellipse is the M31 disk/halo boundary as defined in Racine (1991).
## 3 Observations and data reduction
Our Low-resolution spectroscopic observations were all taken at the 2.16m
optical telescope at Xinglong Site, which belongs to National Astronomical
Observatories, Chinese Academy of Sciences (NAOC), from August 28th to
September 1st, 2011 (Please see Table 1). An OMR (Optomechanics Research Inc.)
spectrograph and a PI 1340${\times}$400 CCD detector were used during this run
with a dispersion of 200 Å mm-1, 4.8 Å pixel-1, and a 3.0 slit. The typical
seeing there was $\sim 2.5$ . The spectra cover the wavelength range of
$3500-8100$ Å at 4 Å resolution. All our spectra have $S/N\geq 40$.
In order to calibrate our 2.16m data onto the Lick system, we also observed
eleven Lick standard stars (HR 6806, HR 6815, HR 7030, HR 7148, HR 7171, HR
7503, HR 7504, HR 7576, HR 7977, HR 8020, HR 8165) near our field, which are
selected from a catalogue of all 25 index measurements and coordinates for 460
stars (ref, available from http://astro.wsu.edu/worthey/html/system.html;
Worthey & Ottaviani, 1997; Worthey et al., 1994a). Most of these standard
stars are luminous ($\sim 5-6$ in V band), hence the exposure time we took was
20 second with the OMR system.
The spectroscopic data were reduced following the standard procedures with
NOAO Image Reduction and Analysis Facility (IRAF v.2.15) software package.
First, the spectra have been bias and flat-field corrected, as well as cosmic-
ray removed. Then the wavelength calibrations were performed based on
Helium/Argon lamps exposed at both the beginning and the end of the
observations in each night. Flux calibrations were performed based on
observations of at least two of the KPNO spectral standard stars (Massey et
al., 1988) each night. The atmospheric extinction was corrected with the mean
extinction coefficients measurements of Xinglong through the Beijing-Arizona-
Taiwan-Connecticut (BATC) multicolor sky survey (H. J. Yan 1995, priv. comm.).
Before the Lick indices were measured, the heliocentric radial velocities
$V_{r}$ were measured by comparing the absorption lines of our spectra with
the templates in various radial velocities. The typical internal velocity
errors on a single measure is $\sim 20$ km s-1. The estimated radial
velocities $V_{r}$ with the associated uncertainties (Col. 2) are listed in
Table 2. The published radial velocities $V_{r}$ (Col. 3) are also listed for
comparisons. The systematic difference between our observed velocity and the
catalogue velocity is found to be $\rm 29\pm 39~{}km~{}s^{-1}$ and the
standard deviation of the differences between our observed velocity and the
catalogue velocity is $\rm 78~{}km~{}s^{-1}$ for the five pairs of the radial
velocities. It suggests that our measurements agree with those listed in RBC
v.4 since the systematic difference between our measurements and the published
values is not significant.
Figure 2 shows the radial velocity $V_{r}$ (corrected for the systemic
velocity of M31) as a function of the projected radii from the galaxy center.
The Left panel is for the all confirmed GCs which have the radial velocity
$V_{r}$ measurements and the Right panel is for the HGCs, which refers to the
GCs in the galaxy halo defined in Figure 1. It can be noted that the radial
velocity distributions are basically symmetric in distributions either for all
the confirmed GC sample or for the HGCs only.
Figure 2: The distributions of radial velocity $V_{r}$ (corrected for the
systemic velocity of M31). Left: all the confirmed GCs. Right: the HGCs only.
Table 2: The radial velocities $V_{r}$ of our sample GCs as well as the
previous results.
ID | our work | RBC v.4
---|---|---
B289 | $-96.81\pm 47.27$ | $-181\pm 30$
B290 | $-488.73\pm 43.14$ | $-381\pm 26$
H11 | $-173.02\pm 39.63$ |
H18 | $-300.48\pm 79.65$ |
SK108A | $-352.17\pm 19.18$ | $-379\pm 38$
SK112A | $-342.68\pm 32.81$ | $-252\pm 46$
MGC1 | $-412.67\pm 17.13$ | $-355\pm 2$
H25 | $-256.49\pm 55.28$ |
We plotted the radial velocities $V_{r}$ versus the projected radii $r_{p}$ in
Figure 3 where the radial velocities have been corrected for the systemic
velocity of M31 galaxy of $300\pm 4$ km s-1(Perrett et al., 2002). The Left
panel is for all the confirmed clusters in RBC v.4 while the right panel is
for the halo clusters which are defined in Figure 1. The points are the
published measurements from RBC v.4 while the open triangles and the filled
circles with errors are the measurements in Paper I and those in our work,
respectively. In the Right panel, the symbols are the same as those in the
Left panel. We find that the dispersion of the velocity becomes smaller when
the GCs are locate further from the center of the galaxy with larger projected
radius $r_{p}$. It can be seen that the dispersion of the radial velocity
becomes smaller when the projected radius $r_{p}$ is larger.
Figure 3: The radial velocity $V_{r}$ (corrected for the systemic velocity of
M31) as a function of the projected radius. Left: all confirmed clusters and
Right: the halo clusters. The filled circles with errors are the halo GCs from
our sample while the points represent the velocities from RBC v.4 catalogue.
Subsequently, all the spectra were shifted to the zero radial velocity and
smoothed to the wavelength dependent Lick resolution with a variable-width
Gaussian kernel following the definition of Worthey & Ottaviani (1997), i.e.
11.5 Å at 4000 Å, 9.2 Å at 4400 Å, 8.4 Å at 4900 Å, 8.4 Å at 5400 Å, 9.8 Å at
6000 Å. Indeed, we measured all the 25 types of Lick indices strictly by using
the parameters and formulae from Worthey et al. (1994a) and Worthey &
Ottaviani (1997). The uncertainty of each index was estimated based on the
analytic formulae (11)$-$(18) of Cardiel et al. (1998).
Figure 4: Calibrations of index measurements from the eleven standard stars of
2.16m raw spectra with those from reference Worthey & Ottaviani (1997);
Worthey et al. (1994a). The linear fit coefficients of Eq. 3 have been derived
to be used for calibrating our raw data to the Lick index system.
Eq. 1 is the linear fit formula for calibrating the raw measurements of our
2.16m data to the standard Lick index system. The eleven standard stars are
utilized for the fitting (Please see Figure 4) and the results are listed in
Table 3.
$\rm EW_{ref}={\it a}+{\it b}\cdot EW_{raw}$ (1)
Table 3: The Linear Fit Coefficients $a$ and $b$ in Eq. 3 for transformations
of the 2.16m data to the Lick index system.
Index | $a$ | $b$
---|---|---
$\rm H\delta_{A}$ (Å) | $-0.15\pm 0.19$ | $1.00\pm 0.04$
$\rm H\delta_{F}$ (Å) | $0.04\pm 0.15$ | $1.15\pm 0.06$
$\rm CN1$ (mag) | $0.04\pm 0.01$ | $0.84\pm 0.07$
$\rm CN2$ (mag) | $0.02\pm 0.01$ | $0.98\pm 0.05$
$\rm Ca4227$ (Å) | $-0.04\pm 0.14$ | $2.73\pm 0.21$
$\rm G4300$ (Å) | $-0.06\pm 0.19$ | $1.05\pm 0.04$
$\rm H\gamma_{A}$ (Å) | $1.73\pm 0.26$ | $0.78\pm 0.03$
$\rm H\gamma_{F}$ (Å) | $0.79\pm 0.16$ | $1.07\pm 0.05$
$\rm Fe4383$ (Å) | $-0.32\pm 0.36$ | $1.46\pm 0.10$
$\rm Ca4455$ (Å) | $0.71\pm 0.56$ | $1.50\pm 1.21$
$\rm Fe4531$ (Å) | $-0.30\pm 0.24$ | $1.33\pm 0.09$
$\rm Fe4668$ (Å) | $-0.16\pm 0.31$ | $1.16\pm 0.06$
$\rm H\beta$(Å) | $0.17\pm 0.16$ | $1.03\pm 0.05$
$\rm Fe5015$ (Å) | $-0.34\pm 1.07$ | $1.44\pm 0.30$
$\rm Mg1$ (mag) | $0.03\pm 0.01$ | $1.18\pm 0.07$
$\rm Mg2$ (mag) | $0.03\pm 0.01$ | $1.04\pm 0.03$
${\rm Mg}b$ (Å) | $-0.12\pm 0.15$ | $1.09\pm 0.04$
$\rm Fe5270$ (Å) | $-0.25\pm 0.11$ | $1.21\pm 0.05$
$\rm Fe5335$ (Å) | $-0.04\pm 0.06$ | $1.23\pm 0.03$
$\rm Fe5406$ (Å) | $-0.10\pm 0.08$ | $1.39\pm 0.07$
$\rm Fe5709$ (Å) | $0.11\pm 0.03$ | $1.30\pm 0.06$
$\rm Fe5782$ (Å) | $0.12\pm 0.10$ | $1.41\pm 0.24$
$\rm NaD$ (Å) | $0.21\pm 0.36$ | $0.91\pm 0.14$
$\rm TiO1$ (mag) | $-0.07\pm 0.01$ | $2.38\pm 0.19$
$\rm TiO2$ (mag) | $-0.07\pm 0.01$ | $2.56\pm 0.14$
## 4 Fitting, analysis and results
### 4.1 Model description
Thomas et al. (2003) provided stellar population models including Lick
absorption line indices for various elemental-abundance ratios, covering ages
from 1 to 15 Gyr and metallicities from 1/200 to $3.5\times$ solar abundance.
These models are based on the standard models of Maraston (1998), with input
stellar evolutionary tracks from Cassisi et al. (1997) and Bono et al. (1997)
and a Salpeter (1955) stellar initial mass function. Thomas et al. (2004)
improved the models by including higher-order Balmer absorption-line indices.
They found that these Balmer indices are very sensitive to changes in the
$\rm\alpha/Fe$ ratio for supersolar metallicities. The latest stellar
population model for Lick absorption-line indices (Thomas et al., 2010) is an
improvement on Thomas et al. (2003) and Thomas et al. (2004). They were
derived from the MILES stellar library, which provides a higher spectral
resolution appropriate for MILES and SDSS spectroscopy, as well as flux
calibration. The models cover ages from 0.1 to 15 Gyr, $\rm[Z/H]$ from $-2.25$
to 0.67 dex, and $\rm[\alpha/Fe]$ from $-0.3$ to 0.5 dex. In our work, we
fitted our absorption indices based on the models of Thomas et al. (2010), by
using the two sets of stellar evolutionary tracks provided, i.e., Cassisi et
al. (1997) and Padova.
### 4.2 Fits with stellar population models and the results
Similar to Sharina et al. (2006) and our Paper I, the $\chi^{2}-$minimization
routine was applied for fitting Lick indices with the SSP models to derive the
physical parameters. As we measured 25 different types of Lick line indices
listed in Table 3, all indices were used for the fitting procedure. As Thomas
et al. (2010) provide only 20 ages, 6 metallicity $\rm[Z/H]$, and 4
$\alpha$-element $\rm[\alpha/Fe]$ for the SSP model, it is necessary to
interpolate the original models to the higher-resolution models for our needs.
We performed the cubic spline interpolations, using equal step lengths, to
obtain a grid of 150 ages from 0.1 to 15 Gyr, 31 $\rm[Z/H]$ values from
$-2.25$ to 0.67 dex, and 51 $\rm[\alpha/Fe]$ from $-0.3$ to $0.5$ dex, which
could make the fitted results smoother and more continuous. Since Worthey
(1994b); Galleti et al. (2009) pointed out the age-metallicity degenaracy for
most of the spectral feature indices measurements, which almost remain the
same when the percentage change ${\rm\Delta age/\Delta}Z=3/2$. Therefore, it
is necessary for us to constrain the metallicity with the metal-sensitive
indices before the fits.
Fortunately, Galleti et al. (2009) provide two ways to measure the metallicity
from the metal-sensitive spectral indices directly. One method is through
combining the absorption line indices Mg and Fe, $\rm[MgFe]$, which is defined
as $\rm[MgFe]=\rm\sqrt{Mg{\it b}\cdot\langle Fe\rangle}$, where $\rm\langle
Fe\rangle=(Fe5270+Fe5335)/2$. Thus, the metallicity can be calculated from the
formula below,
$\rm[Fe/H]_{[MgFe]}=-2.563+1.119[MgFe]-0.106[MgFe]^{2}\pm 0.15.$ (2)
The second way to obtain the metallicity from Mg2 is using a polynomial in the
following,
$\rm[Fe/H]_{Mg2}=-2.276+13.053Mg2-16.462Mg2^{2}\pm 0.15.$ (3)
Finally we obtained $\rm[Fe/H]_{avg}$ with uncertainty in Table 4, which is an
average of the metallicities derived from the metallicity Eqs. 2 and 3,
respectively. The averaged metallicity $\rm[Fe/H]_{avg}$ will be used to
constrain the metallicity in the fits to break the age-metallicity
trends/degeneracy. However, Thomas et al. (2010) model only provide the
metallicity parameters with $\rm[Z/H]$ and $\rm[\alpha/Fe]$, thus we need to
find a relationship between the iron abundance$\rm[Fe/H]$, total metallicity
$\rm[Z/H]$ and $\alpha$-element to iron ratio $\rm[\alpha/Fe]$, which we can
replace $\rm[Fe/H]$ with $\rm[Z/H]$ and $\rm[\alpha/Fe]$ in the fit procedure.
In fact, Thomas et al. (2003) give the relation in Eq. 4.
Table 4: The metallicities $\rm[Fe/H]$ derived from the spectral indices
$\rm[MgFe]$, Mg2.
Name | $\rm[Fe/H]_{avg}$
---|---
B289 | $-1.83\pm 0.27$
B290 | $-0.56\pm 0.63$
H11 | $-0.49\pm 0.58$
H18 | $-1.35\pm 0.65$
SK108A | $-2.35\pm 0.22$
SK112A | $-1.62\pm 0.43$
MGC1 | $-2.06\pm 0.33$
H25 | $-2.74\pm 0.47$
0.86Here we define $\rm[Fe/H]_{avg}=\frac{[Fe/H]_{[MgFe]}+[Fe/H]_{Mg2}}{2}$
$\rm[Z/H]=[Fe/H]+0.94[\alpha/Fe]$ (4)
Here we would like to draw reader’s attention that although the metallicity
$\rm[Fe/H]$ has been determined primarily, there are still many different ways
to combine $\rm[Z/H]$ and $\rm[\alpha/Fe]$ in the parameter grid of the model.
Therefore, we still need to fit the age, $\rm[Z/H]$ and $\rm[\alpha/Fe]$
simultaneously. Here, we would like to constrain the metallicity in the fits
for $\rm|[Fe/H]_{fit}-[Fe/H]_{avg}|\leq 0.3$ dex, which is the typical
metallicity uncertainty for the observations and it will make the fits more
reasonable. Like the Paper I, the physical parameters ages, metallicities
$\rm[Z/H]$, and $\rm[\alpha/Fe]$ can be determined by comparing the
interpolated stellar population models with the observational spectral feature
indices by employing the $\chi^{2}-$minimization method below,
$\chi^{2}_{\rm min}={\rm
min}\left[\sum_{i=1}^{25}\left({\frac{L_{\lambda_{i}}^{\rm
obs}-L_{\lambda_{i}}^{\rm model}(\rm
age,[Z/H],[\alpha/Fe])}{\sigma_{i}}}\right)^{2}\right],$ (5)
where $L_{\lambda_{i}}^{\rm model}(\rm age,[Z/H],[\alpha/Fe])$ is the $i^{\rm
th}$ Lick line index in the stellar population model for age, metallicity
$\rm[Z/H]$, and $\rm[\alpha/Fe]$, while $L_{\lambda_{i}}^{\rm obs}$ represents
the observed calibrated Lick absorption-line indices from our measurements and
the errors estimated in our fitting are given as follows,
$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm model},i}^{2}.$ (6)
Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty while
$\sigma_{{\rm model},i}$ is the uncertainty associated with the models of
Thomas et al. (2010). These two types of uncertainties have been both
considered in our fitting procedure.
Table 5 lists the fitted ages, $\rm[Z/H]$ and $\rm[\alpha/Fe]$ with different
evolutionary tracks of Cassisi et al. (1997) and Padova, respectively. In
addition, we calculated the $\rm[Fe/H]_{cassisi}$ and $\rm[Fe/H]_{padova}$ by
applying the Eq. 4 to the fitted $\rm[Z/H]$ and $\rm[\alpha/Fe]$. For the
reason of keeping consistency with Paper I, we adopted the metallicity
$\rm[Fe/H]_{cassisi}$ in the following statistics and analysis. From Table 5,
we found that the ages, $\rm[Z/H]$ and the $\alpha$-element $\rm[\alpha/Fe]$
fitted from either Cassisi et al. (1997) or Padova tracks are consistant with
each other. Besides, it is worth noting that all of our sample halo GCs are
older than 10 Gyr in both evolutionary tracks except B290 (5.5 to 5.8 Gyr),
which is older than 2 Gyr and it should be identified as the ”old” in Caldwell
et al. (2009). Thus, it indicates that these halo clusters formed at the early
stage of the galaxy formation, which agrees well with the previous findings.
Table 5: The $\chi^{2}-$minimization Fitting Results Using Thomas et al.
(2010) Models with Cassisi et al. (1997) and Padova Stellar Evolutionary
Tracks, respectively.
| Cassisi | Padova
---|---|---
Name | Age | $\rm[Z/H]$ | $\rm[\alpha/Fe]$ | $\rm[Fe/H]$ | Age | $\rm[Z/H]$ | $\rm[\alpha/Fe]$ | $\rm[Fe/H]$
| (Gyr) | (dex) | (dex) | (dex) | (Gyr) | (dex) | (dex) | (dex)
B289 | $10.75\pm 4.15$ | $-1.67\pm 0.23$ | $0.34\pm 0.16$ | $-2.09\pm 0.27$ | $11.70\pm 2.80$ | $-2.07\pm 0.18$ | $-0.12\pm 0.18$ | $-2.13\pm 0.25$
B290 | $5.80\pm 2.40$ | $-0.99\pm 0.05$ | $-0.26\pm 0.05$ | $-0.85\pm 0.07$ | $5.50\pm 0.40$ | $-1.33\pm 0.38$ | $-0.26\pm 0.05$ | $-0.85\pm 0.39$
H11 | $13.75\pm 1.25$ | $0.09\pm 0.32$ | $0.08\pm 0.05$ | $-0.19\pm 0.33$ | $13.60\pm 0.20$ | $-0.10\pm 0.24$ | $0.00\pm 0.06$ | $-0.21\pm 0.24$
H18 | $13.45\pm 1.45$ | $-0.47\pm 0.37$ | $0.48\pm 0.02$ | $-1.07\pm 0.37$ | $13.60\pm 0.20$ | $-0.50\pm 0.24$ | $0.48\pm 0.02$ | $-1.07\pm 0.24$
SK108A | $13.60\pm 0.30$ | $-1.53\pm 0.18$ | $0.28\pm 0.22$ | $-2.09\pm 0.28$ | $13.55\pm 0.45$ | $-1.48\pm 0.23$ | $0.27\pm 0.24$ | $-2.09\pm 0.32$
SK112A | $11.10\pm 3.90$ | $-1.33\pm 0.38$ | $0.25\pm 0.25$ | $-1.35\pm 0.45$ | $11.70\pm 3.30$ | $-1.51\pm 0.47$ | $0.10\pm 0.40$ | $-1.42\pm 0.61$
MGC1 | $13.30\pm 0.80$ | $-1.39\pm 0.14$ | $0.42\pm 0.08$ | $-1.76\pm 0.16$ | $12.90\pm 1.30$ | $-1.39\pm 0.14$ | $0.42\pm 0.08$ | $-1.76\pm 0.16$
H25 | $13.60\pm 0.30$ | $-1.98\pm 0.20$ | $0.50\pm 0.00$ | $-2.45\pm 0.20$ | $13.50\pm 0.50$ | $-2.03\pm 0.05$ | $0.50\pm 0.00$ | $-2.45\pm 0.05$
Actually, Mackey et al. (2010) conclude that the metal abundance of MGC1 is
about $\rm[Fe/H]=-2.3$ and age is 12.5 to 12.7 Gyr through the color-magnitude
diagram fitting. The age estimated agree well with our results while the
metallicity is lower than our estimate $\rm[Fe/H]_{avg}=-2.06\pm 0.33$ in
Table 4 or $\rm[Fe/H]_{cassis}=-1.76\pm 0.16$ in Table 5. Nevertheless, Alves-
Brito et al. (2009) found that the metallicity $\rm[Fe/H]=-1.37\pm 0.15$ by
combining the spectroscopic data and the photometric data, which is higher
than our estimate. Hence, it can be seen that our result is just between the
two results, suggesting that our result agrees with the previous conclusions.
### 4.3 Metallicity Properties of Outer Halo
The metallicity gradient of the halo star clusters and stars are important to
the formation and enrichment processes of their host galaxy. Basically, there
are two possible scenarios for the galaxy formation. One is that the halo
stars and clusters should feature large-scale metallicity gradients if the
enrichment timescale is shorter than the collapse time, which may be due to
the galaxy formation as a consequence of a monolithic, dissipative, and rapid
collapse of a single massive, nearly spherical, spinning gas cloud (Eggen et
al., 1962; Barmby et al., 2000). The other one is a chaotic scheme for early
galactic evolution, when the loosely bound pre-enriched fragments merge with
the protogalaxy during a very long period of time, in which case a more
homogeneous metallicity distribution should develop (Searle & Zinn, 1978).
However, most galaxies are believed to have formed through a combination of
these scenarios.
van den Bergh (1969); Huchra et al. (1982) showed that there is little or no
evidence for a general radial metallicity gradient for GCs within a radius of
50 arcmin. However, studies including Huchra et al. (1991); Perrett et al.
(2002); Fan et al. (2008) support the possible existence of a radial
metallicity gradient for the metal-poor M31 GCs, although the slope is not
very significant. Perrett et al. (2002) suggest that the gradients is $-0.017$
and $-0.015$ dex arcmin-1 for the full sample and inner metal-poor clusters.
More recently, Fan et al. (2008) found that the slope is $-0.006$ and $-0.007$
dex arcmin-1 for the metal-poor subsample and whole sample while the slope
approaches zero for the metal-rich subsample. Nevertheless, all these studies
are based on GCs that are located relatively close to the center of the
galaxy, usually at projected radii of less than 100 arcmin. Recently, Huxor et
al. (2011) investigated the metallicity gradient for 15 halo GCs to $r_{\rm
p}=117$ kpc with the metallicity derived from the CMD fittings Mackey et al.
(2006, 2007, 2010) and the authors found that the metallicity gradient becomes
insignificant if one halo GC H14 is excluded in their Figure 6. We found that
our result is consistent well with the previous findings of Huxor et al.
(2011). In Paper I, we found the slope of metallicity gradient is $-0.018\pm
0.001$ dex kpc -1 for the halo clusters sample extended to $r_{\rm p}\sim 117$
kpc from the galaxy center. Further, the slope turns to be $-0.010\pm 0.002$
dex kpc -1 if only considering the clusters $r_{p}>25$ kpc.
Since we have spectroscopic observations of eight more halo confirmed
clusters, it is interesting to check if the metallicity distribution/spatial
gradient would change with an enlarged halo clusters sample. For the new
observed data, as we recalled in §4.2, only MGC1 have the previous metallicity
measurements from the literatures, which are very different for different
works and our measurement is just the median value. Thus, we adopted our
measurement. Finally we have a metallicity sample of 391 entries in total.
Figure 5 shows the metallicity as a function of projected radius from the
galaxy center for all outer GCs with spectroscopic metallicity with $r_{\rm
p}>25$ kpc from the galaxy center. The slope of a linear fit is $-0.005\pm
0.005$ dex kpc-1, which is marked with a solid black line. However, if the two
highest metallicity star clusters G001 and H11 are excluded, the slope turns
out to be $-0.002\pm 0.003$ dex kpc-1, which is shown with the red dashed
line. Thus, both of the cases suggest there is none metallicity gradient for
the M31 outer halo clusters when $r_{\rm p}>25$ kpc, which agree with the
conclusion of Paper I. Therefore it seems that the “fragments merging”
scenario dominated in the outer halo during the galaxy formation stage.
Figure 5: Metallicities $\rm[Fe/H]$ versus projected radii for the outer halo
GCs with $r_{\rm p}>25$ kpc from the center of the galaxy. The slope of the
linear fitting is $-0.005\pm 0.005$ dex kpc-1 (black solid line). However, if
the two highest metallicity GCs G001 and H11 are excluded, the slope turns out
to be $-0.002\pm 0.003$ dex kpc-1 (red dashed line).
It should be noted that the metallicity gradient is fitted based on the data
of our observations and the literature and the metallicities from different
literature may not the same. For instance, the metallicity of G001 is
$\rm[Fe/H]=-1.08\pm 0.09$ in Huchra et al. (1991) while $\rm[Fe/H]=-0.73\pm
0.15$ in Galleti et al. (2009). Thus we wonder how the slope would change when
the data is changed. We simulated ten sets of random data from $\sigma=-0.5$
to 0.5 and added them to the metallicities that we used in Figure 5 and then
refit the slopes again for ten times separatedly and the results are shown in
Table 6. It shows that the slope dose not change significantly when the
simulated errors were added, suggesting that the slope is stable even the data
from different measures.
Table 6: The slopes of metallicity gradient by adding the random errors to the
data.
No. | $k_{all}$ | $k_{<-1}$
---|---|---
1 | $-0.013\pm 0.010$ | $-0.013\pm 0.011$
2 | $-0.003\pm 0.010$ | $0.000\pm 0.014$
3 | $-0.011\pm 0.012$ | $-0.008\pm 0.012$
4 | $-0.009\pm 0.009$ | $0.000\pm 0.012$
5 | $-0.003\pm 0.013$ | $-0.036\pm 0.021$
6 | $-0.002\pm 0.012$ | $0.004\pm 0.022$
7 | $-0.004\pm 0.013$ | $-0.002\pm 0.022$
8 | $-0.009\pm 0.012$ | $0.009\pm 0.015$
9 | $-0.005\pm 0.010$ | $-0.008\pm 0.017$
10 | $-0.013\pm 0.011$ | $0.003\pm 0.008$
Figure 6 shows the relationship between the metallicities and the radial
velocities $V_{r}$ which have been corrected for the systemic velocity of the
M31 galaxy. The spectroscopic metallicities are from the literature (Huchra et
al., 1991; Barmby et al., 2000; Perrett et al., 2002; Galleti et al., 2009;
Caldwell et al., 2011), Paper I as well as this work and the radial velocities
$V_{r}$ are from the RBC v.4, Paper I and this work. It seems that there is no
any relationship between the metallicities versus the radial velocities
$V_{r}$.
Figure 6: Metallicity $\rm[Fe/H]$ versus radial velocity $V_{r}$ (corrected
for the systemic velocity of M31) for all the GCs with spectroscopic
metallicities and radial velocity. The small points are from the literature;
the squares are from Paper I; the triangles are from our measurement.
Figure 7 shows the metallicities versus ages of the GCs. The metallicities are
from the literature (Huchra et al., 1991; Barmby et al., 2000; Perrett et al.,
2002; Galleti et al., 2009; Caldwell et al., 2011), Paper I as well as this
work and the ages are from the Fan et al. (2010), Paper I and this work. We
would like to see if there is any relationship between the ages and
metallicities for these GCs. Actually we find that the relationships are
different for the GC populations with different age. The slope of the GCs
younger than 7 Gyr is $k=0.035\pm 0.021$ while the slope of the GCs older than
7 Gyr is $k=-0.095\pm 0.034$, which is $\sim 3\sigma$ significant level. It
suggests that for the GCs younger than 7 Gyr, there is no relationship between
the age and metallicity while for the clusters older than 7 Gyr, it seems that
the older GCs are more metal-poor (lower metallicity) and the younger GCs are
more metal-rich (higher metallicity).
Figure 7: Metallicity $\rm[Fe/H]$ versus ages for all the clusters with
spectroscopic metallicity and age estimates. The open triangles are the data
from the literature; the filled circles are the data from Paper I; the filled
triangles are the data from this work. The solid line represents the linear
fit of GCs younger than 7 Gyr while the dashed line is the fit for the GCs
older than 7 Gyr.
Previously, many astronomers found the significant bimodal case in the
metallicity of M31 GC distribution by applying the mixture-model KMM test
(Ashman et al., 1994). Ashman & Bird (1993); Barmby et al. (2000); Perrett et
al. (2002) found the proportion of the metal-poor and metal-rich group is
$\sim 2:1$ to $\sim 3:1$ with the peak positions of $\rm[Fe/H]\approx-1.5$ and
$-0.6$, respectively. Fan et al. (2008) examined the bimodality of metallicity
distribution with a larger sample and the authors found the proportion is
$\sim 1.5:1$ and the the peak positions are $\rm[Fe/H]\sim-1.7$ and
$\sim-0.7$, respectively. However, the recent work of Caldwell et al. (2011)
suggests that there is no significant bimodality or trimodality for
metallicity distribution with a sample of 322 M31 GCs, most of which have
spectroscopic metallicity with high S/N ratio. Since we have new observation
data and a larger spectroscopic data sample, we are able to reexamine the
bimodality of the metallicity distributions of M31 GCs. Figure 8 shows the
metallicity distributions of the GCs and the HGCs, respectively. In the Left
panel, the sample includes all the GCs which have spectroscopic metallicity
from the literature (Huchra et al., 1991; Barmby et al., 2000; Perrett et al.,
2002; Galleti et al., 2009; Caldwell et al., 2011) and Paper I as well as this
work. In total, there are 386 GCs with spectroscopic metallicity in the
distribution. We applied the mixture-model KMM algorithm to the dataset and it
returns an insignificant bimodality with p-value $=0.369$, which means that a
bimodal distribution is preferred over a unimodal one at 63.1% confidence
level. The numbers of the metal-poor group and the metal-rich group are
$N1=196$, $N2=190$, respectively and the mean values of the two groups are
$\rm[Fe/H]_{1}=-1.43$ ($\sigma_{1}^{2}=0.327$) and $\rm[Fe/H]_{2}=-0.73$
($\sigma_{2}^{2}=0.215$), respectively. As we can see from the plot, the
proportion of the metal-poor and metal-rich group is $\sim 1:1$, which is
lower than the published results. The reason why the bimodal case becomes more
insignificant with larger sample may be that more intermediate metallicity
GCs, which is between the two metallicity peaks, have been discovered and
those intermediate metallicity GCs cause the distribution to be unlikely a
bimodal or trimodal distribution. Therefore, the previous works found that the
metallicity distributions of M31 GCs is like that of the Milky Way and more
recent works with more data show that they are less similar to each other,
which may indicate that the formations of the two GC system was substantially
different. In the Right panel, it show the metallicity distribution of the
HGCs and obviously the metal-poor GCs dominate in the distribution.
Figure 8: Metallicity distributions with bin size of 0.3 dex. Left: all the
GCs with spectroscopic metallicities. The mixture-model KMM test was applied
to divide them to two groups. Right: all the HGCs with spectroscopic
metallicities.
As the M31 GCs have been divided into two different groups by the KMM test in
the metallicity distribution of Figure 8, we would like to examine the spatial
distributions of the two groups with different metallicity. Figure 9 plots the
spatial distributions of the metal-rich and metal-poor groups. Note that the
metal-poor group appear to occupy a more extended halo and much more widely
spatially distributed while the metal-rich group is more centrally
concentrated, which is consistent with the conclusions of Perrett et al.
(2002); Fan et al. (2008).
Figure 9: The spatial distributions of HGCs with different metallicities.
Left: metal-rich GCs; Right: metal-poor GCs. The two groups were divided by
the KMM test of Figure 8.
Since we have the age estimates of the halo GCs in M31, we are curious about
whether the spatial distributions of the young and old populations are the
same or not. Here we used the definition of ”old population” for age $>2$ Gyr
and the ”young population” for age $<2$ Gyr as that did in Caldwell et al.
(2009). For the purpose of enlarging our sample, the age estimates for M31 GCs
in Fan et al. (2010) and Paper I are also merged into our sample. Figure 10
plots the young and old population spatial distributions, respectively. It is
obvious that the young population is more centrally concentrated and it traces
the disk shape of the galaxy well. However, the spatial distribution of the
old population is more dispersive and it seems that they do not trace the disk
shape of the galaxy.
Figure 10: The spatial distributions of HGCs with young and old populations,
respectively. Left: young clusters with age $<2$ Gyr; Right: old clusters with
age $>$ 2 Gyr.
## 5 Summary and Conclusions
This is the second paper of our serial works for M31 halo globular clusters.
In Paper I, we mainly focus on the fits method and the metallicity gradient
for the M31 GC sample. In this paper, we focus on the dynamics, metallicity
and age, and their distributions as well as the relationships between these
parameters.
We selected eight more confirmed and bright GCs in the halo of M31 from RBC
v.4 and observed them with the OMR spectrograph on 2.16 m telescope at
Xinglong site of NAOC in the fall of 2011. These star clusters are located in
the halo of galaxy at a projected radius of $\sim 14$ to $\sim 117$ kpc from
the galactic center, where the sky background is dark so that they can be
observed in high signal-to-noise ratio.
For all our sample clusters, we measured all 25 Lick absorption-line indices
(see the definitions in, Worthey et al., 1994a; Worthey & Ottaviani, 1997) and
fitted the radial velocities. We found that distributions of the confirmed GCs
and the halo GCs are basically symmetric to the systematic velocity of the
galalxy.
Similar to Sharina et al. (2006) and our Paper I, we applied the
$\chi^{2}-$minimization method to fit the Lick absorption line indices with
the updated Thomas et al. (2010) stellar population model in two stellar
evolutionary tracks of Cassisi and Padova, separately. The fitting results
show that most of our sample clusters are older than 10 Gyr except B290$\sim
5.5$ Gyr and most of them are metal-poor with metallicity $\rm[Fe/H]<-1$ dex
except H11 and H18, suggesting that these halo star clusters were born at the
early stage of the galaxy’s formation
Again, we would like to study the metallicity gradient of the halo GCs by
merging more spectroscopic metallicity from our work, Paper I and the
literature. We only considered outer halo clusters with $r_{\rm p}>25$ kpc and
the fitted slope is $-0.005\pm 0.005$ dex kpc-1. However, if two metal-rich
outlier clusters G001 and H11 are excluded, the slope is $-0.002\pm 0.003$ dex
kpc-1, which does not change significantly. Furthermore, in order to eliminate
the effect the errors of different observations, we added the random errors
from $\sigma=-0.5$ to $0.5$ to the data and refit the slope agian for ten
times. The result shows that the simulated errors do not affact the slope
much. Thus it seems that metallicity gradient for M31 outer halo clusters dose
not exist, which agrees well with the previous findings (Huxor et al., 2011)
and Paper I. This result may imply that the “fragments merging” scenario is
dominated in the outer halo of the galaxy beyond 25 kpc from the center during
the early stage of the galaxy formation.
We do not find a relationship between metallicity and the radial velocity for
M31 GCs sample. It seems that the metallicity is not a function of age for the
GCs with age $<7$ Gyr while for the old GCs with age $>7$ Gyr there seems to
be a trend that the older ones have lower metallicity. This conclusion is
similar to that of Fan et al. (2006), who found a possible general trend of
the age-metallicity relation with a large scatter. In addition, we plot
metallicity distributions with the largest sample of M31 GCs so far and it
shows the bimodality is not significant compared to the previous work. This is
also found by Caldwell et al. (2011), who used the newly observed
spectroscopic data. We also find that the number of the metal-poor and metal-
rich groups becomes comparable while the previous works show that the number
of metal-poor group is more than that of the metal-rich one. This may be due
to many intermedate metallicty metallicity of Caldwell et al. (2011) have been
merged into our sample for our statistics. The spatial distributions shows
that the metal-rich group is more centrally concentrated while the metal-poor
group is occupy a more extended halo and the young population is centrally
concentrated while the old populaiton is more extended spatially to the outer
halo. This is easy to be understood as the old GCs are usually metal-poor
especially for the halo GCs of M31.
###### Acknowledgements.
We are indebted to an anonymous referee for his/her thoughtfull comments and
insightful suggestions that improved this paper greatly. The authors are also
grateful to the kind staff at the Xinglong 2.16m telescope for the support
during the observations. This research was supported by National Natural
Science Foundation of China through grants Nos. 11003021, 11073027 and
11073032.
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|
arxiv-papers
| 2012-03-08T03:05:52 |
2024-09-04T02:49:28.443814
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhou Fan, Ya-Fang Huang, Jin-Zeng Li, Xu Zhou, Jun Ma, Yong-Heng Zhao",
"submitter": "Zhou Fan",
"url": "https://arxiv.org/abs/1203.1684"
}
|
1203.1760
|
11institutetext: Informatics Research Institute of Albacete,
University of Castilla-La Mancha, Campus Universitario s/n,
02071\. Albacete, SPAIN.
11email: {valentin,jmateo,gregorio}@dsi.uclm.es
# BPEL-RF: A formal framework for BPEL orchestrations integrating distributed
resources
José Antonio Mateo Valentín Valero Gregorio Díaz
###### Abstract
Web service compositions are gaining attention to develop complex web systems
by combination of existing services. Thus, there are many works that leverage
the advantages of this approach. However, there are only few works that use
web service compositions to manage distributed resources. In this paper, we
then present a formal model that combines orchestrations written in BPEL with
distributed resources, by using WSRF.
## 1 Introduction
Software systems are gaining complexity and concurrency with the appearance of
new computational paradigms such as Service-Oriented Computing (SOC), Grid
Computing and Cloud Computing. In this kind of systems, the services provider
needs to ensure some levels of quality and privacy to the final user in a way
that had never been raised. Therefore, it is necessary to develop new models
yielding the advantages of recent approaches as web services compositions, but
applied to these recent scenarios. To this end, we have worked up an
operational semantics to manage web services with associated resources by
using the existing machinery in distributed systems, web services
orchestrations.
The definition of a web service-oriented system involves two complementary
views: Choreography and Orchestration. On the one hand, the choreography
concerns the observable interactions among services and can be defined by
using specific languages, e.g., Web Services Choreography Description Language
(WS-CDL) [15]. On the other hand, the orchestration concerns the internal
behavior of a web service in terms of invocations to other services. Web
Services Business Process Execution Language (WS-BPEL) [1] is usually used to
describe these orchestrations, so this is considered the de facto standard
language for describing web services workflow in terms of web service
compositions.
In this scenario, developers require more standardization to facilitate
additional interoperability among these services. Thus, in January of 2004,
several members of the organization _Globus Alliance_ and the computer
multinational _IBM_ with the help of experts from companies such as _HP, SAP,
Akamai, etc._ defined the basis architecture and the initial specification
documents of a new standard for that purpose, Web Services Resource Framework
(WSRF) [9]. Although the web service definition does not consider the notion
of state, interfaces frequently provide the user with the ability to access
and manipulate states, i.e., data values that persist across, and evolve as a
result of web service interactions. It is then desirable to define web service
conventions to enable the discovery of, introspection on, and interaction with
stateful resources in standard and interoperable ways [4]. These observations
motivated the appearance of the WS-Resource approach to modeling states in web
services.
In WSRF, we can see a WS-Resource as a collection of properties _P_ identified
by an address _EPR_ and with a _timeout_ associated. This timeout represents
the lifetime of the WS-Resource. Without loss of generality, we have reduced
the resource properties set to only one allowing us to use the resource
identifier _EPR_ as the representative of this property. On the BPEL hand, we
have only taken into consideration the root scope avoiding any class of
nesting among scopes and we have only modeled the event and fault handling,
leaving the other handling types as future work.
## 2 Related Work
The use of WS-BPEL has been extensively studied by using different types of
formalism such as Petri nets, Finite State Machines and process algebras.
Regarding the use of WS-BPEL together with WS-RF there are few works, and they
only show a description of this union, without a formalization of the model.
In [14] Slomiski uses BPEL4WS in Grid environments and discusses the benefits
and challenges of extensibility in the particular case of OGSI workflows
combined with WSRF-based Grids. Other two works centered around Grid
environments are [10] and [7]. The first justifies the use of BPEL
extensibility to allow the combination of different GRIDs, whereas Ezenwoye et
al. [7] share their experience on BPEL to integrate, create and manage WS-
Resources that implement the factory/instance pattern.
On the Petri nets hand, Ouyang et al. [12] define the necessary elements for
translating BPEL processes into Petri nets. Thus, they cover all the important
aspects in the standard such as exception handling, dead path elimination and
so on. The model they consider differs from ours in that we formalize the
whole system as a composition of orchestrators with resources associated,
whereas they describe the system as a general scope with nested sub-scopes
leaving aside the possibility of administering resources. Furthermore, we have
also formalized the event handling and notification mechanisms. Another
extensive semantics for BPEL 2.0 is presented in [6] by Dumas et al, which
introduces two new interesting improvements. They define several patterns to
simplify some huge nets and introduce the semantics for the WS-BPEL 2.0 new
patterns. On the $\pi$-calculus hand, Dragoni and Mazzara [5] propose a
theoretical scheme focused on dependable composition for the WS-BPEL recovery
framework. In this approach, the recovery framework is simplified and analyzed
via a conservative extension of $\pi$-calculus. The aim of this approach
clearly differs from ours, but it helps us to have a bigger understanding of
the WS-BPEL recovery framework. Other work focused on the BPEL recovery
framework is [13]. Although this is more interested in the compensation
handler, they describe the corresponding rules that manage a web service
composition. Our work is therefore quite complete as we define rules for
nearly all possible activities. In addition, we also consider time
constraints. Finally, we would like to highlight the works of Farahbod et al.
[8] and Busi et al. [3]. In the first one, the authors extract an abstract
operational semantics for BPEL based on abstract state machines (ASM) defining
the framework BPELAM to manage the agents who perform the workflow activities.
In this approach time constraints are considered, but they do not formalize
the timed model. On the other hand, the goal of the latter one is fairly
similar to ours. They also define a $\pi$-calculus operational semantics for
BPEL and describe a conformance notion. They present all the machinery to
model web service compositions (choreographies and orchestrations). The main
differences with our work are that we are more restrictive with respect to
time constraints and we deal with distributed resources.
## 3 BPEL/WSRF
WS-Resource Framework [2] is a resource specification language developed by
OASIS and some of the most pioneering computer companies, whose purpose is to
define a generic framework for modeling web services with stateful resources,
as well as the relationships among these services in a Grid/Cloud environment.
This approach consists of a set of specifications that define the
representation of the WS-Resource in the terms that specify the messages
exchanged and the related XML documents. These specifications allow the
programmer to declare and implement the association between a service and one
or more resources. It also includes mechanisms to describe the means to check
the status and the service description of a resource, which together form the
definition of a WS-Resource. In Table 1 we show the main WSRF elements.
Name | Describes
---|---
WS-ResourceProperties | WSRF uses a precise specification to define the properties of the WS-Resources.
WS-Basefaults | To standardize the format for reporting error messages.
WS-ServiceGroup | This specification allows the programmer to create groups that share a common set of properties.
WS-ResourceLifetime | The mission of this specification is to standardize the process of destroying a resource and identify mechanisms to monitor its lifetime.
WS-Notification | This specification allows to a _NotificationProducer_ to send _notifications_ to a _NotificationConsumer_ in two ways: without following any formalism or with a predefined formalism.
Table 1: WSRF main elements
On the other hand, web services are becoming more and more important as a
platform for Business-to-Business integration. Web service compositions have
appeared as a natural and elegant way to provide new value-added services as a
combination of several established web services. Services provided by
different suppliers can act together to provide another service; in fact, they
can be written in different languages and can be executed on different
platforms. As we noticed in the introduction, we can use web service
compositions as a way to construct web service systems where each service is
an autonomous entity which can offer a series of operations to the other
services conforming a whole system. In this way, it is fairly necessary to
establish a consistent manner to coordinate the system participants such that
each of them may have a different approach, so it is common to use specific
languages such as WS-BPEL to manage the system workflow. WS-BPEL, for short
BPEL, is an OASIS orchestration language for specifying actions within web
service business processes. These actions are represented by the execution of
two types of activities (_basic_ and _structured_) that perform the process
logic. _Basic activities_ are those which describe elemental steps of the
process behavior and _structured activities_ encode control-flow logic, and
can therefore contain other basic and/or structured activities recursively
[1].
## 4 Operational Semantics
We use the following notation: ORCH is the set of orchestrators in the system,
Var is the set of integer variable names, PL is the set of necessary
partnerlinks, OPS is the set of operations that can be performed, EPRS is the
set of resource identifiers, and A is the set of basic or structured
activities that can form the body of a process. The specific algebraic
language, then, that we use for the activities is defined by the following
BNF-notation:
$\begin{array}[]{l}A::={\it throw}\;|\;{\it receive}(pl,op,v)\;|\;{\it
invoke}(pl,op,v_{1})\;|\\\ {\it
reply}(pl,v)\;|\;{\it\overline{reply}(pl,v_{2})}\;|\;{\it
assign}(expr,v_{1})\;|\;{\it wait}(timeout)\;|\\\ {\it empty}\;|\;{\it
exit}\;|\;\ \,\,A\,;A\,\,\;|\,\;A\,\|\,A\;\,|\,{\it while}(cond,A)\;|\\\ \
{\it pick}(\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A,timeout)\;|\\\ {\it
createResource}(EPR,val,timeout,A_{e{{}_{i}}})\;|\;{\it getProp}(EPR,v)\;|\\\
{\it setProp}(EPR,val)\;|\;{\it setTimeout}(EPR,timeout)\;|\\\ {\it
subscribe}(O,EPR,cond^{\prime},A_{e{{}_{i}}})\end{array}$
where ${\it O\in ORCH,EPR\in EPRS,pl,pl_{i}\in PL,op,op_{i}}$ ${\it\in
OPS,timeout\in{\rm I\\!N},expr}$ is an arithmetic expression constructed by
using the variables in Var and integers; ${\it v,v_{1},v_{2},v_{i}}$ range
over Var, and ${\it val\in\mathbb{Z}}$. A condition ${\it cond}$ is a
predicate constructed by using conjunctions, disjunctions, and negations over
the set of variables ${\it Var}$ and integers, whereas ${\it cond^{\prime}}$
is a predicate constructed by using the corresponding ${\it EPR}$ (as the
resource value) and integers.
BPEL basic activities used in our model are: _invoke_ to request services
offered by service providers, _receive_ and _reply_ to provide services to
partners, _throw_ to signal an internal fault explicitly, _wait_ to specify a
delay, _empty_ to do nothing, _exit_ to end the business process and _assign_
, which is used to copy data from a variable to another. And the _structured
activities_ used are: _sequence_ , which contains two activities that are
performed sequentially, _while_ to provide a repeated execution of one
activity, _pick_ that waits for the occurrence of exactly one event from a set
of events (including an alarm event), and then executes the activity
associated with that event, and, finally, _flow_ to express concurrency.
Another family of control flow constructs in BPEL includes event, fault and
compensation handlers. An event handler is enabled when its associated event
occurs, being executed concurrently with the main orchestrator activity.
Unlike event handlers, fault handlers do not execute concurrently with the
orchestrator main activity [12]. The correspondence among the syntax of WS-
BPEL, WSRF and our model is shown in Table 4.
WS-BPEL Syntax | Metamodel
---|---
$<$process …$>$ |
$<$partnerLinks$>$ … $<$/partnerLinks$>$? |
$<$Variables$>$ … $<$/Variables$>$? |
$<$faultHandlers$>$ … $<$/faultHandlers$>$? |
$<$eventHandlers$>$ … $<$/eventHandlers$>$? |
(activities)* |
$<$/process$>$ |
| (PL,Var,A,Af,$\mathcal{A}_{e}$)
throw/any fault | throw
$<$receive partnerLink=“pl” operation=“op” variable=“v” createInstance=“no”$>$ $<$/receive$>$ | receive(pl,op,v)
$<$reply partnerLink=“pl” variable=“v”$>$ $<$/reply$>$ | reply(pl,v)
$<$invoke partnerLink=“pl” operation=“op”inputVariable=“v1” outputVariable=“v2”$>$ $<$/invoke$>$ | invoke(pl,op,v1); $[\overline{reply}$(pl,op,v2)]
|
$<$empty$>$ … $<$/empty$>$ | empty
$<$exit$>$ … $<$/exit$>$ | exit
$<$assign$>$$<$copy$>$$<$from$>$expr$<$/from$>$$<$to$>$v1$<$/to$>$$<$/copy$>$$<$/assign$>$ | assign(expr,v1)
$<$wait$>$$<$for$>$timeout$<$/for$>$ $<$/wait$>$ | wait(timeout)
$<$sequence$>$ activity1 activity2 $<$/sequence$>$ $<$flow$>$ activity1 activity2 $<$/flow$>$ | A${}_{1}\,;\,$ A2 —————– A${}_{1}\,\|\,$ A2
$<$while$>$$<$condition$>$cond$<$/condition$>$activity1$<$/while$>$ | while(cond,A)
$<$pick createInstance=“no”$>$ $<$onMessage partnerLink=“pl” operation=“op”variable=“v”$>$ activity1 $<$/onMessage$>$ $<$onAlarm$>$$<$for$>$timeout$<$/for$>$activity1$<$/onAlarm$>$ $<$/pick$>$ | pick($\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A$,timeout)
$<$invoke partnerLink=“Factory”operation=“CreateResource” inputVariable=“MessageIn”outputVariable=“MessageOut”$>$ $<$/invoke$>$$<$assign$>$$<$copy$>$$<$from variable=“MessageOut”$>$part=“param” query=“/test:CreateOut/wsa:endpointreference”$<$/from$>$ $<$to$>$ partnerlink=“Factory”$<$/to$>$$<$/copy$>$$<$/assign$>$ | createResource(EPR,val,timeout,A${}_{e{{}_{i}}}$)
$<$wsrp:GetResourceProperty$>$property1$<$/wsrp:GetResourceProperty$>$ | getProp(EPR,v)
$<$wsrp:SetResourceProperties$>$ $<$wsrp:Update$>$ property1 $<$/wsrp:Update$>$ $</$wsrp:SetResourceProperties$>$ | setProp(EPR,val)
$<$wsrl:SetTerminationTime$>$ $<$wsrl:RequestedTerminationTime$>$ timeout $<$/wsrl:RequestedTerminationTime$>$ $<$/wsrl:SetTerminationTime$>$ | setTimeout(EPR,timeout)
$<$wsnt:Subscribe$>$ $<$wsnt:ConsumerReference$>$O$<$/wsnt: ConsumerReference$>$ $<$wsnt:ProducerReference$>$EPR$<$/wsnt: ProducerReference$>$ $<$wsnt:Precondition$>$cond’$<$/Precondition$>$ $<$/wsnt:Subscribe$>$ | subscribe(O,EPR,cond’,A${}_{e{{}_{i}}}$)
$<$wsnt:Notify$>$ $<$wsnt:NotificationMessage$>$ $<$wsnt:SubscriptionReference$>$O$<$/wsnt:SubscriptionReference$>$ $<$wsnt:ProducerReference$>$$EPR$$<$/wsnt:ProducerReference$>$ $<$wsnt:Message$>$ … $<$/wsnt:Message$>$ $<$/wsnt:NotificationMessage$>$ $<$/wsnt:Notify$>$ | Spawn the associated event handler activity A${}_{e{{}_{i}}}$
Table 2: Conversion table
An orchestration is now defined as a tuple ${\it
O=(PL,Var,A,A_{f},\mathcal{A}_{e})}$, where $A$ and $A_{f}$ are activities
defined by the previous syntax and $\mathcal{A}_{e}$ is a set of activities.
Specifically, $A$ represents the normal workflow, $A_{f}$ is the fault
handling activity and $\mathcal{A}_{e}=\\{A_{e_{i}}\\}_{i=0}^{m}$ are the
event handling activities. The operational semantics is, then, defined at
three levels, the internal one corresponds to the evolution of one activity
without notifications. In the second one, we define the orchestration
semantics with notifications, whereas the third level corresponds to the
composition of different orchestrators and resources to conform the
choreography. We first introduce some definitions that are required in order
to define the operational semantics.
###### Definition 1 (States)
We define a state as a pair s=($\sigma,\rho$), where $\sigma$ represents the
variable values and $\rho$ captures the resource state. Thus,
${\it\sigma:Var\rightarrow\mathbb{Z}}$, and
$\it{\rho=\\{(EPR_{i},v_{i},Subs_{i},t_{i},A_{e{{}_{i}}})\\}_{i=1}^{r}}$,
where $r$ is the number of resources in the system. Each resource has its own
identifier, ${\it EPR_{i}}$, and, at each state, has a particular value,
$v_{i}$, and a lifetime, $t_{i}$, initialized with the createResource
function, which can be changed by using the function setTimeout. Moreover,
$\it{Subs_{i}=\\{(O_{i_{j}},cond^{\prime}_{i_{j}},A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}})\\}_{j=1}^{s_{i}}}$
is the set of resource notification subscribers, their associated delivery
conditions and the event handling activity ${\it
A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}}}$ that must be thrown in the case that
${\it cond^{\prime}_{i_{j}}}$ holds; $s_{i}$ is the number of orchestrations
currently subscribed to this resource and ${\it O_{i_{j}}\in ORCH}$ are the
subscriber’s identifiers. The operations are defined as follows: ${\it
OPS=\\{op_{i}|\ op_{i}:\mathbb{Z}^{Var}\rightarrow\mathbb{Z}^{Var}\\}}$. Given
a state $s=(\sigma,\rho)$, a variable $v$ and an expression $e$, we denote by
$s^{\prime}=(\sigma[e/v],\rho)$ the state obtained from $s$ by changing the
value of $v$ for the evaluation of $e$ and ${\it
s{{}^{+}}=(\sigma,\rho^{\prime})}$, where
${\it\rho^{\prime}=\\{(EPR_{i},v_{i},Subs_{i},t_{i}-1,A_{e{{}_{i}}})|t_{i}>1\\}_{i=1}^{r}}$.
Next we define some notation that we use in the operational semantics. We
employ the notation $\it{EPR_{i}\in\rho}$ to denote that there is a tuple
$\it{(EPR_{i},v_{i},Subs_{i},t_{i},A_{e{{}_{i}}})}$ ${\it\in\rho}$,
$i\in[1\ldots r]$. Given a predicate $\it{cond}$, we use the function
$\it{cond(s)}$ to mean the resulting value of this predicate at the state
$\it{s}$. Besides, ${\it\rho[w/EPR]_{1}}$ is used to denote that the new value
in $\rho$ of the resource $\it{EPR}$ is $\it{w}$, $\it{\rho[t/EPR]_{2}}$
denotes a change in the $\it{timeout}$ attribute of the resource in $\rho$ and
$\it{Add\\_subs(\rho,EPR_{i},O_{i_{j}},cond^{\prime}_{i_{j}},A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}})}$
denotes that
$\it{(O_{i_{j}},cond^{\prime}_{i_{j}},A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}})}$
is added to the subscribers of the resource $\it{EPR_{i}\in\rho}$ or ${\it
cond^{\prime}=cond^{\prime}_{i_{j}}}$ in the case that $\it{O_{i_{j}}}$ was
already in ${\it Subs_{i}}$. We need two additional functions. One of them, to
extract the event handling activities that will be launched when the
subscriber condition holds at the current state ${\it s}$: ${\it
N(O,s)=\\{A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}}|(O_{i_{j}},cond^{\prime}_{i_{j}},A_{e{{}_{s}{{}_{{}_{i}{{}_{{}_{j}}}}}}})\in
Subs_{i},O_{i{{}_{j}}}=O,}$ ${\it cond^{\prime}_{i_{j}}=true\\}_{i=1}^{r}}$
and the other one is used to launch the activities when the resource lifetime
expires: ${\it
T(O,s)=\\{A_{e{{}_{r}{{}_{{}_{i}}}}}|(EPR_{i},v_{i},Subs_{i},1,A_{e{{}_{r}{{}_{{}_{i}}}}})\in\rho,O=}$
${\it O_{i{{}_{j}}}\in Subs_{i}\\}_{i=1}^{r}}$. Now, a partnerlink is a pair
$(O_{i},O_{j})$ representing the two roles in communication: sender and
receiver.
###### Definition 2 (Activity Operational semantics)
We specify the activity operational semantics by using two types of
transition:
1. a.
(A,s)$\xrightarrow{a}(A^{\prime},s^{\prime})$, a $\in$ Act (Action
transitions).
2. b.
(A,s)$\xrightarrow{}_{1}(A^{\prime},s^{+})$ (Delay transitions).
where Act is the set of actions that can be performed, namely: $Act=\\{\tau$,
throw, receive(pl,op,v), reply(pl,v), invoke(pl,op,v1),
$\overline{reply}$(pl,v2), assign(e,v1), empty, wait(timeout), exit,
pick({(pli,opi,vi,Ai)}${}_{i=1}^{n}$,A,timeout), while(cond,A),
createResource(EPR,val,timeout,A${}_{e_{i}}$), setProp(EPR,val),
getProp(EPR,v), setTimeout(EPR,timeout), and
subscribe(O,EPR,cond′,A${}_{e{{}_{i}}})$}. Notice that we have included a
${\it\tau}$-action that represents an empty movement.
_Action transitions_ capture a state change by the execution of an action
$a\in Act$, which can be empty ($\tau$). _Delay transitions_ capture how the
system state changes when one time unit has elapsed. In Tables 4,5, we show
the rules of these transitions.
(Throw) ${(throw,s)\xrightarrow{throw}(empty,s)}$ (Exit)
$(exit,s)\xrightarrow{exit}(empty,s)$ (Receive)
${(receive(pl,op,v),s)\xrightarrow{receive(pl,op,v^{\prime})}(empty,s^{\prime})}$
where ${\it v\in Var,v^{\prime}\in\mathbb{Z},op\in OPS,pl\in PL}$, and ${\it
s^{\prime}=(op(\sigma[v^{\prime}/v]),\rho)}$. (Invoke)
${(invoke(pl,op,v_{1}),s)\xrightarrow{invoke(pl,op,v_{1})}(empty,s)}$
${\bf(\overline{Reply})}$
${(\overline{reply}(pl,v_{2}),s)\xrightarrow{\overline{reply}(pl,v^{\prime}_{2})}(empty,s^{\prime})}$
where ${\it v_{2}\in Var,v^{\prime}_{2}\in\mathbb{Z},pl\in PL}$, and ${\it
s^{\prime}=(\sigma[v^{\prime}_{2}/v_{2}],\rho)}$. (Reply)
${(reply(pl,v),s)\xrightarrow{reply(pl,v)}(empty,s)}$ (Assign)
${(assign(expr,v_{1}),s)\xrightarrow{assign(expr,v_{1})}(empty,s^{\prime})}$
where ${\it v_{1}\in Var,expr}$ is an arithmetic expression, and ${\it
s^{\prime}=(\sigma[expr/v_{1}],\rho)}$. (Seq1)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(A^{\prime}_{1},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(A^{\prime}_{1};A_{2},s^{\prime})\end{array}$
(Seq2)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(A_{2},s^{\prime})\end{array}$
(Seq3)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par1)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(A^{\prime}_{1},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(A^{\prime}_{1}||A_{2},s^{\prime})\end{array}$
(Par2)
$\begin{array}[]{c}\displaystyle(A_{2},s)\xrightarrow{a}(A^{\prime}_{2},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(A_{1}||A^{\prime}_{2},s^{\prime})\end{array}$
(Par3)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par4)
$\begin{array}[]{c}\displaystyle(A_{2},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par5)
${(empty||empty,s)\xrightarrow{\tau}(empty,s)}$ (While1)
$\begin{array}[]{c}\displaystyle cond(s)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(while(cond,A),s)\xrightarrow{\tau}(A;(while(cond,A),s))\end{array}$
(While2) $\begin{array}[]{c}\displaystyle\neg cond(s)\\\
\rule{113.81102pt}{0.28453pt}\\\
\displaystyle(while(cond,A),s)\xrightarrow{\tau}(empty,s)\end{array}$ (Pick)
$(pick(\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A,t),s)\xrightarrow{pick(pl_{i},op_{i},v^{\prime}_{i},A_{i})}(A_{i},s^{\prime})$
$\textrm{where}\ {\it t\geq 1,v_{i}\in
Var,v^{\prime}_{i}\in\mathbb{Z},op_{i}\in OPS,pl_{i}\in PL,}\ \textrm{and}\
{\it s^{\prime}=(op_{i}(\sigma[v^{\prime}_{i}/v_{i}]),\rho)}.$ (CR)
$(createResource(EPR,val,t,A_{e{{}_{i}}}),s)\xrightarrow{createResource(EPR,val,t,A_{e{{}_{i}}})}(empty,s^{\prime})$
$\textrm{where}\ {\it t\geq 1,val\in\mathbb{Z}}\ \textrm{and}\
\it{s^{\prime}=(\sigma,\rho\cup\\{EPR,val,\emptyset,t,A_{e{{}_{i}}}\\})},\
\textrm{if}\ {\it EPR\notin\rho}.\ \textrm{Otherwise,
}{\it\rho^{\prime}=\rho}.$ (GetProp) $\begin{array}[]{c}\displaystyle{\it
s=(\sigma,\rho),EPR\in\rho}\\\ \rule{170.71652pt}{0.28453pt}\\\
\displaystyle(getProp(EPR,v),s)\xrightarrow{getProp(EPR,v^{\prime})}(empty,s^{\prime})\end{array}$
where ${\it v\in Var,v^{\prime}\in\mathbb{Z}}$ and ${\it
s^{\prime}=(\sigma[v^{\prime}/v],\rho)}$. (GetProp2)
$\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\notin\rho}\\\
\rule{142.26378pt}{0.28453pt}\\\
\displaystyle(getProp(EPR,v),s)\xrightarrow{throw}(empty,s)\end{array}$
(SetTime) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\in\rho}\\\
\rule{184.9429pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,t),s)\xrightarrow{setTimeout(EPR,t)}(empty,s^{\prime})\end{array}$
where ${\it t\geq 1}$,$\ {\it s^{\prime}=(\sigma,\rho[t/EPR]_{2})}$.
(SetTime2) $\begin{array}[]{c}\displaystyle{\it
s=(\sigma,\rho),EPR\notin\rho}\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,t),s)\xrightarrow{throw}(empty,s)\end{array}$
(SetTime3) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\in\rho}\\\
\rule{170.71652pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,0),s)\xrightarrow{throw}(empty,s)\end{array}$
Table 3: Action and delay transition rules without notifications.
When a resource has used up its lifetime or when a subscription condition
holds, a specific notification is sent to the corresponding resource
subscribers, which is captured by the rules in Table 6. In these rules, the
parallel operator has been extended to spawn some event handling activities,
which must run in parallel with the normal activity of an orchestrator. We
therefore introduce the rules by using the following syntax for the activities
in execution: $\it{(A,\mathcal{A}_{e})}$, where ${\it A}$ represents the
normal system workflow, and ${\it\mathcal{A}_{e}=\\{A_{e_{i}}\\}_{i=0}^{m}}$
are the handling activities in execution. Given any activity ${\it A}$, we
write for short ${\it A||\mathcal{A}_{e}}$ to denote
${\it(A||(A_{e{{}_{1}}}||(\ldots||A_{e{{}_{m}}})))}$. We assume in this
operator that those event handling activities that were already in
$\mathcal{A}_{e}$ will not be spawned twice.
###### Definition 3 (Operational semantics with notifications)
We extend both types of transition to act on pairs ${\it(A,\mathcal{A}_{e})}$.
The transitions have now the following form:
1. a.
$(O:(A,\mathcal{A}_{e}),s)\xrightarrow{a}(O:(A^{\prime},\mathcal{A}^{\prime}_{e}),s^{\prime})$,
a $\in$ Act
2. b.
$(O:(A,\mathcal{A}_{e}),s)\xrightarrow{}_{1}(O:(A^{\prime},\mathcal{A}^{\prime}_{e}),s^{+})$
(Throw) ${(throw,s)\xrightarrow{throw}(empty,s)}$ (Exit)
$(exit,s)\xrightarrow{exit}(empty,s)$ (Receive)
${(receive(pl,op,v),s)\xrightarrow{receive(pl,op,v^{\prime})}(empty,s^{\prime})}$
where ${\it v\in Var,v^{\prime}\in\mathbb{Z},op\in OPS,pl\in PL}$, and ${\it
s^{\prime}=(op(\sigma[v^{\prime}/v]),\rho)}$. (Invoke)
${(invoke(pl,op,v_{1}),s)\xrightarrow{invoke(pl,op,v_{1})}(empty,s)}$
${\bf(\overline{Reply})}$
${(\overline{reply}(pl,v_{2}),s)\xrightarrow{\overline{reply}(pl,v^{\prime}_{2})}(empty,s^{\prime})}$
where ${\it v_{2}\in Var,v^{\prime}_{2}\in\mathbb{Z},pl\in PL}$, and ${\it
s^{\prime}=(\sigma[v^{\prime}_{2}/v_{2}],\rho)}$. (Reply)
${(reply(pl,v),s)\xrightarrow{reply(pl,v)}(empty,s)}$ (Assign)
${(assign(expr,v_{1}),s)\xrightarrow{assign(expr,v_{1})}(empty,s^{\prime})}$
where ${\it v_{1}\in Var,expr}$ is an arithmetic expression, and ${\it
s^{\prime}=(\sigma[expr/v_{1}],\rho)}$. (Seq1)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(A^{\prime}_{1},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(A^{\prime}_{1};A_{2},s^{\prime})\end{array}$
(Seq2)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(A_{2},s^{\prime})\end{array}$
(Seq3)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par1)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(A^{\prime}_{1},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(A^{\prime}_{1}||A_{2},s^{\prime})\end{array}$
(Par2)
$\begin{array}[]{c}\displaystyle(A_{2},s)\xrightarrow{a}(A^{\prime}_{2},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(A_{1}||A^{\prime}_{2},s^{\prime})\end{array}$
(Par3)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par4)
$\begin{array}[]{c}\displaystyle(A_{2},s)\xrightarrow{a}(empty,s),(a=throw\vee
a=exit)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{a}(empty,s)\end{array}$ (Par5)
${(empty||empty,s)\xrightarrow{\tau}(empty,s)}$ (While1)
$\begin{array}[]{c}\displaystyle cond(s)\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(while(cond,A),s)\xrightarrow{\tau}(A;(while(cond,A),s))\end{array}$
(While2) $\begin{array}[]{c}\displaystyle\neg cond(s)\\\
\rule{113.81102pt}{0.28453pt}\\\
\displaystyle(while(cond,A),s)\xrightarrow{\tau}(empty,s)\end{array}$ (Pick)
$(pick(\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A,t),s)\xrightarrow{pick(pl_{i},op_{i},v^{\prime}_{i},A_{i})}(A_{i},s^{\prime})$
$\textrm{where}\ {\it t\geq 1,v_{i}\in
Var,v^{\prime}_{i}\in\mathbb{Z},op_{i}\in OPS,pl_{i}\in PL,}\ \textrm{and}\
{\it s^{\prime}=(op_{i}(\sigma[v^{\prime}_{i}/v_{i}]),\rho)}.$ (CR)
$(createResource(EPR,val,t,A_{e{{}_{i}}}),s)\xrightarrow{createResource(EPR,val,t,A_{e{{}_{i}}})}(empty,s^{\prime})$
$\textrm{where}\ {\it t\geq 1,val\in\mathbb{Z}}\ \textrm{and}\
\it{s^{\prime}=(\sigma,\rho\cup\\{EPR,val,\emptyset,t,A_{e{{}_{i}}}\\})},\
\textrm{if}\ {\it EPR\notin\rho}.\ \textrm{Otherwise,
}{\it\rho^{\prime}=\rho}.$ (GetProp) $\begin{array}[]{c}\displaystyle{\it
s=(\sigma,\rho),EPR\in\rho}\\\ \rule{170.71652pt}{0.28453pt}\\\
\displaystyle(getProp(EPR,v),s)\xrightarrow{getProp(EPR,v^{\prime})}(empty,s^{\prime})\end{array}$
where ${\it v\in Var,v^{\prime}\in\mathbb{Z}}$ and ${\it
s^{\prime}=(\sigma[v^{\prime}/v],\rho)}$. (GetProp2)
$\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\notin\rho}\\\
\rule{142.26378pt}{0.28453pt}\\\
\displaystyle(getProp(EPR,v),s)\xrightarrow{throw}(empty,s)\end{array}$
(SetTime) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\in\rho}\\\
\rule{184.9429pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,t),s)\xrightarrow{setTimeout(EPR,t)}(empty,s^{\prime})\end{array}$
where ${\it t\geq 1}$,$\ {\it s^{\prime}=(\sigma,\rho[t/EPR]_{2})}$.
(SetTime2) $\begin{array}[]{c}\displaystyle{\it
s=(\sigma,\rho),EPR\notin\rho}\\\ \rule{142.26378pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,t),s)\xrightarrow{throw}(empty,s)\end{array}$
(SetTime3) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\in\rho}\\\
\rule{170.71652pt}{0.28453pt}\\\
\displaystyle(setTimeout(EPR,0),s)\xrightarrow{throw}(empty,s)\end{array}$
Table 4: Action and delay transition rules without notifications.
(Subs) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\in\rho}\\\
\rule{227.62204pt}{0.28453pt}\\\
\displaystyle(subscribe(O,EPR,cond^{\prime},A_{e{{}_{i}}}),s)\xrightarrow{subscribe(O,EPR,cond^{\prime},A_{e{{}_{i}}})}(empty,s^{\prime})\end{array}$
where ${\it
s^{\prime}=(\sigma,Add\\_subs(\rho,EPR,O,cond^{\prime},A_{e{{}_{i}}}))}$
(Subs2) $\begin{array}[]{c}\displaystyle{\it s=(\sigma,\rho),EPR\notin\rho}\\\
\rule{227.62204pt}{0.28453pt}\\\
\displaystyle(subscribe(O,EPR,cond^{\prime}),s)\xrightarrow{throw}(empty,s)\end{array}$
(Wait1D) $\begin{array}[]{c}\displaystyle t>1\\\
\rule{113.81102pt}{0.28453pt}\\\
\displaystyle(wait(t),s)\xrightarrow{}_{1}(wait(t-1),s^{+})\end{array}$
(Wait2D) ${(wait(1),s)\xrightarrow{}_{1}(empty,s^{+})}$ (ReceiveD)
${(receive(pl,op,v),s)\xrightarrow{}_{1}(receive(pl,op,v),s^{+})}$ (InvokeD)
${(invoke(pl,op,v_{1},v_{2}),s)\xrightarrow{}_{1}(invoke(pl,op,v_{1},v_{2}),s^{+})}$
(EmptyD) ${(empty,s)\xrightarrow{}_{1}(empty,s^{+})}$ (SequenceD)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{}_{1}(A^{\prime}_{1},s^{+})\\\
\rule{113.81102pt}{0.28453pt}\\\
\displaystyle(A_{1};A_{2},s)\xrightarrow{}_{1}(A_{1}^{\prime};A_{2},s^{+})\end{array}$
(ParallelD)
$\begin{array}[]{c}\displaystyle(A_{1},s)\xrightarrow{}_{1}(A^{\prime}_{1},s^{+})\wedge(A_{2},s)\xrightarrow{}_{1}(A^{\prime}_{2},s^{+})\\\
\rule{142.26378pt}{0.28453pt}\\\
\displaystyle(A_{1}||A_{2},s)\xrightarrow{}_{1}(A^{\prime}_{1}||A^{\prime}_{2},s^{+})\end{array}$
(Pick1D)
${(pick(\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A,1),s)\xrightarrow{}_{1}(A,s^{+})}$
(Pick2D) $\begin{array}[]{c}\displaystyle t>1\\\
\rule{284.52756pt}{0.28453pt}\\\
\displaystyle(pick(\\{(pl_{i},op_{i},v_{i},A_{i})\\}_{i=1}^{n},A,t),s)\xrightarrow{}_{1}(pick(\\{pl_{i},op_{i},v_{i},A_{i}\\}_{i=1}^{n},A,t-1),s^{+})\end{array}$
Table 5: Action and delay transition rules without notifications.
(Notif1)
$\begin{array}[]{c}\displaystyle(A,s)\xrightarrow{a}(A^{\prime},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{170.71652pt}{0.28453pt}\\\
\displaystyle(O:(A,\mathcal{A}_{e}),s)\xrightarrow{a}(O:(A^{\prime},\mathcal{A}_{e}||N(O,s^{\prime})),s^{\prime})\end{array}$
(Notif2)
$\begin{array}[]{c}\displaystyle(A_{e{{}_{i}}},s)\xrightarrow{a}(A^{\prime}_{e{{}_{i}}},s^{\prime}),a\neq
exit,a\neq throw\\\ \rule{170.71652pt}{0.28453pt}\\\
\displaystyle(O:(A,\mathcal{A}_{e}),s)\xrightarrow{a}(O:(A,\mathcal{A}^{\prime}_{e}||N(O,s^{\prime})),s^{\prime})\end{array}$
where $\mathcal{A}^{\prime}_{e}=\\{A^{\prime}_{e{{}_{i}}}\\},\
A^{\prime}_{e{{}_{i}}}=A^{\prime}_{e{{}_{j}}},j\neq i$. (Notif3)
$\begin{array}[]{c}\displaystyle(A,s)\xrightarrow{throw}(empty,s)\\\
\rule{170.71652pt}{0.28453pt}\\\
\displaystyle(O:(A,\mathcal{A}_{e}),s)\xrightarrow{throw}(O:(A_{f},\mathcal{A}_{e})),s)\end{array}$
(Notif4) $\begin{array}[]{c}\displaystyle(A,s)\xrightarrow{exit}(empty,s)\\\
\rule{170.71652pt}{0.28453pt}\\\
\displaystyle(O:(A,\mathcal{A}_{e}),s)\xrightarrow{exit}(O:(empty,empty)),s)\end{array}$
(NotifD)
$\begin{array}[]{c}\displaystyle(A,s)\xrightarrow{}_{1}(A^{\prime},s^{+}),(A_{e{{}_{i}}},s)\xrightarrow{}_{1}(A^{\prime}_{e{{}_{i}}},s^{+}),\forall
i\\\ \rule{170.71652pt}{0.28453pt}\\\
\displaystyle(O:(A,\mathcal{A}_{e}),s)\xrightarrow{}_{1}(O:(A^{\prime},\mathcal{A}^{\prime}_{e}||T(O,s)),s^{+})\end{array}$
Table 6: Action and delay transition rules with notifications.
Finally, the outermost semantic level corresponds to the choreographic level,
which is defined upon the two previously levels. In Table 7, we define the
transition rules related to the evolution of the choreography as a whole.
###### Definition 4 (Choreography operational semantics)
A choreography is defined as a set of orchestrators that run in parallel
exchanging messages: $C=\\{O_{i}\\}_{i=1}^{c}$, where $c$ is the number of
orchestrators presented in the choreography. A choreography state is then
defined as follows:
$S_{c}=\\{(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\\}_{i=1}^{c}$, where
$A_{i}$ is the activity being performed by $O_{i}$ at this state,
${A_{e}}^{i}$ are the event handling activities that are currently being
performed by $O_{i}$, and $s_{i}$ its current state.
(Chor1)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{exit}(O_{i}:(empty,empty),s_{i})\\\
\rule{256.0748pt}{0.28453pt}\\\
\displaystyle\\{(O_{j}:(A_{j},{\mathcal{A}_{e}}^{j}),s_{j})\\}_{j=1}^{c}\xrightarrow{exit}\\{(O_{j}:(empty,empty),s_{j})\\}_{j=1}^{c}\end{array}$
(Chor2)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{a}(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime}_{e}}^{i}),s^{\prime}_{i}),\
a\neq exit,\ a\neq receive,\ a\neq invoke,\\\ \hskip 139.41832pta\neq reply,\
a\neq\overline{reply},\ a\neq pick\\\ \rule{284.52756pt}{0.28453pt}\\\
\displaystyle\\{(O_{j}:(A_{j},{\mathcal{A}_{e}}^{j}),s_{j})\\}_{j=1}^{c}\xrightarrow{a}\\{(O_{j}:(A^{\prime}_{j},{\mathcal{A}^{\prime\prime}_{e}}^{j}),s^{\prime}_{j})\\}_{j=1}^{c}\end{array}$
such that $A^{\prime}_{j}=A_{j},\
{\mathcal{A}^{\prime\prime}_{e}}^{j}={\mathcal{A}_{e}}^{j}||N(O_{j},s^{\prime}_{j}),\forall
j\neq i,j\in\\{1,\ldots,c\\}$. (Chor3)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{}_{1}(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime}_{e}}^{i}),{s_{i}}^{+}),\
\forall i\in\\{1\ldots c\\},\textrm{and rules chor4,\ chor5,}\\\ \hskip
199.16928pt\textrm{chor6 are not applicable}\\\
\rule{284.52756pt}{0.28453pt}\\\
\displaystyle\\{(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\\}_{i=1}^{c}\xrightarrow{}_{1}\\{(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime\prime}_{e}}^{i}),{s_{i}}^{+})\\}_{i=1}^{c}\end{array}$
such that $A^{\prime}_{i}=A_{i},\
{\mathcal{A}^{\prime\prime}_{e}}^{i}={\mathcal{A}_{e}}^{i}||T(O_{i},{s_{i}}^{+})$.
(Chor4)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{invoke(pl,op,v_{1})}(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime}_{e}}^{i}),s_{i}),\
pl=(O_{i},O_{j}),\ s_{i}=(\sigma_{i},\rho_{i}),\\\
(O_{j}:(A_{j},{\mathcal{A}_{e}}^{j}),s_{j})\xrightarrow{receive(pl,op,\sigma_{i}(v_{1}))}(O_{j}:(A^{\prime}_{j},{\mathcal{A}^{\prime}_{e}}^{j}),s^{\prime}_{j})\\\
\rule{284.52756pt}{0.28453pt}\\\
\displaystyle\\{(O_{k}:(A_{k},{\mathcal{A}_{e}}^{k})),s_{k})\\}_{k=1}^{c}\xrightarrow{invoke(pl,op,v_{1})}\\{(O_{k}:(A^{\prime}_{k},{\mathcal{A}^{\prime\prime}_{e}}^{k}),s^{\prime}_{k})\\}_{k=1}^{c}\end{array}$
where $A^{\prime}_{k}=A_{k},\
{\mathcal{A}^{\prime\prime}_{e}}^{k}={\mathcal{A}_{e}}^{k}||N(O_{k},s^{\prime}_{k})$
if $k\neq i,k\neq j$. (Chor5)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{reply(pl,v)}(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime}_{e}}^{i}),s_{i}),\
pl=(O_{i},O_{j}),\ s_{i}=(\sigma_{i},\rho_{i}),\\\
(O_{j}:(A_{j},{\mathcal{A}_{e}}^{j}),s_{j})\xrightarrow{\overline{reply}(pl,\sigma_{i}(v))}(O_{j}:(A^{\prime}_{j},{\mathcal{A}^{\prime}_{e}}^{j}),s^{\prime}_{j})\\\
\rule{284.52756pt}{0.28453pt}\\\
\displaystyle\\{(O_{k}:(A_{k},{\mathcal{A}_{e}}^{k}),s_{k})\\}_{k=1}^{c}\xrightarrow{reply(pl,\sigma_{i}(v))}\\{(O_{k}:(A^{\prime}_{k},{\mathcal{A}^{\prime\prime}_{e}}^{k}),s^{\prime}_{k})\\}_{k=1}^{c}\end{array}$
where $A^{\prime}_{k}=A_{k},\
{\mathcal{A}^{\prime\prime}_{e}}^{k}={\mathcal{A}_{e}}^{k}||N(O_{k},s^{\prime}_{k})$
if $k\neq i,k\neq j$. (Chor6)
$\begin{array}[]{c}\displaystyle(O_{i}:(A_{i},{\mathcal{A}_{e}}^{i}),s_{i})\xrightarrow{invoke(pl,op,v_{1})}(O_{i}:(A^{\prime}_{i},{\mathcal{A}^{\prime}_{e}}^{i}),s_{i}),\
pl=(O_{i},O_{j}),\ s_{i}=(\sigma_{i},\rho_{i}),\\\
(O_{j}:(A_{j},{\mathcal{A}_{e}}^{j}),s_{j})\xrightarrow{pick(pl,op,\sigma_{i}(v_{1}),A)}(O_{j}:(A^{\prime}_{j},{\mathcal{A}^{\prime}_{e}}^{j}),s^{\prime}_{j})\\\
\rule{284.52756pt}{0.28453pt}\\\
\displaystyle\\{(O_{k}:(A_{k},{\mathcal{A}_{e}}^{k}),s_{k})\\}_{k=1}^{c}\xrightarrow{invoke(pl,op,v_{1})}\\{(O_{k}:(A^{\prime}_{k},{\mathcal{A}^{\prime\prime}_{e}}^{k}),s^{\prime}_{k})\\}_{k=1}^{c}\end{array}$
where $A^{\prime}_{k}=A_{k},\
{\mathcal{A}^{\prime\prime}_{e}}^{k}={\mathcal{A}_{e}}^{k}||N(O_{k},s^{\prime}_{k})$
if $k\neq i,k\neq j$.
Table 7: Choreography transition rules.
###### Definition 5 (Labeled transition system)
For a choreography $C$, we define the semantics of $C$ as the labeled
transition system obtained by the application of rules in Table 7, starting at
the state ${s_{0}}_{c}$:
${\it lts}(C)=(\mathcal{Q},{s_{0}}_{c},\rightarrow)$
where $\mathcal{Q}$ is the set of reachable choreography states, and
$\rightarrow\,\,=\,\,\rightarrow_{1}\,\cup\,\\{\stackrel{{\scriptstyle
a}}{{\longrightarrow}}\,|\,$ for all basic activity $a$, or $a=\tau\,\\}$.
###### Example 1
Let us consider the choreography ${\it C=(O_{1},O_{2})}$, where
${\it O_{i}=(PL_{i},Var_{i},A_{i},A_{f{{}_{i}}},\mathcal{A}_{e{{}_{i}}})}$,
i=1, 2,${\it Var_{1}=\\{v_{1},v_{3}\\}}$, ${\it Var_{2}=\\{v_{2},v_{4}\\}}$,
${\it A_{f{{}_{1}}}=exit}$, and ${\it A_{f{{}_{2}}}=exit}$. Suppose that ${\it
s_{0{{}_{1}}}}$ and ${\it s_{0{{}_{2}}}}$ are the initial states of $O_{1}$
and $O_{2}$, respectively, and all the variables are initially $0$. Then,
${\it A_{1}=assign(5,v_{1});}$ ${\it
receive(pl_{1},add,v_{3});reply(pl_{1},v_{3})}$ and ${\it
A_{2}=assign(1,v2);invoke(pl_{1},add,v_{2})}$. In Fig. 1 we show a piece of
the labeled transition system of $C$, where:
$\begin{array}[]{ll}A^{\prime}_{1}=&{\it receive}(pl_{1},add,v_{3});{\it
reply}(pl_{1},v_{3}).\par\\\ A^{\prime}_{2}=&{\it
invoke}(pl_{1},add,v_{2}).\\\
A^{\prime\prime}_{2}=&{\it\overline{reply}}(pl_{1},v_{4}).\par\\\
A^{\prime\prime}_{1}=&{\it reply}(pl_{1},v_{3}).\par\end{array}$
$\\{(O_{1}:(A_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A_{2},\emptyset),s_{0{{}_{2}}})\\}$$\\{(O_{1}:(A^{\prime}_{1},\emptyset),s^{\prime}_{1}),(O_{2}:(A_{2},\emptyset),s_{0{{}_{2}}})\\}$$\\{(O_{1}:(A^{\prime}_{1},\emptyset),s^{\prime}_{1}),(O_{2}:(A^{\prime}_{2},\emptyset),s^{\prime}_{2})\\}$$\\{(O_{1}:(A^{\prime}_{1},\emptyset),{s^{\prime}_{1}}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),{s^{\prime}_{2}})\\}$$\\{(O_{1}:(A^{\prime\prime}_{1},\emptyset),{s^{\prime\prime}_{1}}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),{s^{\prime}_{2}})\\}$$\\{(O_{1}:(empty,\emptyset),s^{\prime\prime}_{1}),(O_{2}:(empty,\emptyset),s^{\prime\prime}_{2})\\}$$\\{(O_{1}:(empty,\emptyset),s^{\prime\prime}_{1}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),s^{\prime}_{2})\\}$assign($5$,$v_{1}$)assign($1$,$v_{2}$)invoke(pl1,add,v2)receive(pl1,add,v3)reply(pl1,v3)$\overline{reply}$(pl1,v4)
Figure 1: A piece of ${\it lts}(C)$ without notifications.
## 5 Case study: Online auction service
The case study concerns a typical online auction process, which consists of
three participants: the online auction system and two buyers, A1 and A2. A
seller owes a good that wants to sell to the highest possible price.
Therefore, he introduces the product in an auction system for a certain time.
Then, buyers (or bidders) may place bids for the product and, when time runs
out, the highest bid wins. In our case, we suppose the resource is the product
for auction, the value of the resource property is the current price (only the
auction system can modify it), the resource subscribers will be the buyers,
their subscription conditions hold when the current product value is higher
than their bid, and the resource lifetime will be the time in which the
auction is active. Finally, when the lifetime has expired, the auction system
sends a notification to the buyers with the result of the process (the
identifier of the winner, $v_{w}$) and, after that, all the processes finish.
Let us consider the choreography ${\it C=(O_{sys},O_{1},O_{2})}$, where ${\it
O_{i}=(PL_{i},Var_{i},A_{i},A_{f{{}_{i}}},\mathcal{A}_{e{{}_{i}}})}$, i=1,2,
${\it Var_{sys}=\\{v_{w},v_{EPR},}$ ${\it end\\_bid\\}}$, ${\it
Var_{1}=\\{v_{1},v_{w{{}_{1}}}\\},\ Var_{2}=\\{v_{2},v_{w_{2}}\\},\
A_{f{{}_{1}}}=exit,}$ and ${\it A_{f{{}_{2}}}=exit}$. Variable $v_{EPR}$
serves to temporarily store the value of the resource property before sending;
$v_{1}$, $v_{2}$, $v_{w}$, $v_{w_{1}}$, $v_{w_{2}}$ are variables used for the
interaction among participants, and, finally, $end\\_bid$ is reset when the
auction lifetime expires. Suppose ${\it s_{0{{}_{sys}}},s_{0{{}_{1}}}}$ and
${\it s_{0{{}_{2}}}}$ are the initial states of $O_{sys}$, $O_{1}$ and
$O_{2}$, respectively, and all the variables are initially $0$:
${\it A_{sys}=assign(1,end\\_bid);createResource(EPR,25,48,A_{not});}$${\it
while(end\\_bid>0,A_{bid})}$. ${\it
A_{1}=subscribe(O_{1},EPR,EPR>=0,A_{cond_{1}});}$ ${\it
while(v_{w{{}_{1}}}==0,A_{pick_{1}})}$ ${\it
A_{2}=subscribe(O_{2},EPR,EPR>=0,A_{cond_{2}});}$${\it
while(v_{w{{}_{2}}}==0,A_{pick_{2}})}$, being: ${\it
A_{not}=assign(0,end\\_bid);(invoke(pl_{3},bid\\_finish_{1},v_{w})||}$${\it
invoke(pl_{4},bid\\_finish_{2},v_{w})})$ ${\it
A_{bid}=pick((pl_{1},cmp,v_{1},setProp(EPR,v_{EPR})),}$${\it(pl_{2},cmp,v_{2},setProp(EPR,v_{EPR})),}$
$\indent\hskip 14.22636pt{\it empty,48)}$ ${\it
A_{cond_{1}}=getProp(EPR,v_{EPR});invoke(pl_{1},{bid\\_up}_{1},v_{EPR})}$
${\it
A_{cond_{2}}=getProp(EPR,v_{EPR});invoke(pl_{2},{bid\\_up}_{2},v_{EPR})}$
${\it A_{pick_{1}}=pick((pl_{1},bid\\_up_{1},v_{1},{\it
invoke}(pl_{1},cmp,v_{1});subscribe(O_{1},EPR,EPR>=}$${\it v_{1},}$
${\it,A_{cond_{1}})),(pl_{3},bid\\_finish_{1},v_{1},empty),empty,48)}$ ${\it
A_{pick_{2}}=pick((pl_{2},bid\\_up_{2},v_{2},{\it
invoke}(pl_{2},cmp,v_{2});subscribe(O_{2},EPR,EPR>=}$${\it v_{2},}$
${\it,A_{cond_{2}})),(pl_{4},bid\\_finish_{2},v_{2},empty),empty,48)}$
In Fig. 2 we show a part of the labeled transition system of $C$, where:
$\begin{array}[]{ll}A^{\prime}_{sys}={\it while}(end\\_bid>0,A_{bid}).\\\
A^{\prime}_{1}={\it while}(v_{w{{}_{1}}}==0,A_{pick_{1}})\\\
A^{\prime}_{2}={\it while}(v_{w{{}_{2}}}==0,A_{pick_{2}})\\\
A^{\prime\prime}_{1}={\it
A_{pick_{1}};while}(v_{w{{}_{1}}}==0,A_{pick_{1}})\\\
A^{\prime\prime}_{2}={\it
A_{pick_{2}};while}(v_{w{{}_{2}}}==0,A_{pick_{2}})\par\\\
A^{\prime\prime}_{sys}={\it A_{bid};while}(end\\_bid>0,A_{bid}).\end{array}$
Let us note that the operations $bid\\_up_{1}$ and $bid\\_up_{2}$ are used to
increase the current bid by adding a random amount to the corresponding
variable $v_{i}$, the operations $bid\\_finish_{1}$, $bid\\_finish_{2}$ reset
the value of $v_{w}$ to finish both buyers. Finally, $cmp$ is an auction
system operation that receives as parameter a variable of the buyers, $v_{i}$,
and if the variable value is greater than the current value of $v_{EPR}$, then
$v_{EPR}$ is modified with this new value. After that, by means of the
activity $setProp(EPR,v_{EPR})$, we can update the value of the resource
property with the new bid.
Chor0Chor1Chor21Chor2Chor3Chor4Chor6Chor5Chor20Chor7Chor8Chor9Chor10Chor11Chor12Chor13Chor14assign(1,end_bid);createResource(EPR,25,48,Anot)subscribe(O1,EPR,EPR$>=$0,A${}_{cond_{1}}$)exitsubscribe(O2,EPR,EPR$>=$0,A${}_{cond_{2}}$)A${}_{cond_{1}};$A${}_{cond_{2}}$A${}_{pick_{1}}||$A${}_{pick_{2}}$Abidsubscribe(O1,EPR,EPR$>=$0,A${}_{cond_{1}}$)A${}_{cond_{1}}$A${}_{pick_{1}}$AbidA${}_{cond_{2}}$A${}_{pick_{2}}$AbidAnotA${}_{pick_{1}}||$A${}_{pick_{2}}$
Figure 2: A piece of ${\it lts}(C)$ for the online auction service.
$\begin{array}[]{ll}Chor_{0}=\\{(O_{sys}:(A_{sys},\emptyset),s_{0{{}_{sys}}}),(O_{1}:(A_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{1}=\\{(O_{sys}:(A^{\prime}_{sys},\emptyset),s^{\prime}_{0{{}_{sys}}}),(O_{1}:(A_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{2}=\\{(O_{sys}:(A^{\prime}_{sys},\emptyset),s^{\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{3}=\\{(O_{sys}:(A^{\prime}_{sys},A_{cond_{1}};A_{cond_{2}}),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{4}=\\{(O_{sys}:(A^{\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{5}=\\{(O_{sys}:(A^{\prime\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{6}=\\{(O_{sys}:(A^{\prime}_{sys},A_{cond_{1}}),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{7}=\\{(O_{sys}:(A^{\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{8}=\\{(O_{sys}:(A^{\prime\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{9}=\\{(O_{sys}:(A^{\prime}_{sys},A_{cond_{2}}),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{10}=\\{(O_{sys}:(A^{\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{11}=\\{(O_{sys}:(A^{\prime\prime}_{sys},\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{12}=\\{(O_{sys}:(A^{\prime}_{sys},A_{not}),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}&Chor_{13}=\\{(O_{sys}:(empty,\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(A^{\prime\prime}_{1},\emptyset),s_{0{{}_{1}}}),(O_{2}:(A^{\prime\prime}_{2},\emptyset),s_{0{{}_{2}}})\\}\\\
Chor_{14}=\\{(O_{sys}:(empty,\emptyset),s^{\prime\prime\prime}_{0{{}_{sys}}}),(O_{1}:(empty,\emptyset),s_{0{{}_{1}}}),(O_{2}:(empty,\emptyset),s_{0{{}_{2}}})\\}&\\\
\end{array}$
## 6 Conclusions and Future Work
We have presented in this paper a formal model for the description of
composite web services with resources associated, and orchestrated by a well-
know business process language (BPEL). The main contribution has therefore
been the integration of WSRF, a resource management language, with BPEL,
taking into account the main structural elements of BPEL, as its basic and
structured activities, notifications, event handling and fault handling.
Furthermore, special attention has been given to timed constraints, as WSRF
consider that resources can only exist for a certain time (lifetime). Thus,
resource leasing is considered in this work, which is a concept that has
become increasingly popular in the field of distributed systems. To deal with
notifications, event handling and fault handling, the operational semantics
has been defined at three levels, the outermost one corresponding to the
choreographic view of the composite web services.
As future work, we plan to extend the language with some additional elements
of BPEL, such as termination and compensation handling. Compensation is an
important topic in web services due to the possibility of faults. We are also
working on a semantics based on timed colored petri nets.
## Acknowledgement
Partially supported by the Spanish Government (co-financed by FEDER funds)
with the project TIN2009-14312-C02-02 and the JCCLM regional project
PEII09-0232-7745.
## References
* [1] T. Andrews et. al. BPEL4WS – Business Process Execution Language for Web Services, Version 1.1, 2003. http://www.ibm.com/developerworks/library/specification/ws-bpel/.
* [2] T. Banks, _Web Services Resource Framework (WSRF) - Primer_ , OASIS, 2006.
* [3] N. Busi, R. Gorrieri, C. Guidi, R. Lucchi and G. Zavattaro, Choreography and Orchestration: A Synergic Approach for System Design. In International Conference of Service Oriented Computing (ICSOC), Lecture Notes in Computer Science, vol. 3826, pp. 228-240, 2005.
* [4] K. Czajkowski, D. Ferguson, I. Foster, J. Frey, S. Graham, I. Sedukhin, D. Snelling, S. Tuecke and W. Vambenepe, _The WS-Resource Framework Version 1.0_ , http://www.globus.org/wsrf/specs/ws-wsrf.pdf, 2004.
* [5] N. Dragoni and M. Mazzara, A formal Semantics for the WS-BPEL Recovery Framework - The $pi$-Calculus Way. In International Workshop on Web Services and Formal Methods (WS-FM). Lecture Notes in Computer Science, vol. 6194, pp. 92-109, 2009.
* [6] M. Dumas, R. Heckel and N. Lohmann, A Feature-Complete Petri Net Semantics for WS-BPEL 2.0. In International Workshop on Web Services and Formal Methods (WS-FM). Lecture Notes in Computer Science, vol. 4937, pp. 77-91, 2008.
* [7] O. Ezenwoye, S.M. Sadjadi, A. Cary, and M. Robinson, Orchestrating WSRF-based GridServices. Technical Report FIU-SCIS-2007-04-01, 2007.
* [8] R. Farahbod, U. Glässer and M. Vajihollahi, A Formal Semantics for the Business Process Execution Language for Web Services. In Joint Workshop on Web Services and Model-Driven Enterprise Information Services (WSMDEIS), pp. 122-133, 2005.
* [9] I. Foster, J. Frey, S. Graham, S. Tuecke, K. Czajkowski, D. Ferguson, F. Leymann, M. Nally, T. Storey and S. Weerawaranna, _Modeling Stateful Resources with Web Services_ , Globus Alliance, 2004.
* [10] F. Leyman. Choreography for the Grid: towards fitting BPEL to the resource framework. Journal of Concurrency and Computation : Practice & Experience, vol. 18, issue 10, pp. 1201-1217, 2006.
* [11] N. Lohmann, E. Verbeek, C. Ouyang and C. Stahl. Comparing and Evaluating Petri Net Semantics for BPEL. Journal of Business Process Integration and Management, vol. 4, issue 1, pp. 60-73, 2009.
* [12] C. Ouyang, E. Verbeek, W.M.P. van der Aalst, S. Breutel, M. Dumas and A.H.M. ter Hofstede. Formal semantics and analysis of control flow in WS-BPEL. Science of Computing Programming, vol. 67, issue 2-3, pp. 162-198, 2007.
* [13] Z. Qiu, S. Wang, G. Pu and X. Zhao. Semantics of BPEL4WS-Like Fault and Compensation Handling. World Congress on Formal Methods (FM), pp. 350-365, 2005.
* [14] A. Slomiski. On using BPEL extensibility to implement OGSI and WSRF Grid workflows. Journal of Concurrency and Computation : Practice & Experience, vol. 18, pp. 1229-1241, 2006.
* [15] Web Services Choreography Description Language Version 1.0 (WS-CDL). http://www.w3.org/TR/ws-cdl-10/.
|
arxiv-papers
| 2012-03-08T11:53:19 |
2024-09-04T02:49:28.463985
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jose Antonio Mateo, Valent{\\i}n Valero, and Gregorio D{\\i}az",
"submitter": "Jose Antonio Mateo",
"url": "https://arxiv.org/abs/1203.1760"
}
|
1203.1984
|
# Physics with the ALICE experiment
Yuri Kharlov, for the ALICE collaboration Institute for High Energy Physics,
Protvino, 142281 Russia
###### Abstract
ALICE experiment at LHC collects data in pp collisions at $\sqrt{s}$=0.9, 2.76
and 7 TeV and in PbPb collisions at 2.76 TeV. Highlights of the detector
performance and an overview of experimental results measured with ALICE in pp
and AA collisions are presented in this paper. Physics with proton-proton
collisions is focused on hadron spectroscopy at low and moderate $p_{\rm t}$.
Measurements with lead-lead collisions are shown in comparison with those in
pp collisions, and the properties of hot quark matter are discussed.
## 1 Introduction
ALICE is a dedicated experiment built to exploit the unique physics potential
of heavy-ion interactions at LHC energies Aamodt:2008zz . Properties of
strongly interacting matter at extreme energy density are explored via a
comprehensive studies of hadron, muon, electron and photon production in the
collisions of heavy nuclei and their comparison with proton-proton collisions.
Presently, the ALICE collaboration consists of about 1600 members from 33
countries. Russian nuclear-physics community takes an active part in ALICE
since the very beginning, now counting 134 members from 12 institutes. Russian
institutes contribute in almost every major sub-detectors of the ALICE
experiment, and also take part in physics analysis of data collected in
2010–2011.
The ALICE experiment has collected a rich sample of data with proton-proton
and lead-lead collisions. In 2010 and beginning of 2011, about $10^{9}$ events
with the minimum bias trigger were recorded, corresponding to the integrated
luminosity $\int{\cal L}dT=16\leavevmode\nobreak\ \mbox{nb}^{-1}$. Rare-event
triggers on muons, jets and photons were dominant in data taking with the
proton beams at collision energy $\sqrt{s}=7$ TeV in the second half of 2011,
with the delivered integrated luminosity $\int{\cal
L}dT=4.9\leavevmode\nobreak\ \mbox{pb}^{-1}$. Limited data samples with the
proton beams at collision energies $\sqrt{s}=0.9$ and 2.76 TeV have been also
recorded with integrated luminosities $\int{\cal
L}dT=0.14\mbox{\leavevmode\nobreak\ and\leavevmode\nobreak\
}1.3\leavevmode\nobreak\ \mbox{nb}^{-1}$ respectively.
Among rare-event triggers used in data taking in 2011, one has to mention the
trigger on the MUON detector selecting events with muons in the high-rapidity
range to enrich statistics for $J/\psi$ and $\Upsilon$ signals (this trigger
was in operation since 2010). A trigger based on the electromagnetic
calorimeter (EMCAL) was selecting events with high-energy photons and jets in
the central barrel. Another ALICE calorimeter, a photon spectrometer PHOS, has
provided a trigger on photons with a moderate energy threshold, to enhance a
data sample for neutral meson and direct photon studies.
The first run with lead-lead beams at collision energy
$\sqrt{s_{{}_{NN}}}=2.76$ TeV was taken with ALICE in November 2010. The
delivered integrated luminosity was $\int{\cal L}dT=10\leavevmode\nobreak\
\mu\mbox{b}^{-1}$. The dominant trigger in 2010 was a minimum bias one. In
November 2011, the LHC has delivered 10 times more data, and the ALICE
experiment has restricted the minimum-bias trigger share in favor of several
rare-event triggers with the total life time 80%. Detector VZERO has deployed
triggers on the most central events with selected centralities $0-10\%$ and
semi-central events with centralities $20-60\%$. A trigger on ultra-peripheral
collisions was realized on SPD and TOF detectors. Other triggers implemented
earlier in pp collisions on EMCAL, PHOS and MUON detectors, were also active
in the PbPb run 2011.
## 2 Hadron production in proton-proton collisions
Measurements of identified hadron spectra are considered as an important test
of various non-perturbative models of hadron production at high energies, as
well as those of perturbative QCD calculations. ALICE performs extensive
studies of hadron production due to its powerful particle identification
capabilities Aamodt:2008zz . Charged particles are identified by several
tracking detectors covering complimentary kinematic ranges. Barrel tracking
detectors are embedded into a solenoidal magnet with magnetic field of 0.5 T.
This is a relatively soft magnetic field which allows to reconstruct charged
tracks at transverse momenta starting from $p_{\rm t}>50$ MeV/$c$. Inner
Tracking System (ITS) and Time Projection Chamber (TPC) can identify charged
particles in the full $2\pi$ azimuthal angle and pseudorapidity range
$|\eta|<0.9$, via measurements of their ionization loss $dE/dx$. Time-of-
flight measurements, provided by the TOF detector in the same solid angle as
ITS and TPC, can discriminate charged pions, kaons and protons in a higher
momentum range. The limited-acceptance High-Momentum Particle Identification
detector (HMPID) is a Cherenkov detector covering a solid angle
$\Delta\phi=60^{\circ}$ and $|\eta|<0.6$ is used to identify charged particles
at a higher momentum range, up to $p=5$ GeV/$c$. Transition Radiation Detector
(TRD) is another barrel detector surrounding TPC, which is designed to
identify electrons and at present covers about a half of the complete
azimuthal angle.
Photons and neutral mesons decaying into photons are detected and identified
by two electromagnetic calorimeters. A precise Photon Spectrometer (PHOS) is a
high-granularity calorimeter built of lead tungstate crystals (PbWO4). Its
small Moliére radius, high density and high light yield allow to detect
photons with the best possible energy resolution in the energy range up to
$E<100$ GeV in the azimuthal angle range $\Delta\phi=60^{\circ}$ and
$|\eta|<0.13$. Its high spatial resolution provides measurements of neutral
pions via invariant mass spectrum at transverse momenta $0.6<p_{\rm t}<50$
GeV/$c$. Another, wide-aperture Electromagnetic Calorimeter (EMCAL) is a
sampling-type calorimeters built of lead-scintillator modules. Its primary
goal is to trigger jets and measure a neutral component of jets. Dynamic range
of EMCAL covers energies up to 250 GeV, and granularity of this calorimeter
allows to reconstruct $\pi^{0}$ mesons at transverse momenta $1<p_{\rm t}<20$
GeV/$c$.
Muon identification is provided in ALICE by the muon arm which is installed in
the forward rapidity range $2.5<y<4$. This muon detector is a magnet
spectrometer consisting of a set of proportional chambers in the dipole
magnetic field. Hadronic background is suppressed by the hadron absorber
installed in front of the muon spectrometer.
Using charged hadron identification in ITS, TPC and TOF, ALICE has measured
production spectra $dN/dp_{\rm t}$ of identified charged hadron ($\pi^{\pm}$,
$K^{\pm}$, $p$, $\bar{p})$ in the minimum bias pp collisions at collision
energies $\sqrt{s}=0.9$ PIDhadron900GeV and $7$ TeV PIDhadron7TeV (Fig.1).
Figure 1: Transverse momentum spectra of $\pi^{-}$, $K^{-}$, $\bar{p}$ in pp
collisions at $\sqrt{s}=7$ TeV. The lines are the Levy-Tsallis fits.
The spectra were fitted with the Tsallis function Tsallis:1987eu
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!E\frac{{\rm
d}^{3}\sigma}{{\rm d}p^{3}}=\displaystyle\frac{\sigma_{pp}}{2\pi}\frac{{\rm
d}N}{{\rm
d}y}\frac{c\cdot(n-1)(n-2)}{nC\left[nC+m(n-2)\right]}\displaystyle\left(1+\frac{m_{\rm
t}-m}{nC}\right)^{-n},$ (1)
where the fit parameters are ${\rm d}N/{\rm d}y$, $C$ and $n$, $\sigma_{\rm
pp}$ is the proton-proton inelastic cross section, $m$ is the meson rest mass
and $m_{\rm t}=\sqrt{m^{2}+p_{\rm t}^{2}}$ is the transverse mass. The
integrated yield at $y=0$, defined by the Tsallis parameter ${\rm d}N/{\rm
d}y$, was evaluated from the ALICE data, and thus the total yields of charged
pions, kaons and protons was found. The ratios of integrated yields
$K^{\pm}/\pi^{\pm}$, $\bar{p}/\pi^{-}$ and $p/\pi^{+}$ in pp collisions at
$\sqrt{s}=0.9$ and $7$ TeV were compared with those measured at lower
collision energies, as shown in Fig.2.
Figure 2: Integrated yield ratio of $K/\pi$ (left) and $\bar{p}/\pi^{-}$
(right) as a function of collision energy.
A trend of slight increase of $K^{\pm}/\pi^{\pm}$ ratio with $\sqrt{s}$ can be
observed. ALICE data also suggest that baryon-antibaryon asymmetry, observed
at RHIC, vanishes at LHC energies, as expected.
Tsallis parameterization allows to find also the mean transverse momentum
$\langle p_{\rm t}\rangle$ and to observe its evolution with collision energy
(Fig.3).
Figure 3: Mean $p_{\rm t}$ for charged $\pi$, $K$ and $p$ at different
collision energy in pp collisions.
Comparison of mean $p_{\rm t}$ of different hadron species measured at
different collision energies indicates that hadron production spectra become
harder at higher $\sqrt{s}$, and also mean $p_{\rm t}$ grows with hadron mass.
ALICE has also measured production spectra of neutral pions and $\eta$ mesons
in pp collisions at $\sqrt{s}=0.9$, $2.76$ and 7 TeV, using the Photon
Spectrometer (PHOS) for real photon detection and central tracking system for
converted photon reconstruction pp-pi0 . Neutral meson reconstruction,
performed via invariant mass spectra of photon pairs, allowed to measure
differential cross section of $\pi^{0}$ and $\eta$ in a wide $p_{\rm t}$
range. In particular, the spectrum of $\pi^{0}$ production at the three
collision energies are shown of the left plot of Fig.4.
Figure 4: Production spectrum of $\pi^{0}$ in pp collisions at $\sqrt{s}=0.9$,
$2.76$ and $7$ TeV (left) and ratio of NLO pQCD calculations to the measured
spectra (right).
Hadron production at high $p_{\rm t}$ can be well calculated in the next-to-
leading orders of perturbative QCD (NLO pQCD). These calculations are based on
parton distribution (PDF) and fragmentation functions (FF) measured at lower
energies. Application of those PDF’s and FF’s to the new energy domain
delivered by LHC, lead to extrapolations of those functions to the kinematic
region where the functions have large uncertainties. The ratio of differential
cross sections of $\pi^{0}$ and $\eta$ mesons in pp collisions, calculated by
NLO pQCD, to the Tsallis fit of the ALICE measurements are shown by curves on
the right plot of Fig.4. Data points on this plot represent the ratio of the
measured cross section to the Tsallis fit to the measurement, which
demonstrates the quality of the data description by the Tsallis
parameterization. This comparison of theoretical calculations and experimental
measurements demonstrates that NLO pQCD at the QCD scale $\mu=p_{\rm t}$
describes well hadron production in pp collisions at $\sqrt{s}=0.9$ TeV, while
significantly overestimate it at $\sqrt{s}=7$ TeV. No common set of pQCD
parameters can be found to describe equally well the spectra of pion
production at all three collision energies.
Strangeness production is one of the most important observables for studying
the strongly interacting matter produced in heavy-ion collisions. That is why
measurements of complete set of strange hadrons in pp collisions is necessary
as a reference for comparison with heavy ion collisions. Besides charged kaons
mentioned earlier, ALICE has measured production spectra of many other strange
hadrons, as well as those of mesons with hidden strangeness ($K^{*}$,
$\Lambda$, $\Sigma$, $\Omega$, $\phi$ and strange resonance baryons).
Production yields of $(\Sigma^{*}+\bar{\Sigma^{*}}^{-})/2$ and $\phi$ mesons
in pp collisions at $\sqrt{s}=7$ TeV are shown in Fig.5 and are compared with
several MC predictions.
Figure 5: Production spectrum of $(\Sigma^{*+}+\bar{\Sigma^{*}}^{-})/2$ and
$\phi$ in pp collisions at $\sqrt{s}=7$ TeV with Monte Carlo predictions by
different models.
Identified hadron spectra measured at LHC energies, in conjunction with
spectra measured by previous experiments at lower collision energies, allow to
observe evolution of hadron production properties with $\sqrt{s}$. Predictions
of various phenomenological models, as well as NLO pQCD calculations were
found to be unable to describe all identified hadron spectra measured by ALICE
in pp collisions
## 3 Heavy ion collisions
Analysis of the first heavy-ion data collected in 2010 brought many results
giving an insight into the properties of strongly interacting matter at the
new energy density regime. Observables characterizing this matter are
classified into several groups which will be reviewed in this section.
### 3.1 Global event properties
As heavy nuclei are extended objects, centrality determination is an essential
point for all heavy-ion measurements. Centrality of the collision, directly
related to the impact parameter and to the number of nucleons $N_{\rm part}$
participating in the collision, allows to study particle production versus the
density of the colliding system. In the ALICE experiment, collision centrality
can be measured by several detectors. The best accuracy of centrality
measurement is achieved with the scintillator hodoscope VZERO covering
pseudorapidity ranges $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$. Distribution of the
sum of amplitudes in VZERO in minimum bias Pb-Pb collisions is shown in Fig.6
(left) bib:PbPb-dNdy . Centrality classes were defined by Glauber model, and
the fit of the Glauber model to the data is shown by a solid line in this
plot. Centrality resolution for all the estimators can be found in Fig.6
(right) bib:ToiaQM2011 which demonstrates that the best resolution is
achieved with the VZERO detector, and is equal to about 0.5% in the most
central events, and varies up to 1.5% in the most peripheral collisions.
Figure 6: Centrality determination in ALICE. Glauber model fit to the VZERO
amplitude with the inset of a zoom of the most peripheral region (left);
Centrality resolution with different detectors (right).
One of the key observable in heavy ion collision is the charged particle
multiplicity and its dependence on the collision centrality. The main detector
used for this measurements in the Silicon Pixel Detector (SPD), two innermost
layers of the barrel tracking system covering the pseudorapidity range
$|\eta|<1.4$. The charged particle density, normalized to the average number
of participants in a given centrality class, $dN_{\rm ch}/d\eta/\left(\langle
N_{\rm part}\rangle\right)$ was measured by ALICE in PbPb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV and compared with similar measurements at lower
energies at RHIC and SPS (Fig.7, left plot) bib:ToiaQM2011 .
Figure 7: Charged track density $dN/d\eta$ in pp and AA collisions vs
collision energy (left) and vs the number of participants (right).
In the most central events (centrality $0-5\%$) at LHC energy the charged
particle density was found to be $dN_{\rm ch}/d\eta=1601\pm 60$ bib:PbPb-dNdy
which is, being normalized to the number of participants, is 2.1 times larger
than the charged particle density measured at RHIC at $\sqrt{s_{{}_{NN}}}=200$
GeV and 1.9 times larger than that in pp collisions at $\sqrt{s}=2.36$ TeV.
The dependence of $dN_{\rm ch}/d\eta$ on the number of participants $N_{\rm
part}$, shown in the right plot of Fig.7, is very similar at LHC
($\sqrt{s_{{}_{NN}}}=2.76$ TeV) and RHIC ($\sqrt{s_{{}_{NN}}}=0.2$ TeV)
energies, provided the RHIC points are scaled by a factor 2.1 to match the LHC
points.
Longitudinal and transverse expansion of the highly compressed strongly-
interacting system created in heavy-ion collisions can be studied
experimentally via intensity interferometry, the Bose-Einstein enhancement of
identical bosons emitted close by in phase space, known as Hanbury Brown-Twiss
analysis (HBT). ALICE has measured the HBT radii and evaluate space-time
propertied on the system generated in Pb-Pb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV bib:HBT . The two-particle correlation function
of the difference $\vec{q}$ of two 3-momenta $\vec{p_{1}}$ and $\vec{p_{2}}$
was measured for like-sign charged pions which allowed to get the Gaussian HBT
radii, $R_{\rm out}$, $R_{\rm side}$ and $R_{\rm long}$. The product of these
3 radii and decoupling time extracted from $R_{\rm long}$, measured by ALICE
at LHC energy, together with this value measured at the AGS, SPS and RHIC, is
shown in Fig.8 (left) as a function of charged track density $dN_{\rm
ch}/d\eta$.
Figure 8: System size (left) and lifetime (right).
This measurements indicate that the homogeneity volume in central PbPb
collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV exceeds that measured at RHIC by a
factor of 2. The increase is present in both longitudinal and transverse
radii. The decoupling time for mid-rapidity pions exceeds 10 fm/c which is 40%
larger than at RHIC (Fig.8, right).
### 3.2 Collective expansion
In non-central collision of nuclei, the overlap region, and hence the initial
matter distribution is anisotropic. During evolution of the matter, the
spatial asymmetry of initial state is converted to an anisotropic momentum
distribution. The azimuthal distribution of the particle yield can be
expressed in terms of the angle between the particle direction $\varphi$ and
the reaction place $\Psi_{\rm RP}$:
$\displaystyle\frac{dN}{d(\varphi-\Psi_{\rm RP})}$ $\displaystyle\propto$
$\displaystyle 1+2\sum_{n=1}v_{n}\cos\left[n(\varphi-\Psi_{\rm RP})\right],$
(3) $\displaystyle v_{2}=\langle\cos\left[n(\varphi-\Psi_{\rm
RP})\right]\rangle.$
The second coefficient of this Fourier series, $v_{2}$, is referred to as
elliptic flow. Theoretical models, based on relativistic hydrodynamics
bib:hydro-v2_Kestin ; bib:hydro-v2_Niemi , successfully described the elliptic
flow observed at RHIC bib:RHIC_v2 and predict its increase at LHC energies
from 10% to 30%.
The first measurements of elliptic flow of charged particles in Pb-Pb
collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV were reported by ALICE in
bib:ALICE-v2 . Charged tracks were detected and reconstructed in the central
barrel tracking system, consisting of ITS and TPC. Elliptic flow integrated
over $p_{\rm t}$ range $0.2<p_{\rm t}<5$ GeV/$c$, for the 2- and 4-particle
cumulant methods, is shown in Fig.9 (left) as a function of centrality.
Figure 9: Azimuthal flow $v_{2}$ of charged particles in Pb-Pb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV vs centrality (left) and $v_{2}$ vs collision
energy (right).
It shows that the integrated elliptic flow increases from central to
peripheral collision and reaches the maximum value $v_{2}\approx 0.1$ in semi-
central collisions in the $40-60\%$ centrality class. Comparison of the
integrated elliptic flow of charged particles, measured at different center-
mass collision energies, shows a smooth increase of $v_{2}$ with
$\sqrt{s_{{}_{NN}}}$, and confirms model expectations that the value of
$v_{2}$ in Pb-Pb collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV increases by
about 30% with respect to $v_{2}$ in Au-Au collisions at
$\sqrt{s_{{}_{NN}}}=0.2$ TeV.
Particle momentum anisotropy is also studied via two-particle correlations
which measure the distributions of azimuthal angles $\Delta\varphi$ and
pseudorapidities $\Delta\eta$ between a ‘‘trigger’’ particle at transverse
momentum $p_{\rm t}^{t}$ and an ‘‘associated’’ particle at $p_{\rm t}^{a}$.
The correlation function $C(\Delta\varphi,\Delta\eta)$ looks differently in
different kinematic regions. At $p_{\rm t}^{t}<3-4$ GeV/$c$, the shape of the
correlation function reveals the ‘‘bulk-dominated’’ regime, where hydrodynamic
modeling has been demonstrated to give a good description of the data from
heavy-ion collisions (see Fig.10, left).
Figure 10: Di-hadron correlations $C(\Delta\varphi,\Delta\eta)$ in central Pb-
Pb collisions in the ‘‘bult-dominated’’ regime (left) and in the ‘‘jet-
dominated’’ regime (right).
At high transverse momenta of both particles, jets become dominating, and the
shape of the correlation function in central Pb-Pb collisions has just a clear
near-side peak centered at $\Delta\varphi=\Delta\eta=0$ and no evident out-
side peak, as shown in Fig.10, right. Harmonic decomposition of two-particle
correlations bib:ALICE-harmonic performed by ALICE, has shown that in the
‘‘bulk-dominated’’ regime a distinct near-side ridge and a doubly-peaked away-
side structure are observed in the most central events, which reflects a
collective response to anisotropic initial conditions.
The results of global event properties and collective expantion studied by
ALICE, indicate that the fireball formed in nuclear collisions at the LHC is
hotter, lives longer, and expands to a larger size at freeze-out as compared
to lower energies.
### 3.3 Strangeness production
Strange particle production has been considered as a probe of strongly
interacting matter by heavy-ion experiments at AGS, SPS and RHIC. We have
already demonstrated that ALICE, due to its powerful particle identification
technique, has measured strange particle spectra in pp collisions. Similar
analysis was performed on the Pb-Pb data collected in 2010. Comparison of
strange meson and baryon production is illustrated by the $\Lambda/K^{0}_{S}$
ratio measured by ALICE in different centrality classes (Fig.11, left). This
ratio in peripheral Pb-Pb collision is similar to that one measured in pp
collisions, but it grows with centrality, increasing the value of 1.5 in the
most central collisions. The qualitative behaviour of this ratio on $p_{\rm
t}$ at the LHC collision energy is similar to the ratio measured at RHIC by
the STAR experiment (Fig.11, right). An enhancement of strange and multi-
strange baryons ($\Omega^{-}$, $\bar{\Omega}^{+}$,
$\Sigma^{-}$,$\bar{\Sigma}^{+}$ ) was obsevred in heavy-ion collisions by
experiments at lower energies, and was confirmed by ALICE at LHC energy ALICE-
Hippolyte . It was also shown that multi-strange baryon enhancement scales
with the number of participants $N_{\rm part}$ and decreases with the
collision energy.
Figure 11: Ratio $\Lambda/K^{0}_{S}$ in Pb-Pb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV in different centralities (left) and comparison
of this ratio at LHC and RHIC in centralities $0-5\%$ and $60-80\%$ (right).
### 3.4 Parton energy loss in medium
Experiments at RHIC reported that hadron production at high transverse
momentum in central Au-Au collisions at a center-of-mass energy per nucleon
pair $\sqrt{s_{{}_{NN}}}=200$ GeV is suppressed by a factor $4-5$ compared to
expectations from an independent superposition of nucleon-nucleon collisions.
This suppression is attributed to energy loss of hard partons as they
propagate through the hot and dence QCD medium. Therefore, a spectrum
suppression of hadron production can be used as a measure of the properties of
the strongly interacting matter.
The strength of suppression of a hadron $h$ is expressed by the nuclear
modification factor $R_{AA}$, defined as a ratio of the particle spectrum in
heavy-ion collision to that in pp, scaled by the number of binary nucleon-
nucleron collisions $N_{\rm coll}$:
$R_{AA}(p_{\rm t})=\frac{(1/N_{AA})d^{2}N_{h}^{AA}/dp_{\rm t}d\eta}{N_{\rm
coll}(1/N_{pp})d^{2}N_{h}^{pp}/dp_{\rm t}d\eta}.$ (4)
At the larger LHC energy, the density of the medium is expected to be higher
than at RHIC, leading to a larger energy loss of high-$p_{\rm t}$ partons.
However, the hadron production spectra are less steeply falling with $p_{\rm
t}$ at LHC than at RHIC which would reduce the value of $R_{AA}$ for a given
value of the parton energy loss.
ALICE has measured the nuclear modification factor $R_{AA}$ for many
particles. All charged particles, detected in the ALICE central tracking
system (ITS and TPC), show a spectrum suppression Otwinowski:2011gq which is
qualitatively similar to that observed at RHIC (Fig.12). However, quantitative
comparison with RHIC demonstrates that the suppression at LHC energy is
stronger which can be interpreted by a denser medium.
Figure 12: Nuclear modification factor $R_{AA}$ of charged particles.
Benefiting from particle identification which has been already mention earlier
in this paper, ALICE has measured suppression of various identified hadrons,
which provides experimental data for studying the flavor and mass dependence
of the spectra suppression.
A nuclear modification factor $R_{AA}$ of charged pion production in mid-
rapidity (Fig.13) has lower values in the range of moderate transverse momenta
($3<p_{\rm t}<7-10$ GeV/$c$) than that of unidentified charged particles, but
at higher $p_{\rm t}$ it coincides with all charged particles.
Figure 13: Nuclear modification factor $R_{AA}$ of charged pions.
To the contrary to charged pions, strange hadrons ($K^{0}_{S}$, $\Lambda$) are
less suppressed in the most central collisions compared to all charged
particles (Fig.14). This is explained by the fact that strange quark
production is enhanced in a hot nuclear medium, and this strangeness
enhancement partially compensates energy loss of strange quarks, such that the
overall $R_{AA}$ value becomes larger than for pions. Lambda hyperons have no
suppression at $p_{\rm t}<3-4$ GeV/$c$, which is interpreted by an additional
baryon enhancement in central heavy-ion collisions.
ALICE has reported also the first measurements of $D$ meson suppression
bib:PbPb-Dmesons in Pb-Pb collisions in two centrality classes, $0-20\%$ and
$40-80\%$, shown in Fig.14. It was shown that the $R_{AA}$ values for $D^{0}$,
$D^{+}$ and $D^{*+}$ are consistent with each other within the statistical and
systematical uncertainties. Although the statistics of the ALICE run 2010 is
marginal for $D$ meson measurement, the obtained result shows a hint that the
$D$ mesons are less suppressed than charged pions.
Figure 14: Nuclear modification factor $R_{AA}$ of charged particles, $K^{0}$,
$\Lambda$, $\pi^{\pm}$, $D^{+}$, $D^{0}$, $D^{*+}$ in central (left) and
peripheral (right) collisions.
## 4 Conclusion
The ALICE collaboration is running an extensive research program with proton-
proton collisions. The domain where ALICE is competitive with other LHC
experiments, covers event characterization and identified particle spectra at
low and medium transverse momenta. Practically all measured spectra in pp
collisions at $\sqrt{s}=7$ TeV show statistically significant deviations from
models which well described lower-energy results. Therefore new experimental
results from pp collision allow to tune various phenomenological models and
pQCD calculations.
A plenty of experimental results produced by the ALICE collaboration from the
first Pb-Pb data gives the first insight on strongly interacting nuclear
matter at the highest achievable collision energy. It is evident that the
quark-gluon matter produced in heavy ion collision at LHC qualitatively has
properties similar to what was observed at RHIC. The matter produced at LHC
has about 3 times larger energy density, twice larger volume of homogeneity
and about 20% larger lifetime. Like at RHIC, the matter at LHC reveals the
properties on an almost perfect liquid. Particle suppression appeared to be
stronger at LHC than at RHIC which is also an evidence of denser medium
produced at LHC. At the end of 2011, LHC has delivered 10 times more data with
Pb-Pb collision at $\sqrt{s_{{}_{NN}}}=2.76$ TeV, which will bring more
precise results.
## References
* (1) K. Aamodt et al. [ALICE Collaboration], JINST 3, S08002 (2008).
* (2) K. Aamodt et al. [ALICE Collaboration], Eur.Phys.J.C 71(6), 1655, 2011.
* (3) R. Preghenella, for the ALICE Collaboration. arXiv:1111.7080v1 [hep-ex].
* (4) C. Tsallis, J. Statist. Phys. 52, 479-487 (1988).
* (5) ALICE collaboration, CERN-PH-EP-2012-001 (2012).
* (6) K.Aamodt et al., ALICE collaboration. Phys.Rev.Lett., 106, 032301 (2011).
* (7) A.Toia for the ALICE collaboration. J. Phys. G: Nucl. Part. Phys. 38 (2011) 124007.
* (8) K.Aamodt et al., ALICE collaboration. Physics Letters B 696 (2011) 328337.
* (9) G. Kestin and U.W. Heinz, Eur. Phys. J. C 61, 545 (2009).
* (10) K.Aamodt et al., ALICE collaboration. Phys.Rev.Lett., 105, 252302 (2010).
* (11) G. Kestin and U.W. Heinz, Eur. Phys. J. C 61, 545 (2009).
* (12) H. Niemi, K. J. Eskola, and P.V. Ruuskanen, Phys. Rev. C 79, 024903 (2009).
* (13) K.Aamodt et al., ALICE collaboration. arXiv:1109.2501
* (14) B.Hippolyte for the ALICE collaboration. arXiv:1112.5803 [nucl-ex].
* (15) J. Otwinowski [ALICE Collaboration], J. Phys. G G 38 (2011) 124112 [arXiv:1110.2985 [hep-ex]].
* (16) A.Grelli for the ALICE collaboration. J. Phys. Conf. Ser. 316 (2011) 012025.
|
arxiv-papers
| 2012-03-09T04:12:33 |
2024-09-04T02:49:28.480458
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuri Kharlov (for the ALICE collaboration)",
"submitter": "Yuri Kharlov",
"url": "https://arxiv.org/abs/1203.1984"
}
|
1203.2016
|
# Gödel-type universes in $f(T)$ gravity
Di Liu, Puxun Wu and Hongwei Yu Center of Nonlinear Science and Department of
Physics, Ningbo University, Ningbo, Zhejiang, 315211 China
###### Abstract
The issue of causality in $f(T)$ gravity is investigated by examining the
possibility of existence of the closed timelike curves in the Gödel-type
metric. By assuming a perfect fluid as the matter source, we find that the
fluid must have an equation of state parameter greater than minus one in order
to allow the Gödel solutions to exist, and furthermore the critical radius
$r_{c}$, beyond which the causality is broken down, is finite and it depends
on both matter and gravity. Remarkably, for certain $f(T)$ models, the perfect
fluid that allows the Gödel-type solutions can even be normal matter, such as
pressureless matter or radiation. However, if the matter source is a special
scalar field rather than a perfect fluid, then $r_{c}\rightarrow\infty$ and
the causality violation is thus avoided.
###### pacs:
04.50.Kd, 04.20.Jb, 98.80.Jk
## I Introduction
General relativity (GR) is established in the framework of the Levi-Civita
connection, therefore there is only curvature rather than torsion in the
spacetime. On the other hand, one can also introduce other connections, such
as the Weitzenböck connection, into the same spacetime where only torsion is
reserved. Thus, there is no such a thing as curvature or torsion of spacetime,
but only curvature or torsion of connection. Basing on the Weitzenböck
connection, Einstein Einstein introduced firstly the Teleparallel Gravity
(TG) in his endeavor to unify gravity and electromagnetism with the
introduction of a tetrad field. TG can, as is well known, show up as a theory
completely equivalent to GR since the difference between their actions (the
actions of TG and GR are the torsion scalar $T$ and Ricci scalar $R$,
respectively) is just a derivative term FNGtnb ; FNGtn1 ; FNGtn2 ; FNGtn3 ;
FNGtn4 ; FNGtne .
Recently, a modification of TG, called $f(T)$ theory Bengochea2009 ;
Ferraro2007 ; Linder ; Zheng2011 ; Ferraro2008 ; Ferraro2011 ; pwhy2011 ;
Wu2011 ; Bamba2011 ; Wu2010a ; Ben2011ab ; Wu2010b ; Zhang2011bb ; FTbe ; FT1
; FT2 ; FT3 ; FT4 ; FT5 ; FT6 ; FT7 ; FT8 ; FT9 ; FT10 ; FT11 ; FT12 ; FT13 ;
FT14 ; FT15 ; FT16 ; FT17 ; FT18 ; FT19 ; FT20 ; FT21 ; FT22 ; FT23 ; FT24 ;
FT25 ; FT26 ; FT27 ; FT28 ; FT29 ; FT30 ; FT31 ; FT32 ; FT33 ; FT34 ; FT35 ;
FT36 ; FT37 ; FT38 ; FTed ; Ferraro2011a ; Li2011aa ; Li2011bb ; Li2011b ;
LiM2011 ; Miao2011 , has spurred an increasing deal of attention, as it can
explain the present accelerated cosmic expansion discovered from observations
(the Type Ia supernova R98 ; P99 , the cosmic microwave background radiation
Spa ; Spb , and the large scale structure T2004 ; E2005 , etc.) without the
need of dark energy. $f(T)$ theory is obtained by generalizing the action $T$
of TG to an arbitrary function $f$ of $T$, which is very analogous to $f(R)$
theory (see FeNoj08 ; FeSot10 ; Felice2010 ; FeNoj11 ; FebCli for recent
review) where the action $R$ of GR is generalized to be $f(R)$. An advantage
of $f(T)$ theory is that its field equation is only second order, while in
$f(R)$ gravity it is forth order.
It has been found that $f(T)$ theory can give an inflation without an inflaton
Ferraro2007 ; Ferraro2008 , avoid the big bang singularity problem in the
standard cosmological model Ferraro2011 , realize the crossing of phantom
divide line for the effective equation of state Wu2011 ; Bamba2011 , and yield
an usual early cosmic evolution Wu2010b ; Zhang2011bb . But, at the same, this
theory lacks the local Lorentz invariance Li2011aa ; Li2011bb , and this
results in the appearance of extra degrees of freedom LiM2011 , the broken
down of the first law of black hole thermodynamic Miao2011 , and the problem
in cosmic large scale structure Li2011b .
In this paper, we plan to study the causality issue of $f(T)$ theory by
examining the possibility of existence of the closed timelike curves in the
Gödel spacetime Godel . The Gödel metric is the first cosmological solution
with rotating matter to the Einstein equation in GR. Since the Gödel solution
is very convenient for studying whether the closed timelike curves exist, it
has been used widely to test the causality issue. For example, Gödel found
that the closed timelike solution cannot be excluded in GR, assuming a
cosmological constant or a perfect fluid with its pressure equal to the energy
density. Gödel’s work has been generalized to include other matter sources,
such as, the vector field Sombe ; SomRe79 ; SomRay80 , scalar field Hiscockbe
; HisCha ; HisPan , spinor field Villalbabe ; VilPim ; VilKre ; VilLea ;
VilHered ; Reboucas1983 and tachyon field Reboucas . In addition, the Gödel-
type universes mjrteibe ; mjrtei1 ; mjrteied ; Reboucas1983 have also been
studied in the framework of other theories of gravitation, such as TG TGGod ,
$f(R)$ gravity RebClif05e ; Reb09b1 ; Reb10ed and string-inspired
gravitational theory stri ; barrow1998 .
Here, assuming that the matter source is the perfect fluid or a scalar field,
we aim to find out the condition for non-violation of causality in $f(T)$
gravity. The paper is organized as follows. We give, in Sec. II, a brief
review of $f(T)$ theory and the vierbein of a general cylindrical symmetry
metric in Sec.III. The Gödel-type universe in $f(T)$ theory is discussed in
Sec. IV. With an assumption of different matter sources, we investigate the
issue of causality in Sec. V. Finally, we present our conclusions in Sec. VI.
## II $f(T)$ gravity
In this section, we give a brief view of $f(T)$ gravity. We use the Greek
alphabet ($\mu$, $\nu$, $\cdots$= 0, 1, 2, 3) to denote tensor indices, that
is, indices related to spacetime, and middle part of the Latin alphabet ($i$,
$j$, $\cdots$= 0, 1, 2, 3) to denote tangent space (local Lorentzian) indices.
TG, instead of using the metric tensor, uses tetrad, $e_{\mu}^{i}$ or
$e_{i}^{\mu}$ (frame or coframe), as the dynamical object. The relation
between frame and coframe is
$e_{i}^{\mu}e_{\mu}^{j}=\delta_{i}^{j}\;,\qquad
e_{i}^{\mu}e_{\nu}^{i}=\delta_{\nu}^{\mu},$ (1)
and the relation between tetrad and metric tensor is
$g_{\mu\nu}=e_{\mu}^{i}e_{\nu}^{j}\eta_{ij}\;,\qquad\eta_{ij}=e_{i}^{\mu}e_{j}^{\nu}g_{\mu\nu}\;,$
(2)
where $\eta_{ij}=diag(1,-1,-1,-1)$ is the Minkowski metric.
Different from GR, the Weitzenböck connection is used in TG
${\Gamma}_{\mu\nu}^{\lambda}=e_{i}^{\lambda}\partial_{\nu}e_{\mu}^{i}=-e_{\mu}^{i}\partial_{\nu}e_{i}^{\lambda}\;.$
(3)
As a result, the convariant derivative, denoted by $D_{\mu}$, satisfies:
$D_{\mu}e_{\nu}^{i}=\partial_{\mu}e_{\nu}^{i}-\Gamma_{\nu\mu}^{\lambda}e_{\lambda}^{i}=0\;.$
(4)
To describe the difference between Weitzenböck and Levi-Civita connections, a
contorsion tensor $K_{\;\;\mu\nu}^{\rho}$ needs to be introduced:
$K_{\;\;\mu\nu}^{\rho}\equiv\Gamma_{\mu\nu}^{\rho}-\overset{\circ}{\Gamma}{}_{\mu\nu}^{\rho}=\frac{1}{2}(T_{\mu}{}^{\rho}{}_{\nu}+T_{\nu}{}^{\rho}{}_{\mu}-T_{\;\;\mu\nu}^{\rho})\;.$
(5)
Here $T_{\;\;\mu\nu}^{\rho}$ is the torsion tensor
$T_{\;\;\mu\nu}^{\rho}={\Gamma}_{\nu\mu}^{\rho}-{\Gamma}_{\mu\nu}^{\rho}=e_{i}^{\rho}(\partial_{\mu}e_{\nu}^{i}-\partial_{\nu}e_{\mu}^{i})\;,$
(6)
and $\overset{\circ}{\Gamma}{}_{\mu\nu}^{\rho}$ denotes the Levi-Civita
connection
$\overset{\circ}{\Gamma}{}_{\mu\nu}^{\rho}=\frac{1}{2}g^{\rho\sigma}(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}).$
(7)
By defining the super-potential $S_{\sigma}^{\;\;\mu\nu}$
$S_{\sigma}^{\;\;\mu\nu}\equiv
K_{\;\;\;\;\sigma}^{\mu\nu}+\delta_{\sigma}^{\mu}T_{\;\;\;\;\;\alpha}^{\alpha\nu}-\delta_{\sigma}^{\nu}T_{\;\;\;\;\;\alpha}^{\alpha\mu}\;,$
(8)
we obtain the torsion scalar $T$
$T\equiv\frac{1}{2}S_{\sigma}^{\;\;\mu\nu}T_{\;\;\mu\nu}^{\sigma}=\frac{1}{4}T^{\alpha\mu\nu}T_{\alpha\mu\nu}+\frac{1}{2}T^{\alpha\mu\nu}T_{\nu\mu\alpha}-T_{\alpha\mu}^{\;\;\;\;\alpha}T_{\;\;\;\;\;\nu}^{\nu\mu}\;.$
(9)
In TG, the Lagrangian density is given by:
$L_{T}=\frac{eT}{2\kappa^{2}}\;,$ (10)
where, $e=\det(e_{\mu}^{i})=\sqrt{-g}\;,\kappa^{2}{\equiv}8\pi G$.
Generalizing $T$ to be an arbitrary function $f$ of $T$ in the above
expression, we obtain the Lagrangian density of $f(T)$ theory
$L_{T}=\frac{ef(T)}{2\kappa^{2}}\;.$ (11)
Adding a matter Lagrangian density $L_{M}$ to Eq. (11), and varying the action
with respect to the vierbein, one finds the following field equation of $f(T)$
theory:
$\displaystyle[e^{-1}\partial_{\mu}(ee^{\rho}_{i}S^{\;\;\nu\mu}_{\rho})-e^{\lambda}_{i}S^{\rho\mu\nu}T_{\rho\mu\lambda}]f_{T}(T)+e^{\rho}_{i}S^{\;\;\nu\mu}_{\rho}\partial_{\mu}(T)f_{TT}(T)$
(12)
$\displaystyle+\frac{1}{2}e^{\nu}_{i}f(T)=\kappa^{2}e^{\rho}_{i}\overset{em}{T}{}^{\;\;\nu}_{\rho}.$
Here $f_{T}=df(T)/dT$, $f_{TT}=d^{2}f(T)/dT^{2}$, and
$\overset{em}{T}{}^{\nu}_{\rho}$ is the matter energy-momentum tensor. In a
coordinate system, this field equation can be rewritten as
$\displaystyle
A_{\mu\nu}f_{T}(T)+S^{\;\;\;\;\;\sigma}_{\nu\mu}(\nabla_{\sigma}T)f_{TT}(T)+\frac{1}{2}g_{\mu\nu}f(T)=\kappa^{2}\overset{em}{T}{}_{\mu\nu}\;,$
(13)
where
$\displaystyle
A_{\mu\nu}=g_{\sigma\mu}e^{i}_{\nu}[e^{-1}\partial_{\xi}(ee^{\rho}_{i}S^{\;\;\sigma\xi}_{\rho})-e^{\lambda}_{i}S^{\rho\xi\sigma}T_{\rho\xi\lambda}]$
(14)
$\displaystyle\qquad=G_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T=-\nabla^{\sigma}S_{\nu\sigma\mu}-S_{\;\;\;\;\mu}^{\rho\lambda}K_{\lambda\rho\nu}\;,$
$G_{\mu\nu}$ is the Einstein tensor, and $\nabla_{\sigma}$ is the covariant
derivative associated with the Levi-Civita connection. The trace of Eq. (12)
or (13), which can be used to simplify and constrain the field equation, can
be expressed as
$\displaystyle-[2e^{-1}\partial_{\sigma}(eT^{\;\;\rho\sigma}_{\rho})+T]f_{T}(T)+S^{\;\;\rho\sigma}_{\rho}(\partial_{\sigma}T)f_{TT}(T)+2f(T)=\kappa^{2}\overset{em}{T}\;,$
(15)
where,
$\overset{em}{T}=\overset{em}{T}{}^{\mu}_{\;\;\mu}=g^{\mu\nu}\overset{em}{T}{}_{\mu\nu}$
is the trace of the energy-momentum tensor. Clearly, in the case of TG,
$f(T)=T$, and Eq. (15) reduces to
$T-2e^{-1}\partial_{\sigma}(eT_{\rho}^{\;\;\rho\sigma})=\kappa^{2}\overset{em}{T}\;,$
(16)
which shows an equivalence between GR and TG since
$-R=T-2e^{-1}\partial_{\sigma}(eT_{\rho}^{\;\;\rho\sigma})\;.$ (17)
## III vierbein for cylindrical symmetry metric
Since the Gödel-type metric is usually expressed in cylindrical coordinates
$[(r,\phi,z)]$, we consider a general cylindrical symmetry metric
$\displaystyle ds^{2}=dt^{2}+2H(r)dtd\phi-dr^{2}-G(r)d\phi^{2}-dz^{2}\;,$ (18)
where $H$ and $G$ are the arbitrary functions of $r$. This metric can be re-
expressed in the following form
$\displaystyle ds^{2}=[dt+H(r)d\phi]^{2}-D^{2}(r)d\phi^{2}-dr^{2}-dz^{2}\;,$
(19)
where
$\displaystyle D(r)=\sqrt{G(r)+H^{2}(r)}\;.$ (20)
Since the local Lorentz invariance is violated in $f(T)$ theory and the
vierbein have six degrees of freedom more than the metric, one should be
careful in choosing a physically reasonable tetrad in terms of Eq.(2). Here,
we choose the tetrad anstaz of the cylindrical symmetry metric to be:
$\displaystyle e^{i}_{\mu}\equiv\left(\begin{array}[]{cccc}1&0&H&0\\\
0&1&0&0\\\ 0&0&D&0\\\
0&0&0&1\end{array}\right)\;,\;\;\;e^{\mu}_{i}\equiv\left(\begin{array}[]{cccc}1&0&-\frac{H}{D}&0\\\
0&1&0&0\\\ 0&0&\frac{1}{D}&0\\\ 0&0&0&1\\\ \end{array}\right)\;.$ (29)
Using Eqs. (3–9), one can find that the Weitzenböck invariant $T$ is
$\displaystyle T=\frac{1}{2}\left(\frac{H^{\prime}}{D}\right)^{2}\;,$ (30)
where a prime presents a derivative with respect to $r$.
Substituting the vierbein given in Eq. (29) into Eq. (13), we obtain the
following non-zero components of the $f(T)$ field equation:
$\nu=0,i=0$
$\displaystyle\bigg{(}T-\frac{D^{\prime\prime}}{D}+\frac{HT^{\prime}}{2H^{\prime}}\bigg{)}f_{T}(T)+\bigg{(}\frac{HT}{H^{\prime}}-\frac{D^{\prime}}{D}\bigg{)}T^{\prime}f_{TT}(T)+\frac{1}{2}f(T)=\kappa^{2}\overset{em}{T}{}^{0}_{\;\;0}$
(31)
$\nu=0,i=2$
$\displaystyle\bigg{(}HT+\frac{T^{\prime}D^{2}}{2H^{\prime}}\bigg{)}f_{T}(T)+\frac{T^{\prime}H^{\prime}}{2}f_{TT}(T)-\frac{H}{2}f(T)=\kappa^{2}\bigg{(}\overset{em}{T}{}^{0}_{\;\;2}-H\overset{em}{T}{}^{0}_{\;\;0}\bigg{)}\;,$
(32)
$\nu=1,i=1$
$\displaystyle-
Tf_{T}(T)+\frac{1}{2}f(T)=\kappa^{2}\overset{em}{T}{}^{1}_{\;\;1}\;,$ (33)
$\nu=2,i=0$
$\displaystyle
T^{\prime}\bigg{[}\frac{1}{2H^{\prime}}f_{T}(T)+\sqrt{\frac{T}{2}}f_{TT}(T)\bigg{]}=\kappa^{2}\overset{em}{T}{}^{2}_{\;\;0}\;,$
(34)
$\nu=2,i=2$
$\displaystyle-
Tf_{T}(T)+\frac{1}{2}f(T)=\kappa^{2}\bigg{(}\overset{em}{T}{}^{2}_{\;\;2}-H\overset{em}{T}{}^{2}_{\;\;0}\bigg{)}\;,$
(35)
$\nu=3,i=3$
$\displaystyle-\frac{D^{\prime\prime}}{D}f_{T}(T)-\frac{T^{\prime}D^{\prime}}{D}T^{\prime}f_{TT}(T)+\frac{1}{2}f(T)=\kappa^{2}\overset{em}{T}{}^{3}_{\;\;3}\;.$
(36)
Apparently, the non-symmetric components of the modified Einstein equation are
consistent with the tetrad anstaz given in Eq. (29). In the above equations,
all other components of $\overset{em}{T}{}^{\mu}_{\;\;\nu}$ must be zero,
which means that, $\overset{em}{T}{}_{\mu\nu}$, has the cylindrical symmetry
as expected. In a Gödel-type spacetime, the energy-momentum tensor in a local
basis, $\overset{em}{T}_{ab}$ given in (50), has a general form:
$\overset{em}{T}{}_{ab}=diag(\rho,p_{1},p_{2},p_{3})$. Using
$\overset{em}{T}{}_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\overset{em}{T}{}_{ab}$, we
have
$\displaystyle\overset{em}{T}{}_{00}=\rho,\;\;\overset{em}{T}{}_{11}=p_{1},\;\;\;\overset{em}{T}{}_{22}=H^{2}\rho+D^{2}p_{2},\;\;\;\overset{em}{T}{}_{33}=p_{3},\;\;\;\overset{em}{T}{}_{02}=\overset{em}{T}{}_{20}=H\rho\;.$
(37)
One can then find easily
$\displaystyle\overset{em}{T}{}^{2}_{\;\;0}=0,\;\;\;\;\overset{em}{T}{}^{0}_{\;\;2}=H\bigg{(}\overset{em}{T}{}^{0}_{\;\;0}-\overset{em}{T}{}^{2}_{\;\;2}\bigg{)}\;\;.$
(38)
Thus, Eq. (32) seems to give an extra constraint on $f(T)$ gravity. This
equation is satisfied automatically in a Gödel-type spacetime, since $T$, as
shown in Eq. (42), is a constant in a Gödel-type universe. Furthermore, it is
easy to see that, in a Gödel-type spacetime, Eq. (32) gives the same
expression as Eq. (35). Four independent field equations are obtained, which
is consistent with the anstaz of tetrad. In addition, one can check that the
field equations (23-28) for the vierbein given in (29) can also be obtained
from an action constructed by replacing the specific form of $T$ (30) with the
general action of $f(T)$ theory. Therefore, the dynamical equations are
consistent, which means that the tetrad anstaz given in Eq. (29) is a good
guess for the Gödel-type spacetime.
## IV Gödel-type universe in $f(T)$ theory
To show the possibility of existence of the closed timelike curves and the
causality feature in $f(T)$ gravity, we consider the Gödel-type metric, which
has the form of Eq. (18) with $H$ and $G$ being:
$\displaystyle
H(r)=\frac{4\omega}{m^{2}}\sinh^{2}\bigg{(}\frac{mr}{2}\bigg{)}\;,$ (39)
$\displaystyle
G(r)=\frac{4}{m^{2}}\sinh^{4}\bigg{(}\frac{mr}{2}\bigg{)}\bigg{[}\coth^{2}\bigg{(}\frac{mr}{2}\bigg{)}-\frac{4\omega^{2}}{m^{2}}\bigg{]}\;,$
(40)
where $\omega$ and $m$ ($-\infty<m^{2}<+\infty,0<\omega^{2}$) are two constant
parameters used to classify different Gödel-type geometries. Thus, we have
$\displaystyle D(r)=\frac{1}{m}\sinh(mr)\;.$ (41)
Substituting the expressions of $H$ and $D$ into Eq. (30), one can obtain
easily
$\displaystyle T=2\omega^{2}\;,$ (42)
which is a positive constant.
If $G(r)<0$, Eq. (18) shows that one type of closed timelike curve, called
noncausal Gödel circle Godel , exists in the case of $t,z,r=const$. This means
a violation of causality. For a particular case of $0<m^{2}<4\omega^{2}$, the
causality violation region, i.e., $G(r)<0$ region, exists if
$\displaystyle\tanh^{2}\frac{mr}{2}<\frac{m^{2}}{4\omega^{2}}\;.$ (43)
Thus, one can define a critical radius $r_{c}$ Godel ; RebClif05e ; Reb09b1 ;
Reb10ed
$\displaystyle\tanh^{2}\frac{mr_{c}}{2}=\frac{m^{2}}{4\omega^{2}}\;,$ (44)
beyond which, $G(r)<0$ and causality is violated. When $m=0$, the critical
radius is $r_{c}=1/\omega$. When $m^{2}=4\omega^{2}$, $r_{c}=+\infty$, which
means that a breakdown of causality is avoided. Thus, the codomain range of
$r_{c}$ is $r_{c}\in(1/\omega,+\infty)$. Therefore, the condition for non-
violation of causality is $m^{2}\geq 4\omega^{2}$ or $r<r_{c}$. For the case
in which $m^{2}=-\mu^{2}<0$, both
$H(r)=\frac{4\omega}{\mu^{2}}\sin^{2}(\frac{\mu r}{2})$ and
$G(r)=\frac{4}{\mu^{2}}\sin^{4}(\frac{\mu r}{2})[\cot^{2}(\frac{\mu
r}{2})-\frac{4\omega^{2}}{\mu^{2}}]$ are periodic functions. Thus, an infinite
circulation of causal and noncausal ranges appears Reb09b1 ; Reb10ed .
It is easy to see that, if one further defines a set of bases
$\\{\theta^{a}\\}$:
$\displaystyle\theta^{0}=dt+H(r)d\phi,\qquad\theta^{1}=dr,$ (45)
$\displaystyle\theta^{2}=D(r)d\phi,\qquad\theta^{3}=dz,$ (46)
the Goödel-type line element can be simplified to be:
$\displaystyle ds^{2}=\eta_{ab}\theta^{a}\theta^{b}\;,$ (47)
where $\eta_{ab}=diag(1,-1,-1,-1)$ is the Minkowski metric. By choosing
$\\{\theta^{a}\\}$ as basis, the $f(T)$ field equation (13) becomes:
$\displaystyle
A_{ab}f_{T}(T)+\frac{1}{2}\eta_{ab}f(T)=\kappa^{2}\overset{em}{T}_{ab}\;.$
(48)
Here, both $f(T)$ and $f_{T}(T)$ are evaluated at $T=2\omega^{2}$. The second
term of Eq. (13) is discarded in obtaining the above equation since the
torsion scalar $T$ is a constant. We find that the nonzero components of
$A_{ab}$ are
$\displaystyle A_{00}=2\omega^{2}-m^{2},\quad A_{11}=A_{22}=2\omega^{2},\quad
A_{33}=m^{2}\;.$ (49)
Thus, we obtain a very simple form of the field equation in $f(T)$ gravity,
which will help us discuss the causality issue.
## V Causality Problem in $f(T)$ theory
One can see, from Eq. (48), that, in order to discuss the causality problem,
the matter source is a very important component. As was obtained in RebClif05e
; Reb09b1 ; Reb10ed , different matter sources may lead to different results.
In this paper, we assume that the matter source consists of two different
components: a perfect fluid and a scalar field. Thus, the energy-momentum
tensor $\overset{em}{T}{}_{ab}$ has the form
$\displaystyle\overset{em}{T}_{ab}=\overset{m}{T}_{ab}+\overset{s}{T}_{ab}\;,$
(50)
where, $\overset{m}{T}_{ab}$ and $\overset{s}{T}_{ab}$ correspond to the
energy-momentum tensors of the perfect-fluid and the scalar field,
respectively. In basis $\\{\theta^{a}\\}$, $\overset{m}{T}_{ab}$ and
$\overset{s}{T}_{ab}$ can be expressed as
$\displaystyle\overset{m}{T}_{ab}=(\rho+p)u_{a}u_{b}-p\eta_{ab}\;,$ (51)
$\displaystyle\overset{s}{T}_{ab}=D_{a}\Phi
D_{b}\Phi-\frac{1}{2}\eta_{ab}D_{c}\Phi D_{d}\Phi\eta^{cd}\;,$ (52)
where $u_{a}=(1,0,0,0)$, $\rho$ and $p$ are the energy density and pressure of
the perfect fluid, respectively, and $p=\text{w}\rho$ with w being the
equation of state parameter. $\Phi$ is the scalar field, and $D_{a}$ denotes
the covariant derivative relative to the local basis $\theta^{a}$. The scalar
field equation is $\square\,\Phi=\eta^{ab}\,\nabla_{a}\nabla_{b}\,\Phi\,=0$.
It is easy to prove that $\Phi(z)=\varepsilon z+\text{const}$ with a constant
amplitude $\varepsilon$ satisfies this field equation Reboucas1983 . Using the
solution $\Phi(z)=\varepsilon z+\text{const}$, one can obtain the nonvanishing
components of $\overset{s}{T}_{ab}$
$\overset{s}{T}_{00}=-\overset{s}{T}_{11}=-\overset{s}{T}_{22}=\overset{s}{T}_{33}=\frac{\varepsilon^{2}}{2}\,,$
(53)
Thus, the energy-momentum tensor of matter source becomes
$\displaystyle\overset{em}{T}_{ab}=diag\bigg{(}\rho+\frac{\varepsilon^{2}}{2}\;,\text{w}\rho-\frac{\varepsilon^{2}}{2}\;,\text{w}\rho-\frac{\varepsilon^{2}}{2}\;,\text{w}\rho+\frac{\varepsilon^{2}}{2}\bigg{)}\;.$
(54)
Substituting Eqs. (49) and (54) into the $f(T)$ field equation (Eq. (48)), we
find
$\displaystyle(2\omega^{2}-m^{2})f_{T}(T)+\frac{1}{2}f(T)=\kappa^{2}(\rho+\frac{\varepsilon^{2}}{2})\;;$
(55) $\displaystyle
2\omega^{2}f_{T}(T)-\frac{1}{2}f(T)=\kappa^{2}(\text{w}\rho-\frac{\varepsilon^{2}}{2})\;;$
(56) $\displaystyle
m^{2}f_{T}(T)-\frac{1}{2}f(T)=\kappa^{2}(\text{w}\rho+\frac{\varepsilon^{2}}{2})\;.$
(57)
Since the effective Newton gravity constant in $f(T)$ gravity becomes
$G_{N,eff}=G_{N}/f_{T}(T)$ Zheng2011 , only the case $f_{T}(T)>0$ will be
considered in the following in order to ensure a positive $G_{N,eff}$. From
Eqs. (55) and (56), one can derive a relation between $m$ and $\omega$:
$\displaystyle
m^{2}=2\omega^{2}\bigg{[}1+\frac{\varepsilon^{2}}{\rho(1+\text{w})+\varepsilon^{2}}\bigg{]}\;,$
(58)
which implies that the critical radius of the Gödel’s circle, Eq. (44),
satisfies
$\displaystyle\tanh^{2}\left(\frac{mr_{c}}{2}\right)=1-\frac{\rho(1+\text{w})}{2[\rho(1+\text{w})+\varepsilon^{2}]}\;.$
(59)
Obviously, different matter sources give rise to different critical radii and
therefore different causality structures, e.g. when $\varepsilon\rightarrow
0$, we have a finite $r_{c}$, while for $\rho\rightarrow 0$, $r_{c}=\infty$.
Therefore, a violation of causality may occur for the case of a perfect fluid
as the matter source, whereas causality is preserved in the case of a scalar
field. In order to show the causality feature in more detail and the
conditions for obtaining the Gödel-type solutions, we will divide our
discussion into two special cases: $\varepsilon\rightarrow 0$ and
$\rho\rightarrow 0$. In addition, a concrete $f(T)$ model will be considered.
### V.1 $\varepsilon^{2}\rightarrow 0$
$\varepsilon^{2}\rightarrow 0$ corresponds to the case that the universe only
contains a perfect fluid. Since $f_{T}(T)>0$, Eqs. (55), (56), and (57) reduce
to:
$\displaystyle m^{2}=2\omega^{2}\;;$ (60) $\displaystyle
Tf_{T}(T)=\kappa^{2}\rho(1+\text{w})\;;$ (61) $\displaystyle
f(T)=2\kappa^{2}\rho\;.$ (62)
From Eqs. (61, 62), it is easy to see that, in the limit of general relativity
without a cosmological constant ($f(T)=T$), $\text{w}=1$ is required to ensure
the existence of the Gödel-type solutions godelnote1978 ; RebClif05e ; Reb09b1
; Reb10ed . This means that a violation of causality in general relativity is
only possible for the so-called stiff fluid ($\text{w}=1$) which is not a
normal fluid in our Universe. In $f(T)$ theory, $Tf_{T}(T)>0$ and $\rho>0$
lead to $\text{w}>-1$. So, the perfect fluid must satisfy the weak energy
condition ($\rho>0$ and $\rho(1+\text{w})>0$). Using the above results, the
equation of state can be expressed as a function of the torsion scalar:
$\displaystyle\text{w}=\frac{2Tf_{T}(T)}{f(T)}-1\;.$ (63)
Different from general relativity that requires $\text{w}=1$ for perfect-fluid
Gödel solutions, the equation of state parameter of the fluid w in $f(T)$
gravity can differ from one and its value is determined by concrete $f(T)$
models. For example, a special $f(T)=\lambda T^{\delta}$ gives
$\text{w}=2\delta-1$, from which one can see that w can be an arbitrary number
for an arbitrary $\delta$. So, even normal matter, such as pressureless matter
or radiation, can lead to a violation of causality in certain $f(T)$ theories.
This indicates that the issue of causality violation seems more severe in
$f(T)$ gravity than in general relativity where only an exotic stiff fluid
allows the existence of Gödel-type solutions. From Eqs. (60), (61) and (62),
and using $T=2\omega^{2}$, we find that the critical radius given in Eq. (59)
becomes
$\displaystyle
r_{c}=2\text{tanh}^{-1}\bigg{(}\frac{1}{\sqrt{2}}\bigg{)}\cdot\sqrt{\frac{f_{T}(T)}{(1+\text{w})\kappa^{2}\rho}}\;,$
(64)
which is dependent both on the specifics of $f(T)$ theory and the properties
of the perfect fluid.
Now, let us consider a concrete power law $f(T)$ model Linder
$\displaystyle f(T)=T-\alpha T_{*}\left(\frac{T}{T_{*}}\right)^{n}\;,$ (65)
where $\alpha$ and $n$ are model parameters, and $T_{*}$ is a special value of
the torsion scalar, which is introduced to make $\alpha$ dimensionless.
$|n|\ll 1$ is required in order to obtain an usual early cosmic evolution
Wu2010b . The current cosmic observations give that
$\alpha=-0.79^{+0.35}_{-0.79}$ and $n=0.04^{+0.22}_{-0.33}$ at the $68.3\%$
confidence level Wu2010a . Thus, a negative $\alpha$ is favored by
observations. In term of Eq. (63), the equation of state of the perfect fluid
becomes
$\displaystyle\text{w}=1-\frac{2\alpha(n-1)T^{1-n}_{*}}{T^{1-n}-\alpha
T^{1-n}}\;.$ (66)
The equation above can be re-expressed as
$\displaystyle\frac{\alpha(2n-1-\text{w})}{1-\text{w}}=\left(\frac{T}{T_{*}}\right)^{1-n}>0\;,$
(67)
where a positive $T/T_{*}$ is considered. Recalling $\alpha<0$ and
$\text{w}>-1$, from Eq. (67) one can obtain the possible ranges of w for the
Gödel-type universes
$\displaystyle 1>\text{w}>-1+2n\quad(1>n>0)\;,\qquad
1>\text{w}>-1\quad(n<0)\;.$ (68)
For this power law model, the critical radius has the form
$\displaystyle
r_{c}=2\left[\frac{\alpha(2n-1-\text{w})}{1-\text{w}}\right]^{\frac{1}{2(n-1)}}\text{tanh}^{-1}(1/\sqrt{2})\;,$
(69)
which is determined completely by the model parameters and the equation of
state of the perfect fluid.
### V.2 $\rho\rightarrow 0$
This is the case of a scalar field as the matter source. Eqs. (55), (56), and
(57) now reduce to
$\displaystyle m^{2}=4\omega^{2}\;,$ (70) $\displaystyle
Tf_{T}(T)=\kappa^{2}\varepsilon^{2}\;,$ (71) $\displaystyle
f(T)=3\kappa^{2}\varepsilon^{2}\;.$ (72)
Note that (71) and (72) combined together admit a relation between $T$ and
$f(T)$:
$\displaystyle 3Tf_{T}(T)-f(T)=0\;,$ (73)
which constrains the class of solutions with no violation of causality. For
the power law model, the causal Gödel-type solution gives that the torsion
scalar should satisfy
$\displaystyle
T=2\omega^{2}=\left[-\frac{(1-3n)\alpha}{2}\right]^{\frac{1}{1-n}}T_{*}\;.$
(74)
Thus, $n<1/3$ is required if the numerator of $\frac{1}{1-n}$ is not even
since the observations show $\alpha<0$.
## VI Conclusions
$f(T)$ theory, a new modified gravity, provides an alternative way to explain
the present accelerated cosmic acceleration with no need of dark energy. Some
problems, including large scale structure, local Lorentz invariance, and so
on, of this modified gravity have been discussed. In this paper, we study the
issue of causality in $f(T)$ theory by examining the possibility of existence
of the closed timelike curves in the Gödel metric. Assuming that the matter
source is a scalar field or a perfect fluid, we examine the existence of the
Gödel-type solutions. For the scalar field case, we find that $f(T)$ gravity
allows a particular Gödel-type solution with $r_{c}\rightarrow\infty$, where
$r_{c}$ is the critical radius beyond which the causality is broken down.
Thus, the violation of causality can be forbidden. In the case of a perfect
fluid as the matter source, we find that the fluid must have an equation of
state parameter greater than minus one and this parameter should satisfy Eq.
(63) for the Gödel-type solutions to exist. For certain $f(T)$ models, the
perfect fluid that allows the Gödel-type solutions can even be normal matter,
such as pressureless matter or radiation. Since the critical radius $r_{c}$ of
perfect fluid Gödel-type solutions which depends on both matter and gravity is
finite, the issue of causality violation seems more severe in $f(T)$ gravity
than in general relativity where only an exotic stiff fluid allows the
existence of Gödel-type solutions.
###### Acknowledgements.
PXW would like to thank Prof. Qingguo Huang for helpful discussions. This work
was supported by the National Natural Science Foundation of China under Grants
Nos. 10935013, 11175093 and 11075083, Zhejiang Provincial Natural Science
Foundation of China under Grants Nos. Z6100077 and R6110518, the FANEDD under
Grant No. 200922, the National Basic Research Program of China under Grant No.
2010CB832803, the NCET under Grant No. 09-0144, and K.C. Wong Magna Fund in
Ningbo University.
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|
arxiv-papers
| 2012-03-09T08:38:59 |
2024-09-04T02:49:28.491578
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Di Liu, Puxun Wu, Hongwei Yu",
"submitter": "Di Liu",
"url": "https://arxiv.org/abs/1203.2016"
}
|
1203.2303
|
# The double charm decays of $B_{c}$ Meson in the Perturbative QCD Approach
Zhou Rui 1,2 Zou Zhitian 1 Cai-Dian Lü1 lucd@ihep.ac.cn 1 Institute of High
Energy Physics and Theoretical Physics Center for Science Facilities, Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China 2 School of
Science, Hebei United University, Tangshan, Hebei 063009, People’s Republic of
China
###### Abstract
We study the double charm decays of $B_{c}$ meson, by employing the
perturbative QCD approach based on $k_{T}$ factorization. In this approach, we
include the non-factorizable emission diagrams and W annihilation diagrams,
which are neglected in the previous naive factorization approach. The former
are important in the color-suppressed modes; while the latter are important in
most $B_{c}$ decay channels due to the large Cabibbo-Kobayashi-Maskawa matrix
elements. We make comparison with those previous naive factorization results
for the branching ratios and also give out the theoretical errors that
previously missed. We predict the transverse polarization fractions of
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0},D^{*+}_{(s)}D^{*0}$ decays for the
first time. A large transverse polarization contribution that can reach
$50\%\sim 60\%$ is predicted in some of the $B_{c}$ meson decays.
###### pacs:
13.25.Hw, 12.38.Bx, 14.40.Nd
## I Introduction
Since the $B_{c}$ meson is the lowest bound state of two different heavy
quarks with open flavor, it is stable against strong and electromagnetic
annihilation processes. The $B_{c}$ meson therefore decays weakly.
Furthermore, the $B_{c}$ meson has a sufficiently large mass, thus each of the
two heavy quarks can decay individually. It has rich decay channels, and
provides a very good place to study nonleptonic weak decays of heavy mesons,
to test the standard model and to search for any new physics signals iiba .
The current running LHC collider will produce much more $B_{c}$ mesons than
ever before to make this study a bright future.
Within the standard model (SM), for the double charm decays of $B_{u,d,s}$
mesons, there are penguin operator contributions as well as tree operator
contributions. Thus the direct CP asymmetry may be present. However, the
double charm decays of $B_{c}$ meson are pure tree decay modes, which are
particularly well suited to extract the Cabibbo-Kobayashi-Maskawa (CKM) angles
due to the absented interference from penguin contributions. As was pointed
out in ref. plb286160 and further elaborated in ref. prd62057503 ; jpg301445
; plb555189 ; prd65034016 , the decays $B_{c}\rightarrow
D_{s}^{+}D^{0},D_{s}^{+}\bar{D}^{0}$ are the gold-plated modes for the
extraction of CKM angle $\gamma$ though amplitude relations because their
decay widths are expected to be at the same order of magnitude. But this needs
to be examined by faithful calculations.
Although many investigations on the decays of $B_{c}$ to double-charm states
have been carried out jpg301445 ; plb555189 ; prd73054024 ; prd564133 ; pan67
; prd61034012 ; prd62014019 ; prd493399 in the literature, there are
uncontrolled large theoretical errors with quite different numerical results.
In fact, all of these old calculations are based on naive factorization
hypothesis, with various form factor inputs. Most of them even did not give
any theoretical error estimates because of the non-reliability of these
models. Recently, the theory of non-leptonic B decays has been improved quite
significantly. Factorization has been proved in many of these decays, thus
allow us to give reliable calculations of the hadronic B decays. It is also
shown that the non-factorizable contributions and annihilation type
contributions, which are neglected in the naive factorization approach, are
very important in these decays cheng .
The perturbative QCD approach (pQCD) prl744388 is one of the recently
developed theoretical tools based on QCD to deal with the non-leptonic B
decays. Utilizing the $k_{T}$ factorization instead of collinear
factorization, this approach is free of end-point singularity. Thus the
Feynman diagrams including factorizable, non-factorizable and annihilation
type, are all calculable. Phenomenologically, the pQCD approach successfully
predict the charmless two-body B decays plb5046 ; prd63074009 . For the decays
with a single heavy $D$ meson in the final states (the momentum of the $D$
meson is $\frac{1}{2}m_{B}(1-r^{2})$, with $r=m_{D}/m_{B}$), it is also proved
factorization in the soft-collinear effective theory scet1 .
Phenomenologically the pQCD approach is also demonstrated to be applicable in
the leading order of the $m_{D}/m_{B}$ expansion 09101424 ; 0512347 for this
kind of decays. For the double charm decays of $B_{c}$ meson, the momentum of
the final state $D$ meson is $\frac{1}{2}m_{B_{c}}(1-2r^{2})$, which is only
slightly smaller than that of the decays with a single D meson final state.
The prove of factorization here is thus trivial. The pQCD approach is
applicable to this kind of decays. In fact, the double charm decays of
$B_{u,d,s}$ meson have been studied in the pQCD approach successfully dd1 ;
dd2 , with best agreement with experiments. In this paper, we will extend our
study to these $B_{c}$ decays in the pQCD approach, in order to give
predictions on branching ratios and polarization fractions for the experiments
to test. Since this study is based on QCD and perturbative expansion, the
theoretical error will be controllable than any of the model calculations.
Our paper is organized as follows: We review the pQCD factorization approach
and then perform the perturbative calculations for these considered decay
channels in Sec.II. The numerical results and discussions on the observables
are given in Sec.III. The final section is devoted to our conclusions. Some
details of related functions and the decay amplitudes are given in the
Appendix.
## II Framework
For the double charm decays of $B_{c}$, only the tree operators of the
standard effective weak Hamiltonian contribute. We can divide them into two
groups: CKM favored decays with both emission and annihilation contributions
and pure emission type decays, which are CKM suppressed. For the former modes,
the Hamiltonian is given by:
$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{uq}[C_{1}(\mu)O_{1}(\mu)+C_{2}(\mu)O_{2}(\mu)],$
$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\beta}\otimes\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\alpha},$
$\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\alpha}\otimes\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\beta},$
(1)
while the effective Hamiltonian of the latter modes reads
$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{ub}^{*}V_{cq}[C_{1}(\mu)O^{\prime}_{1}(\mu)+C_{2}(\mu)O^{\prime}_{2}(\mu)],$
$\displaystyle O^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\beta}\otimes\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\alpha},$
$\displaystyle O^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\alpha}\otimes\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\beta},$
(2)
where $V(q=d,s)$ are the corresponding CKM matrix elements. $\alpha$, $\beta$
are the color indices. $C_{1,2}$ are Wilson coefficients at renormalization
scale $\mu$. $O_{1,2}$ and $O^{\prime}_{1,2}$ are the effective four-quark
operators.
The factorization theorem allows us to factorize the decay amplitude into the
convolution of the hard subamplitude, the Wilson coefficient and the meson
wave functions, all of which are well-defined and gauge invariant. It is
expressed as
$\displaystyle C(t)\otimes
H(x,t)\otimes\Phi(x)\otimes\exp[-s(P,b)-2\int^{t}_{1/b}\frac{d\mu}{\mu}\gamma_{q}(\alpha_{s}(\mu))],$
(3)
where $C(t)$ are the corresponding Wilson coefficients of effective operators
defined in eq.(II,II). Since the transverse momentum of quark is kept in the
pQCD approach, the large double logarithm $\ln^{2}(Pb)$ (with P denoting the
longitudinal momentum, and b the conjugate variable of the transverse
momentum) to spoil the perturbative expansion. A resummation is thus needed to
give a Sudakov factor $\exp[-s(P,b)]$ npb193381 . The term after Sudakov is
from renormalization group running with $\gamma_{q}=-\alpha_{s}/\pi$ the quark
anomalous dimension in axial gauge and $t$ the factorization scale. All non-
perturbative components are organized in the form of hadron wave functions
$\Phi(x)$ (with x the longitudinal momentum fraction of valence quark inside
the meson), which can be extracted from experimental data or other non-
perturbative methods. Since the universal non-perturbative dynamics has been
factored out, one can evaluate all possible Feynman diagrams for the hard
subamplitude $H(x,t)$ straightforwardly, which include both traditional
factorizable and so-called “non-factorizable” contributions. Factorizable and
non-factorizable annihilation type diagrams are also calculable without end-
point singularity.
### II.1 Channels with both emission and annihilation contributions
Figure 1: Feynman diagrams for $B_{c}\rightarrow D^{+}\bar{D}^{0}$ decays.
At leading order, there are eight kinds of Feynman diagrams contributing to
this type of CKM favored decays according to eq.(II). Here, we take the decay
$B_{c}\rightarrow D^{+}\bar{D}^{0}$ as an example, whose Feynman diagrams are
shown in Fig.1. The first line are the emission type diagrams, with the first
two contributing to the usual form factor; the last two so-called “non-
factorizable” diagrams. In fact, the first two diagrams are the only
contributions calculated in the naive factorization approach. The second line
are the annihilation type diagrams, with the first two factorizable; the last
two non-factorizable. The decay amplitude of factorizable diagrams (a) and (b)
in Fig.1 is
$\displaystyle\mathcal{F}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(4)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+2r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
where $r_{b}=m_{b}/M_{B}$, $r_{i}=m_{i}/M_{B}(i=2,3)$ with $m_{2},m_{3}$ are
the masses of the recoiling charmed meson and the emitting charmed meson,
respectively; $C_{F}=4/3$ is a color factor; $f_{3}$ is the decay constant of
the charmed meson, which emitted from the weak vertex. The factorization
scales $t_{a,b}$ are chosen as the maximal virtuality of internal particles in
the hard amplitude, in order to suppress the higher order corrections
prd074004 . The function $h_{e}$ and the Sudakov factor $\exp[-S]$ are
displayed in the Appendix B. $D$ meson distribution amplitude $\phi(x)$ are
given in Appendix C. The factor $S_{t}(x)$ is the jet function resulting from
the threshold resummation, whose definitions can be found in epjc45711 .
The formula for non-factorizable emission diagrams Fig. 1 (c) and (d) contain
the kinematics variables of all the three mesons. Its expression is:
$\displaystyle\mathcal{M}_{e}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})\times$
(5)
$\displaystyle\\{[1-x_{1}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[1-x_{1}-x_{2}+x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$
Generally, for charmless decays of B meson, the non-factorizable contributions
of the emission diagrams are small due to the cancelation between Fig. 1 (c)
and (d). While for double charm decays with the light meson replaced by a
charmed meson, since the heavy $\bar{c}$ quark and the light quark is not
symmetric, the non-factorizable emission diagrams ought to give remarkable
contributions. This has been shown in the pQCD calculation of $B\to D\pi$
decays for a very large branching ratios of color-suppressed modes dpi and
proved by the B factory experiments.
The decay amplitude of factorizable annihilation diagrams Fig. 1 (e) and (f)
involve only the two final states charmed meson wave functions, shown as
$\displaystyle\mathcal{F}_{a}$ $\displaystyle=$ $\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(6)
$\displaystyle\\{[1-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[1-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\}.$
For the non-factorizable annihilation diagrams Fig. 1 (g) and (h), the decay
amplitude is
$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(7)
$\displaystyle\times\\{[x_{1}+x_{3}-1-r_{c}]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{b}-x_{2}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$
where $r_{c}=m_{c}/M_{B}$, with $m_{c}$ the mass of c quark in $B_{c}$ meson.
Finally, the total decay amplitude for $B_{c}\rightarrow D^{+}\bar{D}^{0}$ can
be given by
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}\bar{D}^{0})$ $\displaystyle=$
$\displaystyle
V_{cb}^{*}V_{ud}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}],$
(8)
with the combinations of Wilson coefficients $a_{1}=C_{2}+C_{1}/3$ and
$a_{2}=C_{1}+C_{2}/3$, characterizing the color favored contribution and the
color-suppressed contribution in the naive factorization, respectively. The
total decay amplitudes of $B_{c}\rightarrow D_{s}^{+}\bar{D}^{0}$,
$B_{c}\rightarrow D^{+}\bar{D}^{*0}$ and $B_{c}\rightarrow
D_{s}^{+}\bar{D}^{*0}$ can be obtained from eq.(8) with the following
replacement:
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}\bar{D}^{0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{us}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{D^{+}\rightarrow
D^{+}_{s}},$ $\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}\bar{D}^{*0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{ud}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{\bar{D}^{0}\rightarrow\bar{D}^{*0}},$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}\bar{D}^{*0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{us}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{D^{+}\rightarrow
D^{+}_{s},\bar{D}^{0}\rightarrow\bar{D}^{*0}}.$ (9)
Comparing our eq.(8,II.1) with the formulas of previous naive factorization
approach jpg301445 ; plb555189 ; prd73054024 ; prd564133 ; pan67 ; prd61034012
, it is easy to see that only the first term appearing in eq.(8,II.1) are
calculated in the previous naive factorization approach. The second, third and
fourth terms in these equations, are the corresponding non-factorizable
emission type contribution, factorizable and non-factorizable annihilation
type contributions, respectively, which are all new calculations.
In $B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays, the two vector mesons
in the final states have the same helicity due to angular momentum
conservation, therefore only three different polarization states, one
longitudinal and two transverse for both vector mesons, are possible. The
decay amplitude can be decomposed as
$\displaystyle\mathcal{A}=\mathcal{A}^{L}+\mathcal{A}^{N}\epsilon_{2}^{T}\cdot\epsilon_{3}^{T}+i\mathcal{A}^{T}\epsilon_{\alpha\beta\rho\sigma}n^{\alpha}v^{\beta}\epsilon_{2}^{T\rho}\epsilon_{3}^{T\sigma},$
(10)
where $\epsilon_{2}^{T},\epsilon_{3}^{T}$ are the transverse polarization
vectors for the two vector charmed mesons, respectively. $\mathcal{A}^{L}$
corresponds to the contributions of longitudinal polarization;
$\mathcal{A}^{N}$ and $\mathcal{A}^{T}$ corresponds to the contributions of
normal and transverse polarization, respectively. And the total amplitudes
$\mathcal{A}^{L,N,T}$ have the same structures as eq.(8,II.1). The
factorization formulae for the longitudinal, normal and transverse
polarizations are listed in Appendix A.
For $B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{0}$ decays, only the longitudinal
polarization of $D^{*+}_{(s)}$ meson will contribute, due to the angular
momentum conservation. We can obtain their decay amplitudes from the
longitudinal polarization amplitudes for the $B_{c}\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays with the replacement
$\bar{D}^{*0}\rightarrow\bar{D}^{0}$.
### II.2 Channels with pure emission type decays
Figure 2: Color-suppressed emission diagrams contributing to the
$B_{c}\rightarrow D^{+}D^{0}$ decays.
Figure 3: Color-favored emission diagrams contributing to the
$B_{c}\rightarrow D^{+}D^{0}$ decays.
There are also eight kinds of Feynman diagrams contributing to
$B_{c}\rightarrow D_{(s)}^{(*)+}D^{(*)0}$ decays according to eq.(II), but all
are emission type. Taking the decay $B_{c}\rightarrow D^{+}D^{0}$ as an
example, Fig. 2 are the color-suppressed emission diagrams while Fig. 3 are
the color-favored emission diagrams. We mark the subscript 2 and 3 to denote
the contributions from Fig. 2 and Fig. 3, respectively. The decay amplitude of
factorization emission diagrams $\mathcal{F}_{e2}$, coming from Fig. 2 (a,b),
is similar to eq.(4), but with the replacement $\bar{D}^{0}\rightarrow D^{0}$.
While the decay amplitude of non-factorization emission diagram
$\mathcal{M}_{e2}$, coming from Fig. 2 (c,d), is different from eq.(5), since
the heavy c quark and the light anti-quark are not symmetric. The expression
of the non-factorizable emission diagram is
$\displaystyle\mathcal{M}_{e2}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
(11)
$\displaystyle\times\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$
By exchanging the two final states charmed mesons in Fig. 2, one can obtain
the corresponding decay amplitudes formulae $\mathcal{F}_{e3}$ and
$\mathcal{M}_{e3}$ for Fig. 3. The total decay amplitude of $B_{c}\rightarrow
D^{+}D^{0}$ decay can be written as
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}].$
(12)
If the final recoiling meson is the vector $D^{*}$ meson, the decay amplitudes
of factorization emission diagrams and non-factorization emission diagrams are
given as
$\displaystyle\mathcal{F}^{*}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(13)
$\displaystyle\times\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{*}_{e2}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
$\displaystyle\times\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}+r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$
The total decay amplitudes for other pure emission type decays are then
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}D^{*0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}^{*}_{e3}+C_{1}\mathcal{M}^{*}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{*+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}^{*}_{e2}+C_{2}\mathcal{M}^{*}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}D^{*0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}^{*}_{e3}+C_{1}\mathcal{M}^{*}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{*+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}^{*}_{e2}+C_{2}\mathcal{M}^{*}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}].$
The $B_{c}\rightarrow D^{*+}_{(s)}D^{*0}$ decays have a similar situation to
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$, their factorization formulae are
also listed in Appendix.A.
## III NUMERICAL RESULTS
In this section, we summarize the numerical results and analysis in the double
charm decays of the $B_{c}$ meson. Some input parameters needed in the pQCD
calculation are listed in Table 1.
### III.1 The Form Factors
Table 1: Parameters we used in numerical calculation npp37
Mass(GeV) | $M_{W}=80.399$ | $M_{B_{c}}=6.277$ | $m_{b}=4.2$ | $m_{c}=1.27$
---|---|---|---|---
CKM | $|V_{ub}|=(3.47^{+0.16}_{-0.12})\times 10^{-3}$ | $|V_{ud}|=0.97428^{+0.00015}_{-0.00015}$ | $|V_{us}|=0.2253^{+0.0007}_{-0.0007}$
---|---|---|---
$|V_{cs}|=0.97345^{+0.00015}_{-0.00016}$ | $|V_{cd}|=0.2252^{+0.0007}_{-0.0007}$ | $|V_{cb}|=0.0410^{+0.0011}_{-0.0007}$
Decay constants(MeV) | $f_{B_{c}}=489$ | $f_{D}=206.7\pm 8.9$ | $f_{D_{s}}=257.5\pm 6.1$
---|---|---|---
Lifetime | $\tau_{B_{c}}=0.453\times 10^{-12}\text{s}$
---|---
Table 2: The form factors for $B_{c}\rightarrow D^{(*)}_{(s)}$ at $q^{2}=0$ evaluated in the pQCD approach. The uncertainties are from the hadronic parameters. For comparison, we also cite the theoretical estimates of other models. | This work | Kiselev jpg301445 111 The non-bracket (bracketed) results are evaluated in sum rules (potential model) | IKP plb555189 | WSL prd7905402 | DSV jpg35085002 | DW prd391342 222We quote the result with $\omega=0.7\text{GeV}$
---|---|---|---|---|---|---
$F^{B_{c}\rightarrow D}$ | $0.14^{+0.01}_{-0.02}$ | 0.32 [0.29] | 0.189 | 0.16 | 0.075 | 0.255
$F^{B_{c}\rightarrow D_{s}}$ | $0.19^{+0.02}_{-0.01}$ | 0.45 [0.43] | 0.194 | 0.28 | 0.15 | –
$A_{0}^{B_{c}\rightarrow D^{*}}$ | $0.12^{+0.02}_{-0.01}$ | 0.35 [0.37] | 0.133 | 0.09 | 0.081 | 0.257
$A_{0}^{B_{c}\rightarrow D_{s}^{*}}$ | $0.17^{+0.01}_{-0.01}$ | 0.47 [0.52] | 0.142 | 0.17 | 0.16 | –
The diagrams (a) and (b) in Fig.1 or Fig.3 give the contribution for
$B_{c}\rightarrow D^{(*)}_{(s)}$ transition form factor at $q^{2}=0$ point.
Our predictions of the form factors are collected in Table 2. The error is
from the combined uncertainty in the hadronic parameters: (1) the shape
parameters: $\omega_{B}=0.60\pm 0.05$ for $B_{c}$ meson wave function,
$a_{D}=(0.5\pm 0.1)\text{GeV}$ for $D^{(*)}$ meson and $a_{D_{s}}=(0.4\pm
0.1)\text{GeV}$ for $D_{s}^{(*)}$ meson wave function dd1 ; (2) the decay
constants in the wave functions of charmed mesons, which are given in Table 1.
Since the uncertainties from decay constants of $D_{(s)}$ and the shape
parameters of the wave functions are very small, the relevant uncertainties to
the form factors are also very small. We can see that the $SU(3)$ symmetry
breaking effects between $B_{c}$ to $D^{(*)}$ and $B_{c}$ to $D^{(*)}_{s}$
form factors are large, as the decay constant of $D_{s}$ is about one-fifth
larger than that of the $D$ meson.
In the literature there are already lots of studies on $B_{c}\rightarrow
D^{(*)}_{(s)}$ transition form factors jpg301445 ; prd7905402 ; jpg35085002 ;
plb555189 ; prd391342 , whose results are collected in Table 2. Our results
are generally close to the covariant light-front quark model results of
prd7905402 and the constituent quark model results of plb555189 . However,
other results collected in Table 2, especially for the QCD sum rules (QCDSR)
jpg301445 and the Bauer, Stech and Wirbel (BSW) model jpg35085002 deviate a
lot numerically. The predictions of QCDSR jpg301445 are larger than those in
other works prd7905402 ; jpg35085002 ; plb555189 ; prd391342 . The reason is
that they have taken into account the $\alpha_{s}/v$ corrections and the form
factors are enhanced by 3 times due to the Coulomb renormalization of the
quark-meson vertex for the heavy quarkonium $B_{c}$. The results of BSW model
jpg35085002 are quite small due to the less overlap of the initial and final
states wave functions. Although, the included flavor dependence of the average
transverse quark momentum in the mesons can enhance the form factors for
$B_{c}\rightarrow D^{*}_{(s)}$ transitions, their predictions are still
smaller than other models. The large differences in different models can be
discriminated by the future LHC experiments.
### III.2 Branching Ratios
With the decays amplitudes $\mathcal{A}$ obtained in Sec.II, the branching
ratio $\mathcal{BR}$ reads as
$\displaystyle\mathcal{BR}=\frac{G_{F}\tau_{B_{c}}}{32\pi
M_{B}}\sqrt{(1-(r_{2}+r_{3})^{2})(1-(r_{2}-r_{3})^{2})}|\mathcal{A}|^{2}.$
(16)
As stated in Sec II, the contributions from the penguin operators are absent,
since the penguins add an even number of charmed quarks, while there is
already one from the initial state. There should be no CP violation in these
processes. We tabulate the branching ratios of the considered decays in Table
3 and 4. The processes (1)-(4) in Table 3 have a comparatively large branching
ratios ($10^{-5}$) with the CKM factor $V_{cb}^{*}V_{ud}\sim\lambda^{2}$.
While the branching ratios of other processes are relatively small due to the
CKM factor suppression. Especially for the processes (1)-(4) in Table 4, these
channels are suppressed by CKM element $V_{ub}/V_{cb}$ and $V_{cd}/V_{ud}$.
Thus their branching ratios are three order magnitudes smaller.
Table 3: Branching ratios ($10^{-6}$) of the CKM favored decays with both emission and annihilation contributions, together with results from other models. The errors for these entries correspond to the uncertainties in the input hadronic quantities, from the CKM matrix elements, and the scale dependence, respectively. | channels | This work | Kiselevjpg301445 | IKPplb555189 | IKSprd73054024 | LCprd564133 | CFprd61034012
---|---|---|---|---|---|---|---
1 | $B_{c}\rightarrow D^{+}\bar{D}^{0}$ | $32^{+6+1+2}_{-6-1-4}$ | 53 | 32 | 33 | 86 | 8.4
2 | $B_{c}\rightarrow D^{+}\bar{D}^{*0}$ | $34^{+7+2+3}_{-6-1-3}$ | 75 | 83 | 38 | 75 | 7.5
3 | $B_{c}\rightarrow D^{*+}\bar{D}^{0}$ | $12^{+3+1+0}_{-3-0-1}$ | 49 | 17 | 9 | 30 | 84
4 | $B_{c}\rightarrow D^{*+}\bar{D}^{*0}$ | $34^{+9+2+0}_{-8-1-0}$ | 330 | 84 | 21 | 55 | 140
5 | $B_{c}\rightarrow D_{s}^{+}\bar{D}^{0}$ | $2.3^{+0.4+0.1+0.2}_{-0.4-0.1-0.2}$ | 4.8 | 1.7 | 2.1 | 4.6 | 0.6
6 | $B_{c}\rightarrow D_{s}^{+}\bar{D}^{*0}$ | $2.6^{+0.4+0.1+0.1}_{-0.6-0.1-0.2}$ | 7.1 | 4.3 | 2.4 | 3.9 | 0.53
7 | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{0}$ | $0.7^{+0.1+0.0+0.0}_{-0.2-0.0-0.0}$ | 4.5 | 0.95 | 0.65 | 1.8 | 5
8 | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$ | $2.8^{+0.7+0.1+0.1}_{-0.6-0.1-0.0}$ | 26 | 4.7 | 1.6 | 3.5 | 8.4
Table 4: Branching ratios ($10^{-7}$) of the CKM suppressed decays with pure emission contributions, together with results from other models. The errors for these entries correspond to the uncertainties in the input hadronic quantities, from the CKM matrix elements, and the scale dependence, respectively. | channels | This work | Kiselevjpg301445 | IKPplb555189 | IKSprd73054024
---|---|---|---|---|---
1 | $B_{c}\rightarrow D^{+}D^{0}$ | $1.0^{+0.2+0.1+0.0}_{-0.1-0.0-0.0}$ | 3.2 | 1.1 | 3.1
2 | $B_{c}\rightarrow D^{+}D^{*0}$ | $0.7^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$ | 2.8 | 0.25 | 0.52
3 | $B_{c}\rightarrow D^{*+}D^{0}$ | $0.9^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$ | 4.0 | 3.8 | 4.4
4 | $B_{c}\rightarrow D^{*+}D^{*0}$ | $0.8^{+0.2+0.1+0.2}_{-0.1-0.0-0.0}$ | 15.9 | 2.8 | 2.0
5 | $B_{c}\rightarrow D_{s}^{+}D^{0}$ | $30^{+5+3+1}_{-4-2-1}$ | 66 | 25 | 74
6 | $B_{c}\rightarrow D_{s}^{+}D^{*0}$ | $19^{+3+2+0}_{-3-1-1}$ | 63 | 6 | 13
7 | $B_{c}\rightarrow D_{s}^{*+}D^{0}$ | $25^{+4+2+0}_{-3-2-1}$ | 85 | 69 | 93
8 | $B_{c}\rightarrow D_{s}^{*+}D^{*0}$ | $24^{+3+2+1}_{-3-2-1}$ | 404 | 54 | 45
For comparison, we also cite other theoretical results jpg301445 ; plb555189 ;
prd73054024 ; prd564133 ; prd61034012 for the double charm decays of $B_{c}$
meson in Tables 3 and 4. In general, the results of the various model
calculations are of the same order of magnitude for most channels. However the
difference between different model calculations is quite large. This is
expected from the large difference of input parameters, especially the large
difference of form factors shown in Table 2. As stated in the introduction,
all the calculations of these $B_{c}$ to two D meson decays in the literature
use the same naive factorization approach. Their difference relies only on the
input form factors and decay constants. Therefore the comparison of results
with any of them is straightforward. Larger branching ratios come always with
the larger form factors. As stated in the previous subsection, our results of
form factors are comparable with the relativistic constituent quark model
(RCQM) plb555189 ; prd73054024 , thus our branching ratios in Table 3 are also
comparable with theirs except for the processes $B_{c}\rightarrow
D^{*+}\bar{D}^{*0}$ and $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$. Due to the
sizable contributions of transverse polarization amplitudes, our branching
ratios are larger than those in RCQM model, whose transverse contribution is
negligible.
Since all the previous calculations in the literature are model calculations,
it is difficult for them to give the theoretical error estimations. In our
pQCD approach, the factorization holds at the leading order expansion of
$m_{D}/m_{B}$. At this order, we can do the systematical calculation, so as to
the error estimations in the tables. The first error in these entries is
estimated from the hadronic parameters: (1) the shape parameters:
$\omega_{B}=0.60\pm 0.05$ for $B_{c}$ meson, $a_{D}=(0.5\pm 0.1)\text{GeV}$
for $D^{(*)}$ meson and $a_{D_{s}}=(0.4\pm 0.1)\text{GeV}$ for $D_{s}^{(*)}$
meson dd1 ; (2) the decay constants in the wave functions of charmed mesons,
which are given in Table 1. The second error is from the uncertainty in the
CKM matrix elements, which are also given in Table 1. The third error arises
from the hard scale t varying from $0.75t$ to $1.25t$, which characterizing
the size of next-to-leading order QCD contributions. The not large errors of
this type indicate that our perturbative expansion indeed hold. It is easy to
see that the most important uncertainty in our approach comes from the
hadronic parameters. The total theoretical error is in general around 10% to
30% in size.
The eight CKM favored channels (proportional to $|V_{cb}|$) in Table 3 receive
contributions from both emission diagrams and annihilation diagrams. From
Fig.1, one can find that the contributions from the factorizable emission
diagrams are color-suppressed. The naive factorization approach can not give
reliable predictions due to large non-factorizable contributions fac . As was
pointed out in Sec.II, the non-factorizable emission diagrams give large
contributions in pQCD approach because the asymmetry of the two quarks in
charmed mesons. Thus, the branching ratios of these decays are dominated by
the non-factorizable emission diagrams.
The eight CKM suppressed channels (proportional to $|V_{ub}|$) in Table 4 can
occur only via emission type diagrams. There are two types of emission
diagrams in these decays, one is color-suppressed, one is color favored. It is
expected that the color-favored factorizable amplitude $\mathcal{F}_{e3}$
dominates in eq.(II.2). However, the non-factorizable contribution
$\mathcal{M}_{e2}$, proportional to the large $C_{2}$, is enhanced by the
Wilson coefficient. Numerically it is indeed comparable to the color-favored
factorizable amplitude. This large non-factorizable contribution has already
been shown in the similar $B\to D\pi$ decays theoretically and experimentally
dpi . In all of these channels the non-factorizable contributions play a very
important role, therefore the branching ratios predicted in table 3 and 4 are
not like the previous naive factorization approach calculations jpg301445 ;
plb555189 ; prd73054024 ; prd564133 ; prd61034012 . They are not simply
proportional to the corresponding form factors any more, but with a very
complicated manner, since we have also additional annihilation type
contributions.
From Table III and IV, one can see that as it was expected the magnitudes of
the branching ratios of the decays $B_{c}\rightarrow D^{+}_{s}\bar{D}^{0}$ and
$B_{c}\rightarrow D^{+}_{s}D^{0}$ are very close to each other. In our
numerical results, the ratio of the two decay widths is estimated as
$\frac{\Gamma(B_{c}\rightarrow D_{s}^{+}D^{0})}{\Gamma(B_{c}\rightarrow
D_{s}^{+}\bar{D}^{0})}\approx 1.3$. They are very suitable for extracting the
CKM angle $\gamma$ though the amplitude relations. Hopefully they will be
measured in the experiments soon. However, the decays $B_{c}\rightarrow
D^{+}\bar{D}^{0},D^{+}D^{0}$ are problematic from the methodic point of view
for $\mathcal{BR}(B_{c}\rightarrow D^{+}D^{0})\ll\mathcal{BR}(B_{c}\rightarrow
D^{+}\bar{D}^{0})$. The corresponding ratio in $B_{c}\rightarrow
D^{+}D^{0},D^{+}\bar{D}^{0}$ decays is $\frac{\Gamma(B_{c}\rightarrow
D^{+}D^{0})}{\Gamma(B_{c}\rightarrow D^{+}\bar{D}^{0})}\sim 10^{-3}$, which
confirm the latter decay modes are not useful to determine the angle $\gamma$
experimentally.
Table 5: The transverse polarizations fractions ($\%$) for $B_{c}\rightarrow VV$. The errors correspond to the uncertainties in the hadronic parameters and the scale dependence, respectively. | $B_{c}\rightarrow D^{*+}\bar{D}^{*0}$ | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$ | $B_{c}\rightarrow D^{*+}D^{*0}$ | $B_{c}\rightarrow D_{s}^{*+}D^{*0}$
---|---|---|---|---
$\mathcal{R}_{T}$ | $58^{+3+1}_{-3-0}$ | $68^{+2+1}_{-2-1}$ | $4^{+1+1}_{-1-1}$ | $6^{+1+2}_{-0-1}$
For the $B_{c}$ decays to two vector mesons, the decays amplitudes
$\mathcal{A}$ are defined in the helicity basis
$\displaystyle\mathcal{A}=\sum_{i=0,+,-}|\mathcal{A}_{i}|^{2},\quad$ (17)
where the helicity amplitudes $\mathcal{A}_{i}$ have the following
relationships with $\mathcal{A}^{L,N,T}$
$\displaystyle\mathcal{A}_{0}=\mathcal{A}^{L},\quad\mathcal{A}_{\pm}=\mathcal{A}^{N}\pm\mathcal{A}^{T}.$
(18)
We also calculate the transverse polarization fractions $\mathcal{R}_{T}$ of
the $B_{c}\to D_{(s)}^{*}D^{*}$ decays, with the definition given by
$\displaystyle\mathcal{R}_{T}=\frac{|\mathcal{A}_{+}|^{2}+|\mathcal{A}_{-}|^{2}}{|\mathcal{A}_{0}|^{2}+|\mathcal{A}_{+}|^{2}+|\mathcal{A}_{-}|^{2}}.$
(19)
This should be the first time theoretical predictions in the literature, which
are absent in all the naive factorization calculations. According to the power
counting rules in the factorization assumption, the longitudinal polarization
should be dominant due to the quark helicity analysis. Our predictions for the
transverse polarization fractions of the decays $B_{c}\rightarrow
D^{*+}_{(s)}D^{*0}$, which are given in Table 5, are indeed small, since the
two transverse amplitudes are down by a power of $r_{2}$ or $r_{3}$ comparing
with the longitudinal amplitudes. However, for $B_{c}\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays, the most important contributions for these
two decay channels are from the non-factorizable tree diagrams in Fig. 1(c)
and 1(d). With an additional gluon, the transverse polarization in the non-
factorizable diagrams does not encounter helicity flip suppression. The
transverse polarization is at the same order as longitudinal polarization.
Therefore, we can expect the transverse polarizations take a larger ratio in
the branching ratios, which can reach $\sim 60\%$. The fact that the non-
factorizable contribution can give large transverse polarization contribution
is also observed in the $B^{0}\to\rho^{0}\rho^{0}$, $\omega\omega$ decays
rhorho and in the $B_{c}\rightarrow D_{s}^{*+}\omega$ decay 11121257 .
## IV conclusion
All the previous calculations in the literature for the $B_{c}$ meson decays
to two charmed mesons are based on the very simple naive factorization
approach. The branching ratios predicted in this kind of model calculation
depend heavily on the input form factors. Since all of these modes contain
dominant or large contributions from color-suppressed diagrams, the predicted
branching ratios are also not stable due to the large unknown non-factorizable
contributions. In this paper, we have performed a systematic analysis of the
double charm decays of the $B_{c}$ meson in the pQCD approach based on $k_{T}$
factorization theorem, which is free of end-point singularities. All
topologies of decay amplitudes are calculable in the same framework, including
the non-factorizable one and annihilation type. It is found that the non-
factorizable emission diagrams give a remarkable contribution. There is no CP
violation for all these decays within the standard model, since there are only
tree operators contributions. The predicted branching ratios range from very
small numbers of $\mathcal{O}(10^{-8})$ up to the largest branching fraction
of $\mathcal{O}(10^{-5})$. Since all of the previous naive factorization
calculations did not give the theoretical uncertainty in the numerical
results, it is not easy to compare our results with theirs. The theoretical
uncertainty study in the pQCD approach shows that our numerical results are
reliable, which may be tested in the upcoming experimental measurements. We
predict the transverse polarization fractions of the $B_{c}$ decays with two
vector $D^{*}$ mesons in the final states for the first time. Due to the
cancelation of some hadronic parameters in the ratio, the polarization
fractions are predicted with less theoretical uncertainty. The transverse
polarization fractions are large in some channels, which mainly come from the
non-factorizable emission diagrams.
###### Acknowledgements.
We thank Hsiang-nan Li and Fusheng Yu for helpful discussions. This work is
partially supported by National Natural Science Foundation of China under the
Grant No. 11075168; Natural Science Foundation of Zhejiang Province of China,
Grant No. Y606252 and Scientific Research Fund of Zhejiang Provincial
Education Department of China, Grant No. 20051357.
## Appendix A Factorization formulas for $B_{c}\rightarrow VV$
In the $B_{c}$ decays to two vector meson final states, we use the superscript
L, N and T to denote the contributions from longitudinal polarization, normal
polarization and transverse polarization, respectively. For the CKM favored
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays, the decay amplitudes for
different polarizations are
$\displaystyle\mathcal{F}^{L}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(20)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{L}_{e}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})\times$
(21)
$\displaystyle\\{[1-x_{1}-x_{3}+r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[1-x_{1}-x_{2}+x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{F}^{L}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(22)
$\displaystyle\\{[1-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[1-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{M}^{L}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(23)
$\displaystyle\\{[x_{1}+x_{3}-1-r_{c}]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{b}-x_{2}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$
$\displaystyle\mathcal{F}^{N}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(24)
$\displaystyle\\{[2-r_{b}+r_{2}(4r_{b}-x_{2}-1)]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{T}_{e}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(25)
$\displaystyle\\{[2-r_{b}-r_{2}(1-x_{2})]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{N}_{e}$ $\displaystyle=$
$\displaystyle-\mathcal{M}^{T}_{e}=\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
(26)
$\displaystyle\times\\{[x_{1}+x_{3}-1]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{1}-x_{3}]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{F}^{N}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}r_{2}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(27)
$\displaystyle\\{[2-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[2-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{F}^{T}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}r_{2}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(28)
$\displaystyle\\{x_{2}\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})+$
$\displaystyle
x_{3}\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{M}^{N}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(29)
$\displaystyle\\{[r_{2}^{2}(x_{2}-1)+r_{3}^{2}(x_{3}-1)]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{2}^{2}x_{2}+r_{3}^{2}x_{3}-2r_{2}r_{3}r_{b}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$
$\displaystyle\mathcal{M}^{T}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(30)
$\displaystyle\\{[r_{2}^{2}(x_{2}-1)-r_{3}^{2}(x_{3}-1)]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{2}^{2}x_{2}-r_{3}^{2}x_{3}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\}.$
For the CKM suppressed $B_{c}\rightarrow D^{*+}_{(s)}D^{*0}$ decays, the decay
amplitudes for different polarizations are
$\displaystyle\mathcal{F}^{L}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(31)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{N}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(32)
$\displaystyle\\{[2-r_{b}+r_{2}(4r_{b}-x_{2}-1)]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{T}_{e2}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(33)
$\displaystyle\\{[2-r_{b}-r_{2}(1-x_{2})]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{L}_{e2}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(34)
$\displaystyle\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}+r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{M}^{N}_{e2}$ $\displaystyle=$
$\displaystyle-\mathcal{M}^{T}_{e2}=\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(35)
$\displaystyle\times\\{[r_{3}(x_{1}-x_{3})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]+$
$\displaystyle[2r_{c}-r_{3}(1-x_{1}-x_{3})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$
## Appendix B Scales and related functions in hard kernel
We show here the functions $h_{e}$, coming from the Fourier transform of hard
kernel,
$\displaystyle h_{e}(\alpha,\beta,b_{1},b_{2})$ $\displaystyle=$
$\displaystyle h_{1}(\alpha,b_{1})\times h_{2}(\beta,b_{1},b_{2}),$
$\displaystyle h_{1}(\alpha,b_{1})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}K_{0}(\sqrt{\alpha}b_{1}),&\quad\quad\alpha>0\\\
K_{0}(i\sqrt{-\alpha}b_{1}),&\quad\quad\alpha<0\end{array}\right.$ (38)
$\displaystyle h_{2}(\beta,b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\theta(b_{1}-b_{2})I_{0}(\sqrt{\beta}b_{2})K_{0}(\sqrt{\beta}b_{1})+(b_{1}\leftrightarrow
b_{2}),&\quad\beta>0\\\
\theta(b_{1}-b_{2})J_{0}(\sqrt{-\beta}b_{2})K_{0}(i\sqrt{-\beta}b_{1})+(b_{1}\leftrightarrow
b_{2}),&\quad\beta<0\end{array}\right.$ (41)
where $J_{0}$ is the Bessel function and $K_{0}$, $I_{0}$ are modified Bessel
function with $K_{0}(ix)=\frac{\pi}{2}(-N_{0}(x)+iJ_{0}(x))$. The hard scale t
is chosen as the maximum virtuality of the internal momentum transition in the
hard amplitudes, including $1/b_{i}(i=1,2,3)$:
$\displaystyle t_{a}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{a}|},1/b_{1},1/b_{2}),\quad
t_{b}=\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{b}|},1/b_{1},1/b_{2}),$
$\displaystyle t_{c}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{c}|},1/b_{2},1/b_{3}),\quad
t_{d}=\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{d}|},1/b_{2},1/b_{3}),$
$\displaystyle t_{e}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{e}|},1/b_{2},1/b_{3}),\quad
t_{f}=\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{f}|},1/b_{2},1/b_{3}),$
$\displaystyle t_{g}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{g}|},1/b_{1},1/b_{2}),\quad
t_{h}=\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{h}|},1/b_{1},1/b_{2}),$ (42)
where
$\displaystyle\alpha_{e}$ $\displaystyle=$
$\displaystyle(1-x_{2})(x_{1}-r_{2}^{2})(1-r_{3}^{2})M_{B}^{2},\quad\alpha_{a}=-(1+(r_{3}^{2}-1)x_{2})(1+(r_{2}^{2}-1)x_{3})M_{B}^{2},$
$\displaystyle\beta_{a}$ $\displaystyle=$
$\displaystyle[r_{b}^{2}+(r_{2}^{2}-1)(x_{2}+r_{3}^{2}(1-x_{2}))]M_{B}^{2},\quad\beta_{b}=(1-r_{3}^{2})(x_{1}-r_{2}^{2})M_{B}^{2},$
$\displaystyle\beta_{c}$ $\displaystyle=$
$\displaystyle[r_{c}^{2}-(1-x_{2}(1-r_{3}^{2}))(1-x_{1}-x_{3}(1-r_{2}^{2}))]]M_{B}^{2},$
$\displaystyle\quad\beta_{d}$ $\displaystyle=$
$\displaystyle(1-x_{2})(1-r_{3}^{2})[x_{1}-x_{3}-r_{2}^{2}(1-x_{3})]M_{B}^{2},$
$\displaystyle\beta_{e}$ $\displaystyle=$
$\displaystyle-[1+(r_{3}^{2}-1)x_{2}]M_{B}^{2},\quad\beta_{f}=-[1+(r_{2}^{2}-1)x_{3}]M_{B}^{2},$
$\displaystyle\beta_{g}$ $\displaystyle=$
$\displaystyle[r_{c}^{2}+(1-x_{2}(1-r_{3}^{2}))(x_{1}+x_{3}-1-r_{2}^{2}x_{3})]M_{B}^{2},\quad$
$\displaystyle\beta_{h}$ $\displaystyle=$
$\displaystyle[r_{b}^{2}-x_{2}(r_{3}^{2}-1)(x_{1}-x_{3}(1-r_{2}^{2}))]M_{B}^{2}.$
(43)
The Sudakov factors used in the text are defined by
$\displaystyle S_{ab}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{1})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+\frac{5}{3}\int_{1/b_{1}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{cd}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{3})$
$\displaystyle+\frac{11}{3}\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{3}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{ef}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{3})+2\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{3}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{gh}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{1})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{2}),$
(44)
$\displaystyle+\frac{5}{3}\int_{1/b_{1}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+4\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
where the functions $s(Q,b)$ are defined in Appendix A of epjc45711 .
$\gamma_{q}=-\alpha_{s}/\pi$ is the anomalous dimension of the quark.
## Appendix C Meson Wave functions
In the nonrelativistic limit, the $B_{c}$ meson wave function can be written
as prd81014022
$\displaystyle\Phi_{B_{c}}(x)=\frac{if_{B}}{4N_{c}}[(\hbox
to0.0pt{/\hss}{P}+M_{B_{c}})\gamma_{5}\delta(x-r_{c})]\exp(-\frac{b^{2}\omega_{B}^{2}}{2}),$
(45)
in which the last exponent term represents the $k_{T}$ distribution. Here, we
only consider the dominant Lorentz structure and neglect another contribution
in our calculation epjc28515 .
In the heavy quark limit, the two-particle light-cone distribution amplitudes
of $D_{(s)}/D_{(s)}^{*}$ meson are defined as prd67054028
$\displaystyle\langle D_{(s)}(P_{2})|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle$
$\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2N_{c}}}\int^{1}_{0}dxe^{ixP_{2}\cdot
z}[\gamma_{5}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}})\phi_{D_{(s)}}(x,b)]_{\alpha\beta},$
$\displaystyle\langle
D_{(s)}^{*}(P_{2})|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0}dxe^{ixP_{2}\cdot z}[\hbox
to0.0pt{/\hss}{\epsilon}_{L}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}^{*}})\phi^{L}_{D_{(s)}^{*}}(x,b)$ (46)
$\displaystyle+\hbox to0.0pt{/\hss}{\epsilon}_{T}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}^{*}})\phi^{T}_{D_{(s)}^{*}}(x,b)]_{\alpha\beta},$
with the normalization conditions:
$\displaystyle\int^{1}_{0}dx\phi_{D_{(s)}}(x,0)=\frac{f_{D_{(s)}}}{2\sqrt{2N_{c}}},\quad\int^{1}_{0}dx\phi^{L}_{D_{(s)}^{*}}(x,0)=\int^{1}_{0}dx\phi^{T}_{D_{(s)}^{*}}(x,0)=\frac{f_{D_{(s)}^{*}}}{2\sqrt{2N_{c}}},$
(47)
where we have assumed $f_{D_{(s)}^{*}}=f^{T}_{D_{(s)}^{*}}$. Note that
equations of motion do not relate $\phi^{L}_{D_{(s)}^{*}}$ and
$\phi^{T}_{D_{(s)}^{*}}$. We use the following relations derived from HQET
hqet to determine $f_{D^{*}_{(s)}}$
$\displaystyle
f_{D^{*}_{(s)}}=\sqrt{\frac{m_{D_{(s)}}}{m_{D_{(s)}^{*}}}}f_{D_{(s)}}.$ (48)
The distribution amplitude $\phi^{(L,T)}_{D_{(s)}^{(*)}}$ is taken as 09101424
$\displaystyle\phi^{(L,T)}_{D_{(s)}^{(*)}}=\frac{3}{\sqrt{2N_{c}}}f_{D^{(*)}_{(s)}}x(1-x)[1+a_{D^{(*)}_{(s)}}(1-2x)]\exp(-\frac{b^{2}\omega^{2}_{D_{(s)}}}{2}).$
(49)
We use $a_{D}=0.5\pm 0.1,\omega_{D}=0.1\text{GeV}$ for $D/D^{*}$ meson and
$a_{D}=0.4\pm 0.1,\omega_{D_{s}}=0.2\text{GeV}$ for $D_{s}/D_{s}^{*}$ meson,
which are determined in Ref. dd1 by fitting.
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|
arxiv-papers
| 2012-03-11T02:06:12 |
2024-09-04T02:49:28.507993
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhou Rui, Zou Zhitian and Cai-Dian Lu",
"submitter": "Cai-Dian Lu",
"url": "https://arxiv.org/abs/1203.2303"
}
|
1203.2322
|
# Cauchy’s residue theorem for a class of real valued functions
Branko Sarić Mathematical Institute, Serbian Academy of Sciences and Arts,
Knez Mihajlova 35, 11001 Belgrade, Serbia; College of Technical Engineering
Professional Studies, Svetog Save 65, 32 000 Čačak, Serbia bsaric@ptt.rs
(Date: June 08, 2009)
###### Abstract.
Let $\left[a,b\right]$ be an interval in $\mathbb{R}$ and let $F$ be a real
valued function defined at the endpoints of $[a,b]$ and with a certain number
of discontinuities within $\left[a,b\right]$. Having assumed $F$ to be
differentiable on a set $\left[a,b\right]\backslash E$ to the derivative $f$,
where $E$ is a subset of $\left[a,b\right]$ at whose points $F$ can take
values $\pm\infty$ or not be defined at all, we adopt the convention that $F$
and $f$ are equal to $0$ at all points of $E$ and show that
$\mathcal{KH-}vt\int_{a}^{b}f=F\left(b\right)-F\left(a\right)$, where
$\mathcal{KH-}$ $vt$ denotes the total value of the Kurzweil-Henstock
integral. The paper ends with a few examples that illustrate the theory.
###### Key words and phrases:
The Kurzweil-Henstock integral, Cauchy’s residue theorem
###### 1991 Mathematics Subject Classification:
Primary 26A39; Secondary 26A24, 26A30
The author’s research is supported by the Ministry of Science, Technology and
Development, Republic of Serbia (Project ON144002)
## 1\. Introduction
Let $\left[a,b\right]$ be some compact interval in $\mathbb{R}$. It is an old
result that for an ACGδ function $F:\left[a,b\right]\mapsto\mathbb{R}$ on
$\left[a,b\right]$, which is differentiable almost everywhere on
$\left[a,b\right]$, its derivative $f$ is integrable (in the Kurzweil-Henstock
sense) on $\left[a,b\right]$ and
$\mathcal{KH-}\int_{a}^{b}f=F\left(b\right)-F\left(a\right)$, [3, Theorem
9.17]. The aim of this note is to define a new definite integral named the
total Kurzweil-Henstock integral that can be used to extend the above
mentioned result to any real valued function $F$ defined and differentiable on
$\left[a,b\right]\backslash E$, where $E$ is a certain subset of
$\left[a,b\right]$ at whose points $F$ can take values $\pm\infty$ or not be
defined at all. Unless otherwise stated in what follows, we assume that the
endpoints of $\left[a,b\right]$ do not belong to $E$. Now, define point
functions $F_{ex}:\left[a,b\right]\mapsto\mathbb{R}$ and
$D_{ex}F:\left[a,b\right]\mapsto\mathbb{R}$ by extending $F$ and its
derivative $f$ from $\left[a,b\right]\backslash E$ to $E$ by
$F_{ex}\left(x\right)=0$ and $D_{ex}F\left(x\right)=0$ for $x\in E$, so that
(1.1) $F_{ex}\left(x\right)=\left\\{\begin{array}[]{c}F\left(x\right)\text{,
if }x\in\left[a,b\right]\backslash E\\\ 0\text{, if }x\in
E\end{array}\right.\text{ and}$
$D_{ex}F\left(x\right)=\left\\{\begin{array}[]{c}f\left(x\right)\text{, if
}x\in\left[a,b\right]\backslash E\\\ 0\text{, if }x\in
E\end{array}\right.\text{.}$
## 2\. Preliminaries
A partition $P\left[a,b\right]$ of $\left[a,b\right]\in\mathbb{R}$ is a finite
set (collection) of interval-point pairs
$\left\\{\left(\left[a_{i},b_{i}\right],x_{i}\right)\mid
i=1,...,\nu\right\\}$, such that the subintervals $\left[a_{i},b_{i}\right]$
are non-overlapping,
$\cup_{i\leq\nu}\left[a_{i},b_{i}\right]=\left[a,b\right]$ and
$x_{i}\in\left[a_{i},b_{i}\right]$. The points
$\left\\{x_{i}\right\\}_{i\leq\nu}$ are the tags of $P\left[a,b\right]$, [2].
It is evident that a given partition of $\left[a,b\right]$ can be tagged in
infinitely many ways by choosing different points as tags. If $E$ is a subset
of $\left[a,b\right]$, then the restriction of $P\left[a,b\right]$ to $E$ is a
finite collection of $\left(\left[a_{i},b_{i}\right],x_{i}\right)\in
P\left[a,b\right]$ such that each $x_{i}\in E$. In symbols,
$P\left[a,b\right]\left|{}_{E}\right.=\left\\{\left(\left[a_{i},b_{i}\right],x_{i}\right)\mid
x_{i}\in E,\,i=1,...,\nu_{E}\right\\}$. Let $\mathcal{P}\left[a,b\right]$ be
the family of all partitions $P\left[a,b\right]$ of $\left[a,b\right]$. Given
$\delta:\left[a,b\right]\mapsto\mathbb{R}_{+}$, named a gauge, a partition
$P\left[a,b\right]\in$ $\mathcal{P}\left[a,b\right]$ is called $\delta$-fine
if
$\left[a_{i},b_{i}\right]\subseteq\left(x_{i}-\delta\left(x_{i}\right),x_{i}+\delta\left(x_{i}\right)\right)$.
By Cousin’s lemma the set of $\delta$-fine partitions of $\left[a,b\right]$ is
nonempty, [4].
The collection $\mathcal{I}\left(\left[a,b\right]\right)$ is the family of
compact subintervals $I$ of $\left[a,b\right]$. The Lebesgue measure of the
interval $I$ is denoted by $\left|I\right|$. Any real valued function defined
on $\mathcal{I}\left(\left[a,b\right]\right)$ is an interval function. For a
function $f:\left[a,b\right]\mapsto\mathbb{R}$, the associated interval
function of $f$ is an interval function
$f:\mathcal{I}\left(\left[a,b\right]\right)\mapsto\mathbb{R}$, again denoted
by $f$, [5]. If $f\equiv 0$ on $\left[a,b\right]$ then its associated interval
function is trivial.
A function $f:\left[a,b\right]\mapsto\mathbb{R}$ is said to be Kurzweil-
Henstock integrable to a real number $A$ on $\left[a,b\right]$ if for every
$\varepsilon>0$ there exists a gauge
$\delta_{\varepsilon}:\left[a,b\right]\mapsto\mathbb{R}_{+}$ such that
$\left|\sum_{i\leq\nu}[f\left(x_{i}\right)\left|\left[a_{i},b_{i}\right]\right|]-A\right|<\varepsilon$,
whenever $P\left[a,b\right]$ is a $\delta_{\varepsilon}$-fine partition of
$\left[a,b\right]$. In symbols, $A=\mathcal{KH-}\int_{a}^{b}f$.
## 3\. Main results
In what follows we will use the following notations
(3.1)
$\Xi_{f}\left(P\left[a,b\right]\right)=\sum_{i\leq\nu}[f\left(x_{i}\right)\left|b_{i}-a_{i}\right|]\text{
and
}\Sigma_{\Phi}\left(P\left[a,b\right]\right)=\sum_{i\leq\nu}[\Phi\left(b_{i}\right)-\Phi\left(a_{i}\right)]\text{.}$
Now, we are in a position to introduce the total Kurzweil-Henstock integral.
###### Definition 1.
For any compact interval $\left[a,b\right]\in\mathbb{R}$ let $E$ be a non-
empty subset of $\left[a,b\right]$. A function
$f:\left[a,b\right]\mapsto\mathbb{R}$ is said to be totally Kurzweil-Henstock
integrable to a real number $\Im$ on $\left[a,b\right]$ if there exists a
nontrivial interval function
$\Phi:\mathcal{I}\left(\left[a,b\right]\right)\mapsto\mathbb{R}$ with the
following property: for every $\varepsilon>0$ there exists a gauge
$\delta_{\varepsilon}$ on $\left[a,b\right]$ such that
$\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)\right|<\varepsilon$ and
$\Sigma_{\Phi}\left(P\left[a,b\right]\right)=\Im$, whenever
$P\left[a,b\right]\in\mathcal{P}\left[a,b\right]$ is a
$\delta_{\varepsilon}$-fine partition and
$P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash E}\right.$ is its
restriction to $\left[a,b\right]\backslash E$. In symbols,
$\mathcal{KH-}vt\int_{a}^{b}f=\Im$.
###### Definition 2.
Let $E$ be a non-empty subset of $\left[a,b\right]$. Then, an interval
function $\Phi:\mathcal{I}\left(\left[a,b\right]\right)\mapsto\mathbb{R}$ is
said to be basically summable $($BS${}_{\delta_{\varepsilon}})$ to the sum
$\mathbf{\Re}$ on $E$ if there exists a real number $\mathbf{\Re}$ with the
following property: given $\varepsilon>0$ there exists a gauge
$\delta_{\varepsilon}$ on $\left[a,b\right]$ such that
$\left|\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{E}\right.\right)-\Re\right|<\varepsilon$,
whenever $P\left[a,b\right]\in\mathcal{P}\left[a,b\right]$ is a
$\delta_{\varepsilon}$-fine partition and
$P\left[a,b\right]\left|{}_{E}\right.$ is its restriction to $E$. If $E$ can
be written as a countable union of sets on each of which the interval function
$\Phi$ is BS${}_{\delta_{\varepsilon}}$, then $\Phi$ is said to be
BSG${}_{\delta_{\varepsilon}}$ on $E$.
Our main result reads as follows.
###### Theorem 1.
For any compact interval $\left[a,b\right]\in\mathbb{R}$ let $E$ be a non-
empty subset of $\left[a,b\right]$ at whose points a real valued function $F$
can take values $\pm\infty$ or not be defined at all. If $F$ is defined and
differentiable on the set $\left[a,b\right]\backslash E$, then $D_{ex}F$ is
totally Kurzweil-Henstock integrable on $\left[a,b\right]$ and
(3.2)
$\mathcal{KH-}vt\int_{a}^{b}D_{ex}F=F\left(b\right)-F\left(a\right)\text{.}$
If the associated interval function of $F_{ex}$ defined by (1.1) is in
addition basically summable $($BS${}_{\delta_{\varepsilon}})$ to the sum
$\mathbf{\Re}$ on $E$, then
(3.3)
$F\left(b\right)-F\left(a\right)=\mathcal{KH-}\int_{a}^{b}D_{ex}F+\mathbf{\Re}.$
Before starting with the proof we give the following lemma.
###### Lemma 1.
Let $E$ be a non-empty subset of $\left[a,b\right]$. If a function
$f:\left[a,b\right]\mapsto\mathbb{R}$ is totally Kurzweil-Henstock integrable
on $\left[a,b\right]$ and $\Phi$ is basically summable
$($BS${}_{\delta_{\varepsilon}})$ to the sum $\mathbf{\Re}$ on $E$, then $f$
is Kurzweil-Henstock integrable on $\left[a,b\right]$ and
(3.4)
$\mathcal{KH-}vt\int_{a}^{b}f=\mathcal{KH-}\int_{a}^{b}f+\mathbf{\Re}\text{.}$
###### Proof.
Given $\varepsilon>0$ we will construct a gauge for $f$ as follows. Since $f$
is totally Kurzweil-Henstock integrable on $\left[a,b\right]$ it follows from
Definition 1 that there exist a real number $\Im$ and an interval function
$\Phi$ with the following property: for every $\varepsilon>0$ there exists a
gauge $\delta_{\varepsilon}^{\ast}$ on $\left[a,b\right]$ such that
$\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)\right|<\varepsilon$ and
$\Sigma_{\Phi}\left(P\left[a,b\right]\right)=\Im$, whenever
$P\left[a,b\right]\in\mathcal{P}\left[a,b\right]$ is a
$\delta_{\varepsilon}^{\ast}$-fine partition and
$P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash E}\right.$ is its
restriction to $\left[a,b\right]\backslash E$. Choose a gauge
$\delta_{\varepsilon}^{\star}\left(x\right)$ as required in Definition 2
above. The function
$\delta_{\varepsilon}=min\left(\delta_{\varepsilon}^{\ast},\delta_{\varepsilon}^{\star}\right)$
is a gauge on $\left[a,b\right]$. We now let
$P\left[a,b\right]=\left\\{\left(\left[a_{i},b_{i}\right],x_{i}\right)\mid
i=1,...,\nu\right\\}$ be a $\delta_{\varepsilon}$-fine partition of
$\left[a,b\right]$. It is readily seen that
$\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Im+\mathbf{\Re}\right|=$
$=\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Im+\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{E}\right.\right)-[\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{E}\right.\right)-\mathbf{\Re]}\right|\leq$
$\leq\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)\right|+\left|\Sigma_{\Phi}\left(P\left[a,b\right]\left|{}_{E}\right.\right)-\Re\right|<2\varepsilon\text{.}$
Therefore, $f$ is Kurzweil-Henstock integrable on $\left[a,b\right]$ and
$\mathcal{KH-}\int_{a}^{b}f=\Im-\mathbf{\Re}$, that is
$\mathcal{KH-}vt\int_{a}^{b}f=\mathcal{KH-}\int_{a}^{b}f+\mathbf{\Re}\text{.}$
We now turn to the proof of Theorem 1.
###### Proof.
Given $\varepsilon>0$. By definition of $f$ at the point
$x\in\left[a,b\right]\backslash E$, given $\varepsilon>0$ there exists
$\delta_{\varepsilon}\left(x\right)>0$ such that if
$x\in\left[u,v\right]\subseteq\left[x-\delta_{\varepsilon}\left(x\right),x+\delta_{\varepsilon}\left(x\right)\right]$
and $x\in\left[a,b\right]\backslash E$, then
$\left|F\left(v\right)-F\left(u\right)-f\left(x\right)\left(v-u\right)\right|<\varepsilon\left(v-u\right)\text{.}$
For $F_{ex}$ defined by (1.1) let
$F_{ex}:\mathcal{I}\left(\left[a,b\right]\right)\mapsto\mathbb{R}$ be its
associated interval function. We now let
$P\left[a,b\right]=\left\\{\left(\left[a_{i},b_{i}\right],x_{i}\right)\mid
i=1,...,\nu\right\\}$ be a $\delta_{\varepsilon}$-fine partition of
$\left[a,b\right]$. Since
$F\left(b\right)-F\left(a\right)=\sum_{i=1}^{\nu}\left[F_{ex}\left(b_{i}\right)-F_{ex}\left(a_{i}\right)\right]$
and (remember if $x\in E$, then $D_{ex}F=0$)
$\left|\Xi_{f}\left(P\left[a,b\right]\right)-\Sigma_{F}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)\right|=$
$=\left|\Xi_{f}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)-\Sigma_{F}\left(P\left[a,b\right]\left|{}_{\left[a,b\right]\backslash
E}\right.\right)\right|<\varepsilon\left(b-a\right)\text{,}$
it follows from Definition 1 that $D_{ex}F$ is totally Kurzweil-Henstock
integrable on $\left[a,b\right]$ and
$\mathcal{KH-}vt\int_{a}^{b}D_{ex}F=F\left(b\right)-F\left(a\right)\text{.}$
Finally, based on the result of Lemma 1
$F\left(b\right)-F\left(a\right)=\mathcal{KH-}\int_{a}^{b}D_{ex}F+\mathbf{\Re}\text{.}$
By Definition 2 one can easily see that if $\mathbf{\Re}=0$ then $F$ has
negligible variation on $E$, [1, Definition 5.11]. So, we now in position to
define a residual function of $F$.
###### Definition 3.
Let $F:\left[a,b\right]\mapsto\mathbb{R}$. A function
$\mathcal{R}:\left[a,b\right]\mapsto\mathbb{R}$ is said to be a residual
function of $F$ on $\left[a,b\right]$ if given $\varepsilon>0$ there exists a
gauge $\delta_{\varepsilon}$ on $\left[a,b\right]$ such that
$\left|F\left(b_{i}\right)-F\left(a_{i}\right)-\mathcal{R}\left(x_{i}\right)\right|<\varepsilon$,
whenever $P\left[a,b\right]\in\mathcal{P}\left[a,b\right]$ is a
$\delta_{\varepsilon}$-fine partition.
###### Definition 4.
Let $E$ be a non-empty subset of $\left[a,b\right]$ and let
$F:\left[a,b\right]\mapsto\mathbb{R}$ be a function whose associated interval
function $F:\mathcal{I}\left(\left[a,b\right]\right)\mapsto\mathbb{R}$ is
BS${}_{\delta_{\varepsilon}}$ $($BSG${}_{\delta_{\varepsilon}})$ to the sum
$\mathbf{\Re}$ on $E$. Then, a residual function
$\mathcal{R}:\left[a,b\right]\mapsto\mathbb{R}$ of $F$ is said to be also
BS${}_{\delta_{\varepsilon}}$ $($BSG${}_{\delta_{\varepsilon}})$ to the same
sum $\mathbf{\Re}$ on $E$. In symbols, $\sum_{x\in
E}\mathcal{R}\left(x\right)=\mathbf{\Re}$.
Clearly, Definition 4 establishes a causal connection between Definitions 2
and 3. If $E$ is a countable set, the causality is so obvious. However, if $E$
is an infinite set, then this connection is not necessarily a causal
connection. Namely, if $F:\left[a,b\right]\mapsto\mathbb{R}$ has negligible
variation on some subset $E$ of $\left[a,b\right]$, which is a countably
infinite set, then its residual function $\mathcal{R}$ vanishes identically on
$E$, so that the sum $\sum_{x\in E}\mathcal{R}\left(x\right)$ is reduced to
the so-called indeterminate expression $\infty\cdot 0$ that have, in this
case, the null value. On the contrary, if $F$ has no negligible variation on
$E$, and its residual function $\mathcal{R}$ also vanishes identically on $E$,
as in the case of the Cantor function, then the sum $\sum_{x\in
E}\mathcal{R}\left(x\right)$ is reduced to the indeterminate expression
$\infty\cdot 0$ that actually have, in Cantor’s case, the numerical value of
$1$. By Definition 4, we may rewrite (3.3) as follows,
(3.5)
$F\left(b\right)-F\left(a\right)=\mathcal{KH-}\int_{a}^{b}D_{ex}F+\sum_{x\in
E}\mathcal{R}\left(x\right)\text{.}$
If $f$ in addition vanishes identically on $\left[a,b\right]\backslash E$,
then
(3.6) $F\left(b\right)-F\left(a\right)=\sum_{x\in
E}\mathcal{R}\left(x\right).$
The previous result is an extension of Cauchy’s residue theorem result in
$\mathbb{R}$.
## 4\. Examples
For an illustration of (3.5) and (3.6) we consider the Heaviside unit function
defined by
(4.1) $F\left(x\right)=\left\\{\begin{array}[]{c}0\text{, if }a\leq x\leq 0\\\
1\text{, if }0<x\leq b\end{array}\right.\text{.}$
In this case, if $a<0$, then $\mathcal{KH-}vt\int_{a}^{b}D_{ex}F=1$, in spite
of the fact that $D_{ex}F\equiv 0$ on $\left[a,b\right]$. Accordingly, it
follows from (3.5) and (3.6) that $\mathcal{R}\left(0\right)=1$, since
(4.2) $f\left(x\right)=\left\\{\begin{array}[]{c}+\infty\text{, if }x=0\\\
0\text{, otherwise}\end{array}\right.\text{,}$
where $f$ is the derivative of $F$, and $\mathcal{KH-}\int_{a}^{b}D_{ex}F=0$.
Let $\left[a,b\right]\subset\mathbb{R}$ be an arbitrary compact interval
within which is the point $x=0$. For an illustration of the result (3.2) of
Theorem 1 we consider the real valued function $F\left(x\right)=1/x$ that is
differentiable to $f\left(x\right)=-\left(1/x^{2}\right)$ at all but the
exceptional set $\left\\{0\right\\}$ of $\left[a,b\right]$. In spite of the
fact that $f$ is not Kurzweil-Henstock integrable on $\left[a,b\right]$ it
follows from (3.2) that
$\mathcal{KH-}vt\int_{a}^{b}D_{ex}F=\left(a-b\right)/\left(ab\right)$. In this
case, $\mathcal{R}\left(x\right)$ is not defined at the point $x=0$, that is
(4.3) $\mathcal{R}\left(x\right)=\left\\{\begin{array}[]{c}+\infty\text{, if
}x=0\\\ 0\text{, otherwise}\end{array}\right.\text{,}$
and $\mathcal{KH-}vt\int_{a}^{b}D_{ex}F$ is reduced to the so-called
indeterminate expression $\infty-\infty$ (in the sense of the difference of
limits) that actually have, in this situation, the real numerical value of
$\left(a-b\right)/\left(ab\right)$.
## References
* [1] R. G. Bartle: A Modern Theory of Integration. Graduate Studies in Math., Vol. 32, AMS, Providence, 2001. Zbl 0968.26001
* [2] I. J. L. Garces, P. Y. Lee: Convergence theorem for the $H_{1}$-integral. Taiw. J. Math. Vol. 4 No. 3 (2000), 439–445. Zbl 0973.26008
* [3] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron and Henstock, Graduate Studies in Math., Vol. 4, AMS, Providence, 1994. Zbl 0807.26004
* [4] A. Macdonald: Stokes’ theorem, Real Analysis Exchange 27 (2002), 739–747. Zbl 1059.26008
* [5] V. Sinha, I. K. Rana: On the continuity of associated interval functions, Real Analysis Exchange 29(2) (2003/2004), 979–981. Zbl 1073.26005
|
arxiv-papers
| 2012-03-11T10:37:56 |
2024-09-04T02:49:28.516894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Branko Sari\\'",
"submitter": "Branko Saric",
"url": "https://arxiv.org/abs/1203.2322"
}
|
1203.2420
|
# Recent results from ALICE
Yuri Kharlov, for the ALICE collaboration Institute for High Energy Physics,
Protvino, 142281 Russia
###### Abstract
The ALICE experiment at the LHC has collected wealthy data in proton-proton
and lead-lead collisions. An overview of recent ALICE results is given in this
paper. Hadron spectra measured in pp collisions at $\sqrt{s}=0.9$, 2.76 and 7
TeV are discussed. Properties of hot nuclear matter produced in Pb-Pb
collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV, revealed via many observables
measured with the ALICE experiment, are shown.
## 1 Introduction
The ALICE experiment was designed to study interactions of heavy ions at the
LHC. This goal determines the unique performance of the ALICE detectors to
reconstruct events with very high multiplicity and to measure spectra of
identified hadrons, electrons, photons, muons in a wide energy range.
The data collected with the ALICE experiment with pp collisions at
$\sqrt{s}=7$ TeV in 2010–2011, consist of a minimum-bias sample with
integrated luminosity $\int{\cal L}dT=16\leavevmode\nobreak\ \mbox{nb}^{-1}$
and a sample recorded with rare-event triggers with integrated luminosity
$\int{\cal L}dT=4.9\leavevmode\nobreak\ \mbox{pb}^{-1}$. Rare-event triggers
implemented pp collisions were based in EMCAL, PHOS and MUON detectors.
Limited data samples with the proton beams at collision energies
$\sqrt{s}=0.9$ and 2.76 TeV have been also recorded with integrated
luminosities $\int{\cal L}dT=0.14\mbox{\leavevmode\nobreak\
and\leavevmode\nobreak\ }1.3\leavevmode\nobreak\ \mbox{nb}^{-1}$ respectively.
The LHC has delivered heavy-ion collisions at the center-of-mass energy
$\sqrt{s_{{}_{NN}}}=2.76$ TeV to the ALICE experiment in 2010 with integrated
luminosity $\int{\cal L}dT=10\leavevmode\nobreak\ \mu\mbox{b}^{-1}$, and the
data set of 2011 exceeded the previous one by an order of magnitude. Data
taking of heavy ion collisions recorded in 2010 was dominated by the minimum
bias trigger. In 2011, a fraction of minimum bias events was suppressed in
favor of the triggers on the most central and semi-central events, as well as
rare events which selected events with high-energy clusters in the
electromagnetic calorimeters, muon tracks in the muon spectrometer, ultra-
peripheral collisions.
## 2 QCD tests in proton-proton collisions
Properties of hot nuclear matter produced in heavy ion collisions are studied
via a comprehensive set of observables. As a reference, similar observables
are measured in proton-proton collisions. ALICE is performing detailed studies
of hadron production spectra in pp collisions at all center-mass energies
provided by the LHC. Apart of being a reference for heavy ion collisions, pp
collisions is considered as a powerful tool for QCD studying. Advance particle
identification capabilities Aamodt:2008zz and a moderate magnetic field
($B=0.5$ T) allow to measure a variety of hadron spectra in a wide momentum
range.
Identified charged hadron production in mid-rapidity are measured by the
central tracking system consisting of the Inner Tracking System detector
(ITS), Time Projection Chamber (TPC), Time-of-Flight detector (TOF) and a
Cherenkov High-Momentum Particle Identification detector (HMPID). Each of
these detectors provide particle identification in different complimentary
momentum ranges, which allows to measure the spectra in a wide $p_{\rm t}$
range. ALICE has already published production spectra of $\pi^{\pm}$,
$K^{\pm}$, p, $\bar{\mbox{p}}$ in pp collisions at $\sqrt{s}=0.9$ TeV
PIDhadron900GeV , and has reported preliminary results on those spectra in pp
collisions at $\sqrt{s}=7$ TeV PIDhadron7TeV . Similar to charged hadrons,
ALICE is able to measure neutral meson spectra by complimentary and redundant
methods which ensures the result validity. Neutral pion and $\eta$-meson
spectra were measured in pp collisions at all three LHC energies by the Photon
Spectrometer (PHOS) which detected real photons and by the central tracking
system which identifies photons converted to $e^{+}e^{-}$ pairs on the medium
of the inner ALICE detectors pp-pi0 . Combined analysis of all ALICE detectors
allowed to measure spectra of resonance production, in particular strange and
charmed hadrons.
Results obtained by the ALICE on hadron production in pp collisions show a
gradual increase of the mean transverse momentum with collision energy.
Observed ratio between antiprotons and protons suggests that baryon-antibaryon
asymmetry is restoring at high energies. Comparison of all measured spectra
with Monte Carlo event generators and with next-to-leading perturbative QCD
calculations demonstrates that no model can correctly describe spectra of
hadron production at LHC energies.
## 3 Global event properties in Pb-Pb collisions
Centrality of the collision, directly related to the impact parameter and to
the number of nucleons $N_{\rm part}$ participating in the collision, allows
to study particle production versus density of the colliding system. Various
ALICE detectors measure collision centrality, among them the best accuracy is
achieved with the scintillator hodoscope VZERO covering pseudorapidity ranges
$2.8<\eta<5.1$ and $-3.7<\eta<-1.7$. Distribution of the sum of amplitudes in
VZERO in minimum bias Pb-Pb collisions is shown in Fig.1 bib:PbPb-dNdy .
Centrality classes were defined by the Glauber model, and the fit of the
Glauber model to the data is shown by a solid line in this plot.
Figure 1: Centrality determination in ALICE. Glauber model fit to the VZERO
amplitude with the inset of a zoom of the most peripheral region.
Charged particle multiplicity in Pb-Pb collisions at $\sqrt{s_{{}_{NN}}}=2.76$
TeV and its dependence on the collision centrality was measured with the
Silicon Pixel Detector (SPD), two innermost layers of the barrel tracking
system covering the pseudorapidity range $|\eta|<1.4$. The charged particle
density, normalized to the average number of participants in a given
centrality class, $dN_{\rm ch}/d\eta/\left(\langle N_{\rm
part}\rangle\right)$, measured by ALICE, was compared with similar
measurements at lower energies at RHIC and SPS (Fig.2) bib:PbPb-dNdy_central .
Figure 2: Charged track density $dN/d\eta$ in pp and AA collisions vs
collision energy.
In the most central events (centrality $0-5\%$) at LHC energy the charged
particle density was found to be $dN_{\rm ch}/d\eta=1601\pm 60$ which is,
being normalized to the number of participants, is 2.1 times larger than the
charged particle density measured at RHIC at $\sqrt{s_{{}_{NN}}}=200$ GeV and
1.9 times larger than that in pp collisions at $\sqrt{s}=2.36$ TeV.
## 4 Collective phenomena in heavy ion collisions
The initial anisotropy of nuclei non-central collisions leads to anisotropic
distribution on initial matter in the overlapping reagion. During evolution of
the matter, the spatial asymmetry of initial state is converted to an
anisotropic momentum distribution. The azimuthal distribution of the particle
yield can be described by a Fourier series of a function of the angle between
the particle direction $\varphi$ and the reaction place $\Psi_{\rm RP}$. The
second coefficient of this series, $v_{2}$, is referred to as elliptic flow.
Theoretical models, based on relativistic hydrodynamics bib:hydro-v2_Kestin ;
bib:hydro-v2_Niemi , successfully described the elliptic flow observed at RHIC
and predict its increase at LHC energies from 10% to 30%.
The first measurements of elliptic flow of charged particles in Pb-Pb
collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV were reported by ALICE in
bib:ALICE-v2 . Charged tracks were detected and reconstructed in the central
barrel tracking system, consisting of ITS and TPC. Comparison of elliptic flow
integrated over $p_{\rm t}$ measured by the ALICE and lower-energy experiments
is shown in Fig.3.
Figure 3: Azimuthal flow $v_{2}$ of charged particles measured by the ALICE in
Pb-Pb collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV in comparison with the
lower-energy experiment results.
The observed trend of $v_{2}$ vs $\sqrt{s_{{}_{NN}}}$ confirms model
expectations that the value of $v_{2}$ in Pb-Pb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV increases by about 30% with respect to $v_{2}$
in Au-Au collisions at $\sqrt{s_{{}_{NN}}}=0.2$ TeV.
The results of global event properties and collective expantion studied by
ALICE, studied via the azimuthal anisotropy and intensity interferometry of
identical particles bib:HBT , indicate that the fireball formed in nuclear
collisions at the LHC is hotter, lives longer, and expands to a larger size at
freeze-out as compared to lower energies.
## 5 Strangeness production in heavy ion collisions
Strange particle production has been considered as a probe of strongly
interacting matter by heavy-ion experiments at AGS, SPS and RHIC. We have
already demonstrated that ALICE, due to its powerful particle identification
technique, has measured strange particle spectra in pp collisions. Similar
analysis was performed on the Pb-Pb data collected in 2010. Comparison of
strange meson and baryon production is illustrated by the $\Lambda/K^{0}_{S}$
ratio measured by ALICE in different centrality classes (Fig.4, left). This
ratio in peripheral Pb-Pb collision is similar to that one measured in pp
collisions, but it grows with centrality, increasing the value of 1.5 in the
most central collisions. The qualitative behaviour of this ratio on $p_{\rm
t}$ at the LHC collision energy is similar to the ratio measured at RHIC by
the STAR experiment (Fig.4, right). An enhancement of strange and multi-
strange baryons ($\Omega^{-}$, $\bar{\Omega}^{+}$,
$\Sigma^{-}$,$\bar{\Sigma}^{+}$ ) was obsevred in heavy-ion collisions by
experiments at lower energies, and was confirmed by ALICE at LHC energy ALICE-
Hippolyte . It was also shown that multi-strange baryon enhancement scales
with the number of participants $N_{\rm part}$ and decreases with the
collision energy.
The large yield of strange, and especially multi-strange baryons in heavy-ion
collision was observed earlier at SPS and RHIC. This effect supports
predictions of quark-gluon plasma formation which assumed that strange
antiquarks are as abundant as light antiquarks in quark matter. The strange
quark phase space becomes fully equilibrated, and therefore all strange
hadrons are produced more abundantly. An overview of strangeness production in
heavy ion collisions can be found in bib:Muller2011 .
Figure 4: Ratio $\Lambda/K^{0}_{S}$ in Pb-Pb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV in different centralities (left) and comparison
of this ratio at LHC and RHIC in centralities $0-5\%$ and $60-80\%$ (right).
## 6 Parton energy loss in medium
Final-state partons produced at the initial stage of nucleus-nucleus
collision, pass through medium with multiple secondary interactions. Energy
loss by partons depends on density and temperature of the QCD medium. Hadrons
produced in fragmentation of these partons should be suppressed compared to
expectations from an independent superposition of nucleon-nucleon collisions.
The strength of suppression of a hadron $h$ is expressed by the nuclear
modification factor $R_{AA}$, defined as a ratio of the particle spectrum in
heavy-ion collision to that in pp, scaled by the number of binary nucleon-
nucleon collisions $N_{\rm coll}$:
$R_{AA}(p_{\rm t})=\frac{(1/N_{AA})d^{2}N_{h}^{AA}/dp_{\rm t}d\eta}{N_{\rm
coll}(1/N_{pp})d^{2}N_{h}^{pp}/dp_{\rm t}d\eta}.$ (1)
Experiments at RHIC reported that hadron production at high transverse
momentum in central Au-Au collisions at a center-of-mass energy per nucleon
pair $\sqrt{s_{{}_{NN}}}=200$ GeV is suppressed by a factor $4-5$ with respect
to pp collisions. At the larger LHC energy, the density of the medium is
expected to be higher than at RHIC, leading to a larger energy loss of
high-$p_{\rm t}$ partons. However, the hadron production spectra are less
steeply falling with $p_{\rm t}$ at LHC than at RHIC which would reduce the
value of $R_{AA}$ for a given value of the parton energy loss.
ALICE has measured the nuclear modification factor $R_{AA}$ for many
particles. All charged particles, detected in the ALICE central tracking
system (ITS and TPC), show a spectrum suppression Otwinowski:2011gq which is
qualitatively similar to that observed at RHIC (Fig.5).
Figure 5: Nuclear modification factor $R_{AA}$ of charged particles.
However, quantitative comparison with RHIC demonstrates that the suppression
at LHC energy is stronger which can be interpreted by a denser medium.
Benefiting from particle identification which has been already mention earlier
in this paper, ALICE has measured suppression of various identified hadrons,
which provides experimental data for studying the flavor and mass dependence
of the spectra suppression.
A nuclear modification factor $R_{AA}$ of charged pion production in mid-
rapidity (Fig.6) has lower values in the range of moderate transverse momenta
($3<p_{\rm t}<7-10$ GeV/$c$) than that of unidentified charged particles, but
at higher $p_{\rm t}$ it coincides with all charged particles.
Figure 6: Nuclear modification factor $R_{AA}$ of charged pions.
To the contrary to charged pions, strange hadrons ($K^{0}_{S}$, $\Lambda$) are
less suppressed in the most central collisions compared to all charged
particles (Fig.7). This is explained by the fact that strange quark production
is enhanced in a hot nuclear medium, and this strangeness enhancement
partially compensates energy loss of strange quarks, such that the overall
$R_{AA}$ value becomes larger than for pions. Lambda hyperons have no
suppression at $p_{\rm t}<3-4$ GeV/$c$, which is interpreted by an additional
baryon enhancement in central heavy-ion collisions.
ALICE has reported also the first measurements of $D$ meson suppression
bib:PbPb-Dmesons in Pb-Pb collisions in two centrality classes, $0-20\%$ and
$40-80\%$, shown in Fig.7. It was shown that the $R_{AA}$ values for $D^{0}$,
$D^{+}$ and $D^{*+}$ are consistent with each other within the statistical and
systematical uncertainties. Although the statistics of the ALICE run 2010 is
marginal for $D$ meson measurement, the obtained result shows a hint that the
$D$ mesons are less suppressed than charged pions.
Figure 7: Nuclear modification factor $R_{AA}$ of charged particles, $K^{0}$,
$\Lambda$, $\pi^{\pm}$, $D^{+}$, $D^{0}$, $D^{*+}$ in central (left) and
peripheral (right) collisions.
Nuclear modification factor $R_{AA}$ is $J/\psi$ production in Pb-Pb
collisions was measured by the ALICE in two kinematic regions: in the forward
rapidity with the muon spectrometer and in mid-rapidity deploying the central
tracking system bib:PbPb-Jpsi . The result of $R_{AA}$ measurement in forward
rapidity shows almost no dependence on collision centrality with the average
value $R_{AA}=0.545\pm 0.032\leavevmode\nobreak\ \mbox{(stat.)}\pm
0.084\mbox{(syst.)}$ which is significantly different from the RHIC results.
## 7 Conclusion
The ALICE collaboration is performing QCD studies via hadron production
measurements in proton-proton collisions. Obtained results in pp collisions at
$\sqrt{s}=0.9,2.76$ and 7 TeV show statistically significant deviations from
models which well described lower-energy results. Therefore new experimental
results from pp collision allow to tune various phenomenological models and
pQCD calculations.
Comprehensive studies of heavy-ion collisions, performed by the ALICE
experiment show that the properties of strongly interacting nuclear matter
produced at the LHC energy, qualitatively similar to those observed at RHIC
and reveal smooth evolution with collision energy. The matter produced at LHC
has about 3 times larger energy density, twice larger volume of homogeneity
and about 20% larger lifetime. Like at RHIC, the matter at LHC reveals the
properties on an almost perfect liquid. Particle suppression appeared to be
stronger at LHC than at RHIC which is also an evidence of denser medium
produced at LHC.
This work was parially supported by the RFBR grant 10-02-91052.
## References
* (1) K. Aamodt et al. [ALICE Collaboration], JINST 3, S08002 (2008).
* (2) K. Aamodt et al. [ALICE Collaboration], Eur.Phys.J.C 71(6), 1655, (2011).
* (3) R. Preghenella, for the ALICE Collaboration. arXiv:1111.7080v1 [hep-ex]. B.Guerzoni, this issue.
* (4) ALICE collaboration, CERN-PH-EP-2012-001 (2012).
* (5) K.Aamodt et al., ALICE collaboration. Phys.Rev.Lett., 106, 032301 (2011).
* (6) K.Aamodt et al., ALICE collaboration. Phys.Rev.Lett., 105, 252301 (2010).
* (7) G. Kestin and U.W. Heinz, Eur. Phys. J. C 61, 545 (2009).
* (8) H. Niemi, K. J. Eskola, and P.V. Ruuskanen, Phys. Rev. C 79, 024903 (2009).
* (9) K.Aamodt et al., ALICE collaboration. Phys.Rev.Lett., 105, 252302 (2010).
* (10) K.Aamodt et al., ALICE collaboration. Physics Letters B 696, 328 (2011).
* (11) B.Hippolyte for the ALICE collaboration. arXiv:1112.5803 [nucl-ex].
* (12) B.Muller. arXiv:1112.5382v1 [nucl-th].
* (13) J. Otwinowski [ALICE Collaboration], J. Phys. G G 38, 124112 (2011). [arXiv:1110.2985 [hep-ex]].
* (14) A.Grelli for the ALICE collaboration. J. Phys. Conf. Ser. 316, 012025 (2011). R.Averbeck for the ALICE collaboration, this issue.
* (15) F.Bossu for the ALICE collaboration, this issue.
|
arxiv-papers
| 2012-03-12T07:58:18 |
2024-09-04T02:49:28.526084
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuri Kharlov (for the ALICE collaboration)",
"submitter": "Yuri Kharlov",
"url": "https://arxiv.org/abs/1203.2420"
}
|
1203.2503
|
# The expected value under the Yule model of the squared path-difference
distance
Gabriel Cardona gabriel.cardona@uib.es Arnau Mir arnau.mir@uib.es Francesc
Rosselló cesc.rossello@uib.es Department of Mathematics and Computer Science,
University of the Balearic Islands, E-07122 Palma de Mallorca, Spain
###### Abstract
The path-difference metric is one of the oldest and most popular distances for
the comparison of phylogenetic trees, but its statistical properties are still
quite unknown. In this paper we compute the expected value under the Yule
model of evolution of its square on the space of fully resolved rooted
phylogenetic trees with $n$ leaves. This complements previous work by
Steel–Penny and Mir–Rosselló, who computed this mean value for fully resolved
unrooted and rooted phylogenetic trees, respectively, under the uniform
distribution.
###### keywords:
Phylogenetic tree, Nodal distance, Path-difference metric, Yule model, Sackin
index
## 1 Introduction
The definition and study of metrics for the comparison of rooted phylogenetic
trees on the same set of taxa is a classical problem in phylogenetics [6, Ch.
30]. A classical and popular family of such metrics is based on the
comparison, by different methods, of the vectors of lengths of the
(undirected) paths connecting all pairs of taxa in the corresponding trees [4,
5, 14, 20]. These metrics are generically called _nodal distances_ , although
some of them have also specific names. For instance, the metric defined
through the euclidean distance between path-lengths vectors is called _path-
difference metric_ [18], or _cladistic difference_ [4].
In contrast with other metrics, the statistical properties of these nodal
distances are mostly unknown. Actually, the only statistical property that has
been established so far for any one of them is the expected, or mean, value of
the square of the path-difference metric for unrooted [18] and rooted [11]
fully resolved phylogenetic trees under the uniform distribution (that is,
when all phylogenetic trees with the same number of taxa are equiprobable).
The knowledge of the expected value of a metric is useful, because it provides
an indication about the significance of the similarity of two individuals
measured through this metric [18].
But phylogeneticists consider also other probabilistic distributions on the
space of phylogenetic trees on a fixed set of taxa, defined through stochastic
models of evolution [6, Ch. 33]. The most popular such model is Yule’s [7,
21], defined by an evolutionary process where, at each step, each currently
extant species can give rise, with the same probability, to two new species.
Under this model, different phylogenetic trees with the same number of leaves
may have different probabilities. Formal details on this model are given in
the next section.
In this paper we compute the expected value of the square of the path-
difference metric for rooted fully resolved phylogenetic trees under this Yule
model. Besides the aforementioned application of this value in the assessment
of tree comparisons, the knowledge of formulas for this expected value under
different models may allow the use of the path-difference metric to test
stochastic models of tree growth, a popular line of research in the last years
which so far has been mostly based on shape indices [13].
## 2 Preliminaries
In this paper, by a _phylogenetic tree_ on a set $S$ of taxa we mean a fully
resolved, or binary, rooted tree with its leaves bijectively labeled in $S$.
We understand such a rooted tree as a directed graph, with its arcs pointing
away from the root. To simplify the language, we shall always identify a leaf
of a phylogenetic tree with its label. We shall also use the term
_phylogenetic tree with $n$ leaves_ to refer to a phylogenetic tree on the set
$\\{1,\ldots,n\\}$. We shall denote by $\mathcal{T}(S)$ the space of all
phylogenetic trees on $S$ and by $\mathcal{T}_{n}$ the space of all
phylogenetic trees with $n$ leaves.
Whenever there exists a directed path from $u$ to $v$ in a phylogenetic tree
$T$, we shall say that $v$ is a _descendant_ of $u$. The _distance_
$d_{T}(u,v)$ between two nodes $u,v$ in a phylogenetic tree $T$ is the length
(in number of arcs) of the unique undirected path connecting $u$ and $v$. The
_depth_ $\delta_{T}(v)$ of a node $v$ in $T$ is the distance from the root $r$
of $T$ to $v$. The _path-difference distance_ [4, 5] between a pair of trees
$T,T^{\prime}\in\mathcal{T}_{n}$ is
$d_{\nu}(T,T^{\prime})=\sqrt{\sum_{1\leqslant i<j\leqslant
n}(d_{T}(i,j)-d_{T^{\prime}}(i,j))^{2}}.$
The _Yule_ , or _Equal-Rate Markov_ , model of evolution [7, 21] is a
stochastic model of phylogenetic trees’ growth. It starts with a node, and at
every step a leaf is chosen randomly and uniformly and it is splitted into two
leaves. Finally, the labels are assigned randomly and uniformly to the leaves
once the desired number of leaves is reached. Under this model, if $T$ is a
phylogenetic tree with $n$ leaves and set of internal nodes $V_{int}(T)$, and
if for every internal node $v$ we denote by $\ell_{T}(v)$ the number of its
descendant leaves, then the probability of $T$ is [1, 17]
$P_{Y}(T)=\frac{2^{n-1}}{n!}\prod_{v\in V_{int}(T)}\frac{1}{\ell_{T}(v)-1}.$
For every $n\geqslant 1$, let $H_{n}=\sum_{i=1}^{n}1/i$ and
$H_{n}^{(2)}=\sum_{i=1}^{n}1/i^{2}$. Let, moreover, $H_{0}=H_{0}^{(2)}=0$.
$H_{n}$ is called the $n$-th _harmonic number_ , and $H_{n}^{(2)}$, the $n$-th
_generalized harmonic number of power $2$_.
## 3 Main results
Let $N_{n}^{2}$ the random variable that chooses independently a pair of trees
$T,T^{\prime}\in\mathcal{T}_{n}$ and computes
$d_{\nu}(T,T^{\prime})^{2}=\sum_{1\leqslant i<j\leqslant
n}(d_{T}(i,j)-d_{T^{\prime}}(i,j))^{2}.$
In this section we establish the following result.
###### Theorem 1.
The expected value of $N_{n}^{2}$ under the Yule model is
$E_{Y}(N_{n}^{2})=\frac{2n}{n-1}\big{(}2(n^{2}+24n+7)H_{n}+13n^{2}-46n+1-16(n+1)H_{n}^{2}-8(n^{2}-1)H_{n}^{(2)}\big{)}.$
To prove this formula, we shall use the following auxiliary random variables:
* 1.
$D_{n}$ is the random variable that chooses a tree $T\in\mathcal{T}_{n}$ and
computes $D(T)=\sum\limits_{1\leqslant i<j\leqslant n}d_{T}(i,j)$.
* 2.
$D_{n}^{(2)}$ is the random variable that chooses a tree $T\in\mathcal{T}_{n}$
and computes $D^{(2)}(T)=\sum\limits_{1\leqslant i<j\leqslant
n}d_{T}(i,j)^{2}$.
The connection between $E_{Y}(N_{n}^{2})$ and the expected values under the
Yule model of $D_{n},D_{n}^{(2)}$ is given by the following result.
###### Proposition 2.
$E_{Y}(N_{n}^{2})=2\big{(}E_{Y}(D^{(2)}_{n})-E_{Y}(D_{n})^{2}/\binom{n}{2}\big{)}.$
###### Proof.
Let us develop $E_{Y}(N_{n}^{2})$ from its raw definition:
$\begin{array}[]{l}\displaystyle
E_{Y}(N_{n}^{2})=\sum_{T,T^{\prime}\in\mathcal{T}_{n}}d_{\nu}(T,T^{\prime})^{2}p_{Y}(T)p_{Y}(T^{\prime})=\sum_{T,T^{\prime}\in\mathcal{T}_{n}}\Big{(}\sum_{1\leqslant
i<j\leqslant
n}(d_{T}(i,j)-d_{T^{\prime}}(i,j))^{2}\Big{)}p_{Y}(T)p_{Y}(T^{\prime})\\\
\quad\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\sum_{T,T^{\prime}}(d_{T}(i,j)^{2}+d_{T^{\prime}}(i,j)^{2}-2d_{T}(i,j)d_{T^{\prime}}(i,j))p_{Y}(T)p_{Y}(T^{\prime})\\\
\quad\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\Big{(}\sum_{T,T^{\prime}}d_{T}(i,j)^{2}p_{Y}(T)p_{Y}(T^{\prime})+\sum_{T,T^{\prime}}d_{T^{\prime}}(i,j)^{2}p_{Y}(T)p_{Y}(T^{\prime})\\\
\quad\qquad\qquad\qquad\displaystyle-2\sum_{T,T^{\prime}}d_{T}(i,j)d_{T^{\prime}}(i,j)p_{Y}(T)p_{Y}(T^{\prime})\Big{)}\\\
\quad\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\Big{(}\sum_{T}d_{T}(i,j)^{2}p_{Y}(T)+\sum_{T^{\prime}}d_{T^{\prime}}(i,j)^{2}p_{Y}(T^{\prime})-2\Big{(}\sum_{T}d_{T}(i,j)p_{Y}(T)\Big{)}\Big{(}\sum_{T^{\prime}}d_{T^{\prime}}(i,j)p_{Y}(T^{\prime})\Big{)}\Big{)}\\\
\quad\displaystyle=\sum_{1\leqslant i<j\leqslant
n}\Big{(}2\sum_{T}d_{T}(i,j)^{2}p_{Y}(T)-2\Big{(}\sum_{T}d_{T}(i,j)p_{Y}(T)\Big{)}^{2}\Big{)}\\\
\quad\displaystyle=2\sum_{T}\Big{(}\sum_{1\leqslant i<j\leqslant
n}d_{T}(i,j)^{2}\Big{)}p_{Y}(T)-2\sum_{1\leqslant i<j\leqslant
n}\Big{(}\sum_{T}d_{T}(i,j)p_{Y}(T)\Big{)}^{2}\\\
\quad\displaystyle=2E_{Y}(D_{n}^{(2)})-2\binom{n}{2}\Big{(}\sum_{T}{d}_{T}(1,2)p_{Y}(T)\Big{)}^{2}\end{array}$
and now
$E_{Y}(D_{n})=\sum_{T\in\mathcal{T}_{n}}\sum_{1\leqslant i<j\leqslant
n}d_{T}(i,j)p_{Y}(T)=\sum_{1\leqslant i<j\leqslant
n}\sum_{T}d_{T}(i,j)p_{Y}(T)=\binom{n}{2}\sum_{T}d_{T}(1,2)p_{Y}(T)$
from where we deduce that
$\Big{(}\sum\limits_{T}{d}_{T}(1,2)p_{Y}(T)\Big{)}^{2}={E_{Y}(D_{n})^{2}}/{\binom{n}{2}^{2}}$,
and the formula in the statement follows. ∎
Now, it is known that the expected value under the Yule model of $D_{n}$ is
$E_{Y}(D_{n})=2n(n+1)H_{n}-4n^{2}\qquad\mbox{\cite[cite]{[\@@bibref{Number}{MRR}{}{}]}}.$
As far as $E_{Y}(D_{n}^{(2)})$ goes, its value is given by the following
result. We postpone the proof until the appendix at the end of the paper.
###### Theorem 3.
$E_{Y}(D_{n}^{(2)})=8n(n+1)(H_{n}^{2}-H_{n}^{(2)})-2n(15n+7)H_{n}+45n^{2}-n$
Then, replacing in the expression for $E_{Y}(N_{n}^{2})$ given in Proposition
2 , $E_{Y}(D_{n})$ and $E_{Y}(D_{n}^{(2)})$ by their values, we obtain the
formula for $E_{Y}(N_{n}^{2})$ given in Theorem 1.
## 4 Conclusions
In this paper we have computed the expected value $E_{Y}(N_{n}^{2})$ of the
square of the path-difference metric for rooted fully resolved phylogenetic
trees under the Yule model:
$E_{Y}(N_{n}^{2})=\frac{2n}{n-1}\big{(}2(n^{2}+24n+7)H_{n}+13n^{2}-46n+1-16(n+1)H_{n}^{2}-8(n^{2}-1)H_{n}^{(2)}\big{)}.$
This complements the computation of this expected value under the uniform
distribution carried out in [11], which turned out to be
$E_{U}(N_{n}^{2})=2\binom{n}{2}\left(4(n-1)+2-\frac{2^{2(n-1)}}{\binom{2(n-1)}{n-1}}-\left(\frac{2^{2(n-1)}}{\binom{2(n-1)}{n-1}}\right)^{2}\right)$
The proof of the formula for $E_{Y}(N_{n}^{2})$ consists of several long
algebraic manipulations of sums of sequences. Since it is not difficult to
slip some mistake in such long algebraic computations, to double-check our
result we have directly computed the value of $E_{Y}(N_{n}^{2})$ for
$n=3,\ldots,7$ and confirmed that our formula gives the right figures. The
Python scripts used to compute them and the results obtained are available in
the Supplementary Material web page
http:/bioinfo.uib.es/~recerca/phylotrees/nodaldistYule/.
The formulas for $E_{Y}(N_{n}^{2})$ and $E_{U}(N_{n}^{2})$ grow in different
orders: $E_{Y}(N_{n}^{2})$ is in $O(n^{2}\ln(n))$, while $E_{U}(N_{n}^{2})$ is
in $O(n^{3})$. Therefore, they can be used to test the Yule and the uniform
models as null stochastic models of evolution for collections of phylogenetic
trees reconstructed by different methods. This kind of analysis has only been
performed so far through shape indices of single trees, not by means of the
comparison of pairs of trees. We shall report on it elsewhere.
## Acknowledgements
The research reported in this paper has been partially supported by the
Spanish government and the UE FEDER program, through projects MTM2009-07165
and TIN2008-04487-E/TIN. We thank J. Miró for several comments on a previous
version of this work.
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## Appendix
In this appendix we prove Proposition 3, as well as of some preliminary
lemmas. To begin with, the following identities on harmonic numbers will be
systematically used in the next proofs, usually without any further notice.
###### Lemma.
For every $n\geqslant 2$:
1. (1)
$\displaystyle\sum_{k=1}^{n-1}H_{k}=n(H_{n}-1)$
2. (2)
$\sum\limits_{k=1}^{n-1}kH_{k}=\frac{1}{4}n(n-1)(2H_{n}-1)$
3. (3)
$\sum\limits_{k=1}^{n-1}{H_{k}}/({k+1})=\frac{1}{2}(H_{n}^{2}-H_{n}^{(2)})$
4. (4)
$\sum\limits_{k=1}^{n-1}kH_{k}H_{n-k}=\binom{n+1}{2}(H_{n+1}^{2}-H_{n+1}^{(2)}-2H_{n+1}+2)$
5. (5)
$\sum\limits_{k=1}^{n-1}(H_{k}^{2}-H_{k}^{(2)})=n(H_{n}^{2}-H_{n}^{(2)})-2n(H_{n}-1)$
6. (6)
$\sum\limits_{k=1}^{n-1}k(H_{k}^{2}-H_{k}^{(2)})=\binom{n}{2}(H_{n}^{2}-H_{n}^{(2)})-\frac{1}{4}n(n-1)(2H_{n}-1)$
###### Proof.
Identities (1)–(3) are well known and easily proved by induction on $n$: see,
for instance, [Knuth2, §6.3, 6.4] and [10, §1.2.7]. Identity (4) is proved in
[19, Thm. 2]. We shall prove (5) and (6) using the technique introduced in
[3]. The main ingredient is _Abel’s lemma on summation by parts_ : for every
two sequences $(a_{k})_{k}$ and $(b_{k})_{k}$,
$\sum_{k=1}^{n-1}(a_{k+1}-a_{k})b_{k}=-\sum_{k=1}^{n-1}(b_{k+1}-b_{k})a_{k+1}+a_{n}b_{n}-a_{1}b_{1}.$
To prove (5), take $a_{k}=k$ and $b_{k}=H_{k}^{2}-H_{k}^{(2)}$, so that
$a_{k+1}-a_{k}=1$ and $b_{k+1}-b_{k}={2H_{k}}/({k+1})$. Then, by Abel’s lemma
$\sum_{k=1}^{n-1}(H_{k}^{2}-H_{k}^{(2)})=-\sum_{k=1}^{n-1}(k+1)\frac{2H_{k}}{k+1}+n(H_{n}^{2}-H_{n}^{(2)})=n(H_{n}^{2}-H_{n}^{(2)})-2\sum_{k=1}^{n-1}H_{k}=n(H_{n}^{2}-H_{n}^{(2)})-2n(H_{n}-1).$
To prove (6), take $a_{k}=\binom{k}{2}$, so that $a_{k+1}-a_{k}=k$, and
$b_{k}=H_{k}^{2}-H_{k}^{(2)}$. Then, again by Abel’s lemma,
$\begin{array}[]{rl}\displaystyle\sum_{k=1}^{n-1}k(H_{k}^{2}-H_{k}^{(2)})&\displaystyle=-\sum_{k=1}^{n-1}\binom{k+1}{2}\frac{2H_{k}}{k+1}+\binom{n}{2}(H_{n}^{2}-H_{n}^{(2)})\\\
&\displaystyle=\binom{n}{2}(H_{n}^{2}-H_{n}^{(2)})-2\sum_{k=1}^{n-1}kH_{k}=\binom{n}{2}(H_{n}^{2}-H_{n}^{(2)})-\frac{1}{4}n(n-1)(2H_{n}-1).\end{array}$
∎
Let us consider now the following two random variables:
* 1.
$S_{n}$, that chooses a tree $T\in\mathcal{T}_{n}$ and computes its _Sackin
index_ [15] $S(T)=\sum\limits_{i=1}^{n}\delta_{T}(i)$.
* 2.
$S_{n}^{(2)}$, that chooses a tree $T\in\mathcal{T}_{n}$ and computes
$S^{(2)}(T)=\sum\limits_{1\leqslant i<j\leqslant n}\delta_{T}(i)^{2}$.
It is known that the expected value under the Yule model of $S_{n}$ is
$E_{Y}(S_{n})=2n(H_{n}-1)\qquad\qquad\mbox{\cite[cite]{[\@@bibref{Number}{KiSl:93}{}{}]}}.$
We shall compute now the expected values under this model of $S_{n}^{(2)}$ and
$D_{n}^{(2)}$: the first will be used in the computation of the second. To do
this, we shall use the following recursive expressions for
$S^{(2)}(T\,\widehat{\ }\,T^{\prime})$ and $D^{(2)}(T\,\widehat{\
}\,T^{\prime})$.
###### Lemma.
Let $T,T^{\prime}$ be two phylogenetic trees on disjoint sets of taxa
$S,S^{\prime}$, with $|S|=k$ and $|S^{\prime}|=n-k$. Then:
1. (1)
$S^{(2)}(T\,\widehat{\
}\,T^{\prime})=S^{(2)}(T)+S^{(2)}(T^{\prime})+2(S(T)+S(T^{\prime}))+n$
2. (2)
$D^{(2)}(T\,\widehat{\
}\,T^{\prime})=D^{(2)}(T)+D^{(2)}(T^{\prime})+(n-k)(S^{(2)}(T)+4S(T))+k(S^{(2)}(T^{\prime})+4S(T^{\prime}))+2S(T)S(T^{\prime})+4k(n-k)$
###### Proof.
Let us assume, without any loss of generality, that $S=\\{1,\ldots,k\\}$ and
$S^{\prime}=\\{k+1,\ldots,n\\}$ . Then, as far as (1) goes, we have that
$\delta_{T\,\widehat{\
}\,T^{\prime}}(i)^{2}=\left\\{\begin{array}[]{ll}(\delta_{T}(i)+1)^{2}&\mbox{
if $1\leqslant i\leqslant k$}\\\ (\delta_{T^{\prime}}(i)+1)^{2}&\mbox{ if
$k+1\leqslant i\leqslant n$}\end{array}\right.$
and therefore
$\begin{array}[]{l}S^{(2)}(T\,\widehat{\
}\,T^{\prime})\displaystyle=\sum_{i=1}^{n}\delta_{T\,\widehat{\
}\,T^{\prime}}(i)^{2}=\sum_{i=1}^{k}(\delta_{T}(i)+1)^{2}+\sum_{i=k+1}^{n}(\delta_{T^{\prime}}(i)+1)^{2}\\\
\quad\displaystyle=\sum_{i=1}^{k}(\delta_{T}(i)^{2}+2\delta_{T}(i)+1)+\sum_{i=k+1}^{n}(\delta_{T^{\prime}}(i)^{2}+2\delta_{T^{\prime}}(i)+1)=S^{(2)}(T)+2S(T)+S^{(2)}(T^{\prime})+2S(T^{\prime})+n.\end{array}$
As far as (2) goes, we have that
$d_{T\,\widehat{\
}\,T^{\prime}}(i,j)^{2}=\left\\{\begin{array}[]{ll}d_{T}(i,j)^{2}&\mbox{ if
$1\leqslant i<j\leqslant k$}\\\ d_{T^{\prime}}(i,j)^{2}&\mbox{ if
$k+1\leqslant i<j\leqslant n$}\\\
(\delta_{T}(i)+\delta_{T^{\prime}}(j)+2)^{2}&\mbox{ if $1\leqslant i\leqslant
k<j\leqslant n$}\end{array}\right.$
and therefore
$\begin{array}[]{l}\displaystyle D^{(2)}(T\,\widehat{\
}\,T^{\prime})=\sum_{1\leqslant i<j\leqslant n}d_{T\,\widehat{\
}\,T^{\prime}}(i,j)^{2}=\sum_{1\leqslant i<j\leqslant
k}d_{T}(i,j)^{2}+\sum_{k+1\leqslant i<j\leqslant
n}d_{T^{\prime}}(i,j)^{2}+\sum_{1\leqslant i\leqslant k\atop k+1\leqslant
j\leqslant n}(\delta_{T}(i)+\delta_{T^{\prime}}(j)+2)^{2}\\\
\displaystyle\quad=D^{(2)}(T)+D^{(2)}(T^{\prime})+\sum_{1\leqslant i\leqslant
k\atop k+1\leqslant j\leqslant
n}(\delta_{T}(i)^{2}+\delta_{T^{\prime}}(j)^{2}+2\delta_{T}(i)\delta_{T^{\prime}}(j)+4\delta_{T}(i)+4\delta_{T^{\prime}}(j)+4)\\\
\displaystyle\quad=D^{(2)}(T)+D^{(2)}(T^{\prime})+(n-k)\sum_{i=1}^{k}(\delta_{T}(i)^{2}+4\delta_{T}(i))+k\sum_{j=k+1}^{n}(\delta_{T^{\prime}}(j)^{2}+4\delta_{T^{\prime}}(j))\\\
\displaystyle\quad\qquad\qquad\qquad+2\Big{(}\sum_{i=1}^{k}\delta_{T}(i)\Big{)}\Big{(}\sum_{j=k+1}^{n}\delta_{T^{\prime}}(j)\Big{)}+4k(n-k)\\\
\displaystyle\quad=D^{(2)}(T)+D^{(2)}(T^{\prime})+(n-k)(S^{(2)}(T)+4S(T))+k(S^{(2)}(T^{\prime})+4S(T^{\prime}))+2S(T)S(T^{\prime})+4k(n-k).\end{array}$
∎
Now we can compute explicit formulas for $E_{Y}(S_{n}^{(2)})$ and
$E_{Y}(D_{n}^{(2)})$
###### Proposition.
$E_{Y}(S_{n}^{(2)})=4n(H_{n}^{2}-H_{n}^{(2)})-6n(H_{n}-1)$.
###### Proof.
We compute $E_{Y}(S_{n}^{(2)})$ using its very definition:
$\begin{array}[]{l}E_{Y}(S_{n}^{(2)})\displaystyle=\sum_{T\in\mathcal{T}_{n}}S^{(2)}(T)\cdot
p_{Y}(T)=\frac{1}{2}\sum_{k=1}^{n-1}\sum_{S_{k}\subsetneq\\{1,\ldots,n\\}\atop|S_{k}|=k}\sum_{T_{k}\in\mathcal{T}(S_{k})}\sum_{T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})}S^{(2)}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})\cdot p_{Y}(T_{k}\widehat{\ }\,{}T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\binom{n}{k}\sum_{T_{k}\in\mathcal{T}_{k}}\sum_{T^{\prime}_{n-k}\in\mathcal{T}_{n-k}}\Big{(}S^{(2)}(T_{k})+S^{(2)}(T_{n-k}^{\prime})+2(S(T_{k})+S(T_{n-k}^{\prime}))+n\Big{)}\\\
\displaystyle\quad\qquad\cdot\dfrac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S^{(2)}(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S^{(2)}(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}nP_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\bigg{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}\sum_{T_{k}}S^{(2)}(T_{k})P_{Y}(T_{k})+\sum_{T^{\prime}_{n-k}}S^{(2)}(T^{\prime}_{n-k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+2\sum_{T_{k}}S(T_{k})P_{Y}(T_{k})+2\sum_{T^{\prime}_{n-k}}S(T_{n-k}^{\prime})P_{Y}(T^{\prime}_{n-k})+n\bigg{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}E_{Y}(S^{(2)}_{k})+E_{Y}(S^{(2)}_{n-k})+2E_{Y}(S_{k})+2E_{Y}(S_{n-k})+n\bigg{)}\\\
\quad\displaystyle=\frac{2}{n-1}\sum_{k=1}^{n-1}E_{Y}(S^{(2)}_{k})+\frac{4}{n-1}\sum_{k=1}^{n-1}E_{Y}(S_{k})+n\end{array}$
In particular
$E_{Y}(S_{n-1}^{(2)})=\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(S^{(2)}_{k})+\frac{4}{n-2}\sum_{k=1}^{n-2}E_{Y}(S_{k})+n-1$
and therefore
$\begin{array}[]{l}E_{Y}(S_{n}^{(2)})\displaystyle=\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(S^{(2)}_{k})+\frac{2}{n-1}E_{Y}(S^{(2)}_{n-1})\\\
\displaystyle\quad\qquad\quad\qquad+\frac{n-2}{n-1}\cdot\frac{4}{n-2}\sum_{k=1}^{n-2}E_{Y}(S_{k})+\frac{4}{n-1}E_{Y}(S_{n-1})+\frac{n-2}{n-1}\cdot(n-1)+2\\\
\displaystyle\quad=\frac{n-2}{n-1}E_{Y}(S_{n-1}^{(2)})+\frac{2}{n-1}E_{Y}(S^{(2)}_{n-1})+\frac{4}{n-1}E_{Y}(S_{n-1})+2=\frac{n}{n-1}E_{Y}(S_{n-1}^{(2)})+8H_{n-1}-6\end{array}$
Setting $x_{n}=E_{Y}(S_{n}^{(2)})/n$, this recurrence becomes
$x_{n}=x_{n-1}+\frac{8H_{n-1}}{n}-\frac{6}{n}.$
Since $S^{(2)}$ applied to a single node is 0, $x_{1}=E_{Y}(S_{1}^{(2)})=0$,
and the solution of this recursive equation with this initial condition is
$x_{n}=\sum_{k=2}^{n}\Big{(}\frac{8H_{k-1}}{k}-\frac{6}{k}\Big{)}=8\sum_{k=1}^{n-1}\frac{H_{k}}{k+1}-6\sum_{k=2}^{n}\frac{1}{k}=4(H_{n}^{2}-H_{n}^{(2)})-6(H_{n}-1)$
from where we deduce that
$E_{Y}(S_{n}^{(2)})=nx_{n}=4n(H_{n}^{2}-H_{n}^{(2)})-6n(H_{n}-1)$
as we claimed. ∎
###### Theorem 3.
$E_{Y}(D_{n}^{(2)})=8n(n+1)(H_{n}^{2}-H_{n}^{(2)})-2n(15n+7)H_{n}+45n^{2}-n$
###### Proof.
Again, we compute $E_{Y}(D_{n}^{(2)})$ using its very definition:
$\begin{array}[]{l}E_{Y}(D_{n}^{(2)})\displaystyle=\sum_{T\in\mathcal{T}_{n}}D^{(2)}(T)\cdot
p_{Y}(T)\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\sum_{S_{k}\subsetneq\\{1,\ldots,n\\}\atop|S_{k}|=k}\sum_{T_{k}\in\mathcal{T}(S_{k})}\sum_{T^{\prime}_{n-k}\in\mathcal{T}(S_{k}^{c})}D^{(2)}(T_{k}\widehat{\
}\,{}T^{\prime}_{n-k})\cdot p_{Y}(T_{k}\widehat{\ }\,{}T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\binom{n}{k}\sum_{T_{k}\in\mathcal{T}_{k}}\sum_{T^{\prime}_{n-k}\in\mathcal{T}_{n-k}}\Big{(}D^{(2)}(T_{k})+D^{(2)}(T^{\prime}_{n-k})+2S(T_{k})S(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+(n-k)(S^{(2)}(T_{k})+4S(T_{k}))+k(S^{(2)}(T_{n-k}^{\prime})+4S(T_{n-k}^{\prime}))+4k(n-k))\Big{)}\cdot\dfrac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}D^{(2)}(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}D^{(2)}(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+2\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S(T_{k})S(T^{\prime}_{n-k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+(n-k)\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S^{(2)}(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+4(n-k)\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S(T_{k})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+k\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S^{(2)}(T_{n-k}^{\prime})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+4k\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}S(T_{n-k}^{\prime})P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})+4\sum_{T_{k}}\sum_{T^{\prime}_{n-k}}k(n-k)P_{Y}(T_{k})P_{Y}(T^{\prime}_{n-k})\bigg{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}\sum_{T_{k}}D^{(2)}(T_{k})P_{Y}(T_{k})+\sum_{T^{\prime}_{n-k}}D^{(2)}(T^{\prime}_{n-k})P_{Y}(T^{\prime}_{n-k})\\\
\displaystyle\quad\qquad+2\Big{(}\sum_{T_{k}}S(T_{k})P_{Y}(T_{k})\Big{)}\Big{(}\sum_{T^{\prime}_{n-k}}S(T_{n-k}^{\prime})P_{Y}(T^{\prime}_{n-k})\Big{)}+(n-k)\sum_{T_{k}}S^{(2)}(T_{k})P_{Y}(T_{k})\\\
\displaystyle\quad\qquad+4(n-k)\sum_{T_{k}}S(T_{k})P_{Y}(T_{k})+k\sum_{T^{\prime}_{n-k}}S^{(2)}(T_{n-k}^{\prime})P_{Y}(T^{\prime}_{n-k})+4k\sum_{T^{\prime}_{n-k}}S(T_{n-k}^{\prime})P_{Y}(T^{\prime}_{n-k})+4k(n-k)\bigg{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\bigg{(}E_{Y}(D^{(2)}_{k})+E_{Y}(D^{(2)}_{n-k})+2E_{Y}(S_{k})E_{Y}(S_{n-k})+(n-k)E_{Y}(S^{(2)}_{k})\\\
\displaystyle\quad\qquad+4(n-k)E_{Y}(S_{k})+kE_{Y}(S^{(2)}_{n-k})+4kE_{Y}(S_{n-k})+4k(n-k)\bigg{)}\\\
\quad\displaystyle=\frac{2}{n-1}\sum_{k=1}^{n-1}\bigg{(}E_{Y}(D^{(2)}_{k})+E_{Y}(S_{k})E_{Y}(S_{n-k})+(n-k)E_{Y}(S^{(2)}_{k})+4(n-k)E_{Y}(S_{k})\bigg{)}+\frac{2}{3}n(n+1)\\\
\end{array}$
In particular
$\begin{array}[]{l}E_{Y}(D_{n-1}^{(2)})\displaystyle=\frac{2}{n-2}\sum_{k=1}^{n-2}\bigg{(}E_{Y}(D^{(2)}_{k})+E_{Y}(S_{k})E_{Y}(S_{n-1-k})+(n-1-k)E_{Y}(S^{(2)}_{k})\\\
\displaystyle\quad\qquad+4(n-1-k)E_{Y}(S_{k})\bigg{)}+\frac{2}{3}n(n-1)\end{array}$
and therefore
$\begin{array}[]{l}E_{Y}(D_{n}^{(2)})\displaystyle=\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(D^{(2)}_{k})+\frac{2}{n-1}E_{Y}(D^{(2)}_{n-1})\\\
\displaystyle\quad\qquad+\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{Y}(S_{k})E_{Y}(S_{n-1-k})+\frac{2}{n-1}\bigg{(}\sum_{k=1}^{n-1}E_{Y}(S_{k})E_{Y}(S_{n-k})-\sum_{k=1}^{n-2}E_{Y}(S_{k})E_{Y}(S_{n-1-k})\Bigg{)}\\\
\displaystyle\quad\qquad+\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}(n-1-k)E_{Y}(S^{(2)}_{k})+\frac{2}{n-1}\bigg{(}\sum_{k=1}^{n-1}(n-k)E_{Y}(S^{(2)}_{k})-\sum_{k=1}^{n-2}(n-1-k)E_{Y}(S^{(2)}_{k})\Bigg{)}\\\
\displaystyle\quad\qquad+\frac{n-2}{n-1}\cdot\frac{8}{n-2}\sum_{k=1}^{n-2}(n-1-k)E_{Y}(S_{k})+\frac{8}{n-1}\bigg{(}\sum_{k=1}^{n-1}(n-k)E_{Y}(S_{k})-\sum_{k=1}^{n-2}(n-1-k)E_{Y}(S_{k})\Bigg{)}\\\
\displaystyle\quad\qquad+\frac{n-2}{n-1}\cdot\frac{2}{3}n(n-1)+\frac{2}{3}n(n+1)-\frac{n-2}{n-1}\cdot\frac{2}{3}n(n-1)\\\\[6.45831pt]
\quad\displaystyle=\frac{n-2}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{2}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{Y}(S_{k})(E_{Y}(S_{n-k})-E_{Y}(S_{n-k-1}))\\\
\displaystyle\quad\qquad+\frac{2}{n-1}\sum_{k=1}^{n-1}E_{Y}(S_{k}^{(2)})+\frac{8}{n-1}\sum_{k=1}^{n-1}E_{Y}(S_{k})+2n\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{8}{n-1}\sum_{k=1}^{n-2}k(H_{k}-1)H_{n-k-1}+\frac{2}{n-1}\sum_{k=1}^{n-1}(4k(H_{k}^{2}-H_{k}^{(2)})-6k(H_{k}-1))\\\
\displaystyle\quad\qquad+\frac{16}{n-1}\sum_{k=1}^{n-1}k(H_{k}-1)+2n\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}-\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{n-k-1}+\frac{8}{n-1}\sum_{k=1}^{n-1}k(H_{k}^{2}-H_{k}^{(2)})\\\
\displaystyle\quad\qquad+\frac{4}{n-1}\sum_{k=1}^{n-1}k(H_{k}-1)+2n\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}-\frac{8}{n-1}\sum_{k=1}^{n-2}(n-k-1)H_{k}+\frac{8}{n-1}\sum_{k=1}^{n-1}k(H_{k}^{2}-H_{k}^{(2)})\\\
\displaystyle\quad\qquad+\frac{4}{n-1}\sum_{k=1}^{n-1}kH_{k}-\frac{4}{n-1}\sum_{k=1}^{n-1}k+2n\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+\frac{8}{n-1}\sum_{k=1}^{n-2}kH_{k}H_{n-k-1}-8\sum_{k=1}^{n-2}H_{k}+\frac{8}{n-1}\sum_{k=1}^{n-1}k(H_{k}^{2}-H_{k}^{(2)})\\\
\displaystyle\quad\qquad+\frac{12}{n-1}\sum_{k=1}^{n-1}kH_{k}-8H_{n-1}\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+4n(H_{n}^{2}-H_{n}^{(2)}-2H_{n}+2)-8(n-1)(H_{n-1}-1)+4n(H_{n}^{2}-H_{n}^{(2)})\\\
\displaystyle\quad\qquad-2n(2H_{n}-1)+3n(2H_{n}-1)-8H_{n-1}\\\
\quad\displaystyle=\frac{n}{n-1}E_{Y}(D^{(2)}_{n-1})+8n(H_{n}^{2}-H_{n}^{(2)})-14nH_{n-1}+15n-14\end{array}$
Setting $x_{n}=E_{Y}(D_{n}^{(2)})/n$, this recurrence becomes
$x_{n}=x_{n-1}+8(H_{n}^{2}-H_{n}^{(2)})-14H_{n-1}+15-\frac{14}{n}.$
The solution of this recursive with $x_{1}=E_{Y}(D_{1}^{(2)})=0$ is
$\begin{array}[]{rl}x_{n}&\displaystyle=\sum_{k=2}^{n}\Big{(}8(H_{k}^{2}-H_{k}^{(2)})-14H_{k-1}+15-\frac{14}{k}\Big{)}\\\
&\displaystyle=8\sum_{k=1}^{n}(H_{k}^{2}-H_{k}^{(2)})-14\sum_{k=1}^{n-1}H_{k}+15(n-1)-14\sum_{k=2}^{n}\frac{1}{k}\\\
&\displaystyle=8(n+1)(H_{n+1}^{2}-H_{n+1}^{(2)})-16(n+1)(H_{n+1}-1)-14n(H_{n}-1)+15(n-1)-14(H_{n}-1)\\\
&\displaystyle=8(n+1)(H_{n}^{2}-H_{n}^{(2)})-2(15n+7)H_{n}+45n-1\end{array}$
from where we deduce that
$E_{Y}(D_{n}^{(2)})=nx_{n}=8n(n+1)(H_{n}^{2}-H_{n}^{(2)})-2n(15n+7)H_{n}+45n^{2}-n$
as we claimed. ∎
|
arxiv-papers
| 2012-03-12T14:33:52 |
2024-09-04T02:49:28.535471
|
{
"license": "Public Domain",
"authors": "Gabriel Cardona, Arnau Mir, Francesc Rossello",
"submitter": "Francesc Rossell\\'o",
"url": "https://arxiv.org/abs/1203.2503"
}
|
1203.2543
|
# Biclique-colouring verification complexity and biclique-colouring power
graphs††thanks: An extended abstract published in: Proceedings of Cologne
Twente Workshop (CTW) 2012, pp. 134–138. Research partially supported by
FAPERJ–Cientistas do Nosso Estado, and by CNPq-Universal.
Hélio B. Macêdo Filho COPPE, Universidade Federal do Rio de Janeiro Simone
Dantas IME, Universidade Federal Fluminense
Raphael C. S. Machado Inmetro — Instituto Nacional de Metrologia, Qualidade e
Tecnologia. Celina M. H. Figueiredo COPPE, Universidade Federal do Rio de
Janeiro
###### Abstract
Biclique-colouring is a colouring of the vertices of a graph in such a way
that no maximal complete bipartite subgraph with at least one edge is
monochromatic. We show that it is co$\mathcal{NP}$-complete to check whether a
given function that associates a colour to each vertex is a biclique-
colouring, a result that justifies the search for structured classes where the
biclique-colouring problem could be efficiently solved. We consider biclique-
colouring restricted to powers of paths and powers of cycles. We determine the
biclique-chromatic number of powers of paths and powers of cycles. The
biclique-chromatic number of a power of a path $P_{n}^{k}$ is $\max(2k+2-n,2)$
if $n\geq k+1$ and exactly $n$ otherwise. The biclique-chromatic number of a
power of a cycle $C_{n}^{k}$ is at most 3 if $n\geq 2k+2$ and exactly $n$
otherwise; we additionally determine the powers of cycles that are 2-biclique-
colourable. All proofs are algorithmic and provide polynomial-time biclique-
colouring algorithms for graphs in the investigated classes.
††footnotetext:
## 1 Introduction
Let $G=(V,E)$ be a simple graph with order $n=|V|$ vertices and $m=|E|$ edges.
A _clique_ of $G$ is a maximal set of vertices of size at least 2 that induces
a complete subgraph of $G$. A _biclique_ of $G$ is a maximal set of vertices
that induces a complete bipartite subgraph of $G$ with at least one edge. A
_clique-colouring_ of $G$ is a function $\pi$ that associates a colour to each
vertex such that no clique is monochromatic. If the function uses at most $c$
colours we say that $\pi$ is a _$c$ -clique-colouring_. A _biclique-colouring_
of $G$ is a function $\pi$ that associates a colour to each vertex such that
no biclique is monochromatic. If the function $\pi$ uses at most $c$ colours
we say that $\pi$ is a _$c$ -biclique-colouring_. The _clique-chromatic
number_ of $G$, denoted by $\kappa(G)$, is the least $c$ for which $G$ has a
$c$-clique-colouring. The _biclique-chromatic number_ of $G$, denoted by
$\kappa_{B}(G)$, is the least $c$ for which $G$ has a $c$-biclique-colouring.
Both clique-colouring and biclique-colouring have a “hypergraph colouring
version.” Recall that a hypergraph $\mathcal{H}=(V,\mathcal{E})$ is an ordered
pair where $V$ is a set of vertices and $\mathcal{E}$ is a set of hyperedges,
each of which is a set of vertices. A colouring of hypergraph
$\mathcal{H}=(V,\mathcal{E})$ is a function that associates a colour to each
vertex such that no hyperedge is monochromatic. Let $G=(V,E)$ be a graph and
let $\mathcal{H}_{C}(G)=(V,\mathcal{E}_{C})$ and
$\mathcal{H}_{B}(G)=(V,\mathcal{E}_{B})$ be the hypergraphs in which
hyperedges are, respectively, $\mathcal{E}_{C}=\\{K\subseteq V\mid K\mbox{ is
a clique of }G\\}$ and $\mathcal{E}_{B}=\\{K\subseteq V\mid K\mbox{ is a
biclique of }G\\}$ — hypergraphs $\mathcal{H}_{C}(G)$ and $\mathcal{H}_{B}(G)$
are called, resp., the _clique-hypergraph_ and the _biclique-hypergraph_ of
$G$. A clique-colouring of $G$ is a colouring of its clique-hypergraph
$\mathcal{H}_{C}(G)$; a biclique-colouring of $G$ is a colouring of its
biclique-hypergraph $\mathcal{H}_{B}(G)$.
Clique-colouring and biclique-colouring are analogous problems in the sense
that they refer to the colouring of hypergraphs arising from graphs. In
particular, the hyperedges are subsets of vertices that are clique (resp.
biclique). The clique is a classical important structure in graphs, hence it
is natural that the clique-colouring problem has been studied for a long time
— see [1, 13, 21, 25]. Only recently the biclique-colouring problem started to
be investigated [19].
Many other problems, initially stated for cliques, have their version for
bicliques [3, 20], such as _Ramsey number_ and _Turán’s theorem_. The
combinatorial game called on-line Ramsey number also has a version for
bicliques [12]. Although complexity results for complete bipartite subgraph
problems are mentioned in [16] and the (maximum) biclique problem is shown to
be $\mathcal{NP}$-hard in [32], only in the last decade the (maximal)
bicliques were rediscovered in the context of counting problems [17, 28],
enumeration problems [14, 27], and intersection graphs [18].
Clique-colouring and biclique-colouring have similarities with usual vertex-
colouring. A proper vertex-colouring is also a clique-colouring and a
biclique-colouring — in other words, both the clique-chromatic number and the
biclique-chromatic number are bounded above by the vertex-chromatic number.
Optimal vertex-colourings and clique-colourings coincide in the case of
$K_{3}$-free graphs, while optimal vertex-colourings and biclique-colourings
coincide in the (much more restricted) case of $K_{1,2}$-free graphs — notice
that the triangle $K_{3}$ is the minimal complete graph that includes the
graph induced by one edge ($K_{2}$), while the $K_{1,2}$ is the minimal
complete bipartite graph that includes the graph induced by one edge
($K_{1,1}$). But there are also essential differences. Most remarkably, it is
possible that a graph has a clique-colouring (resp. biclique-colouring), which
is not a clique-colouring (resp. biclique-colouring) when restricted to one of
its subgraphs. Subgraphs may even have a larger clique-chromatic number (resp.
biclique-chromatic number) than the original graph.
Clique-colouring and biclique-colouring also have similarities on complexity
issues. It is known [1] that it is co$\mathcal{NP}$-complete to check whether
a given function that associates a colour to each vertex is a clique-colouring
by a reduction from $3DM$. Later, an alternative $\mathcal{NP}$-completeness
proof was obtained by a reduction from a variation of $3SAT$, in order to
construct the complement of a bipartite graph [13]. Based on this, we open
this paper providing a corresponding result regarding the biclique-colouring
problem: it is co$\mathcal{NP}$-complete to check whether a given function
that associates a colour to each vertex is a biclique-colouring. The
co$\mathcal{NP}$-completeness holds even when the input is a
$\\{C_{4},K_{4}\\}$-free graph.
We select two structured classes for which we provide linear-time biclique-
colouring algorithms: powers of paths and powers of cycles. The choice of
those classes has also a strong motivation since they have been recently
investigated in the context of well studied variations of colouring problems.
For instance, for a power of a path $P_{n}^{k}$, its $b$-chromatic number is
$n$, if $n\leq k+1$; $k+1+\lfloor\frac{n-k-1}{3}\rfloor$, if $k+2\leq n\leq
4k+1$; or $2k+1$, if $n\geq 4k+2$; whereas, for a power of a cycle
$C_{n}^{k}$, its $b$-chromatic number is $n$, if $n\leq 2k+1$; $k+1$, if
$n=2k+2$; at least $\min(n-k-1,k+1+\lfloor\frac{n-k-1}{3}\rfloor)$, if
$2k+3\leq n\leq 3k$; $k+1+\lfloor\frac{n-k-1}{3}\rfloor$, if $3k+1\leq n\leq
4k$; or $2k+1$, if $n\geq 4k+2$ [15]. Moreover, other well studied variations
of colouring problems when restricted to powers of cycles have been
investigated: chromatic number [29], chromatic index [26], total chromatic
number [8], choice number [29], and clique-chromatic number [9]. It is known,
for a power of a cycle $C_{n}^{k}$, that the chromatic number and the choice
number are both $k+1+\lceil r/q\rceil$, where $n=q(k+1)+t$ with $q\geq 1$,
$0\leq t\leq k$ and $n\geq 2k+1$, that the chromatic index is the maximum
degree of $C_{n}^{k}$ if, and only if, $n$ is even, that the total chromatic
number is at most the maximum degree of $C_{n}^{k}$ plus 2, when $n$ is even
and $n\geq 2k+1$, and that the clique-chromatic number is $2$, when $n\leq
2k+1$, and is at most 3, when $n\geq 2k+2$. Particularly, in the latter case,
the clique-chromatic number is 3, when $n$ is odd and $n\geq 5$; otherwise, it
is 2. Note that total colouring is an open and difficult problem and remains
unsolved for powers of cycles [8]. Other significant works have been done in
power graphs [7, 10] and, in particular, in powers of paths and powers of
cycles [5, 6, 22, 23, 24, 31].
## 2 Complexity of biclique-colouring
The biclique-colouring problem is a variation of the clique-colouring problem.
Hence, it is natural to investigate the complexity of biclique-colouring based
on the tools that were developed to determine the complexity of clique-
colouring. We show that, similarly to the case of clique-colouring, it is
co$\mathcal{NP}$-complete to check whether a given function that associates a
colour to each vertex of a graph is a biclique-colouring. To achieve a result
in this direction, we prove the $\mathcal{NP}$-completeness of the following
problem: of deciding whether there exists a biclique of a graph $G$ contained
in a given subset of vertices of $G$. Indeed, a function that associates a
colour to each vertex of a given graph $G$ is a biclique-colouring if, and
only if, there is no biclique of $G$ contained in a subset of the vertices of
$G$ associated with the same colour.
We call Biclique Containment the problem that decides whether there exists a
biclique of a graph $G$ contained in a given subset of vertices of $G$.
###### Problem 2.1.
Biclique Containment
Instance: Graph $G=(V,E)$ and $V^{\prime}\subset V$
Question: Does there exist a biclique $B$ of $G$ such that $B\subseteq
V^{\prime}$?
In order to show that Biclique Containment is $\mathcal{NP}$-complete, we use
in Theorem 1 a reduction from 3SAT problem.
###### Theorem 1.
The Biclique Containment problem is $\mathcal{NP}$-complete, even if the input
graph is $\\{K_{4},C_{4}\\}$-free.
###### Proof.
Deciding whether a graph has a biclique in a given subset of vertices is in
$\mathcal{NP}$: a biclique is a certificate and verifying this certificate is
trivially polynomial.
We prove that Biclique Containment problem is $\mathcal{NP}$-hard by reducing
3SAT to it. The proof is outlined as follows. For every formula $\phi$, a
graph $G$ is constructed with a subset of vertices denoted by $V^{\prime}$,
such that $\phi$ is satisfiable if, and only if, there exists a biclique $B$
of $G$ such that $B\subseteq V^{\prime}$.
Let $n$ (resp. $m$) be the number of variables (resp. clauses) in formula
$\phi$. We define the graph $G$ as follows.
* •
For each variable $x_{i}$, $1\leq i\leq n$, there exist two adjacent vertices
$x_{i}$ and $\overline{x_{i}}$. Let
$L=\\{x_{1},\dots,x_{n},\overline{x_{1}},\dots,\overline{x_{n}}\\}$.
* •
For each clause $c_{j}$, $1\leq j\leq m$, there exists a vertex $c_{j}$.
Moreover, each $c_{j}$, $1\leq j\leq m$, is adjacent to a vertex
$l\in\\{x_{1},\dots,x_{n},\overline{x_{1}},\dots,\overline{x_{n}}\\}$ if, and
only if, the literal corresponding to $l$ is in the clause corresponding to
vertex $c_{j}$. Let $C=\\{c_{1},\ldots,c_{m}\\}$.
* •
There exists a universal vertex $u$ adjacent to all $x_{i}$,
$\overline{x_{i}}$, $1\leq i\leq n$, and to all $c_{j}$, $1\leq j\leq m$.
We define the subset of vertices $V^{\prime}$ as
$\\{u,x_{1},\dots,x_{n},\overline{x_{1}},\dots,\overline{x_{n}}\\}$. Refer to
Figure 1 for an example of such construction given a formula
$\phi=(x_{1}\vee\overline{x_{2}}\vee
x_{4})\wedge(x_{2}\vee\overline{x_{3}}\vee\overline{x_{5}})\wedge(x_{1}\vee
x_{3}\vee x_{5})$.
Figure 1: Example for $\phi=(x_{1}\vee\overline{x_{2}}\vee
x_{4})\wedge(x_{2}\vee\overline{x_{3}}\vee\overline{x_{5}})\wedge(x_{1}\vee
x_{3}\vee x_{5})$
We claim that formula $\phi$ is satisfiable if, and only if, there exists a
biclique of $G[V^{\prime}]$ that is also a biclique of $G$.
Each biclique $B$ of $G[V^{\prime}]$ containing vertex $u$ corresponds to a
choice of precisely one vertex of $\\{x_{i},\overline{x_{i}}\\}$, for each
$1\leq i\leq n$, and so $B$ corresponds to a truth assignment $v_{B}$ that
gives true value to variable $x_{i}$ if, and only if, the corresponding vertex
$x_{i}\in B$.
Notice that we may assume three properties on the 3SAT instance.
* •
A variable and its negation do not appear in the same clause. Else, any
assignment of values (true or false) to such a variable satisfies the clause.
* •
A variable appears in at least one clause. Else, any assignment of values
(true or false) to such a variable is indifferent to formula $\phi$.
* •
Two distinct clauses have at most one literal in common. Else, we can modify
the instance as follows. For each clause $(l_{i},l_{j},l_{k})$, we replace it
by clauses $(l_{i},x^{\prime}_{1},x^{\prime}_{2})$,
$(l_{j},x^{\prime}_{1},\overline{x^{\prime}_{2}})$,
$(l_{j},\overline{x^{\prime}_{1}},x^{\prime}_{3})$, and
$(l_{k},\overline{x^{\prime}_{1}},\overline{x^{\prime}_{3}})$ with variables
$x^{\prime}_{1}$, $x^{\prime}_{2}$, and $x^{\prime}_{3}$. Clearly, the number
of variables and clauses created is upper bounded by 7 times the number of
clauses in the original instance. Moreover, the original formula is
satisfiable if, and only if, the new formula is satisfiable.
We consider the bicliques of $G[V^{\prime}]$ according to two cases.
1. 1.
Biclique $B$ does not contain vertex $u$. Then, the biclique is precisely
formed by a pair of vertices, say $x_{i}$ and $\overline{x_{i}}$, where $1\leq
i\leq n$. Now, our assumption says that there exists a $c_{j}$ adjacent to one
precise vertex in $\\{x_{i},\overline{x_{i}}\\}$ which implies that $B$ is not
a biclique of $G$.
2. 2.
Biclique $B$ contains vertex $u$. Then, the biclique is precisely formed by
vertex $u$ and one vertex of $\\{x_{i},\overline{x_{i}}\\}$, for each $1\leq
i\leq n$. $B$ is a biclique of $G$ if, and only if, for each $1\leq j\leq m$,
there exists a vertex $l\in L\cap B$ such that $c_{j}$ is adjacent to $l$,
which in turn occurs if, and only if, the truth assignment $v_{B}$ satisfies
$\phi$. Therefore, $B$ is a biclique of $G$ if, and only if, $v_{B}$ satisfies
$\phi$.
Now, we still have to prove that $G$ is $\\{K_{4},C_{4}\\}$-free.
For the sake of contradiction, suppose that there exists a $K_{4}$ in $G$, say
$K$. There are no two distinct vertices of $C$ in $K$, since $C$ is an
independent set. There are no three distinct vertices of $L$ in $K$, since
there is a non-edge between two of these three vertices. Hence, $K$ precisely
contains vertex $u$, one vertex of $C$, and two vertices of $L$. Since $K$ is
a complete set, the two vertices in $L\cap K$ are adjacent and the vertex of
$C\cap K$ is adjacent to both vertices of $L\cap K$. This contradicts our
assumption that a variable and its negation do not appear in the same clause.
For the sake of contradiction, suppose there exists a $C_{4}$ in $G$, say $H$.
The universal vertex $u$ cannot belong to $H$. Since $C$ is an independent
set, $H$ contains at most two vertices of $C$. Now, if $H$ contains two
vertices of $C$, then the other two vertices of $H$ must be two literals,
which contradicts our assumption that two distinct clauses have at most one
literal in common. Since $L$ induces a matching, $H$ is not contained in $L$.
Therefore, $H$ contains one vertex of $C$ and three vertices of $L$, which by
the construction of $G$ gives the final contradiction. ∎
###### Corollary 2.
Let $G$ be a $\\{C_{4},K_{4}\\}$-free graph. It is co$\mathcal{NP}$-complete
to check if a colouring of the vertices of $G$ is a biclique-colouring.
## 3 Powers of paths, powers of cycles, and their bicliques
A _power of a path_ $P_{n}^{k}$, for $k\geq 1$, is a simple graph with
$V(G)=\\{v_{0},\dots,v_{n-1}\\}$ and $\\{v_{i},v_{j}\\}\in E(G)$ if, and only
if, $|i-j|\leq k$. Note that $P_{n}^{1}$ is the induced path $P_{n}$ on $n$
vertices and $P_{n}^{k}$, $n\leq k+1$, is the complete graph $K_{n}$ on $n$
vertices. In a power of a path $P_{n}^{k}$, the _reach_ of an edge
$\\{v_{i},v_{j}\\}$ is $|i-j|$. A _power of a cycle_ $C_{n}^{k}$, for $k\geq
1$, is a simple graph with $V(G)=\\{v_{0},\dots,v_{n-1}\\}$ and
$\\{v_{i},v_{j}\\}\in E(G)$ if, and only if, $\min\\{(j-i)\bmod n,(i-j)\bmod
n\\}\leq k$. Note that $C_{n}^{1}$ is the induced cycle $C_{n}$ on $n$
vertices and $C_{n}^{k}$, $n\leq 2k+1$, is the complete graph $K_{n}$ on $n$
vertices. In a power of a cycle $C_{n}^{k}$, we take $(v_{0},\dots,v_{n-1})$
to be a _cyclic order_ on the vertex set of $G$ and we always perform
arithmetic modulo $n$ on vertex indices. The _reach_ of an edge
$\\{v_{i},v_{j}\\}$ is $\min\\{(i-j)\bmod n,(j-i)\bmod n\\}$. The definition
of reach is extended to an induced path to be the sum of the reach of its
edges. A _block_ is a maximal set of consecutive vertices. The _size_ of a
block is the number of vertices in the block.
All power graphs considered in the present work contain a polynomial number of
bicliques, a sufficient condition for the Biclique Containment problem to be
polynomial. In what follows, we explicitly identify the bicliques of a power
of a path and the bicliques of a power of a cycle. We say that a biclique of
size 2 is a $P_{2}$ biclique and that a biclique of size 3 is a $P_{3}$
biclique. Notice that, for each value of $n$ in the considered range, every
biclique in Lemmas 3 and 4 always exists. We refer to Figure 2 to illustrate
the distinct biclique structures for each considered case of non-complete
powers of cycles.
(a) Power of a cycle $C_{11}^{4}$
($2k+2\leq n\leq 3k+1$)
(b) Power of a cycle $C_{11}^{3}$
($3k+2\leq n\leq 4k$)
(c) Power of a cycle $C_{11}^{2}$
($n\geq 4k+1$)
Figure 2: For each case of non-complete powers of cycles according to Lemma 4,
we highlight in bold the distinct biclique structures.
###### Lemma 3.
The bicliques of a power of a path $P_{n}^{k}$ are precisely: $P_{2}$
bicliques, if $n\leq k+1$; $P_{2}$ bicliques and $P_{3}$ bicliques, if
$k+2\leq n\leq 2k$; and $P_{3}$ bicliques if $n\geq 2k+1$.
###### Proof.
A power of a path is $K_{1,3}$-free and $C_{4}$-free. Thus, the bicliques of a
power of a path are possibly $P_{2}$ or $P_{3}$ bicliques.
Let $P_{n}^{k}$ be a power of a path with $n\leq k+1$. Since
$P_{n}^{k}=K_{n}$, every pair of vertices is a $P_{2}$ biclique.
Let $P_{n}^{k}$ be a power of a path with $k+2\leq n\leq 2k$. Since $n>k+1$
and $k>n-1-k$, the edge $\\{v_{n-1-k},v_{k}\\}$ exists and both vertices
$v_{n-1-k}$ and $v_{k}$ are adjacent to every other vertex of $P_{n}^{k}$.
This implies that they define a $P_{2}$ biclique. Clearly, vertices $v_{0}$,
$v_{k}$, and $v_{k+1}$ are distinct and define a $P_{3}$ biclique.
Now, let $P_{n}^{k}$ be a power of a path with $n\geq 2k+1$. We claim that
always exists only $P_{3}$ biclique. Let $v_{i}$ and $v_{j}$ be two adjacent
vertices in $P_{n}^{k}$, such that $i<j$. If $j\leq k$, $v_{i},v_{j},v_{j+k}$
induce a $P_{3}$, since $v_{i}$ is not adjacent to $v_{j+k}$. Otherwise $j\geq
k+1$ and $v_{j-(k+1)},v_{i},v_{j}$ induce a $P_{3}$, since $v_{j-(k+1)}$ is
not adjacent to $v_{j}$. We conclude that every $P_{2}$ is contained in a
$P_{3}$, and so every biclique in $P_{n}^{k}$ is a $P_{3}$ biclique. ∎
###### Lemma 4.
The bicliques of a power of a cycle $C_{n}^{k}$ are precisely: $P_{2}$
bicliques, if $n\leq 2k+1$; $C_{4}$ bicliques, if $2k+2\leq n\leq 3k+1$;
$P_{3}$ bicliques and $C_{4}$ bicliques, if $3k+2\leq n\leq 4k$; and $P_{3}$
bicliques, if $n\geq 4k+1$.
###### Proof.
A power of a cycle is $K_{1,3}$-free. Thus, the bicliques of a power of a
cycle are possibly $P_{2}$, $P_{3}$ or $C_{4}$ bicliques. Let $C_{n}^{k}$ be a
power of a cycle with $n\leq 2k+1$. Since $C_{n}^{k}=K_{n}$, every pair of
vertices is a $P_{2}$ biclique. Otherwise, $n\geq 2k+2$, and every $P_{2}$ is
properly contained in a $P_{3}$, as we explain next. Let $v_{i}$ and $v_{j}$
be two adjacent vertices in $C_{n}^{k}$ such that $i<j$ (indices are taken
modulo $n$). Let $v_{\ell}$ be the last consecutive vertex after $v_{j}$
adjacent to $v_{i}$ along the cyclic order. It follows that $v_{\ell+1}$ is
not adjacent to $v_{i}$ but $v_{\ell+1}$ is adjacent to $v_{j}$ and that
vertices $v_{i}$, $v_{j}$, and $v_{\ell+1}$ define a $P_{3}$. Thus, in what
follows, each biclique is possibly $P_{3}$ or $C_{4}$ biclique.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $2k+2\leq n\leq 4k$. Since
$2k+2\leq n\leq 4k$, the subset of vertices
$H=\\{v_{0},v_{\lceil\frac{n}{4}\rceil},v_{\lceil\frac{n}{2}\rceil},v_{\lceil\frac{3n}{4}\rceil}\\}$
is a $C_{4}$ biclique. Hence, $G$ has a $C_{4}$ biclique.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $n\geq 4k+1$. Suppose
$P=\\{v_{h},v_{s},v_{r}\\}$ is a $P_{3}$. If the missing edge is
$\\{v_{h},v_{r}\\}$, then, by symmetry, we may assume $h<s<r$. Since $n\geq
4k+1$, vertices $v_{h}$ and $v_{r}$ have no common neighbor with index at most
$h-1$ and at least $r+1$. Hence, $G$ does not have a $C_{4}$ biclique.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $2k+2\leq n\leq 3k+1$. Suppose
$P^{\prime}=\\{v_{h^{\prime}},v_{s^{\prime}},v_{r^{\prime}}\\}$ is a $P_{3}$.
If the missing edge is $\\{v_{h^{\prime}},v_{r^{\prime}}\\}$, then, by
symmetry, we may assume $h^{\prime}<s^{\prime}<r^{\prime}$. Since $2k+2\leq
n\leq 3k+1$, vertices $v_{h^{\prime}}$ and $v_{r^{\prime}}$ have a common
neighbor with index at most $h^{\prime}-1$ and at least $r^{\prime}+1$ which
is not a neighbor of $v_{s^{\prime}}$. We conclude that every $P_{3}$ is
contained in a $C_{4}$ and therefore $G$ contains only $C_{4}$ biclique.
Now, let $G$ be a power of a cycle $C_{n}^{k}$ with $n\geq 3k+2$. Consider the
$P_{3}$ induced by vertices $v_{0}$, $v_{k}$, and $v_{k+1}$. Since $n\geq
3k+2$, vertices $v_{0}$ and $v_{k+1}$ have no common neighbor with index at
least $k+2$. Hence, $G$ has a $P_{3}$ biclique. ∎
## 4 Determining the biclique-chromatic number of $P_{n}^{k}$
The extreme cases are easy to compute: the densest case occurs when $n\leq
k+1$, which implies that a power of a path $P_{n}^{k}$ is the complete graph
$K_{n}$ whose biclique-chromatic number is its order $n$, whereas for the non-
complete case, the sparsest case $P_{n}^{k}$ occurs when $k=1$, which implies
that a power of a path $P_{n}^{k}$ is the chordless path $P_{n}$ whose
biclique-chromatic number is 2. According to Lemma 3, we consider other two
cases: the less dense case $n\in[k+2,2k]$, and the sparse case
$n\in[2k+1,\infty)$. The proof of Theorem 5 (resp. Theorem 6) additionally
yields an efficient $2k+2-n$-biclique-colouring (resp. 2-biclique-colouring)
algorithm for the less dense case (resp. for the sparse case).
###### Theorem 5.
A power of a path $P_{n}^{k}$, when $k+2\leq n\leq 2k$, has biclique-chromatic
number $2k+2-n$.
###### Proof.
Let $G$ be a power of a path $P_{n}^{k}$ with $k+2\leq n\leq 2k$. Each of the
vertices $v_{n-1-k},\ldots,v_{k}$ is universal and any pair of vertices in
$\\{v_{n-1-k},\ldots,v_{k}\\}$ induces a $P_{2}$ biclique in the graph. Hence,
we are forced to give distinct colours to each of the vertices
$v_{n-1-k},\ldots,v_{k}$ and we have $\kappa_{B}(G)\geq 2k+2-n$.
We define $\pi:V(G)\rightarrow\\{1,\ldots,2k+2-n\\}$ by giving (arbitrarily)
distinct colours $3,\ldots,2k+2-n$ to vertices $v_{n-k},\ldots,v_{k-1}$. Now,
use colour $1$ in the uncoloured vertices before $v_{n-k}$ and colour $2$ in
the uncoloured vertices after $v_{k-1}$. Every monochromatic edge contains
either both end vertices before $v_{n-k}$ or both end vertices after
$v_{k-1}$. By symmetry, consider $\\{v_{i},v_{j}\\}$ a monochromatic edge such
that $i<j<n-k$. Now, vertices $v_{i},v_{j},v_{j+k}$ induce a $P_{3}$ biclique.
Since any choice of three vertices either before $v_{n-k}$ or after $v_{k-1}$
defines a triangle, $\pi$ is a biclique-colouring of $G$.
We refer to Figure 3a to illustrate the given $(2k+2-n)$-biclique-colouring. ∎
###### Theorem 6.
A power of a path $P_{n}^{k}$, when $n\geq 2k+1$, has biclique-chromatic
number 2.
###### Proof.
Let $G$ be a power of a path $P_{n}^{k}$ with $n\geq 2k+1$. Let $n=ak+t$, with
$0\leq t<k$. We define $\pi:V(G)\rightarrow\\{blue,red\\}$ as follows. A
number of $a$ monochromatic-blocks of size $k$ switching colours _red_ and
_blue_ alternately, followed by a monochromatic-block of size $t$ with _red_
colour if $a$ is even or _blue_ colour if $a$ is odd. We refer to Figure 3b to
illustrate the given 2-biclique-colouring.
Lemma 3 says that every biclique of $G$ is a $P_{3}$. Thus, every biclique is
polychromatic, since it contains vertices from two consecutive monochromatic-
blocks (with distinct colours by the given colouring). ∎
(a) $(2k+2-n)$-biclique-colouring, when $k+1\leq n\leq 2k$.
(b) $2$-biclique-colouring, when $n\geq 2k+2$ and $0\leq t<k$.
Figure 3: Biclique-colouring of powers of paths
## 5 Determining the biclique-chromatic number of $C_{n}^{k}$
The extreme cases are easy to compute: the densest case occurs when $n\leq
2k+1$, which implies that a power of a cycle $C_{n}^{k}$ is the complete graph
$K_{n}$ whose biclique-chromatic number is its order $n$, whereas for the non-
complete case, the sparsest case $C_{n}^{k}$ occurs when $k=1$, which implies
that a power of a cycle $C_{n}^{k}$ is the chordless cycle $C_{n}$ whose
biclique-chromatic number is 2. According to Lemma 4, we consider other two
cases: the less dense case $n\in[2k+2,3k+1]$, whose biclique-chromatic number
is 2, and the sparse case $n\in[3k+2,\infty)$.
The division algorithm says that any natural number $a$ can be expressed using
the equation $a=bq+t$, with a requirement that $0\leq t<b$. We shall use the
following version where $b$ is even and $0\leq t<2k$.
###### Theorem 7 (Division algorithm).
Given two natural numbers $n$ and $k$, with $n\geq 2k$, there exist unique
natural numbers $a$ and $t$ such that $n=ak+t$, $a\geq 2$ is even, and
$0~{}\leq~{}t~{}<~{}2k$.
Given a non-complete power of a cycle, Lemma 8 shows that there exists a
3-colouring of its vertices such that no $P_{3}$ is monochromatic. Since every
biclique contains a $P_{3}$, Lemma 8 provides an upper bound of 3 for the
biclique-chromatic number of a power of a cycle — the proof of Lemma 8
additionally yields an efficient 3-biclique-colouring algorithm using the
version of the division algorithm stated in Theorem 7. Moreover, this upper
bound of 3 to the biclique-chromatic number is tight. Please refer to Figure 4
for an example of a graph not 2-biclique-colourable.
###### Lemma 8.
Let $G$ be a power of a cycle $C_{n}^{k}$, where $n\geq 2k+2$. Then, $G$
admits a 3-colouring of its vertices such that $G$ has no monochromatic
$P_{3}$.
###### Proof.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $n\geq 2k+2$. Theorem 7 says
that $n=ak+t$ for natural numbers $a$ and $t$, $a\geq 2$ is even, and $0\leq
t<2k$. If $0\leq t\leq k$, we define $\pi:V(G)\rightarrow\\{blue,red,green\\}$
as follows. An even number $a$ of monochromatic-blocks of size $k$ switching
colours _red_ and _blue_ alternately, followed by a monochromatic-block of
size $t$ with colour _green_. Otherwise, i.e. $k<t<2k$, we define
$\pi:V(G)\rightarrow\\{blue,red,green\\}$ as follows. An odd number $a+1$ of
monochromatic-blocks of size $k$ switching colours _red_ and _blue_
alternately, followed by a monochromatic-block of size $k$ with colour _green_
, a monochromatic-block of size $k$ with colour _blue_ , and a monochromatic-
block of size $t-k$ with colour _green_. We refer to Figure 5a to illustrate
the former 3-biclique-colouring and to Figure 5b to illustrate the latter
3-biclique-colouring.
Consider any three vertices $v_{i}$, $v_{j}$ and $v_{\ell}$ with the same
colour. Then, either they are in the same monochromatic-block — and induce a
triangle — or two of them are not in consecutive monochromatic-blocks – and
induce a disconnected graph. In both cases, $v_{i}$, $v_{j}$ and $v_{\ell}$ do
not induce a $P_{3}$. ∎
###### Theorem 9.
A power of a cycle $C_{n}^{k}$, when $n\geq 2k+2$, has biclique-chromatic
number at most 3.
Figure 4: Power of a cycle $C_{11}^{3}$ with biclique-chromatic number 3. We
highlight in bold a $P_{3}$ biclique of reach $4$ and a $C_{4}$ biclique.
(a) 3-biclique-colouring, when $n\geq 2k+2$ and $0\leq t\leq k$.
(b) 3-biclique-colouring, when $n\geq 2k+2$ and $k<t<2k$.
(c) 2-biclique-colouring when $2k+2\leq n\leq 3k+1$
(d) 2-biclique-colouring of a 2-biclique-colourable graph, when $n\geq 3k+2$
Figure 5: Biclique-colouring of powers of cycles
As a consequence of Theorem 9, every non-complete power of a cycle has
biclique-chromatic number 2 or 3, and it is a natural question how to decide
between the two values. We first settle this question in the less dense case
$n\in[2k+2,3k+1]$. In fact, we show that all powers of cycles in the less
dense case $n\in[2k+2,3k+1]$ are 2-biclique-colourable — the proof of Theorem
10 additionally yields an efficient 2-biclique-colouring algorithm.
###### Theorem 10.
A power of a cycle $C_{n}^{k}$, when $2k+2\leq n\leq 3k+1$, has biclique-
chromatic number 2.
###### Proof.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $2k+2\leq n\leq 3k+1$. We
define $\pi:V(G)\rightarrow\\{blue,red\\}$ as follows. A monochromatic-block
of size $k$ with colour _red_ followed by a monochromatic-block of size $n-k$
with colour _blue_. We refer to Figure 5c to illustrate the given 2-biclique-
colouring.
Recall that every biclique of $G$ is a $C_{4}$ biclique. For the sake of
contradiction, suppose that there exists a monochromatic set $H$ of four
vertices. If $H$ is contained in the block of size $k$, then $H$ induces a
$K_{4}$ and cannot be a $C_{4}$. Otherwise, $H$ is contained in the block of
size $n-k\leq 2k+1$ and there exists a subset of $H$ which induces a triangle,
so that $H$ cannot be a $C_{4}$ biclique. ∎
The sparse case $n\geq 3k+2$ is more tricky. Let $G$ be a power of a cycle
$C_{n}^{k}$ with $n\geq 3k+2$. Following Lemma 4, there always exists a
$P_{3}$ biclique in $G$. Clearly, a biclique-colouring of $G$ has every
$P_{3}$ biclique polychromatic, but we may think that there exists some
monochromatic $P_{3}$ (not biclique). Nevertheless, we prove that $G$ has
biclique-chromatic number 2 if, and only if, there exists a 2-colouring of $G$
such that no $P_{3}$ is monochromatic, which happens exactly when there exists
a 2-colouring of $G$ where every monochromatic-block has size $k$ or $k+1$.
(a) vertices $v_{i-1}$, $v_{i+k-x}$, and $v_{i+k}$ induce a monochromatic
$P_{3}$ with reach $k+1$
(b) vertices $v_{i}$, $v_{i+k}$, and $v_{i+k+1}$ induce a monochromatic
$P_{3}$ with reach $k+1$
(c) vertices $v_{i-1}$, $v_{i+k-x}$, and $v_{i+k+1}$ induce a monochromatic
$P_{3}$ with reach $k+2$
Figure 6: A monochromatic-block of size $x\neq k,k+1$ in a power of a cycle
$C_{n}^{k}$, with $n\geq 2k+2$, implies a monochromatic $P_{3}$ with reach
$k+1$ or $k+2$.
###### Lemma 11.
Let $G$ be a power of a cycle $C_{n}^{k}$, where $n\geq 2k+2$, and consider a
2-colouring of its vertices. If every monochromatic-block has size $k$ or
$k+1$, then $G$ has no monochromatic $P_{3}$. Otherwise, if not every
monochromatic-block has size $k$ or $k+1$, then $G$ has a monochromatic
$P_{3}$ with reach $k+1$ or $k+2$; in particular, when $n=3k+2$, $G$ has a
monochromatic $P_{3}$ with reach $k+1$ or $G$ has a monochromatic $C_{4}$.
###### Proof.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $n\geq 2k+2$. Consider a
2-colouring $\pi$ of the vertices of $G$ such that every monochromatic-block
has size $k$ or $k+1$.
Consider any three vertices $v_{i}$, $v_{j}$ and $v_{\ell}$ with the same
colour. Then, either they are in the same monochromatic-block — and induce a
triangle — or two of them have indices that differ by at least $k+1$ with
respect to the third vertex — and the three vertices induce a disconnected
graph. In both cases, $v_{i}$, $v_{j}$ and $v_{\ell}$ do not induce a $P_{3}$.
Hence, no $P_{3}$ is monochromatic.
Now, consider a 2-colouring $\pi$ of the vertices of $G$ such that there
exists a monochromatic-block of size $x\neq k,k+1$. Consider a monochromatic-
block of size $p\geq k+2$ with vertices $v_{i}$, $v_{i+1}$, $v_{i+2}$,
$\ldots$, $v_{i+k+1}$, $\ldots$, and $v_{i+p-1}$. Notice that vertices
$v_{i}$, $v_{i+1}$, and $v_{i+k+1}$ induce a $P_{3}$. So, we may assume that
there exists a monochromatic-block with vertices $v_{i}$, $v_{i+1}$,
$v_{i+2}$, $\ldots$, $v_{i+k+1}$, $\ldots$, $v_{i+k-x-1}$, where $x>0$. By
symmetry, consider that $v_{i}$ has blue colour. Notice that vertices
$v_{i-1}$ and $v_{i+k-x}$ are adjacent and with red colour. Please refer to
Figure 6. Suppose that vertex $v_{i+k}$ has red colour. Then, vertices
$v_{i-1}$, $v_{i+k-x}$, and $v_{i+k}$ induce a monochromatic $P_{3}$ with
reach $k+1$ (see Figure 6a). Now, consider vertex $v_{i+k}$ has blue colour.
Suppose that vertex $v_{i+k+1}$ has blue colour, then vertices $v_{i+k}$,
$v_{i+k+1}$, and $v_{i}$ induce a monochromatic $P_{3}$ with reach $k+1$ (see
Figure 6b). Now, consider vertex $v_{i+k+1}$ has red colour and vertices
$v_{i-1}$, $v_{i+k-x}$, and $v_{i+k+1}$ induce a monochromatic $P_{3}$ with
reach $k+2$ (see Figure 6c).
Now, consider the case $n=3k+2$. We know that $G$ has a monochromatic $P_{3}$
of reach $k+1$ or $k+2$. In the first case, we are done, so we assume that $G$
has a monochromatic $P_{3}$ $v_{i-1}$, $v_{i+k-x}$, and $v_{i+k+1}$ of red
colour. Moreover, vertex $v_{i}$ (resp. vertex $v_{i+k}$) has blue colour,
otherwise vertices $v_{i}$, $v_{i+k-x}$, and $v_{i+k+1}$ (resp. vertices
$v_{i-1}$, $v_{i+k-x}$, and $v_{i+k}$) would induce a monochromatic $P_{3}$
with reach $k+1$. Vertices $v_{i-1}$, $v_{i+k-x}$, $v_{i+k+1}$, and
$v_{i+2k+1}$ induce the unique $C_{4}$ that includes vertices $v_{i-1}$,
$v_{i+k-x}$, and $v_{i+k+1}$. Please refer to Figure 7. Suppose vertex
$v_{i+2k+1}$ has red colour, then vertices $v_{i-1}$, $v_{i+k-x}$,
$v_{i+k+1}$, and $v_{i+2k+1}$ induce a monochromatic $C_{4}$ (see Figure 7a).
Now, consider vertex $v_{i+2k+1}$ has blue colour. Suppose that vertex
$v_{i+2k}$ (resp.$v_{i+2k+2}$) has blue colour, then vertices $v_{i+k}$,
$v_{i+2k}$, and $v_{i+2k+1}$ (resp. $v_{i+2k+1}$, $v_{i+2k+2}$, and
$v_{i+3k+2}$) induce a monochromatic $P_{3}$ with reach $k+1$ (see Figure 7b).
Now, consider vertices $v_{i+2k}$ and $v_{i+2k+2}$ have red colour. Vertices
$v_{i+k+1}$, $v_{i+2k}$, and $v_{i+2k+2}$ induce a monochromatic $P_{3}$ with
reach $k+1$ (see Figure 7c). ∎
(a) vertices $v_{i-1}$, $v_{i+k-x}$, $v_{i+k+1}$, and $v_{i+2k+1}$ induce a
monochromatic $C_{4}$
(b) vertices $v_{i+k}$, $v_{i+2k}$, and $v_{i+2k+1}$ (resp. $v_{i+2k+1}$,
$v_{i+2k+2}$, and $v_{i}$) induce a monochromatic $P_{3}$ with reach $k+1$
(c) vertices $v_{i+k+1}$, $v_{i+2k}$, and $v_{i+2k+2}$ induce a monochromatic
$P_{3}$ with reach $k+1$
Figure 7: A monochromatic-block of size $x\neq k,k+1$ in a power of a cycle
$C_{n}^{k}$, with $n=3k+2$, implies a monochromatic $P_{3}$ with reach $k+1$
or a monochromatic $C_{4}$.
###### Theorem 12.
A power of a cycle $C_{n}^{k}$, when $n\geq 3k+2$, has biclique-chromatic
number 2 if, and only if, there exist natural numbers $a$ and $b$, such that
$n=ak+b(k+1)$ and $a+b\geq 2$ is even.
###### Proof.
Let $G$ be a power of a cycle $C_{n}^{k}$ with $n\geq 3k+2$.
First, consider natural numbers $a$ and $b$, such that $n=ak+b(k+1)$ and
$a+b\geq 2$ is even. Then, there exists a 2-colouring $\pi$ such that every
monochromatic-block has size $k$ or $k+1$. Lemma 11 says that $G$ has no
monochromatic $P_{3}$ and therefore $\pi$ is a 2-biclique-colouring. We refer
to Figure 5d to illustrate such 2-biclique-colouring.
For the converse, suppose that there are no such $a$ and $b$, which implies
that any 2-colouring $\pi^{\prime}$ of the vertices of $G$ is such that there
exists a monochromatic-block of size $x\neq k,k+1$. Consider $n=3k+2$. Lemma
11 says that such 2-colouring of the vertices of $G$ has a monochromatic
$P_{3}$ with reach $k+1$ or a monochromatic $C_{4}$. Every $P_{3}$ with reach
$k+1$ is a biclique and every $C_{4}$ is a biclique, which implies that
$\pi^{\prime}$ is not a 2-biclique-colouring, which is a contradiction. Now,
consider $n>3k+2$. Lemma 11 says that such 2-colouring of the vertices of $G$
has a monochromatic $P_{3}$ with reach $k+1$ or $k+2$. Every $P_{3}$ with
reach $k+1$ or $k+2$ is a $P_{3}$ biclique, which implies that $\pi^{\prime}$
is not a 2-biclique-colouring, which is a contradiction. ∎
There exists an efficient algorithm that verifies if the system of equations
of Theorem 12 has a solution. If so, it also computes values of $a$ and $b$ –
the proof of Theorem 13 yields Algorithm 1 to determine if the biclique-
chromatic number is 2 or 3 and also computes values of $a$ and $b$. When the
biclique-chromatic number is 2, we define a 2-biclique-colouring
$\pi:V(G)\rightarrow\\{blue,red\\}$ as follows. A number $a$ of monochromatic-
blocks of size $k$ plus a number $b$ of monochromatic-blocks of size $k+1$
switching colours _red_ and _blue_ alternately. We refer to Figure 5d to
illustrate the given 2-biclique-colouring.
###### Theorem 13.
There exists an algorithm that computes the biclique-chromatic number of a
power of a cycle $C_{n}^{k}$, when $n\geq 3k+2$.
###### Proof.
Theorem 9 states that the biclique-chromatic number of a power of a cycle
$C_{n}^{k}$ is at most 3 and Theorem 12 states that a power of a cycle
$C_{n}^{k}$ with $n\geq 3k+2$ has biclique-chromatic number 2 if, and only if,
there exist natural numbers $a$ and $b$, such that $n=ak+b(k+1)$ and $a+b\geq
2$ is even.
Let $c=a+b$. We show that there exist natural numbers $b$ and $c$, such that
$n=ck+b$, $b\leq c$, and $c$ is even if, and only if, natural numbers
$c_{0}=\left\lfloor\frac{n}{k}\right\rfloor$ and $b_{0}=n-c_{0}k$ have the
following properties: $c_{0}$ is even and $b_{0}\leq c_{0}$; or natural
numbers $c_{1}=\left\lfloor\frac{n}{k}\right\rfloor-1$ and $b_{1}=n-c_{1}k$
have the following properties: $c_{1}$ is even and $b_{1}\leq c_{1}$.
Clearly, $b_{0}$ and $c_{0}$ (resp. $b_{1}$ and $c_{1}$) are natural numbers
such that $n=c_{0}k+b_{0}$ (resp. $n=c_{1}k+b_{1}$), $b_{0}\leq c_{0}$ (resp.
$b_{1}\leq c_{1}$), $c_{0}$ (resp. $c_{1}$) is even, and $c_{0}\geq 2$ (resp.
$c_{1}\geq 2$) since $n\geq 2k+2$.
For the converse, suppose that there exist natural numbers $a$ and $b$, such
that $n=ck+b$ and $c$ is even. Let $b^{\prime}=b$ and $c^{\prime}=c$. While
$b^{\prime}\geq 2k$, do $c^{\prime}:=c^{\prime}+2$ and
$b^{\prime}:=b^{\prime}-2k$. Clearly, in the end of the loop, we have
$c^{\prime}$ even, $b^{\prime}\geq 0$, and $c^{\prime}\geq b^{\prime}$.
Moreover, we consider two cases.
* •
$b^{\prime}<k$ in the end of the loop. Then,
$c^{\prime}=\left\lfloor\frac{n}{k}\right\rfloor$ and
$b^{\prime}=n-c^{\prime}k$.
* •
$k\leq b^{\prime}<2k$ in the end of the loop. Then,
$c^{\prime}=\left\lfloor\frac{n}{k}\right\rfloor-1$ and
$b^{\prime}=n-c^{\prime}k$.
∎
As a remark, in Theorem 13, we let $c=a+b$ and rewrite the equation
$n=ak+b(k+1)$ as $n=ck+b$, very similar to the Division Algorithm formula.
Nevertheless, there is a rather subtle difference: in the Division Algorithm
formula, the choice for the value of the remainder is bounded by the value of
the divisor, while in the equation $n=ck+b$, the choice for the value of the
remainder is bounded by the choice for the value of the quotient (recall
$b\leq c$). This subtle difference may change drastically the behavior of the
equation. More precisely, given two natural numbers $n$ and $k$, with $n\geq
2k+2$, it is not necessarily true that there exist natural numbers $b$ and $c$
such that $n=ck+b$, $c\geq 2$ is even, and $b\leq c$. For instance, there do
not exist natural numbers $b$ and $c$ such that $11=3c+b$, $c\geq 2$ is even,
and $b\leq c$.
input : $C_{n}^{k}$, a power of a cycle with $n\geq 3k+2$
output : $\kappa_{B}(C_{n}^{k})$, the biclique-chromatic number of
$C_{n}^{k}$.
1 begin
2 $c\longleftarrow\left\lfloor\frac{n}{k}\right\rfloor$;
3 $b\longleftarrow n-ck$;
4 if _$c\bmod 2=0$ and $c\geq b$_ then
_5_ _ _ return _$2$ ;_
6
7 else
8 $c\longleftarrow\left\lfloor\frac{n}{k}\right\rfloor-1$;
9 $b\longleftarrow n-ck$;
10 if _$c\bmod 2=0$ and $c\geq b$_ then
_11_ _ _ return _$2$ ;_
12
13 else
_14_ _ _ return _$3$ ;_
15
16
17
18
Algorithm 1 To compute the biclique-chromatic number of a power of a cycle
$C_{n}^{k}$ with $n\geq 3k+2$
## 6 Final considerations
The reader should notice the structure differences between the two considered
classes of power graphs and observe the similarities on giving lower and upper
bounds on the biclique-chromatic number. For instance, the lower bound on the
biclique-chromatic number in both cases when $n\leq 2k$ is a consequence of
the existence of a set of $K_{2}$ bicliques whose union induces a complete
graph — in the case of powers of cycles, this can happen only when such union
is the whole vertex set, but in the case of powers of paths such union can be
the whole vertex set (when $n\leq k+1$) or a vertex subset of size $2k+2-n$
(when $k+2\leq n\leq 2k$). When $n\geq 2k+1$, monochromatic-blocks are the key
step to construct optimal colourings. Nevertheless, in the given colourings,
for powers of paths, vertices $v_{0}$ and $v_{n-1}$ may have the same colour,
which is not the case for powers of cycles.
Table 1 highlights the exact values for the biclique-chromatic number of the
power graphs settled in this work. In Figures 8 and 9, we illustrate the
biclique-chromatic number for a fixed value of $k$ and an increasing $n$ of
powers of paths and powers of cycles, respectively.
Figure 8: The biclique-chromatic number of a non-complete power of a path for
a fixed value of $k$ and an increasing $n$ Figure 9: The biclique-chromatic
number of a non-complete power of a cycle for a fixed value of $k$ and an
increasing $n$
As a corollary of Theorem 12, every non-complete power of a cycle $C_{n}^{k}$
with $n\geq 2k^{2}$ has biclique-chromatic number 2. Thus, the biclique-
chromatic number of a power of a cycle $C_{n}^{k}$, for a fixed value of $k$
and an increasing $n\geq 3k+2$, does not oscillate forever.
###### Corollary 14.
A non-complete power of a cycle $C_{n}^{k}$ with $n\geq 2k^{2}$ has biclique-
chromatic number 2.
###### Proof.
Theorem 7 says that $n=a^{\prime}k+t$ for natural numbers $a^{\prime}$ and
$t$, $a^{\prime}\geq 2$ is even, and $0\leq t<2k$. If we can rewrite
$n=ak+b(k+1)$ with natural numbers $a$ and $b$, such that $a+b\geq 2$ is even,
then Theorem 12 says that a power of a cycle $C_{n}^{k}$ with $n\geq 2k^{2}$
has biclique-chromatic number 2. Since $0\leq t\leq 2k$, $n\geq 2k^{2}$, and
$a^{\prime}$ is an even natural number, we have
$\displaystyle n=a^{\prime}k+t$ $\displaystyle\geq$ $\displaystyle 2k^{2}$
$\displaystyle a^{\prime}k$ $\displaystyle\geq$ $\displaystyle 2k^{2}-2k+1$
$\displaystyle a^{\prime}$ $\displaystyle\geq$ $\displaystyle 2k-1$
$\displaystyle a^{\prime}$ $\displaystyle\geq$ $\displaystyle 2k$
Let $a=a^{\prime}-t$ and $b=t$. Clearly, $a$ and $b$ are natural numbers.
Moreover, $a+b\geq 2$ is even. ∎
Groshaus, Soulignac, and Terlisky have recently proposed a related hypergraph
colouring, called _star-colouring_ [19], defined as follows. A _star_ is a
maximal set of vertices that induces a complete bipartite graph with a
universal vertex and at least one edge. The definition of star-colouring
follows the same line as clique-colouring and biclique-colouring: a _star-
colouring_ of a graph $G$ is a function that associates a colour to each
vertex such that no star is monochromatic. The _star-chromatic number_ of a
graph $G$, denoted by $\kappa_{S}(G)$, is the least number of colours $c$ for
which $G$ has a star-colouring with at most $c$ colours. Many of the results
of biclique-colouring achieved in the present work are naturally extended to
star-colouring. Since the constructed graph of Corollary 2 is $C_{4}$-free and
the bicliques in a $C_{4}$-free graph are precisely the stars of the graph, we
can restate Corollary 2 as follows below.
###### Corollary 15.
Let $G$ be a $\\{C_{4},K_{4}\\}$-free graph. It is co$\mathcal{NP}$-complete
to check if a colouring of the vertices of $G$ is a star-colouring.
About star-colouring and the investigated classes of power graphs, we also
have some few remarks. On one hand, the bicliques of a power of a path
$P_{n}^{k}$ are the stars of the graph and, consequently, all results obtained
for biclique-colouring powers of paths hold to star-colouring powers of paths.
On the other hand, a power of a cycle $C_{n}^{k}$ is not necessarily
$C_{4}$-free, and there are examples of powers of cycles with $P_{3}$ stars
that are not bicliques due to the fact that such $P_{3}$ stars are contained
in $C_{4}$ bicliques of the graph. This happens for instance in the case
$n\in[2k+2,3k+1]$ and one such example is graph $C_{11}^{4}$ exhibited in
Figure 10. Notice that the highlighted vertices form a monochromatic $P_{3}$
star, so that the colouring is not a 2-star-colouring. The three highlighted
vertices together with vertex $u$, on the other hand, form a polychromatic
$C_{4}$ biclique — indeed, the exhibited colouring is a 2-biclique-colouring.
We summarize the results about star-colouring powers of paths and powers of
cycles in the following theorems and also in Table 1. Please refer to the line
of the table where we consider a power of a cycle with $n\in[2k+2,3k+1]$ to
check the difference between the biclique-chromatic number (which is always 2)
and the star-chromatic number (which depends on $n$ and $k$).
Figure 10: Power of a cycle $C_{11}^{4}$ with a 2-biclique-colouring which is
not a 2-star-colouring. Notice that there exists a monochromatic $P_{3}$ star
highlighted in bold.
###### Theorem 16.
For any power of a path, the star-chromatic number is equal to the biclique-
chromatic number.
###### Theorem 17.
A power of a cycle $C_{n}^{k}$, when $n\leq 2k+1$ or $n\geq 3k+2$, has star-
chromatic number equal to the biclique-chromatic number. If $2k+2\leq n\leq
3k+1$, then $C_{n}^{k}$ has star-chromatic number 2 if, and only if, there
exist natural numbers $a$ and $b$, such that $n=ak+b(k+1)$ and $a+b\geq 2$ is
even. If there does not exist such natural numbers, it has star-chromatic
number 3.
Graph $G$ | Range of $n$ | $\kappa_{B}(G)$ | $\kappa_{S}(G)$
---|---|---|---
$P_{n}^{k}$ | $[1,k+1]$ | $n$ | $n$
$[k+2,2k]$ | $2k+2-n$ | $2k+2-n$
$[2k+1,\infty[$ | $2$ | $2$
$C_{n}^{k}$ | $[1,2k+1]$ | $n$ | $n$
$[2k+2,3k+1]$ | $2$ |
$[3k+2,2k^{2}[$ | $2$, if there exist natural numbers $a$ and $b$,
such that $n~{}=~{}ak~{}+~{}b(k+1)$
and $a+b\geq 2$ is even;
| $3$, otherwise.
| $[2k^{2},\infty[$ | $2$ | $2$
Table 1: Biclique- and star-chromatic numbers of powers of paths and powers of
cycles
A _distance graph_ $P_{n}(d_{1},\dots,d_{k})$ is a simple graph with
$V(G)=\\{v_{0},\dots,v_{n-1}\\}$ and $E(G)=E^{d_{1}}\cup\dots\cup E^{d_{k}}$,
such that $\\{v_{i},v_{j}\\}\in E^{d_{\ell}}$ if, and only if, it has reach –
in the context of a power of a path – $d_{\ell}$. Notice that a distance graph
$P_{n}(d_{1},\dots,d_{k})$ is a power of a path if $d_{1}=1$,
$d_{i}=d_{i-1}+1$, and $d_{k}<n-1$. A _circulant graph_
$C_{n}(d_{1},\dots,d_{k})$ has the same definition as the distance graph,
except by the reach, which, in turn, is in the context of a power of a cycle.
Notice that a circulant graph $C_{n}(d_{1},\dots,d_{k})$ is a power of a cycle
if $d_{1}=1$, $d_{i}=d_{i-1}+1$, and $d_{k}<\lfloor\frac{n}{2}\rfloor$.
Circulant graphs have been proposed for various practical applications [4]. We
suggest, as a future work, to biclique colour the classes of distance graphs
and circulant graphs, since colouring problems for distance graphs and for
circulant graphs have been extensively investigated [2, 30, 33]. Moreover,
some results of intractability have been obtained, e.g. determining the
chromatic number of circulant graphs in general is an $\mathcal{NP}$-hard
problem [11].
## Acknowledgments
The authors would like to thank Renan Henrique Finder for the discussions on
the algorithm to compute the biclique-chromatic number of a power of a cycle
$C_{n}^{k}$, when $n\geq 3k+2$; and to thank Vinícius Gusmão Pereira de Sá and
Guilherme Dias da Fonseca for discussions on the complexity of numerical
problems. At last, but not least, we thank Vanessa Cavalcante for the careful
proofreading of this paper.
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|
arxiv-papers
| 2012-03-12T16:45:21 |
2024-09-04T02:49:28.542818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H\\'elio B. Mac\\^edo Filho, Simone Dantas, Raphael C. S. Machado, and\n Celina M. H. de Figueiredo",
"submitter": "H\\'elio Mac\\^edo Filho",
"url": "https://arxiv.org/abs/1203.2543"
}
|
1203.2593
|
# Supergranules as Probes of Solar Convection Zone Dynamics
David H. Hathaway NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
david.hathaway@nasa.gov
###### Abstract
Supergranules are convection cells seen at the Sun’s surface as a space
filling pattern of horizontal flows. While typical supergranules have
diameters of about 35 Mm, they exhibit a broad spectrum of sizes from $\sim
10$ Mm to $\sim 100$ Mm. Here we show that supergranules of different sizes
can be used to probe the rotation rate in the Sun’s outer convection zone. We
find that the equatorial rotation rate as a function of depth as measured by
global helioseismology matches the equatorial rotation as a function of
wavelength for the supergranules. This suggests that supergranules are
advected by flows at depths equal to their wavelengths and thus can be used to
probe flows at those depths. The supergranule rotation profiles show that the
surface shear layer, through which the rotation rate increases inward, extends
to depths of $\sim 50$ Mm and to latitudes of at least $70\arcdeg$. Typical
supergranules are well observed at high latitudes and have a range of sizes
that extend to greater depths than those typically available for measuring
subsurface flows with local helioseismology. These characteristics indicate
that probing the solar convection zone dynamics with supergranules can
complement the results of helioseismology.
Sun: convection, Sun: rotation
## 1 INTRODUCTION
Supergranules were discovered in the 1950s by Hart (1954) but were best
characterized in the 1960s by Leighton et al. (1962) who gave them their name.
Leighton et al. (1962) showed that this cellular pattern of horizontal flows
covers the solar surface and that the boundaries of the cells coincide with
the chromospheric/magnetic network. Typical supergranules have diameters of
$\sim 35$ Mm and maximum flow speeds of $\sim 500$ m s-1. While the kinetic
energy spectrum has a distinct peak at wavelengths of $\sim 35$ Mm, the
spectrum includes cells at least three times larger and extends to much
smaller cells where the supergranule spectrum blends into the granulation
spectrum (Hathaway et al., 2000).
Larger cells live longer than smaller cells. Typical supergranules with
diameters of $\sim 30$ Mm live for $\sim 24$ hr (Simon & Leighton, 1964; Wang
& Zirin, 1989). Typical granules with diameters of $\sim 1$ Mm only live for
$\sim 5$ min (Title et al., 1989). Cells of intermediate size ($\sim 5-10$ Mm)
have intermediate lifetimes of $\sim 2$ hr (November et al., 1981).
The rotation rate of the supergranule pattern was first measured by Duvall
(1980) who cross-correlated the Doppler velocity pattern from equatorial
spectral scans obtained over several days. He found that the pattern rotates
about 3% faster than the photospheric plasma and faster rates are found for
the 24-hr time lags from day-to-day than for the 8-hr time lags from the
beginning to end of an observing day. He concluded that larger cells dominate
the longer time lags and that the observations are consistent with
supergranules embedded in a surface shear layer in which the rotation rate
increases with depth.
The presence of this shear layer was first suggested by Foukal & Jokipii
(1975) as a consequence of the conservation of angular momentum by convective
elements moving inward and outward in the near surface layers. Global
helioseismology inversions (Thompson et al., 1996; Schou et al., 1998)
indicate that the shear layer extends to depths of 35-50 Mm near the equator
but may disappear or reverse at latitudes above $\sim 55\arcdeg$. While local
helioseismology (Basu et al., 1999; Corbard & Thompson, 2002) does not probe
as deeply, it produces similar results which suggest that the rotation rate
may not continue to increase inward at higher latitudes.
Beck & Schou (2000) measured the rotation of the supergranule pattern using a
Fourier technique with space-based Doppler data from the ESA/NASA Solar and
Heliospheric Observatory (SOHO) Michelson Doppler Imager (MDI) (Scherrer et
al., 1995). They mapped the data onto heliographic coordinates, took the
Fourier transform in longitude of data from equatorial latitudes, and then the
Fourier transform in time of those spectral coefficients over 6 10-day
intervals in 1996 which had continuous coverage at a 15-min cadence. They
found that the larger cells do indeed rotate more rapidly than the smaller
cells, but with rotation rates that exceeded the peak internal rotation rate
at the base (50 Mm depth) of the surface shear layer as determined from global
helioseismology (Schou et al., 1998). This discovery led them to conclude that
supergranules must have wave-like properties in order to rotate faster than
the flows they are embedded in.
However, Hathaway et al. (2006) showed that line-of-sight projection effects
on a rigidly rotating fixed velocity pattern could reproduce the excess
rotation velocities found by Beck & Schou (2000). In projecting the vector
velocities onto the line-of-sight the function $\sin\phi$ (where $\phi$ is the
heliographic longitude relative to the central meridian) multiplies the
longitudinal flow velocities near the equator. Since the flows are largely
horizontal the $\sin\phi$ multiplier effectively pushes the peaks in the
Doppler pattern away from the central meridian and makes the pattern appear to
rotate more rapidly.
In fact, Schou (2003) largely removed the line-of-sight projection effects
from the equatorial Doppler data by dividing the data by a function that
approximated the function $\sin\phi$ and found that rotation velocities were
much more in line with those from global helioseismology (but noted that there
were still motions relative to this that suggested wave-like properties for
supergranules). Recently, Hathaway et al. (2010) found that the rotation
profiles as functions of latitude determined by the cross-correlation
technique used by Duvall (1980) for time lags from 2-hr to 16-hr could be
reproduced by cellular patterns that are advected by a differential rotation
with a peak velocity consistent with that found in the Sun’s surface shear
layer by global helioseismology.
Here we measure the rotation of the pattern of supergranules by analyzing the
same data used by Beck & Schou (2000) and by Schou (2003). We execute a series
of 2D Fourier transform analyses. We remove the line-of-sight projection
effects near the equator as was done by Schou (2003) and repeat the Fourier
analysis done by Beck & Schou (2000) to show that the equatorial rotation rate
of supergranules as a function of wavelength matches the equatorial rotation
rate as a function of depth determined from global helioseismology (Schou et
al., 1998). This “de-projection” can only be done at the equator and is only
approximate since the flows are not purely horizontal. We determine the
rotation rates at other latitudes by repeating the Fourier transform analysis
on the raw Doppler data (without the removal of projections effects) and using
a data simulation to support our conclusions.
## 2 DATA PREPARATION
The data consist of $1024^{2}$ pixel images of the line-of-sight velocity
determined from the Doppler shift of a spectral line due to the trace element
nickel in the solar atmosphere. The images are acquired at a 1 min cadence and
cover the full visible disk of the Sun. We average the data over 31 min with a
Gaussian weighting function which filters out variations on time scales less
than about 16 min, and sample that data at 15 min intervals. We then map these
temporally filtered images onto a $1024^{2}$ grid in heliographic latitude
from pole to pole and in longitude $\pm 90\arcdeg$ from the central meridian
(Figure 1). This mapping accounts for the position angle of the Sun’s rotation
axis relative to the imaging CCD and the tilt angle of the Sun’s rotation axis
toward or away from the spacecraft. Both of these angles include modification
in line with the most recent determinations of the orientation of the Sun’s
rotation axis (Beck & Giles, 2005; Hathaway & Rightmire, 2010). We analyze
data obtained during a 60 day period of continuous coverage in 1996 from May
24 to July 22.
Line-of-sight projection effects influence the results so we also generate and
analyze simulated data to assist in our determination of the actual rotation
as a function of latitude and depth in the Sun’s outer convection zone. We
construct the simulated data from an evolving spectrum of vector spherical
harmonics in such a manner as to reproduce the spatial, spectral, and temporal
behavior of the observed cellular flows. The amplitudes of the spectral
coefficients are constrained by matching the observed velocity spectrum
(Hathaway et al., 2000) with the radial flow component constrained by the disk
center to limb variation in the RMS Doppler signal(Hathaway et al., 2002).
Figure 1: Heliographic map details of the line-of-sight (Doppler) velocity
from SOHO/MDI (top) and from the data simulation (bottom). Each map detail
extends $90\arcdeg$ in longitude from the central meridian on the left and
about $35\arcdeg$ in latitude from the equator (the thick horizontal line).
The mottled pattern is the Doppler signal (blue for blue shifts and red for
red shifts) due to the supergranules.
The cells are given finite lifetimes and made to rotate by adding changes to
the phases of the spectral coefficients (Hathaway et al., 2010). The rotation
rates are constrained by matching the observed rotation rates as functions of
latitude and wavelength.
The rotation rates of the cells in the simulated data are given by a fairly
simple function with the latitudinal, $\theta$, variation separated from the
wavelength, $\lambda$, variation such that
$\Omega(\theta,\lambda)/2\pi=f(\theta)[1+g(\lambda)]$ (1)
with
$f(\theta)=454-51\sin^{2}\theta-92\sin^{4}\theta\ \rm{nHz}$ (2)
and
$g(\lambda)=0.045\tanh{\lambda\over 31}\left[2.3-\tanh{(\lambda-65)\over
20}\right]/3.3$ (3)
where the wavelength, $\lambda$, is given in Mm.
## 3 EQUATORIAL ROTATION RATE
We determine the equatorial rotation rate of the supergranules by 2D Fourier
transforms with and without removing line-of-sight projection effects. The
line-of-sight projection effects can be minimized using the method described
by Schou (2003). The mapped Doppler velocities near the equator are divided by
a function, ${\rm sgn}(\phi)\sqrt{(}\sin^{2}\phi+0.01),$ which approximates
the geometric factor, $\sin\phi$, that multiplies the longitudinal velocity in
producing the Doppler signal. This “de-projected” signal and the raw Doppler
signal are both then apodized near the limb and then Fourier transformed over
longitude for the 50 latitude positions that straddle the equator. These
spectral coefficients are then Fourier transformed in time over six 10-day
intervals.
The rotation rate as a function of wavelength is determined by first finding
the temporal frequency of the centroid of the spectral power for that
wavelength. This temporal frequency is then divided by the longitudinal
wavenumber to give the synodic rotation rate which is then converted to a
sidereal rotation rate by adding a correction based on the rate of change of
the ecliptic longitude during the observations ($\sim 1\arcdeg\ {\rm
day}^{-1}$).
The results of these analyses are shown in Fig. 2 along with the rotation rate
with depth from a global helioseismology analysis by Schou et al. (1998). The
sidereal rotation rate of the de-projected supergranules as a function of
longitudinal wavelength matches the rotation rate as a function of depth
through the outer half of the solar convection zone and the simulation matches
the MDI observations. The raw Doppler data give faster rotation rates than the
de-projected data at all wavelengths (in both the MDI and the simulated data)
with larger increases for larger cells.
Figure 2: The equatorial rotation rate for the Doppler pattern as a function
of longitudinal wavelength. Results for the raw data are shown in the upper
panel. Results for the de-projected data are shown in the lower panel. MDI
results are shown by the black dots (with $2\sigma$ error bars for wavelengths
longer than 20 Mm). Simulated data results are shown by open circles. The
equatorial rotation rate as a function of depth from global helioseismology
(Schou et al., 1998) is shown by the large red dots while the rotation profile
used in the simulation is shown by the solid black lines.
The rotation rates seen with both de-projected datasets fall slightly below
both the helioseismology results and the input profile for the simulated data.
Experiments with the simulated data suggest that this can be attributed to the
presence of a small radial flow component. This component is projected into
the line-of-sight along the equator by multiplying by a projection factor
$\cos\phi$. When the Doppler signal is de-projected by dividing by $\sin\phi$
this small signal can become large and influence the results in this manner.
We also see that the rotation rates for the raw simulated data fall
systematically below the MDI results for wavelengths $>50$ Mm. This too may be
a result of the radial flows.
These results do however lead to a key conclusion - that supergranules with
sizes from 10 Mm to 100 Mm are advected by flows within the convection zone at
depths equal to their widths.
## 4 ROTATION PROFILES
We determine the rotation rate of the supergranules as functions of latitude
and wavelength (depth) by repeating the 2D Fourier transforms on the raw
Doppler data for a series of latitude strips. Each strip is 11 pixels or $\sim
2\arcdeg$ high in latitude and offset from the previous strip by 5 pixels. We
repeat the procedure for 172 positions between $\pm 75\arcdeg$ latitude and
for each of six 10-day intervals for both MDI and simulated data. Latitudinal
rotation profiles are obtained for a series of cell wavelengths by averaging
the profiles for all waveumbers that produce wavelengths within 5 Mm of the
target wavelength. These profiles are then averaged between hemispheres and
smoothed with a 9-point binomial smoothing kernal which then limits the data
to $\pm 70\arcdeg$ latitude. The results for averages from the MDI datasets
and the from simulated data are shown in Fig. 3.
Figure 3: The differential rotation profiles (average longitudinal velocity
relative to a frame of reference rotating at the Carrington sidereal rate of
456 nHz) for the cells with 30 Mm wavelengths are shown in the upper panel
with a solid line for the MDI data, a dashed line for the simulated data, and
$2\sigma$ error limits on the MDI data indicated by thin solid lines. The
differences between the differential rotation at the other wavelengths from
that found with the 30 Mm wavelength cells are shown in the lower panel using
colors for the different wavelengths - violet for 10 Mm, blue for 20 Mm, green
for 50 Mm, and red for 70 Mm.
The rotation velocity appears to increase at all latitudes with increasing
wavelength. However, the small increase at 70 Mm wavelength (in particular
near the equator) can be attributed to projection effects. The actual rotation
rate of the convection cells in the simulation is given by equations 1-3 and
the solid lines in Fig 2. which give a rotation rate that increases with
wavelength to a maximum at $\sim 50$ Mm at all latitudes.
The smallest cells, $10\pm 5$ Mm, are difficult to resolve at high latitudes
but clearly show slower rotation than larger cells over their observed
latitude range. Cells with wavelengths of $20\pm 5$ Mm can be resolved at all
latitudes and have significantly lower rotation rates than the larger cells.
While this method with the raw Doppler data is subject to systematic offsets
due to line-of-sight projection effects, the same offsets are present in the
simulated data and are much smaller for the smaller cells. We conclude that
the surface shear layer, in which the rotation rate increases inward, extends
to latitudes of at least $70\arcdeg$.
## 5 CONCLUSIONS
We conclude that supergranules are anchored or steered in the subsurface flows
at depths equal to their wavelengths. This is a simple explanation for the
match with the rotation rate with depth from global helioiseismology (Figure 2
lower panel) and is based on well known physics. This is consistent with
numerical simulations of convection in the outermost 16 Mm of the Sun by Stein
et al. (2011) who show that flow structures at different depths have diameters
about equal to the depth itself.
This conclusion is, however, somewhat surprising given the much smaller
estimates for the depth of typical supergranules previously determined from
the visibility of their internal flows using local helioseismology. Duvall
(1998) estimated a depth of 8 Mm for typical supergranules while Zhao &
Kosovichev (2003) estimated a depth of 15 Mm. This suggests that local
helioseismology is less sensitive to these deeper (and slower) flows and that
this new method of probing the convection zone with supergranules can probe
flows at greater depths.
We also conclude that the surface shear layer extends to a depth of $\sim 50$
Mm at all latitudes. The increase in rotation rate with depth has long been
suggested by observations and is attributed to the conservation of angular
momentum for fluid elements moving inward and outward in the near surface
layers (Foukal & Jokipii, 1975; Hathaway, 1982). Measurements of this rotation
rate increase from helioseismology (Schou et al., 1998; Corbard & Thompson,
2002) indicate that it follows this critical gradient to depths of 10-15 Mm
and reaches a maximum rotation rate at depths of 35-50 Mm. However, many
helioseismology results also suggest that the shear layer disappears at
latitudes above about $50\arcdeg$. The results reported here, using
supergranules, indicate that the shear layer extends to the highest latitudes
probed with this MDI data - $\sim 70\arcdeg$.
The rotation increase with depth given by Equation 3 follows the critical
gradient (with $\partial\ln\Omega/\partial\ln r=-1$) given by angular momentum
mixing near the surface but then drops below that value at greater depths as
the cell turn-over times become longer and the convective flows adjust to the
solar rotation.
Global helioseismology can measure the internal rotation rate to great depths
but it gives less reliable results at high latitudes. Local helioseismology
can measure the non-axisymmetric flows (as well as the axisymmetric meridional
flow) but only in the near surface layers. Using supergranules of different
sizes to probe the flows in the Sun’s convection zone extends these
measurements to greater depths and higher latitudes. This new method of
probing solar convection zone dynamics should provide information
complementary to that obtained with helioseismology.
The author would like to thank NASA for its support of this research through
grants from the Heliophysics Causes and Consequences of the Minimum of Solar
Cycle 23/24 Program and the Living With a Star Program to NASA Marshall Space
Flight Center. He is indebted to Ron Moore, Lisa (Rightmire) Upton, and an
anonymous referee whose comments greatly improved the manuscript. He would
also like to thank the American taxpayers who support scientific research in
general and this research in particular. SOHO, is a project of international
cooperation between ESA and NASA.
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|
arxiv-papers
| 2012-03-12T19:16:24 |
2024-09-04T02:49:28.554277
|
{
"license": "Public Domain",
"authors": "David H. Hathaway",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1203.2593"
}
|
1203.2661
|
# Non-classicality criteria from phase-space representations and information-
theoretical constraints are maximally inequivalent
Alessandro Ferraro Department of Physics and Astronomy, University College
London, Gower Street, London WC1E 6BT, UK Matteo G. A. Paris
matteo.paris@fisica.unimi.it Dipartimento di Fisica dell’Università degli
Studi di Milano, I-20133 Milano, Italy.
###### Abstract
We consider two celebrated criteria for defining the non-classicality of
bipartite bosonic quantum systems, the first stemming from information
theoretic concepts and the second from physical constraints on the quantum
phase-space. Consequently, two sets of allegedly classical states are singled
out: i) the set $\cal{C}$ composed of the so called classical-classical (CC)
states—separable states that are locally distinguishable and do not possess
quantum discord; ii) the set $\cal{P}$ of states endowed with a positive
P-representation (P-classical states)—mixture of Glauber coherent states that,
e.g., fail to show negativity of their Wigner function. By showing that
$\cal{C}$ and $\cal{P}$ are almost disjoint, we prove that the two defining
criteria are maximally inequivalent. Thus, the notions of classicality that
they put forward are radically different. In particular, generic CC states
show quantumness in their P-representation and, viceversa, almost all
P-classical states have positive quantum discord, hence are not CC. This
inequivalence is further elucidated considering different applications of
P-classical and CC states. Our results suggest that there are other quantum
correlations in nature than those revealed by entanglement and quantum
discord.
###### pacs:
03.65.Ta,03.65.Ud
The question of whether a quantum system exhibits a behaviour without
classical analogue has been of interest since the early days of quantum
mechanics. Considering bosonic systems, a major framework for attacking this
question has been established more then half a century ago, stemming from the
notions of quantum phase-space and quasi-probability distributions Wig32 ;
Gla63 . There, physical constraints expressing classical behaviour impose
criteria of non-classicality that have been experimentally tested in a variety
of quantum systems Smi93 ; Lei96 ; Hof09 . On the other hand, in the last two
decades non-classical correlations have been the subject of a renewed
interest, mainly due to the general belief that they are a fundamental
resource for quantum information processing. Within this perspective, a
different approach to non-classicality have emerged, which bases its ground on
the information-theoretic aspects of quantum correlations. In particular,
rigorous criteria to define non-classicality of correlations have been put
forward Wer89 ; OZ01 ; HV01 ; PHH , giving rise to well established concepts
like entanglement or quantum discord.
Here we compare these two approaches, investigating in particular whether
physical constraints emerging from the former can bring new insight in the
assessment of quantum correlations beyond the purely information-theoretic
aspects of the latter. We have found that this is indeed the case: the notion
of non-classical correlations springing from physical considerations on the
quantum phase-space is inequivalent to that emerging from information-
theoretic arguments. In a sense that will be specified in the following, these
two notions of non-classicality are maximally inequivalent. This, in
particular, suggests that there are other quantum correlations in nature than
those revealed by entanglement and quantum discord.
Non-classicality in the phase-space—The uncertainty relations make the notion
of phase-space in quantum mechanics problematic. Following the seminal
investigations of Wigner Wig32 , an abundance of quantum mechanical phase-
space quasi-distributions were introduced, ranging from the Husimi function to
the Glauber-Sudarshan P-function HOSW84 . Besides the fundamental aspect,
investigations on quasi-distributions boosted the development of efficient
theoretical tools in various fields of modern physics, e.g. quantum optics and
quantum chemistry HOSW84 ; qchem . These functions cannot, however, be
interpreted as probability distributions over a classical phase-space because
for some quantum states they may be negative or singular. Consistently, it is
commonly accepted that such features underpin a good notion of
nonclassicality. Supporting this interpretation, fundamental links between
quasi-probability functions and the notions of nonlocality BW99 and
contextuality Spe08 have been recognised.
In this framework, possibly the most accepted definition of non-classicality
has been introduced by Glauber in terms of the P-function Gla63 . For
concreteness, let us consider the Hilbert space ${\cal H}={\cal
H}_{A}\otimes{\cal H}_{B}$ of a bipartite system made of two modes $a$ and $b$
of a bosonic field ($[a,a^{\dagger}]=[b,b^{\dagger}]=1$). Considering
$\alpha,\beta\in{\mathbb{C}}$, let us denote with $|\alpha\rangle$ and
$|\beta\rangle$ the Glauber coherent states of the systems, that is the
eigenstates of the annihilation operators
($a|\alpha\rangle=\alpha|\alpha\rangle$ and
$b|\beta\rangle=\beta|\beta\rangle$). Any state $\varrho$ of the system can be
expressed in terms of a diagonal mixture of coherent states:
$\varrho=\int\\!\\!\\!\int
d^{2}\alpha\,d^{2}\beta\>P(\alpha,\beta)\>|\alpha\rangle\langle\alpha|\otimes|\beta\rangle\langle\beta|$
where $P(\alpha,\beta)$ is the P-function of $\varrho$. When the P-function is
a well-behaved probability density function, then $\varrho$ can be expressed
as a statistical mixture of coherent states Bon66 . Thus, we have the
following classicality criterion:
Criterion P (P-classical states). A state of a bipartite bosonic system is
P-classical if it can be written as
$\displaystyle\varrho_{p}=\int\\!\\!\\!\int_{\mathbb{C}}d^{2}\alpha\,d^{2}\beta\>P(\alpha,\beta)\>|\alpha\rangle\langle\alpha|\otimes|\beta\rangle\langle\beta|\,,$
(1)
where $P(\alpha,\beta)$ is a positive, non-singular, and normalised function.
This Criterion represents the most conservative notion of non-classicality in
the quasi-probability setting, since when the P-function is well-behaved so
are all other quasi-probabilities. The success of using quasi-probabilities to
characterise the quantumness of a state or its space-time correlations WM08
is also, loosely speaking, related to their ability to capture the difficulty
in generating and manipulating quantum states. In particular in quantum
optics, the easiest states to generate in a lab are coherent and thermal
states, characterized by a well-behaved P-function. On the other hand
squeezed, photon-subtracted, photon-added, and number states, characterized by
increasingly ill-behaved P-functions, happen to be much more difficult to
generate. In this sense, the P-function captures the physical constraints of
producing increasingly-more-quantum states. Notice, however, that different
coherent states are not orthogonal, hence even when $P(\alpha,\beta)$ behaves
like a true probability density, it does not describe probabilities of
mutually exclusive events.
Nonclassicality and information theory—The first rigorous attempt to address
the classification of quantum correlations from an information theoretical
viewpoint was pioneered by Werner Wer89 , who put on firm basis the elusive
concept of quantum entanglement Hor09 ; Guh09 . A state of a bipartite system
is called entangled if it cannot be written as follows:
$\displaystyle\varrho_{\scriptstyle AB}=\sum p_{k}\sigma_{{\scriptstyle
A}k}\otimes\sigma_{{\scriptstyle B}k},$ (2)
where $\sigma_{{\scriptstyle A}k}$ and $\sigma_{{\scriptstyle B}k}$ are
generic density matrices describing the states of the two subsystems. The
definition above has an immediate operational interpretation: Unentangled
(separable) states can be prepared by local operations and classical
communication between the two parties. One might have thought that such
classical information exchange cannot bring any quantum character to the
correlations in the state. In this sense separability has often been regarded
as a synonymous of classicality in this information theoretical framework.
On the other hand, as it has been extensively discussed in the last decade
OZ01 ; HV01 ; Luo08 ; Ale10 ; Maz10 , this may not be the case. An entropic
measure of correlations—quantum discord—has been introduced as the mismatch
between the quantum analogues of two classically equivalent expressions of the
mutual information. For pure entangled states, quantum discord coincides with
the entropy of entanglement. However, quantum discord can be different from
zero also for (mixed) separable states. In other words, classical
communication can give rise to quantum correlations. This can be understood by
considering that the states $\sigma_{{\scriptstyle A}k}$ and
$\sigma_{{\scriptstyle B}k}$ in Eq. (2) may be physically indistinguishable,
and thus not all the information about them can be locally retrieved. This
phenomenon has no classical counterpart, thus accounting for the quantumness
of the correlations in separable state with positive discord. Few explicit
formulas have been derived for the quantum discord of some states Luo08 ;
Gio10 ; Lu11 , and more general entropic measures of nonclassical correlations
have been also discussed Ani11 . Discord finds an operational meaning in terms
of quantum state merging Cav11 , and its role has been studied in quantum
information processing with mixed states, where there are computational and
communication tasks which are seemingly impossible to achieve classically, and
yet can be attained using little or no entanglement Kni98 ; Dat08 ; Div04 .
More recently, monogamy properties of discord have been investigated glg11 ,
and it has been shown that quantum correlations in separable states may be
activated into distillable entanglement Pia11 . Discord is also related to the
minimum entanglement generated between system and apparatus in a partial
measurement process Str11 .
Remarkably, even states with zero discord can show non-classical correlations.
In order to see this effect in details let us recall that discord is
asymmetric in the two modes and that a bipartite state with zero A-discord can
be written in the form $\varrho_{{\scriptstyle
AB}}=\sum_{k}p_{k}|\theta_{k}\rangle\langle\theta_{k}|\otimes\sigma_{{\scriptstyle
B}k}\,,$ where the $|\theta_{k}\rangle$’s form an orthonormal basis and the
$\sigma_{{\scriptstyle B}k}$’s are a set of generic non-orthogonal states.
These states—dubbed quantum-classical states—cannot be cloned locally (locally
broadcasted), despite having zero discord PHH . This security against local
broadcasting is not featured by any correlated state of a classical system,
thus revealing the quantumness of this type of zero discord states. The set of
states that can be locally broadcasted has been shown to be equivalent to a
set of states called classical-classical (CC) PHH . Any member of such set can
be written as
$\displaystyle\varrho_{AB}=\sum_{ks}p_{ks}|\theta_{k}\rangle\langle\theta_{k}|\otimes|\eta_{s}\rangle\langle\eta_{s}|,$
(3)
where $|\theta_{k}\rangle$ and $|\eta_{s}\rangle$ are basis for the Hilbert
spaces of the two subsystems. These states are now commonly regarded as purely
classical correlated states Pia11 . The reason for this is based on
information theoretic arguments. All the information encoded in a CC state can
be locally retrieved and stored in a classical register. Indeed, states
appearing in (3) are perfectly distinguishable by local quantum measurements.
In this sense, CC states simply accommodate the joint probability $p_{ks}$ in
a quantum formalism, thus putting forward the most conservative notion of non-
classicality in an information-theoretical setting. However, we will show in
the following that also this class of states can exhibit quantum correlations
that cannot be featured by systems that admit a classical description in the
quantum phase-space.
Definition (3) was introduced in the context of finite-dimensional systems and
it needs to be slightly generalized in order to fully take into account some
subtleties of bosonic systems for which there exists basis that are unitarily
inequivalent footnote1 . Considering $x,y\in{\mathbb{R}}$, let us denote with
$|x\rangle$ and $|y\rangle$ two generic basis of $A$ and $B$ respectively. We
introduce the following classical criterion:
Criterion C (classical-classical states). A state of a bipartite bosonic
system is CC if it can be written as
$\displaystyle\varrho_{c}=\int\\!\\!\\!\int_{\mathbb{R}}dxdy\>F(x,y)\>|x\rangle\langle
x|\otimes|y\rangle\langle y|$ (4)
and $F(x,y)$ is a positive, non-singular, and normalised function. Notice
that, in general, the joint probability distribution $F(x,y)$ spans over a
continuous set. Clearly, one recovers Eq. (3) if $F(x,y)$ is non-zero only
over a discrete set.
Number correlated states—In the following we show that the foregoing criteria
of non-classicality are maximally inequivalent. However, before proceeding
with a formal proof, let us discuss a specific example. Consider the two-mode
P-classical states introduced in Eq. (1), and define the observable
$O_{\scriptstyle D}=a^{\dagger}a-b^{\dagger}b$, which detects the difference
between the number of quanta of the two modes. Since for coherent states
$\langle z|a^{\dagger}a|z\rangle=|z|^{2}$ and $\langle
z|(a^{\dagger}a)^{2}|z\rangle=|z|^{4}+|z|^{2}$, for any P-classical state
(different from the vacuum) we have
$\displaystyle\Delta O^{2}_{\scriptstyle
D}=|\alpha_{0}|^{2}+|\beta_{0}|^{2}+\mathrm{Tr}\,C\geq|\alpha_{0}|^{2}+|\beta_{0}|^{2}>0$
(5)
being $\alpha_{0}$, $\beta_{0}$ and $C$ the mean values and the covariance
matrix of $P(\alpha,\beta)$ respectively. The observable $O_{\scriptstyle D}$
detects correlations between the number of quanta in the two modes. The above
inequality captures the intuition behind the idea that the behaviour of a
classical state should be that of a mixture of coherent states: each mode has
a fluctuating number of quanta and the difference should fluctuate
accordingly. In other words, for a classical two-mode system the amount of
intensity correlations between two modes is bounded.
Let us now consider the two modes prepared in the state
$\varrho_{nc}=\sum_{n}p_{n}|n\rangle\langle n|\otimes|n\rangle\langle n|$,
where $a^{\dagger}a\left|n\right\rangle=n\left|n\right\rangle$. This is the
state generated by, say, a pair of machine guns, each producing a random but
equal number of bullets $n$ according to the distribution $p_{n}$. The state
$\varrho_{nc}$ is separable and, according to the terminology introduced
above, CC. Yet it shows perfect correlations in the number of quanta.
Actually, the product states $|n\rangle\langle n|\otimes|n\rangle\langle n|$
are the projectors over the degenerate eigenspace of $O_{\scriptstyle D}$ with
eigenvalue zero. In other words, for any choice of the distribution
$\\{p_{n}\\}$ we have $\Delta O^{2}_{\scriptstyle D}=0$ for $\varrho_{nc}$,
which in turn violates the inequality (5). Thus the family of number
correlated states $\varrho_{nc}$ gives an example of states that obey
Criterion C while violating Criterion P. We will now proceed to prove that the
two criterion are not only inequivalent, but that their inequivalence is
maximal. Specifically we will show that generic states obeying Criterion P
violates Criterion C and vice-versa.
Generic P-classical states are not CC—Consider the following property of any
CCstate (necessary condition for CC states): any two states of system $A$
conditioned to a measurement on $B$ commute. This can be seen by considering
the definition in Eq. (4) and applying any POVM on $B$. It immediately follows
that any state of $A$ conditioned on any outcome at $B$ will remain diagonal
in the original basis. Thus, all possible conditioned states of $A$ will
mutually commute.
Consider now a generic P-classical state and the following two convenient
conditioned states of $A$:
$\varrho_{A}=\mathrm{Tr}_{B}{[\varrho_{p}]}=\int\\!\\!d^{2}\alpha\>P(\alpha)\>|\alpha\rangle\langle\alpha|$,
and $\varrho_{0}=\mathrm{Tr}_{B}{[\varrho_{p}\left|0\right\rangle\left\langle
0\right|]}=\int\\!\\!d^{2}\alpha\>P_{0}(\alpha)\>|\alpha\rangle\langle\alpha|$,
where $P(\alpha)=\int d^{2}\beta\>P(\alpha,\beta)$, $P_{0}(\alpha)=\int
d^{2}\beta\>P(\alpha,\beta)e^{-|\beta|^{2}}$, and
$\left|0\right\rangle\left\langle 0\right|$ is the vacuum. Calculating the
commutator between the above states and evaluating it on the vacuum, one has
$\displaystyle\left\langle
0\right|[\varrho_{A},\varrho_{0}]\left|0\right\rangle=\int$ $\displaystyle
d^{2}\alpha\,d^{2}\alpha^{\prime}\>P(\alpha)P_{0}(\alpha^{\prime})$
$\displaystyle
e^{-|\alpha|^{2}}e^{-|\alpha^{\prime}|^{2}}(e^{\alpha\overline{\alpha^{\prime}}}-c.c.).$
(6)
Imposing that the commutator above is identical to zero yields the following
nontrivial constraint on the P-function $P(\alpha,\beta)$: $\int
d^{2}\alpha\,d^{2}\alpha^{\prime}d^{2}\beta\,d^{2}\beta^{\prime}\>P(\alpha,\beta)P(\alpha^{\prime},\beta^{\prime})\times\\\
e^{-|\alpha|^{2}}e^{-|\alpha^{\prime}|^{2}}e^{-|\beta^{\prime}|^{2}}(e^{\alpha\overline{\alpha^{\prime}}}-c.c.)=0\,.$
A generic (well-behavied) P-function does not satisfy the above constraint.
This, in turn, implies that almost all P-classical states are not CC.
Equivalently, generic P-classical states violate Criterion C. Notice that the
proof works as well for $A$-discord states, thus showing that almost all
P-classical states have positive discord.
Generic CC states are not P-classical—We first need to show that the set
${\cal P}$ of single mode P-classical states is nowhere dense in the bosonic
space. By definition, ${\cal P}$ is nowhere dense if its closure
${\cal\overline{P}}$ has no interior points. Denoting by $\partial{\cal P}$
the frontier of ${\cal P}$ (namely, the set of its accumulation points), one
has that ${\cal\overline{P}}={\cal P}\cup\partial{\cal P}$. The P-function of
any operator $\delta\in\partial{\cal P}$ must be positive since it is the
limit of positive functions. In addition, it cannot be singular everywhere in
the phase space, given that it is the limit of normalizable functions. As a
consequence any operator ${\overline{\varrho}}\in{\cal\overline{P}}$ is such
that its P-function is positive and not everywhere singular. Let us now show
that no ${\overline{\varrho}}$ can be an interior point of
${\cal\overline{P}}$. First, given any ${\overline{\varrho}}$ denote by
${\overline{\alpha}}$ a point in the phase space where the P-function of
${\overline{\varrho}}$ is non-singular (i.e.,
$P_{{\overline{\varrho}}}({\overline{\alpha}})<\infty$). Then define a
convenient perturbation of ${\overline{\varrho}}$:
$\varrho=(1-\epsilon){\overline{\varrho}}+\epsilon
D({\overline{\alpha}})\varrho_{1}D^{\dagger}({\overline{\alpha}})$, where
$0<\epsilon<1$,
$D({\overline{\alpha}})=\exp[{\overline{\alpha}}a^{\dagger}-{\overline{\alpha}}^{*}a]$
is the displacement operator, and
$\varrho_{1}=\left|1\right\rangle\left\langle 1\right|$ is a single excitation
state. One has that the P-function of $\varrho$ is given by:
$P_{\varrho}(\alpha)=(1-\epsilon)P_{\overline{\varrho}}(\alpha)+\epsilon
P_{\varrho_{1}}(\alpha-{\overline{\alpha}})$. Since the P-function of the
single excitation state is negative and singular at the origin, one has that
$P_{\varrho}(\alpha)$ is non-positive (and singular in ${\overline{\alpha}}$).
For what shown above, this means that (for any $\epsilon$)
$\varrho\notin{\cal\overline{P}}$, hence ${\overline{\varrho}}$ is not an
interior point of ${\cal\overline{P}}$. Since this holds true for any
${\overline{\varrho}}$, one has that ${\cal\overline{P}}$ has no interior
points. As a consequence $P$ is nowhere dense in the space of single mode
bosonic systems.
Consider now the set ${\cal P}_{2}$ of two-mode P-classical states. Based on
the above considerations one can show that P-classical states
$\varrho_{p}\in{\cal P}_{2}$ are nowhere dense in the set ${\cal C}$ of CC
states. First, recall that the partial trace of any P-classical state is a
P-classical state (necessary condition for P-classical states). This implies
that the partial trace of any
${\overline{\varrho}}_{p}\in{\cal\overline{P}}_{2}$ must have a non-negative
P-function. Then, using the same arguments as above (technical details are
omitted), one can build a CC state $\varrho^{\prime}$ that, despite being an
infinitesimal perturbation of ${\overline{\varrho}}_{p}$, does not belong to
${\cal\overline{P}}_{2}$. This implies that ${\cal\overline{P}}_{2}$ has no
interior point in ${\cal C}$, hence P-classical states are nowhere dense in
the set of CC states. Equivalently, generic CC states violate Criterion P.
Discussion—The foregoing arguments show that the set of states simultaneously
obeying Criteria P and C is negligible, both in a metrical and topological
sense note . In other words, the two criteria considered here put forward two
radically different notions of classicality of correlations. Criterion C looks
at the correlations between the information of A and B, as encoded in their
states and regardless the quantumness of the states themselves, whereas
Criterion P takes into account physical constraints on those as well.
Referring to the example of number correlated states $\varrho_{nc}$: creating
Fock states with the same number of quanta does correspond to establishing
quantum correlations between the modes, irrespectively from the fact that the
information needed to perform this action may be of purely classical (local)
origin. It has been often argued that a suitable quantity to reveal quantum
correlations in bipartite systems, beyond the presence of entanglement, should
be related to the joint information carried by the state. For example, quantum
discord focus on this and can be used to assess states for application in
quantum communication. On the other hand, from a fundamental physical point of
view, discord (and information-theoretical quantities more in general) appears
unable to account for the very physical constraints involved in the
establishment of correlations. Ultimately, this means that allegedly classical
correlations established between systems prepared in states with no classical
analogue are quantum in nature.
Operationally, the fact that P-classical states violate Criterion C allows to
use them as an experimentally cheap resource in communication protocols that
require security against local broadcasting. On the other hand, the
nonclassicality of CC states like $\varrho_{nc}$ may find an operational
characterization in terms of conditional measurements. Consider a generic
bipartite state and perform a measurement described by the POVM
$\\{\Pi_{x}\\}$ on one mode, say mode $1$. If the state is P-classical then
the P-function of the conditional state
$\varrho_{px}=\hbox{Tr}_{1}[\varrho_{p}\,\Pi_{x}\otimes\mathbb{I}]/p_{x}$ may
be written as
$P(\beta)=\frac{1}{p_{x}}\int\\!d^{2}\alpha\,P(\alpha,\beta)\,\langle\alpha|\Pi_{x}|\alpha\rangle\,.$
This is a well behaved probability density function, and thus the state
$\varrho_{px}$ is classical. In other words, only states violating Criterion P
may lead to the conditional generation of genuine quantum states with no
classical analogue fer04 ; bon07 .
Conclusions— In the last two decades the fruitful exchange of notions between
information science and quantum physics led to the emergence of radically new
concepts and applications. The slogan information is physical lan93 has
become increasingly popular, emphasizing the role of physical constraints in
quantum information processing dvl98 . Our results reinforce this position,
however also present an unusual case in which the information-theoretical and
physical perspectives appear fundamentally conflicting. Specifically, by
addressing the notion of non-classicality as it emerges from physical
considerations, we have shown that there exist other genuinely quantum
correlations than those revealed by information-theoretic arguments. This
indicates that the slogan should be complemented by a second part illustrating
that information-theoretic considerations cannot substitute physical
constraints, thus suggesting that information is physical, and physics is not
merely information.
MGAP thanks Paolo Giorda, Sabrina Maniscalco, Kavan Modi and Jyrki Piilo for
interesting discussions. This work has been supported by MIUR (FIRB LiCHIS-
RBFR10YQ3H) and by the Finnish Cultural Foundation.
## References
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* (10) See e.g., M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
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* (22) X.-M. Lu, J. Ma, Z. Xi, X. Wang, Phys. Rev. A 83, 012327 (2011); D. Girolami, G. Adesso, Phys. Rev. A 83, 052108 (2011); F. Galve, G. L. Giorgi, R. Zambrini, EPL 96, 40005 (2011).
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* (31) Namely, basis that cannot be inter-converted via a unitary transformation, such as the position $|q\rangle$ [with $\hat{q}|q\rangle=q|q\rangle$ and $\hat{q}=(a+a^{\dagger})/\sqrt{2}$] and number $|n\rangle$ [with $a^{\dagger}a|n\rangle=n|n\rangle$] basis.
* (32) Notice that, as expected, fully factorized P-classical states are identified as classical by Criterion C as well. An interesting question that still remains open is whether, besides those factorized states, there exist also correlated states satisfying both criteria simultaneously.
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|
arxiv-papers
| 2012-03-12T21:35:57 |
2024-09-04T02:49:28.562306
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alessandro Ferraro and Matteo G. A. Paris",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1203.2661"
}
|
1203.2704
|
# A model and framework for reliable build systems
Derrick Coetzee
Anand Bhaskar
University of California, Berkeley
{dcoetzee,bhaskar,necula}@eecs.berkeley.edu George Necula
###### Abstract
Reliable and fast builds are essential for rapid turnaround during development
and testing. Popular existing build systems rely on correct manual
specification of build dependencies, which can lead to invalid build outputs
and nondeterminism. We outline the challenges of developing reliable build
systems and explore the design space for their implementation, with a focus on
non-distributed, incremental, parallel build systems. We define a general
model for resources accessed by build tasks and show its correspondence to the
implementation technique of _minimum information libraries_ , APIs that return
no information that the application doesn’t plan to use. We also summarize
preliminary experimental results from several prototype build managers.
††footnotetext: Also available as Technical Report No. UCB/EECS-2012-27. All
rights to this work are released under the Creative Commons Zero Waiver (CC0).
It may be used by anyone for any purpose without permission or condition. This
is not a peer-reviewed work. Published 2012 February 17.
## 1 Introduction
Large software projects often reach thousands of files and millions of lines
of source code. Build automation systems, or _build systems_ for short, are
responsible for automating the execution of build tools such as compilers in
order to process all the source code and produce the final, executable output.
The time required to execute a build is a critical factor in a number of
software engineering metrics such as: developer cycle time, frequency of
continuous integration testing, throughput of check-in verification systems,
and time to ship a critical patch; yet a 2003 survey showed that more than
half of the 30 surveyed commercial projects had a clean, sequential build time
of 5-10 hours. [11] This motivates the development of builds that can run
faster than a clean build.
To address this need, existing build systems provide two features: _parallel
builds_ , in which multiple build tasks are executed simultaneously, and
_incremental builds_ , in which results of previous builds are reused and only
a subset of build tasks are run, based on what build inputs have changed. In
both types of builds, the developer must explicitly specify _dependencies_ for
each build task, describing other build tasks which must run before it. For
example, in a C project, C source files must be compiled into object files
before the object files can be linked into an executable binary. If even one
dependency is omitted, the soundness of both parallel and incremental builds
is compromised: build tasks may be run out of order, leading to incorrect re-
use of out-of-date results, build failure due to missing results, and race
conditions due to concurrent access to files. Whether a failure occurs, and
which failure occurs, depends on which input files have changed and the build
schedule selected by the build system. As a consequence, “[m]ost organizations
run their builds completely sequentially or with only a small speedup, in
order to keep the process as reliable as possible.” [11] If developers and
organizations viewed their parallel, incremental builds as highly reliable,
they could use them consistently throughout the development, testing, and
release process, accelerating these processes and offloading the mental burden
of build management.
Incomplete dependencies arise naturally whenever a developer change introduces
a new dependency, but fails to correctly update the dependency information. As
a simple example, consider the build described by this makefile:
all: generated.h foo
generated.h: config
./gen config -o generated.h
foo: foo.c
gcc foo.c -o foo
Here, a tool called _gen_ is run to generate the header file generated.h from
a file _config_ ; then the binary foo is compiled from the C source file
foo.c. Now suppose the developer modified foo.c to include the header file
generated.h, and also modified _config_. A serial build will still produce the
expected result, since generated.h is listed before foo in the “all” target;
but an incremental or parallel build may run the _gcc_ action before, or
simultaneously with, the _gen_ action, leading to incorrect output or build
failure.
This work explores background and existing work in build systems and obstacles
and design options for reliable build systems. It also presents a formal model
for build system analysis and discusses some early experimental results with
several prototypes.
## 2 Background
Dependencies in a build are described by a dependency graph, a directed
acyclic graph (DAG) where build tasks (typically, invocations of a build tool)
are vertices, and an edge from A to B indicates that B depends on A. Given
such a graph and a uniform set of processors, deciding which tasks to run at
what time is an instance of the _DAG scheduling problem_ , which is studied in
the context of static scheduling of processes in high-performance computing.
It is NP-complete even in the restricted case where there are two processors,
no dependencies, and the run time of every task is known (closely related to
the _partition problem_), but a number of effective heuristics are available
in practice.
A single-node build can be scheduled using a topological sort, which can be
computed by a simple online algorithm: at each step, select an arbitrary
vertex with no incoming edges to run, and when it completes, delete it. A
similar algorithm can schedule parallel builds: whenever at least one
processor is free, run an arbitrary task with no incoming edges, and whenever
a build step finishes delete its vertex. It is possible that all tasks have
incoming edges, in which case processors may remain idle until more tasks
complete. This algorithm, used by _make_ , is a version of Graham’s classical
online list scheduling algorithm, [4] and has the advantage of not requiring
task runtimes, but does not take into account the critical path (the path of
largest total time).
The technique can be improved by assigning priorities to nodes, using any of a
number of heuristics, and then selecting the node with the highest priority at
each step. [16] Effective priority assignment requires task runtime estimates,
which can be inferred from previous builds and/or a runtime model. This
approach has not been yet tried.
### 2.1 Shared state and resources
To model builds we define the _shared state space_ $S$, typically representing
the filesystem and other state visible to multiple tasks as well as the task
input (e.g. command line, environment). A _resource_ is a function $r$ with
domain $S$. Intuitively, a resource is anything that may be returned by a
library function. Resources can range from simple predicates (“does this file
exist?”) to values (“what are the contents of the file at this path?”) to
complex operations (“what is the abstract syntax tree obtained after parsing
the source file at this path?”). A resource can also encompass many files
(such as the contents of all files in a subdirectory). Prior to starting the
build, a fixed (typically infinite) _resource space_ is selected—no build
process may access resources outside that set.
A build task performs a sequence of _accesses_ (reads or writes) to resources.
During a parallel build, accesses by many tasks may be interleaved to form an
access sequence, subject to the constraint that if $g$ depends on $f$, all
accesses by $f$ precede all accesses by $g$. Reads make the current value of a
resource accessible to the task executing it, while writes update the shared
state in such a way that one or more resources are set to a new given value.
Any resources not written to during a write must remain unmodified.
Build tasks must be _deterministic_ , in the sense that their accesses
(including type, resource, and value written) depend only on the results of
prior reads. Two tasks are said to _conflict_ (during a particular build) if
one of them writes a resource that the other reads or writes. A given build is
_valid_ if, for any pair of conflicting tasks, there is a directed path from
one to the other in the dependency graph. It can be proven that if a given
build is valid, it produces the same final result as any other parallel
schedule, given the same initial shared state (see appendix A). This allows us
to meaningfully define a _valid configuration_ as a pair (dependency graph,
start state) that produces valid builds.
To model an incremental build, suppose we start with initial state $s_{i}$,
perform a build resulting in state $s_{f}$, modify the shared state to get
$s^{\prime}_{f}$, and then perform another build. For now, we assume that for
every task $f$, $f$ has no effect when acting on $s_{f}$ — that is, right
after the first build is complete, re-running any one step will change nothing
(in practice, this typically means retaining and not reusing intermediate
files). Define the special task $d$ updating state $s_{f}$ to
$s^{\prime}_{f}$, representing the actions of the developer, and add edges
from $d$ to all tasks that $d$ conflicts with. Now we assign $d$ the lowest
priority and create a DAG schedule. This will move all nodes that don’t
conflict with $d$ before $d$, where they will have no effect, since they are
acting on $s_{f}$. Effectively, this means the only part of the graph that
needs to be scheduled is the transitive closure of $d$.
### 2.2 Selecting a resource space
There is a tradeoff in the choice of the resource space: if resources
encompass too much state, there will be spurious conflicts. For example, a
trivial resource space has a single resource returning the entire shared
state. In this space, all reads conflicts with all writes, and the build must
run sequentially.
On the other hand if resources are too fine-grained, the result will be that
processes read and write a very large number of resources, resulting in
excessive overhead for build management and a large dependency graph. For
example, if every byte of every file had its own resource, a typical build
task would access many thousands of resources.
One straightforward strategy is to create a single resource for the contents
of each file on the disk. To account for the creation and deletion of files,
there is a resource for every possible filepath, with a special value
indicating the file does not exist or is inaccessible, analogous to a “read
file contents” library function that returns NULL on failure. This simple
resource system is similar to that used by _make_ and is sufficient for many
builds.
Many applications require a notion of a collection/set resources, such as a
directory. A naive representation would have a resource for the contents of
each collection; but then two tasks creating files in the same directory would
conflict. Such a collection is best represented as an infinite set of
resources, one for each potential element of the collection, indicating
whether or not that element is present (in the case of a directory, one for
each filename, indicating whether that file exists in that directory). A
process that reads the collection (e.g. listing the files in the directory)
reads all of these resources (note that this requires a concise representation
for certain infinite resource sets). A process that adds or removes items from
the collection may only affect a few of them.
Although files are by far the most common resource, there are many examples of
other resources that are useful. For example, the Linux kernel build has a
single header containing all configuration options which is included by all
source files. In order to make incremental builds useful in the event of
configuration option changes, the Linux build tracks each option as a separate
resource.
A set of resources in a resource space may be _contracted_ to form a merged
resource which yields a tuple of all the resources used to form it. Such
contracted resources allow a gradual tradeoff between the number of resources
accessed and the number of conflicts that occur during the build—see section 8
for more details.
### 2.3 Hidden resources
There are resources that are used in practice by many tools but are not
tracked by existing build managers, either by convention or because supporting
them is difficult. These include:
* •
Compiler flags and tool configuration: if a build is done, and then tool
configuration is altered, for example to enable debugging flags, all files
must be rebuilt. If it is changed back, there is no need to rebuild everything
again. Visual Studio implements solution configurations with separate output
directories to cope with this, but these are rarely used for more than two
configurations. Vesta [6] records outputs of many previous builds in its
derived file cache.
* •
Nonexistent files: A C source file reading ”#include <stdio.h>” will search
the system include path in order to find the header. Developers often add
project directories to this path. If a file named ”stdio.h” were ever created
along this path, it would change the result of the task, but most extant tools
would not detect the need to rebuild. Vesta [6] and _scons_ [8] track
dependencies on nonexistent files.
* •
Build configuration file: determining which part of a build needs to be
rebuilt after changing the build configuration file itself (e.g. Makefile) is
a difficult problem. Even small changes may affect all tasks or only a few,
and determining which may require analyzing structural changes since the
previous version.
* •
Build tools, libraries, and system headers: upgrading build tools or libraries
used by build tools, or copying a source tree to a machine with different
tools, can dramatically alter build output, but these are usually untracked.
Sometimes this results in an incompatible combination of files generated by
different versions of tools. This motivates the common industry practice of
including all build tools in the version control repository. As mentioned in
section 8, it often makes sense to treat these files as a single aggregate
resource.
* •
Non-file resources: accesses to network resources, peripheral devices, the
time, and so on are usually untracked. Some real-world builds retrieve files
during the build from remote sources, query remote databases, or even do web
service queries. These should be tracked as resources, even if coarsely.
* •
Special files: some files like those under ”/proc” and ”/dev” in UNIX may fail
to update their last modified time, or even change each time they are read.
* •
Operating system: the results of system calls made to the kernel by build
tools may affect build output. These results may vary depending on the
specific operating system, operating system version and patches, filesystem
and drivers, or even kernel configuration options. These are untracked by all
extant systems, and largely benign given a carefully designed resource space
and a standards-compliant operating system.
The choice of how to handle hidden resources depends on the resource space,
the application, and build platform variability. Some applications may not use
certain types of resources or may be built only on a fixed build server. In
some cases, like the build configuration file, merely detecting any change and
triggering a full rebuild may be sufficient in practice. In other cases, where
changes are frequent, fine-grained resource tracking is needed.
## 3 Related work
### 3.1 Build systems
A small number of build systems dominate in practice today, most of them based
on _make_ , created by Stuart Feldman in 1977 at Bell Labs. [9] With _make_ ,
the developer uses a domain-specific language to specify a series of targets,
and each target may declare explicit dependencies on other targets and/or
source files. Each target has an associated shell command that builds the
target. This explicit representation of the dependency graph facilitates both
incremental and parallel builds. However, dependencies must be specified
correctly; if they are not, incremental builds may fail to rebuild portions of
the application, leading to incorrect results with unpredictable behavior, and
parallel builds may produce different outputs nondeterministically. Make is
designed for use on a single machine, and build results are not shared between
developers. A number of important dependencies are either difficult to
represent or omitted by convention, such as the ones mentioned in section
2.3—changes in these may require a complete rebuild. Even incremental builds
in _make_ take time proportional to the size of the build as a whole due to
the need to process all targets and scan all input files for changes. This
process can be accelerated by using file timestamps to detect changes, at the
expense of correctness, since this is not reliable in general. Although some
build systems like Apache Ant and MSBuild adopt XML build description files in
place of _make_ ’s domain-specific language, facilitating greater
extensibility, they still inherit all of these issues.
One of the most developed research build systems is Compaq/Digital Systems
Research Center’s Vesta, developed in the late 1990s and released under the
GNU LGPL in 2001. [6] Although Vesta does not support parallel builds, it
provides incremental builds reliable enough to be used in practice for product
releases (“every build is incremental”). It tracks dependencies that extant
tools like _make_ incorrectly ignore, such as dependencies on build
description files, compiler flags, nonexistent files, and build tools. Through
the derived file cache, compilation outputs are easily reused between
developers. Change detection and inferrence of dependencies is implemented
using a custom filesystem, so that the filesystem does not need to be scanned
to find modified source files, and a sophisticated functional build
description language allows large portions of the build to be reused. [7]
Using its derived file cache, Vesta can reuse results not only from the
previous build but from all previous builds, by treating tool executions as
functions and memoizing their results (see their _runtool cache_).
Vesta was deployed by large product teams at Compaq and Intel, but has not
achieved widespread use. This can be attributed to several factors. One is
that Vesta is a “package deal,” requiring teams who use it to also use Vesta’s
custom filesystem and version control, both of which are not as mature,
featureful, or well-supported as existing systems. Migration of existing
projects to Vesta while preserving change histories is difficult or
impossible, and requires translating existing build description files into
Vesta’s very different language. Modern builds are done in parallel, even on
single nodes, and large builds are done on clusters, neither of which Vesta
supports. Finally, the cost of incorrect incremental builds is hidden: it is
difficult to measure the time spent by developers resolving incorrect builds,
or the time that might have been saved by building product releases
incrementally.
A central feature of Vesta was _repeatability_ , in which all source files
used in a particular build can always be retrieved at a later time, and used
to repeat the same build. Although this feature is valuable (e.g. for
isolating source changes leading to behavior changes), it is separable from
the other features and depends critically on integration with version control,
so it is disregarded in this report.
A very different approach to build systems was taken by Electric Cloud, [11]
which disregarded incremental builds in favor of using clusters of machines
with parallel processors to speed up full builds as much as possible,
currently deployed as an enterprise commercial product. A network filesystem
infers dependencies, and visualization tools facilitate the identification of
bottlenecks. Although fast and well-supported, Electric Cloud is not suitable
for routine developer builds, does not scale down effectively to small
projects, and is too expensive for many applications such as open-source
development.
More recently, in 2012, Electric Cloud has released ElectricAccelerator
Developer Edition, [3] which is designed to run on a single machine, infers
dependencies, and implements accurate incremental and parallel builds, scaling
up to four cores. Although this product effectively accomplishes the primary
goals set out in this report, it chooses a single design and leaves room for
improvement in numerous directions, such as tool cooperation, sharing of
derived files, custom resources, and so on.
### 3.2 Build augmentation
A number of more practical efforts have sought to augment existing build tools
by providing services to accelerate them or improve their reliability.
The GNU Make manual illustrates how to use the “-M” flag of gcc (the GNU C
Compiler) to generate _make_ dependencies for C/C++ builds on-the-fly and keep
them up-to-date automatically. These dependencies are incomplete, including
only header and source files, but greatly increase reliability and reduce
maintenance effort compared to manual specification for this specific type of
build.
The _ccache_ tool, [15] based on _compilercache_ , [14] caches results of
invocations of standard compiler tools like _gcc_ , even if the intermediate
files are later deleted or overwritten. It can dramatically improve
incremental build times for C/C++ projects, but does not generalize to other
tools. _scons_ [8] provides similar functionality.
Google relies on conventional distributed builds with coarse-grained tasks and
manually-specified dependencies. Their efforts have focused on dramatically
reducing the runtime of important build tools, such as the C/C++ linker, which
is a bottleneck in large parallel builds because it is used in the final step
to combine all results. [13]
## 4 Build specification
Systems like _make_ lean heavily on build specification via an explicit
dependency graph. This has certain advantages: dynamic scheduling of
incremental, parallel builds is straightforward as outlined above, and it’s
also intuitive to create build description files that include multiple targets
and allow the developer to choose to build only a subset of them (and these
targets may share dependencies).
One of the simplest ways to specify a build is with a sequence of shell
commands, a basic shell script. Any sequential build is equivalent to such a
script. Both incremental and parallel builds can be implemented in this
setting by inferring dependencies from previous builds (see sections 7.3,
7.4). This scheme can be extended to include nonrecursive function calls and
variables without adding significant complexity. It has the advantage of being
intuitive and familiar to procedural programmers, but unlike explicit
dependency graphs becomes less intuitive when building a subset of targets.
The most general type of build specification is the build program or build
script. Such a script is written in a general-purpose language and may employ
sophisticated abstraction mechanisms, algorithms, and data structures. Vesta’s
functional build language [7] and _scons_ ’s Python build descriptions [8] are
examples. In some cases it may even be integrated into the application being
built, allowing the application to generate source code and rebuild itself or
portions of itself. Incrementalism can be extracted using memoization, as in
Vesta, and parallelism can be extracted using futures. Although the most
flexible option, automatically extracting incremental and parallelism from a
general build program is challenging and in some cases infeasible.
Some practical tools mix these approaches; _make_ for example incorporates
basic variables and conditionals while remaining primarily based on dependency
graphs. Other hybrids may be possible, such as a Makefile-like language where
both dependency lists and actions can be program fragments in a general-
purpose language. A major goal of future work is to design a build description
language that can concisely represent typical builds in practice, minimizing
opportunities for error, but remain flexible and scalable enough to
accommodate large and complex builds.
## 5 Capturing access to shared state
Standard tools such as _make_ rely on the developer to manually specify all
shared state which is accessed by each task, making the system unable to
distinguish a valid build from an invalid one. There are several techniques
for reliably, automatically capturing access to shared state.
### 5.1 File system filtering
It is straightforward to implement a filesystem or network filesystem server
which acts as a proxy, monitoring all file operations and mapping them onto an
underlying filesystem. Some filesystem subsystems, as in Windows NT, have
explicit support for filters to capture all file operations, for use by virus
scanners and backup utilities. To detect conflicts, the system must know which
build task is performing each file operation, usually inferred from the
process ID. The technique extends easily to distributed build systems.
This approach was the primary means of capturing dependencies in Vesta, and is
simple to deploy (although it typically requires superuser access). Its main
disadvantages are that it only captures operations on files and only at whole-
file granularity, it must be applied to every filesystem a build process could
possibly access, and that the file API is typically at an inappropriate level
of abstraction, yielding too much information on each call.
### 5.2 System call interception
On typical MMU-based systems, all access to shared state by a process passes
through system calls, which can be intercepted either through binary rewriting
or through kernel support for system call interception such as _ptrace_.
Unlike file system filtering, system call interception can capture all access
to shared state including all filesystems, the network, and kernel data
structures (with some minor exceptions like RDTSC, which can be disabled).
One obstacle with system call interception is that typical build tools
generate very high volumes of system calls, many of which are unimportant for
dependency tracking. In experiments, handling all system calls with a central
_ptrace_ monitor process led to crippling overhead. Binary rewriting suffers
from a different performance issue: load-time rewriting is too expensive for
short-lived processes, necessitating on-disk caching of instrumented binaries.
Kernel patches (for ptrace) or in-process filtering (for binary rewriting) can
reduce the number of system calls, but is more difficult to implement and
deploy and less flexible than minimum information libraries.
A more fundamental obstacle with system call interception is that applications
routinely invoke system calls that return more information than they require.
For example, UNIX applications testing for the existence of a file routinely
use the _stat()_ system call, which also returns the last modified and last
accessed time of the file, which change frequently. Another daunting case is
environment variables, which are passed to new processes as a complete array;
there is no way to determine which ones are used through the system call
interface. Similarly, an application may read in a database file just to use
one row of a table, or (as was observed in some open-source tools) cache the
contents of a directory to accelerate future queries. To ensure correctness,
the build system must assume all the information available to the process is
used by it, which leads to unacceptable performance. Dynamic taint tracking,
[5] used to track the flow of untrusted data in security applications, could
be used to trace the flow of system call results in-process, but has high
overhead, and may fail to accurately track complex cases, such as an array of
environment variables being transformed into a hash table data structure.
### 5.3 Minimum information libraries
A _minimum information library_ is a library designed to supply the minimum
information that will be used by the caller and no other information, even in
case of error. For example, whereas a POSIX application may use _stat_ or
_fopen_ to determine if a file exists, a minimum information library would
supply a _fileExists_ method returning a boolean. It would only return true if
the file exists, or false if it doesn’t exist or is inaccessible. Similarly,
environment variables would be accessed through _get_ and _set_ functions
instead of by parsing the environment block. These expose fine-grained
dependencies in the application while still making the same number of system
calls under the covers.
Minimum information libraries have a natural correspondence to resources as
defined in this work: every resource can have an associated call in the
library that reads and (where applicable) writes that resource. Other calls
may read or write multiple resources.
A minimum information library can be easily instrumented to acquire one or
more resources with every call, or to acquire a single resource to serve many
calls, avoiding a proliferation of acquisitions. It can either save this
information for later analysis, or contact a central build manager process to
acquire a lock on the resource. By eliminating or wrapping all library calls
that invoke the kernel, all access to shared state can be directed through the
minimum information library, ensuring that all dependencies are systematically
tracked.
When an application is written against a minimum information library,
dependency tracking is simplified, but for many build tools that are either
binary-only or managed by third parties, porting to another runtime library is
a poor investment. For cases like these, a promising alternative is the _build
wrapper_ , a small tool using a minimum information library that replaces the
tool and acquires any needed resources, then invokes the underlying tool
normally. Such a wrapper often requires only a small subset of the
functionality implemented by the full build tool.
Unlike the other solutions above, minimum information libraries require some
work to be done for every build tool, including application-specific build
tools, and bugs in this code can lead to build unreliability. However, the
number of build tools in a build is very small compared to the number of build
tasks, typically ranging from 1 to 50. For widely-used tools like _gcc_ , the
work can be shared among many users of the tool and developed to maturity,
while application-specific build tools tend to be very simple, with
dependencies inferrable from the command line alone.
## 6 Change detection
The change detection problem is the problem of capturing changes to shared
state _between builds_ , for the purpose of implementing incremental builds.
Traditional build systems like _make_ rely on comparing timestamps between
task input and output files to determine if a task needs to be re-run. This is
overconservative, in that unmodified files may have updated timestamps;
incorrect, in that tasks may not be run if timestamps travel backwards (as
when restoring from a backup); and inefficient, in that all tasks and all
their input files must be examined even for a small incremental build. New
build managers like _scons_ [8] rely on hashes of file contents to detect
changes, fixing the first two problems at the expense of even more
inefficiency. Moreover, both these approaches are ineffective for resources
other than simple files.
Ideally, change detection should log exactly which resources in the chosen
resource space are modified at the moment they are modified, making their
retrieval trivial. This would be straightforward if all applications were
written against the same minimum information library as the build tools, but
this is infeasible in practice because development tools are generally third
party and difficult to wrap due to being interactive and long-lived.
Some kernels support keeping a log of all modified files, including NTFS’s USN
change journal [10] and Linux with Stefan Büttcher’s fschange patch. [2]
Combined with a resource database that tracks old values of resources, these
can be used to detect changes to filesystem-based resources as soon as they
occur. ZFS uses Merkle trees to efficiently track hashes of the contents of
all files at all times, for integrity and de-duplication, but this information
is not user-accessible without a patch. Network-based resources can be
intercepted by packet sniffers, at some overhead.
## 7 Specifying and inferring dependencies
### 7.1 Manual dependency specification
Although primitive, manual dependency specification offers a transparency and
flexibility difficult to achieve with other methods. If coarse-grained tasks
are used (see section 8), dependencies don’t have to be updated too often,
easing the maintenance burden. In this scenario, the primary function of the
build manager is to detect invalid builds, with error messages suggesting how
to repair the build description file. It can also optionally warn about
redundant dependencies.
### 7.2 Phased dependency specification
An extension of manual dependency specification is to have a build that
proceeds in phases, where earlier phases generate dependencies used by later
phases. A simple example of this is the typical integration of _make_ with gcc
-M, where dependency files are generated from source files in the first phase,
and in the second phase source files are compiled using those dependencies.
This can be extended to more phases in scenarios where tools must first be
built to generate dependencies. Because each phase can be parallelized and
incrementalized separately, this approach can be similar in performance to the
manual approach. Some degree of interleaving may be possible, but caution is
required to ensure that no dependencies become available after the point where
they are needed (or alternatively, rollback may be used in this case—see
section 7.4 below).
### 7.3 Offline dependency graph augmentation
An alternate strategy is to infer dependencies based on the conflicts observed
in an invalid build. If two tasks conflict but there is no directed path
between them, the system can add an edge between them, but needs more
information to infer the direction of the edge. One simple way to supply this
information is to give a serial ordering of all tasks—then if A and B
conflict, whichever comes earlier in the serial order is run first. In the
case of dynamically scheduled tools, such a serial order can be inferred after
the fact from any deterministic walk over the task execution tree of the
build. Once the graph is updated, the build is re-executed (invalidating the
conflicting tasks to force them to re-execute), and this process is repeated
until a valid build is observed. Termination is guaranteed because eventually
the dependency graph will contain a path through all tasks, and so necessarily
be valid.
Inferred dependencies are stored as derived files that can be shared between
developers (via a derived file cache, or simply through version control). For
this reason, invalid builds are expected to occur infrequently, only when
source files change in a way that adds dependencies.
Because the serial ordering is used to direct dependencies, the parallel build
that results from this algorithm will produce the same final result as a
sequential build of the serial ordering (per the theorem of Appendix A). Such
a build is predictable, easy to test, and easy to conceptualize for the
developer. Compared to manual dependency specification, dependency inferrence
allows more concise build description files that require less frequent
updating. However, unforeseen conflicts may lead to excessive edges and build
bottlenecks.
A challenging problem for this strategy is determining when to remove inferred
dependencies. The build can easily detect when there is no conflict between
two tasks, but it is difficult to establish whether the lack of conflict is a
short-lived or long-lived phenomenon. For example, in a C++ project, there may
be a certain header file which is only included in debug builds, resulting in
dependencies that appear in debug builds but not in release builds. One simple
strategy is to periodically erase all inferred dependencies and re-run the
build to reproduce them.
### 7.4 Transaction-based task synchronization
Another strategy is to prevent any invalid builds from occurring by inferring
dependencies on-the-fly at runtime. Using concepts from database transactions,
we lock resources before accessing them by submitting a lock request to the
build manager process. If the resource is already locked, the task is blocked
until it is available. Tools with build wrappers can lock all necessary
resources before invoking the real tool. However, once locks are in use
deadlock is possible, and to make progress tasks must support abort and
rollback, which kills the task and undoes its previous effects to the shared
state.
By itself, this algorithm will yield an unpredictable ordering of conflicting
tasks, leading to nondeterminism in build outputs. Suppose we wish instead to
produce the same final output as the sequential serial build. In this case, we
can employ a version of multiversion timestamp concurrency control [1],
placing each task inside a transaction with a virtual timestamp equal to its
order in the serial build. If a task observes a value that was written by a
task with a later timestamp, this is termed _physically unrealizable behavior_
, and forces an abort and rollback of the reader and any tasks influenced by
its writes directly or indirectly (ordinary multiversion timestamping rolls
back the writer, but in our scenario this can lead to a failure to make
progress). Unlike the pessimistic locking strategy above this is an optimistic
strategy, and so avoids blocking tasks at the cost of more frequent restarts.
## 8 Task and resource granularity
Fine-grained tasks allow incremental builds to avoid redundant work and
parallel builds to run more tasks in parallel. Generally the most fine-grained
task possible is an execution of a build process, since such tasks cannot be
easily subdivided. However, the intuitive association of a single process with
a task may be counterproductive: a large number of processes leads to a large
number of tasks and a large dependency graph which takes more time to
construct and analyze. By partitioning this graph and collapsing each
partition to a single task, the graph size can be dramatically reduced with
only a modest increase in incremental build times. There is also little to no
decrease in parallelism in practice, either because the reduced build is still
capable of saturating the hardware’s parallelism capacity, or because
individual build tools support parallel execution. One typical strategy for
accomplishing this is switching from a “file-based” compilation method to a
“module-based” method, where entire directories are compiled into
static/shared libraries or binaries in a single step. Some build tools, like
the Microsoft Visual C# compiler, exclusively use this approach.
Along with a decrease in graph size, the frequency of updates to the
dependency graph is lowered, making manual graph maintenance more feasible and
leading to a smaller number of rebuilds.
Similarly, the intuitive fine-grained association of resources with individual
files can be counterproductive. For example, every task has a set of “owned”
resources that only that task depends on, which can be collapsed into a single
resource without increasing build times. If the tasks are coarse-grained, this
can substantially reduce graph size. Another important case is the set of
system resources, such as build tool executables, that are rarely updated and
used by nearly all tasks. By collapsing rarely-updated, widely-used resources
into a single resource, an enormous number of dependency edges are eliminated,
and long incremental builds are only needed during a system update—at which
time a full rebuild is needed anyway.
Decreasing graph size decreases overhead differently depending on the system
used. In a lock-based system with a central build manager, it results in less
lock and unlock operations and less interprocess communication. In a system
that logs dependencies, it leads to fewer and smaller log files and less time
loading them. In a system that performs static DAG scheduling, the scheduling
algorithm runtime is reduced and an improved schedule may become feasible.
These optimizations are essential to ensure that build overhead does not
dominate build time.
In order to achieve these gains, the partitioning must be known and available
to all tasks before the build begins. Both task and resource partitioning can
be inferred by analysis of the dependency graphs of previous builds. Task
partitioning can also be specified implicitly by describing each task using a
command sequence or script that performs all necessary actions for that task.
Resource partitioning can be specified manually, e.g. by using directory
patterns to distinguish application and system resources. A promising hybrid
approach that both limits incremental build time and keeps graph size small is
to automatically use smaller partitions for resources that are modified
frequently (e.g. the module the developer is currently working on) and
increasingly larger partitions for resources that are modified less
frequently.
## 9 Preliminary experimental results
Three prototype build systems were constructed.
In the first, a ptrace-based prototype that could only perform full builds, a
pessimistic locking scheme was used where build processes took locks on any
files they accessed. Processes also took “predicted locks” on any files they
accessed during previous builds; predicted locks cause processes with later
timestamps (which occur later in the serial build) to block if they attempt to
lock the file. This allows cascading rollback to be avoided. Given enough
concurrent processes, this build scaled to 85% the time of a parallel _make_
build of the Linux kernel. However, it was not a complete system, as it was
unable to handle unexpected new dependencies, could not perform incremental
builds, and inferred its list of processes to execute from a prior _make_ run,
making it necessary to rerun the _make_ build whenever this process sequence
was changed.
The major performance bottleneck in this prototype was the necessity for the
central build monitor process to sequentially handle all ptrace messages. A
variant of this prototype used binary rewriting based on Jockey [12] to track
system calls without the use of ptrace. Jockey rewrites binaries at load time
by searching for system calls, and also keeps a cache of patches to apply for
binaries it’s seen before. In practice, even with caching, the system added
too much overhead to be practical due to the Linux kernel build’s enormous
number of short-lived processes like _cp_ and _mkdir_. This is less likely to
be an issue in a more monolithic build system.
The second prototype was based on multiversion timestamping and was able to
handle process hierarchies. Instead of replacing _make_ , _make_ is run
sequentially and children of _make_ are run speculatively, pretending to
succeed so that _make_ will continue and begin the next process. Rollback was
implemented by performing all writes in a temporary location and then
committing them after a process completes, which can be accomplished by
rewriting results of system calls (a simple form of filesystem
virtualization). Although the system was able to run real-world builds, and
was powerful enough to complete builds even given no initial dependency
information at all, the overhead of its transaction management and filesystem
virtualization prevented it from scaling to larger builds, particularly since
the build manager ran sequentially. On very small builds with few
dependencies, it could outperform a sequential _make_ build by 30% while
offering the same results and reliability, but even on medium-sized builds
this performance advantage was lost. In neither case could it compete with
parallel _make_ builds.
The system also supports reliable incremental builds: it keeps a cache similar
to Vesta’s _runtool cache_ , and whenever a process is re-executed with the
same inputs as in a prior run, it skips running the process and commits its
cached results. Although its incremental builds are much faster than its full
builds, they are not competitive with incremental builds by _make_ , for
several reasons: the main _make_ process is still run as it would be in a full
build, input files have to be hashed to implement the cache reliably, and the
filesystem virtualization (particularly committing cached results) is
expensive.
The third prototype abandoned transactions and system call tracing in favor of
cooperation with build tools. A variety of open-source build tools were
instrumented to declare their dependencies at runtime using a C library called
_deptracker_ , which then wrote them out to an XML file when the process
exited. An offline analysis step would then load all of these, detect
conflicts, and (together with sequential build order information logged by an
instrumented _make_ tool) generate a supplementary Makefile to augment the
existing dependency graph. Initial performance evaluation with small builds
showed that the time needed to load and process the XML files was a
substantial portion of build time, as much as 30% of the build, suggesting
that a coarser granularity of tasks and/or resources is needed to accelerate
this stage.
Another challenge for this prototype was the impracticality of maintaining a
forked and instrumented codebase for every build tool used by a build,
including many like _gcc_ with much larger builds themselves than the build
under evaluation—effective build wrappers could mitigate this problem.
## 10 Conclusion and future work
This work discussed design options for constructing a reliable build system
and highlighted tradeoffs between them, but many of the ideas remain untested.
A clear next step is building a complete build manager that can handle a real-
world large build, including change detection and dependency inferrence, and
measure overhead compared to existing solutions. Developing a meaningful
performance testing method for incremental builds is another challenge.
Expanding the model and giving design options to support distributed builds
would be valuable.
Incorporating features of Vesta, such as a shared derived file cache and
repeatable builds via integration with existing version control systems would
be another intriguing direction. Taking this to extremes, it may be valuable
to have a “cloud cache” that shares derived files for building open-source
projects among developers throughout the world.
Some of the concepts that are useful for reliable build systems can also be
applied in other domains. For example, because minimum information libraries
allow resource dependencies of code segments to be reliably and precisely
identified, they can be used to compute information transfer from one portion
of a program to another through shared state, which is often overlooked by
dynamic analysis tools.
Finally, there is a great deal of practical work needed to get a functional
reliable build system into the hands of everyday users, including supporting
major tools and environments, providing an expressive build description
language, and pushing for better change detection support in mainstream
kernels.
## 11 Acknowledgements
The authors wish to thank: the Compaq research team, for providing a
fundamentally new design in the space of build systems; David Wagner, for
advice regarding coping with concurrent accesses to files and for suggesting
other faculty; Maria Welborn, for advice regarding system call virtualization
via system call rewriting; Ras Bodik for providing assistance with funding;
Philip Reames, for ideas on reusing concepts in other domains; Eric Brewer,
for providing suggestions regarding file operation interception and feedback
on evaluation; and software developers at Microsoft, UC Berkeley, and other
organizations for providing feedback regarding their experiences with build
systems.
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* [11] Ousterhout, J., Delmas, S., Graham-Cumming, J., Melski, J. E., Muzaffar, U., and Stanton, S. U.S. Patent #7,676,788: architecture and method for executing program builds, Filed 2003 March 25, approved 2010 March 9.
* [12] Saito, Y. Jockey: a user-space library for record-replay debugging. In Proceedings of the sixth international symposium on Automated analysis-driven debugging (New York, NY, USA, 2005), AADEBUG’05, ACM, pp. 69–76.
* [13] Taylor, I. L., and Team, S. S. gold: Google releases new and improved gcc linker. http://google-opensource.blogspot.com/2008/04/gold-google-releases-new-%and-improved.html, 2002-2010.
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* [15] Tridgell, A., Rosdahl, J., et al. ccache — a fast C/C++ compiler cache. http://ccache.samba.org/, 2002-2010.
* [16] Wu, M.-Y., Shu, W., and Gu, J. Efficient local search for dag scheduling. IEEE Trans. on Parallel and Distributed Systems 2001 (2001), 617–627.
## Appendix A Well-definedness of a valid configuration
A _configuration_ specifies the dependency graph and initial shared state for
a build. Recall that a valid build is one where, for any pair of conflicting
tasks, there is a directed path from one to the other in the dependency graph.
We begin by showing a lemma:
###### Lemma A.1.
If a given build is valid, any other valid build with the same configuration
produces the same final result.
###### Proof.
Define the canonical access sequence as the sequence obtained by fixing some
topological order and executing each task sequentially in that order. Given a
valid build’s access sequence, we will perform a series of swaps to transform
it into the canonical sequence.
Suppose two tasks are interleaved (neither performs all its accesses before
those of the other). Then there is not a directed path between them in the
dependency graph, and since the build is valid, they must not conflict. Hence
we can safely swap accesses to ensure that the two tasks are no longer
interleaved. By doing this for all pairs of tasks, we get a sequential
schedule which performs all of each task in some order
$(t_{1},t_{2},\ldots,t_{n})$ which is a topological sort of the dependency
graph.
Any two tasks in a topological sort can be swapped unless there is an edge
between them, and the result is still a topological sort. Such swaps can be
used to transform the sequence into any other topological sort while
preserving the final output, including the canonical sequence. Hence any valid
build’s access sequence produces the same final result: the result produced by
the canonical sequence. ∎
We now generalize this to the stronger result:
###### Theorem A.1.
If a given build is valid, all builds with the same configuration are valid
and produce the same final result. If a given build is invalid, all builds
with the same configuration are invalid.
###### Proof.
By Lemma A.1, if all builds are valid, they all produce the same final result.
It remains to show a single configuration cannot generate both a valid and an
invalid build.
Suppose we have an invalid build $(a_{1},a_{2},\ldots,a_{n})$ and a valid
build $(b_{1},b_{2},\ldots,b_{n})$, both with a given access sequence. We will
gradually transform the first into the second.
We find the first point at which they diverge $a_{i}\neq b_{i}$, locate
$a_{j}$ such that $a_{j}=b_{i}$, and move it up to the $i$th position by a
series of swaps. If $a_{j}$ did not conflict with any of
$a_{i+1},\ldots,a_{j-1}$, then the behavior of all tasks is preserved: the new
access sequence is a feasible build, and is valid if and only if the previous
sequence was valid.
Suppose on the other hand $a_{j}$ does conflict with at least one of
$a_{i+1},\ldots,a_{j-1}$; let the first be $a_{m}$. Because swapping
$a_{j},a_{m}$ may change task behavior, the build must be conceptually re-
executed starting after $a_{m}$ to get a feasible new access sequence. In the
previous iteration, $a_{j}$ followed $a_{m}$, whereas in the current iteration
$a_{j}$ precedes $a_{m}$; this implies the two tasks owning these accesses
have no directed path between them in the dependency graph. But $a_{j},a_{m}$
conflict, so the new build is invalid.
In either case, the common prefix of the two builds grows by at least one
access with each iteration, and eventually the build $(b_{k})$ is reached.
However, in both cases the invalidity of the original build is preserved, so
$(b_{k})$ is invalid as well. This is a contradiction, so there cannot be both
an invalid and a valid build. ∎
This means the definition of a _valid configuration_ as a pair (dependency
graph, start state) producing valid builds is well-defined.
|
arxiv-papers
| 2012-03-13T03:28:55 |
2024-09-04T02:49:28.570813
|
{
"license": "Public Domain",
"authors": "Derrick Coetzee, Anand Bhaskar, George Necula",
"submitter": "Derrick Coetzee",
"url": "https://arxiv.org/abs/1203.2704"
}
|
1203.2787
|
040002 2012 J-P. Hulin 040002
We determine the flow structure in an axisymmetric diffuser or expansion
region connecting two cylindrical pipes when the inlet flow is a solid body
rotation with a uniform axial flow of speeds $\Omega$ and $U$, respectively. A
quasi-cylindrical approximation is made in order to solve the steady Euler
equation, mainly the Bragg–Hawthorne equation. As in our previous work on the
cylindrical region downstream [R González et al., Phys. Fluids 20, 24106
(2008); R. González et al., Phys. Fluids 22, 74102 (2010), R González et al.,
J. Phys.: Conf. Ser. 296, 012024 (2011)], the steady flow in the transition
region shows a Beltrami flow structure. The Beltrami flow is defined as a
field $\mathbf{v}_{B}$ that satisfies
$\mbox{\boldmath${\omega}$}_{B}=\nabla\times\mathbf{v}_{B}=\gamma\mathbf{v}_{B}$,
with $\gamma=constant$. We say that the flow has a Beltrami flow structure
when it can be put in the form ${\mathbf{v}}=U{\mathbf{e}}_{z}+\Omega
r{\mathbf{e}}_{\theta}+{\mathbf{v}}_{B}$, being $U$ and $\Omega$ constants,
i.e it is the superposition of a solid body rotation and translation with a
Beltrami one. Therefore, those findings about flow stability hold. The quasi-
cylindrical solutions do not branch off and the results do not depend on the
chosen transition profile in view of the boundary conditions considered. By
comparing this with our earliest work, we relate the critical Rossby number
$\vartheta_{cs}$ (stagnation) to the corresponding one at the fold
$\vartheta_{cf}$ [J. D. Buntine et al., Proc. R. Soc. Lond. A 449, 139
(1995)].
# Beltrami flow structure in a diffuser. Quasi-cylindrical approximation
Rafael González [inst1, inst3] Ricardo Page E-mail: rgonzale@ungs.edu.ar
[inst2] Andrés S. Sartarelli[inst1]
(29 August 2011; 29 February 2012)
††volume: 4
99 inst1 Instituto de Desarrollo Humano, Universidad Nacional de General
Sarmiento, Gutierrez 1150, 1613 Los Polvorines, Pcia de Buenos Aires,
Argentina. inst3 Departamento de Física FCEyN, Universidad de Buenos Aires,
Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina . inst2
Instituto de Ciencias, Universidad Nacional de General Sarmiento, Gutierrez
1150, 1613 Los Polvorines, Pcia de Buenos Aires, Argentina.
## 1 Introduction
We have recently conducted studies on the formation of Kelvin waves and some
of their features when an axisymmetric Rankine flow experiences a soft
expansion between two cylindrical pipes [1, 2]. One of the significant
characteristics of this phenomenon is that the downstream flow shows a Rankine
flow superposing a Beltrami flow (Beltrami flow structure [4])). Yet, upstream
and downstream cylindrical geometries were considered without taking into
account the flow in the expansion. This work considered that the base upstream
flow, formed by a vortex core surrounded by a potential flow, would have the
same Beltrami structure at the expansion and downstream. Nevertheless, the
flow at the expansion was not determined. However, it has been seen that this
flow is only possible when no reversed flow is present and if its parameters
do not take the values where a vortex breakdown appears [6, 7, 8]. The
starting point in the study of the expansion flow is an axysimmetric steady
state resulting from the Bragg–Hawthorne equation [7, 9, 10, 11] for both the
vortex breakdown and the formation of waves. Therefore, the solution behavior,
whether it branches off or shows a possible stagnation point on the axis, will
be determinant to delimit both phenomena.
Our previous research focused on the formation of Kelvin waves with a Beltrami
flow structure downstream [1, 2, 3], when the upstream flow was a Rankine one.
This present investigation considers only a solid body rotation flow with
uniform axial flow at the inlet. As a first step in the study of the flow at
the expansion, we only study the rotational flow. However, comparisons with
our previous work [1] will be drawn.
The aim of this present work is to obtain the steady flow structure at the
expansion, considering a quasi-cylindrical approximation when the inlet flow
is a solid body rotation with uniform axial flow of speeds $\Omega$ and $U$,
respectively. If $a$ is the radius of the cylindrical region upstream, a
relevant parameter is the Rossby number $\vartheta=\frac{U}{\Omega a}$. Thus,
we would like to determine how this flow depends on the Rossby number, on the
geometrical parameters of the expansion and on the critical values of the
parameters. We focus on finding the parameter values for which a stagnation
point emerges on the axis, or for which the solution of the Bragg–Hawthorne
equation branches off. We take them as the conditions for the vortex breakdown
to develop.
First, this paper presents the inlet flow and the corresponding
Bragg–Hawthorne equation written for the transition together with the boundary
conditions in section II. Second, it works on the quasi-cylindrical
approximation for the Bragg–Hawthorne equation and its solution is developed
in section III. Third, results and discussions are offered in section IV
together with a comparison with our previous work [1]. Finally, conclusions
are presented in section V.
## 2 The Bragg–Hawthorne equation
We assume an upstream flow in a pipe of radius $a$ as an inlet flow in an
axisymmetric expansion of length $L$ connecting to another pipe with radius
$b$, $b>a$. The inlet flow filling the pipe consists of a solid body rotation
of speed $\Omega$ with a uniform axial flow of speed $U$:
$\displaystyle{\mathbf{v}}=U{\mathbf{e}}_{z}+\Omega r{\mathbf{e}}_{\theta},$
(1)
$U$ and $\Omega$ being constants. The equilibrium flow in the whole region is
determined by the steady Euler equation which can be written as the
Bragg–Hawthorne equation [10]
$\displaystyle\frac{\partial^{2}\psi}{\partial
z^{2}}+r\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial\psi}{\partial r}\right)+r^{2}\frac{\partial
H}{\partial\psi}+C\frac{\partial C}{\partial\psi}=0,$ (2)
where $\psi$ is the defined stream function
$\displaystyle v_{r}=-\frac{1}{r}\frac{\partial\psi}{\partial z},\
v_{z}=\frac{1}{r}\frac{\partial\psi}{\partial r},$ (3)
and $H(\psi),C(\psi)$ are the total head and the circulation, respectively
$\displaystyle
H(\psi)=\frac{1}{2}(v_{r}^{2}+v_{\theta}^{2}+v_{z}^{2})+\frac{p}{\rho},\
C(\psi)=rv_{\theta}.$ (4)
To solve Eq. (2), the boundary conditions must be established. These consist
of giving the inlet flow, of being both the centerline and the boundary wall,
streamlines, and of being the axial velocity positive ($v_{z}>0$). For the
upstream flow, the stream function is $\psi=\frac{1}{2}U{r}^{2}$, and
$H(\psi),C(\psi)$ are given by
$\displaystyle H(\psi)=\frac{1}{2}U^{2}+\Omega\gamma\psi,\
C(\psi)=\gamma\psi,$ (5)
$\gamma=\frac{2U}{\Omega}$ being the eigenvalue of the flow with Beltrami
structure [3]. Thus, by considering the inlet flow, Eqs. (5) are valid for the
whole region. The second condition regarding the streamlines implies the
following relations
$\displaystyle\psi(r=0,z)=0,$
$\displaystyle\psi(r=\sigma(z),z)=\frac{1}{2}U{a}^{2},\ 0\leq z\leq L$ (6)
where $r=\sigma(z)$ gives the axisymmetric profile of the pipe expansion.
Deducing from Eq. (6), the boundary conditions are determined by the inlet
flow. Additionally, curved profiles are considered, so
$\displaystyle\frac{\partial\psi}{\partial z}(r,z=L)=0,\ 0\leq r\leq b.$ (7)
## 3 Quasi-cylindrical approximation
If we consider that $\frac{\partial^{2}\psi}{\partial z^{2}}=0$, the solutions
to Eqs. (2) and (5) for the cylindrical regions are given by [10]
$\displaystyle\psi=\frac{1}{2}U{r}^{2}+ArJ_{1}[\gamma r],$ (8)
where $A$ is a constant. The quasi-cylindrical approximation consists of
taking the dependence of $A(z)$ on $z$ but with the condition
$\frac{\partial^{2}\psi}{\partial z^{2}}\approx 0$ compared with the remaining
terms of (2). The amplitude $A(z)$ is then obtained by imposing the boundary
conditions (6) which depend on the wall profile $r=\sigma(z)$, giving
$\displaystyle
A(z)=\frac{1}{2}\frac{U\left(a^{2}-\sigma^{2}(z)\right)}{\sigma(z)J_{1}[\gamma\sigma(z)]}.$
(9)
By using the dimensionless quantities $\tilde{r}=\frac{r}{a}$,
$\tilde{z}=\frac{z}{a}$, $\tilde{v}=\frac{v}{U}$ the stream function in the
quasi-cylindrical approximation can be written as
$\displaystyle\tilde{\psi}=\frac{1}{2}{\tilde{r}}^{2}+\tilde{A}(\tilde{z})\tilde{r}J_{1}[\frac{2}{\vartheta}\tilde{r}],$
$\displaystyle\tilde{A}(\tilde{z})=\frac{1}{2}\frac{\left(1-\tilde{\sigma}^{2}(\tilde{z})\right)}{\tilde{\sigma}(\tilde{z})J_{1}[\frac{2}{\vartheta}\tilde{\sigma}(\tilde{z})]},$
(10)
where $\vartheta=\frac{U}{\Omega a}$ is the Rossby number. Hence the velocity
field becomes
$\displaystyle\tilde{v}_{r}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle-\tilde{A}^{{}^{\prime}}(\tilde{z})J_{1}[\frac{2}{\vartheta}\tilde{r}]$
(11) $\displaystyle\tilde{v}_{\theta}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle\frac{1}{\vartheta}\tilde{r}+\frac{2}{\vartheta}\tilde{A}(\tilde{z})J_{1}[\frac{2}{\vartheta}\tilde{r}]$
(12) $\displaystyle\tilde{v}_{z}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle
1+\frac{2}{\vartheta}\tilde{A}(\tilde{z})J_{0}[\frac{2}{\vartheta}\tilde{r}],$
(13)
where
$\tilde{A}^{{}^{\prime}}(\tilde{z})={d\tilde{A}(\tilde{z})}/{d\tilde{z}}$.
Finally, it is necessary to give the wall profile $\tilde{\sigma}(z)$ to
completely determine the flow. Two kinds of profiles were seen:
i-
conical profile
$\displaystyle\tilde{\sigma}(\tilde{z})=1+\left(\frac{\eta-1}{\tilde{L}}\right)\tilde{z},$
$\displaystyle 0\leq\tilde{z}\leq\tilde{L}\text{ and }\eta=\frac{b}{a}.$ (14)
ii-
curved profile
$\displaystyle\tilde{\sigma}(\tilde{z})$
$\displaystyle=\frac{1+\eta}{2}-\left(\frac{\eta-1}{2}\right)\cos\left(\frac{\pi\tilde{z}}{\tilde{L}}\right),$
$\displaystyle 0\leq\tilde{z}\leq\tilde{L}.$ (15)
The latter meets the boundary condition (7) as well. Therefore, Eqs. (11-15)
together with the boundary conditions (6,7) allow to determine the flow
structure for both the conical and curved wall profile.
## 4 Results and discussion
We note that the flow keeps a Beltrami flow structure in the quasi-cylindrical
approximation. Effectively, giving (11-13)
$\displaystyle\tilde{v}_{r}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle\tilde{v}_{Br}(\tilde{r},\tilde{z})$ (16)
$\displaystyle\tilde{v}_{\theta}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle\frac{1}{\vartheta}\tilde{r}+\tilde{v}_{B\theta}(\tilde{r},\tilde{z})$
(17) $\displaystyle\tilde{v}_{z}(\tilde{r},\tilde{z})$ $\displaystyle=$
$\displaystyle 1+\tilde{v}_{Bz}(\tilde{r},\tilde{z}),$ (18)
it is easy to see that under this approximation
$\nabla\times\mathbf{v}_{B}(\tilde{r},\tilde{z})=\frac{2}{\vartheta}\mathbf{v}_{B}(\tilde{r},\tilde{z})$
and so, the whole flow is the sum of a solid body rotation flow with a uniform
axial flow plus a Beltrami flow, given the latter in a system with uniform
translation velocity $\mathbf{U}=1.\mathbf{\hat{z}}$ and uniform rigid
rotation velocity
$\mathbf{V}=\frac{1}{\vartheta}\tilde{r}\mbox{\boldmath${\hat{\theta}}$}$.
Given the flow field and its structure, the parameters are considered by
evaluating the behavior of $\tilde{v}_{z}(\tilde{r},\tilde{z_{0}})$ with
$\tilde{z_{0}}=\tilde{L}$ i.e., taken at outlet, and with $\tilde{L}=1$. In
order to do so, a wall profile is selected (14 or 15) and three different
values of the expansion parameter are taken, mainly $\eta_{1}=1.1,\
\eta_{2}=1.2$ and $\eta_{3}=1.3$.
Figure 1: Contour flow in the transition region for conical and curved
profiles for $\eta_{1}=1.1$, $\vartheta_{1}=0.695$. Figure 2: Contour flow in
the transition region for conical and curved profiles for $\eta_{1}=1.1$,
$\vartheta_{1}=0.68$. The broken lines represent points with
$\tilde{v}_{z}=0$.
The first step is to analyze the flow dependence on the Rossby number. In Fig.
1, the contour flows corresponding to the conical and curved profiles for
$\eta_{1}=1.1$, $\vartheta_{1}=0.695$ are shown. Graphics in Fig. 2 represent
the same configuration but for $\vartheta=0.68<\vartheta_{1}$. The broken
lines represent points for which $\tilde{v}_{z}=0$. Inflow and recirculation
are present but it is not a real flow because the model fails when considering
inflow. It can be seen that for $\vartheta_{1}=0.695$, $\tilde{v}_{z}=0$ at
the outlet, on the axis. For the Rossby numbers with
$\vartheta\geq\vartheta_{c}$, the azimuthal flow vorticity is negative
($\omega_{\phi}<0$), resulting in an increase in the axial velocity with the
radius, and so having a minimum on the axis where the stagnation point appears
[6]. Therefore, the critical Rossby number can be defined $\vartheta_{c}$ as
the value where $\tilde{v}_{z}$ is zero at the outlet on the axis i.e., where
the flow shows a stagnation point. This is the necessary condition to produce
a vortex breakdown [6]. We find the same critical Rossby number for both wall
profiles and so we will not treat them separately from now on. The critical
Rosssby values for $\eta_{2}=1.2$ and $\eta_{3}=1.3$ are $\vartheta_{2}=0.869$
and $\vartheta_{3}=1.052$, respectively.
Given the previous analysis, the second step is to show the behavior of
$\tilde{v}_{z}$ on the axis at the outlet as a function of $\vartheta$ for
each $\eta$ in order to study the existence of folds in the Rossby number-
continuation parameter (equivalent to the swirl parameter in [7, 11, 5]);
indeed, we have seen that $\tilde{v}_{z}$ has the minimum on the axis.
Besides, when using Eq. (13) when $r=0$, it is easy to see that
$\tilde{v}_{z}$ decreases with $z$ and so it reaches the minimum at the outlet
being $\tilde{v}_{z}\geq 0$. In Fig. 3, the radial dependence of
$\tilde{v}_{z}$ is plotted at the outlet for $\eta_{1}$,$\eta_{2}$,$\eta_{3}$
and its variation with $\vartheta$ when it is slightly shifted from
$\vartheta_{1}$. In Fig. 4, it can be seen that the minimum of $\tilde{v}_{z}$
on the axis increases with $\vartheta$ so there is no fold of
$\tilde{v}_{zmin}$ as defined by Buntine and Saffman in a similar
approximation [5].
Figure 3: (a) $\tilde{v}_{z}$ at the outlet as a function of $r$ for
$\eta_{1}$,$\eta_{2}$,$\eta_{3}$ and the corresponding critical Rossby numbers
$\vartheta_{1}$,$\vartheta_{2}$,$\vartheta_{3}$. (b) $\tilde{v}_{z}$ at the
outlet as a function of $r$ for $\vartheta_{1}$ and for values of $\vartheta$
slightly shifted from $\vartheta_{1}$ . In each case, the minimum of
$\tilde{v}_{z}$ is reached on the axis. Figure 4: $\tilde{v}_{z}$ at the
outlet on the axis as a function of the Rossby number $\vartheta$ for
$\eta_{1}=1.1,\eta_{2}=1.2$, $\eta_{3}=1.3$. Here
$\vartheta_{1}=0.695,\vartheta_{2}=0.869$ and $\vartheta_{3}=1.052$ correspond
to stagnation points.
The dependence of the results on $L$ is analyzed. It can be seen that when
$z=L$ in Eqs. (14) and (15), $\tilde{\sigma}(\tilde{L})=\eta$ is obtained. By
replacing this in Eq. (13) for $z=L$ and $r=0$ it gives
$\displaystyle\tilde{v}_{z}min=1+\frac{\left(1-\eta^{2}\right)}{\vartheta\eta
J_{1}[\frac{2}{\vartheta}\eta]},$ (19)
and so $\vartheta_{c}$ is obtained as a function of $\eta$ by solving the last
equation when $\tilde{v}_{z}min=0$, as shown in Fig. 5. This result seems to
be surprising, but it is not so if it is considered as derived from the quasi-
cylindrical approximation: the dependence of the flow on $z$ is obtained
through the boundary conditions expressed by Eq. (6). At the same time, these
boundary conditions depend on the inlet flow and on the parameter $\eta$. This
explains the fact that the same results, for both conical and curved profiles,
have been obtained and that the condition given by Eq. (7) at the outlet has
not influenced them.
Figure 5: Critical Rossby number $\vartheta_{c}$ as a function of $\eta$.
Differences with Batchelor’s seminal work should be marked [10]. Mainly, he
works in cylindrical geometry and does not consider the dependence of the flow
on $z$ . We introduce this $z$ dependence through the quasi-cylindrical
approximation. This, therefore, allows us to find the structure of the flow in
the transition together with the Rossby critical number defined by considering
this structure and by showing that the minimum of $\tilde{v}_{z}$ is reached
at the outlet on the axis. Nevertheless, once the flow reaches the pipe
downstream, the analysis coincides because, as shown, the problem depends on
the inlet flow and on the parameter expansion $\eta$. This allows us to
consider the issue of the vortex core that we have not considered at the inlet
flow. As we know the structure of the flow in the downstream cylindrical
region [1] and by assuming a quasi-cylindrical approximation for the vortex
core in the transition region, the minimum of ${{{v_{core}}}_{z}}$ at the
outlet on the axis is given by
$\displaystyle{{{v_{core}}}_{z}}_{min}=1+\frac{\left(1-\hat{\eta}^{2}\right)}{\hat{\vartheta}\hat{\eta}J_{1}[\frac{2}{\hat{\vartheta}}\hat{\eta}]},$
(20)
where $\hat{\vartheta}=\frac{\vartheta}{\iota}$,
$\hat{\eta}=\frac{\xi}{\iota}$ and $\xi$ and $\iota$ are the dimensionless
radius of the core downstream and upstream, respectively. We note that
$\hat{\eta}$ is the expansion parameter of the core. Hence Eqs. (19) and (20)
have the same structure. In the present work, we have not found any fold in
the Rossby number-continuation parameter of $\tilde{v}_{z}$, as found in our
previous work [1] where the fold was associated with a critical Rossby number
called $\vartheta_{cf}$ by Buntine and Saffman [5]. As we have already done,
we define the Rossby critical number for which ${{{v_{core}}}_{z}}_{min}=0$
where there is a stagnation point, and we will call it ${\vartheta}_{cs}$. In
[1], for $\iota=0.272$ and pipe expansion parameters
$\eta_{1}$,$\eta_{2}$,$\eta_{3}$, we have found that $\vartheta_{cf}$ were
$0.35$, $0.44$ and $0.53$, respectively, while the core expansion parameters
$\hat{\eta}$ were $1.25$, $1.47$ and $1.65$, respectively.
By replacing these values in Eq. (20) when ${{{v_{core}}}_{z}}_{min}$ is zero,
we get the corresponding $\hat{\vartheta}_{cs}$ and then ${\vartheta}_{cs}$
for the vortex core. These are respectively $0.26$, $0.38$ and $0.49$. That is
to say that in all the cases we have ${\vartheta}_{cs}<\vartheta_{cf}$.
Therefore, at the fold $\tilde{v}_{z}>0$. This coincides with the results
found by Buntine and Saffman [5] in their analysis using a three-parameter
family inlet flow.
## 5 Conclusions
The main conclusions drawn from the previous sections are:
1. 1.
In the quasi-cylindrical approximation, the steady flow in the transition
expansion region corresponding to a solid body rotation with uniform axial
flow as inlet flow has the same Beltrami flow structure as in the pipe
downstream, which is compatible with the boundary conditions. Therefore,
findings from our previous work on stability [1, 2, 3] can hold.
2. 2.
For fixed values of $\eta$ and $\vartheta\geq\vartheta_{c}$, $\omega_{\phi}<0$
and then $\tilde{v}_{z}$ in the transition region is an increasing function of
$r$ and a decreasing function of $z$ reaching its the minimum on the axis at
the outlet.
3. 3.
For fixed values of $\eta$, the minimum of $\tilde{v}_{z}$ on the axis is an
increasing function of $\vartheta$ (Fig. 4), where the stagnation point
corresponds to $\vartheta_{c}$.
4. 4.
As a consequence, no branching off takes place for the solutions of
Bragg–Hawthorne equation.
5. 5.
The critical Rossby number $\vartheta_{c}$ corresponding to stagnation is an
increasing function of $\eta$ (Fig. 5).
6. 6.
The whole picture can be reached by putting together these results with those
obtained in [1], where there is a branching owing to the boundary conditions
at the frontier between the vortex and the irrotational flow. Moreover, since
the results in [1] for the rotational flow depend on the inlet flow as well as
on the rotational expansion parameter $\hat{\eta}$ defined in Eq. (20), given
a quasi-cylindrical approximation, it can be concluded that this expression is
the minimum of ${v}_{z}$ in the core. Therefore, we can get the critical
Rossby number $\vartheta_{cs}$ and compare it with that corresponding to the
fold $\vartheta_{cf}$. This present work verifies that
${\vartheta}_{cs}<\vartheta_{cf}$, in accordance with Buntine and Saffman’s
results [5].
7. 7.
In the quasi-cylindrical approximation, previous results do not depend on the
chosen profile. This can be explained by the boundary conditions chosen
depending on the inlet flow and on the parameter expansion.
###### Acknowledgements.
We would like to thank Unversidad Nacional de General Sarmiento for its
support for this work, and our colleague Gabriela Di Gesú for her advice on
the English version of this paper.
## References
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* [2] R González, A Costa, E S Santini, On a variational principle for Beltrami flows, Phys. Fluids 22, 74102 (2010).
* [3] R González, E S Santini, The dynamics of beltramized flows and its relation with the Kelvin waves, J. Phys.: Conf. Ser. 296, 012024 (2011).
* [4] The Beltrami flow is defined as a field $\mathbf{v}_{B}$ that satisfies $\mbox{\boldmath${\omega}$}_{B}=\nabla\times\mathbf{v}_{B}=\gamma\mathbf{v}_{B}$, with $\gamma=constant$. We say that the flow has a beltrami flow structure when it can be put in the form ${\mathbf{v}}=U{\mathbf{e}}_{z}+\Omega r{\mathbf{e}}_{\theta}+{\mathbf{v}}_{B}$, being $U$ and $\Omega$ constants, i.e it is the superposition of a solid body rotation and translation with a Beltrami one. For a potential flow $\gamma=0$.
* [5] J D Buntine, P G Saffman, Inviscid swirling flows and vortex breakdown, Proc. R. Soc. Lond. A 449, 139 (1995).
* [6] G L Brown, J M Lopez, Axisymmetric vortex breakdown Part 2. Physical mechanisms, J. Fluid Mech. 221, 573 (1990).
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* [9] S L Bragg, W R Hawthorne, Some exact solutions of the flow through annular cascade actuator discs, J. Aero. Sci. 17, 243 (1950).
* [10] G K Batchelor, An introduction to fluids dynamics, Cambridge University Press, Cambridge (1967).
* [11] S V Alekseenko, P A Kuibin, V L Okulov, Theory of concentrated vortices. An introduction, Springer-Verlag, Berlin Heidelberg (2007).
|
arxiv-papers
| 2012-03-13T12:53:38 |
2024-09-04T02:49:28.582155
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rafael Gonz\\'alez, Ricardo Page, Andr\\'es Salvador Sartarelli",
"submitter": "Rafael Gonz\\'alez",
"url": "https://arxiv.org/abs/1203.2787"
}
|
1203.2805
|
# The molecular emissions and the infall motion in the high-mass young stellar
object G8.68-0.37
Zhiyuan Ren1, Yuefang Wu1, Ming Zhu2, Tie Liu1, Ruisheng Peng3, Shengli, Qin4,
and Lixin Li1,5
1Department of Astronomy, Peking University, 100871, Beijing China,
E-mail:rzy,ywu@pku.edu.cn
2National Astronomical Observatory of China, 20A Datun Road, Chaoyang
District, Beijing, China
3Caltech Submillimeter Observatory
4I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937
5The Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He
Yuan Lu 5, Hai Dian Qu, Beijing 100871, P. R. China
###### Abstract
We present a multi-wavelength observational study towards the high-mass young
stellar object G8.68-0.37. A single massive gas-and-dust core is observed in
the (sub)millimeter continuum and molecular line emissions. We fitted the
spectral energy distribution (SED) from the dust continuum emission. The best-
fit SED suggests the presence of two components with temperature of $T_{\rm
d}=20$ K and 120 K, respectively. The core has a total mass of up to
$1.5\times 10^{3}$ $M_{\odot}$ and bolometric luminosity of $2.3\times
10^{4}~{}L_{\odot}$. Both the mass and luminosity are dominated by the cold
component ($T_{\rm d}=20$ K). The molecular lines of C18O, C34S, DCN, and
thermally excited CH3OH are detected in this core. Prominent infall signatures
are observed in the 12CO $(1-0)$ and $(2-1)$. We estimated an infall velocity
of 0.45 km s-1 and mass infall rate of $7\times 10^{-4}~{}M_{\odot}$ year-1.
From the molecular lines, we have found a high DCN abundance and relative
abundance ratio to HCN. The overabundant DCN may originate from a significant
deuteration in the previous cold pre-protostellar phase. And the DCN should
now be rapidly sublimated from the grain mantles to maintain the overabundance
in the gas phase.
###### keywords:
stars: pre-main sequence — ISM: molecules — ISM: kinematics and dynamics —
ISM: individual (G8.68-0.37) — stars: formation
††pagerange: The molecular emissions and the infall motion in the high-mass
young stellar object G8.68-0.37–LABEL:lastpage††pubyear: 2010
## 1 Introduction
Gravitational infall, or core collapse can take place in high mass young
stellar objects (YSOs) at early stages and continue all the way to the stage
of Ultra Compact (UC) Hii regions (Keto, 2003; Sollins et al., 2005). As shown
by theoretical works(Jijina & Adams, 1996; Yorke & Sonnhalter, 2002; Gong &
Ostrike, 2009, etc.), the infall motion is critical for initiating the high
mass star formation and maintaining the accretion flow to feed the stellar
mass during the subsequent evolutionary stages. However, further observations
are still needed to better constrain the physical properties of the infall,
including its spatial distribution, mass infall rate, chemical effect, and to
understand its relation with other dynamical processes, including outflow,
disk accretion, and core fragmentation. In the recent decade, extensive
spectroscopic surveys (e.g. Wu & Evans, 2003; Fuller et al., 2005; Wyrowski et
al., 2006; Purcell et al., 2006; Wu et al., 2007) have been performed towards
the potential high mass YSOs throughout the Milky Way. As a result, many
infall candidates have been identified based on their spectral signatures.
These sources can serve as good candidates to study the massive star birth and
gas dynamics in the molecular cores. In the mean time, strong outflows are
also widely detected towards those massive cores (e.g. Beuther et al., 2002;
Wu et al., 2004; Zhang et al., 2007). The infall and outflow motions should be
closely related and interacting with each other throughout the star formation
history.
G8.68-0.37 (G8.68 here after) is a young high-mass star forming region at a
distance of 4.5 kpc (Mueller et al., 2002). In this region, compact multiple
gas-and-dust clumps has been discovered by Longmore et al. (2011, L11 here
after). The dusty core is associated with strong 6.7 GHz methanol masers
(Walsh et al., 1998), but has no radio continuum emission, indicating that
high mass stars are already formed, but have not yet ionized its surrounding
gas. L11 also detected a bi-polar outflow in CO $(2-1)$. The outflow may be
responsible for the shock interaction traced by the extended 4.5 $\micron$
emission (Figure 2 therein). In the mean time, the observation in HCO+ $(1-0)$
suggests a plausible infall motion (Purcell et al., 2006) which should be
examined quantitatively. To improve the understanding in physical and chemical
properties of this source, we performed a multi-wavelength study using both
the single dish antennae and the interferometers. The next section introduces
the observations and data reduction, Section 3 presents the general
observational results. Section 4 describes the dust continuum and molecular
line emissions, wherein the infall signature is specifically described in
Section 4.3. A summary is given in Section 5.
## 2 Observations and Data Reduction
### 2.1 The single dishes
In Figure 1 we show the central positions and the beam sizes of all the
observations. We have observed G8.68 using three different single-dish
telescopes. In 2005, we observed HCN (3-2) and H13CO+ $(3-2)$ from the James
Clerk Maxwell Telescope111JCMT is operated by the JAC, Hawaii, on behalf of
the UK PPARC, the Netherlands OSR, and the Canadian NRC, see
http://www.jach.hawaii.edu/JCMT/ (JCMT). The pointing center was adopted to be
the coordinate of the strongest methanol maser (Walsh et al., 1998, with a
position accuracy of $1.8^{\prime\prime}$), which is close to the continuum
emission peak of L11 (cross in Figure 1). The mapping step is
$10^{\prime\prime}$ (corresponding to 1/2 beam size), as shown in Figure 2b.
In November 2009, we observed $J=1-0$ line of 12CO, 13CO, and C18O using the
13.7 m telescope at the Purple Mountain
Observatory222http://www.dlh.pmo.cas.cn/ (PMO). The PMO observation contains a
grid-mapping with a coverage of several arc minutes over the region of G8.68.
In this paper, we only use the spectra at one point which is closest to the
continuum peak, as shown in Figure 1. The PMO beam is much larger than the CSO
and JCMT, and is significantly deviated from the continuum peak. Nevertheless,
the beam has well covered the emission regions of the continuum and molecular
lines. The data can thus be used to trace the gas motion on a larger scale
near the core.
In May 2011, the $J=2-1$ lines of the three CO isotopologues were observed
from the 10 m telescope at the Caltech Submillimeter
Observatory333http://www.submm.caltech.edu/cso/ (CSO). The 12CO $(2-1)$ is
observed at five symmetric points around the continuum peak with an offset of
$\pm 23^{\prime\prime}$ in the R.A. and Dec. directions. Their positions are
shown in Figure 2a. All the single-dish spectra are discussed in detailed in
Section 3.2.2.
In Table 1, we present the basic observational parameters and weather
conditions for the three instruments. In Table 2, we shows the more specific
observational parameters for the molecular lines. All the observations were
performed in good weather conditions, with pointing accuracies better than
$5^{\prime\prime}$. GILDAS software package444http://iram.fr/IRAMFR/GILDAS/ is
used for the data reduction and image plot.
To measure the flux densities at different wavelengths, we also retrieved the
Spitzer archival images at four IRAC bands from the GLIMPSE survey555Available
at http://irsa.ipac.caltech.edu/, see also Benjamin et al. (2003)., and the 24
and 70 $\micron$ images from MIPSGAL666http://irsa.ipac.caltech.edu/, and
JCMT/SCUBA images at 450 and 850 $\micron$ bands, which are available at the
Canadian Astronomy Data Center (CADC) repository of the SCUBA Legacy
Fundamental Object Catalogue777http://www4.cadc-ccda.hia-iha.nrc-cnrc.gc.ca.
### 2.2 The Submillimeter Array
The Submillimeter Array888The Submillimeter Array is a joint project between
the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of
Astronomy and Astrophysics and is funded by the Smithsonian Institution and
the Academia Sinica, see http://www.cfa.harvard.edu/sma/ (SMA) observations
towards G8.68 are taken from the released SMA data archive. The observations
are made in three epochs in the year 2007, 2008, and 2009, respectively. The
observational parameters, including the calibration sources for each epoch are
presented in Table 3. In all three epochs, the compact array was used, and the
phase tracking center is R.A.(J2000)=18h06m23.23s,
Dec.(J2000)=$-21^{\circ}37^{\prime}14.19^{\prime\prime}$. The three
observations have similar beam sizes for the synthesized and primary beams. In
Figure 1, we only show the beams of the 2008 observation in order to have a
clear appearance. The observed gas-and-dust structures (Figure 4) turn out to
be smaller than the SMA primary beam. Thus the beam-edge weakening is not
significant. The calibration and imaging were performed in Miriad1. The
absolute flux level has an uncertainty of $\sim 15\%$. The continuum emission
was subtracted from the line-free channels in each sideband. The gain solution
is self-calibrated for the continuum image and then exported to the spectral
line data.
We note that among the SMA data, the 2008 observation has the longest on-
source integration time hence the lowest noise level. In addition, in 2008 all
eight antennae of the SMA were at work, while the 2007 observation (280 GHz)
only employed seven antennae. As a result, despite its higher frequency, the
2007 data has a lower angular resolution than the 2008 data (as indicated by
their synthesized beam sizes, in Table 3). We therefore used the 2008 data
(frequency centered at 225 GHz, or 1.3 mm) to analyze the dust continuum
emission. The continuum was averaged from the line-free channels and then
subtracted from the side-band spectrum. The continuum data of the two
sidebands were averaged on the (u,v) plane and then converted to the image
domain. After Clean and Self-calibration, the 1.3 mm continuum image has an
rms noise level ($1~{}\sigma$) of 3.6 mJy beam-1 (corresponding to a
brightness temperature of $T_{\rm b}=0.0025$ K).
## 3 Results
### 3.1 Dust continuum emission
In Figure 4, we show the continuum emissions of G8.68 from infrared to
(sub)millimeter wavebands, including the IRAC 3.6, 4.5, and 8 $\micron$
emissions (RGB color image), the SMA 1.3 mm continuum emission (white
contours), and the SCUBA 450 $\micron$ continuum (blue dashed contours).
Figure 4 is centered at the 1.3 mm continuum peak, the coordinates of which
are R.A.(J2000)$=18^{\rm h}06^{\rm m}23.52^{\rm s}$,
Decl.(2000)$=-21^{\circ}37^{\prime}11^{\prime\prime}$. It is close in
projection to the SMA phase tracking center (labeled with the red cross).
After deconvolution with the synthesized beam, the core has an angular size of
$11^{\prime\prime}\times 6^{\prime\prime}$ for the 4 $\sigma$ contour
($0.24\times 0.13$ pc at a distance of 4.5 kpc). It is elongated in the north-
south direction ($PA=-10^{\circ}$ for the major axis), reasonably coherent
with the 4.5 $\micron$ emission (green color). Since the 4.5 $\micron$
emission traces the shock interaction between the outflow and the envelope
gas, it is possible that the outflow and shocks are also affecting the dust
distribution, causing its observed elongation. We did not find any evidence
for multiple sub-cores either in our 1.3 mm continuum or any molecular lines
(Figure 5). Therefore the gas-and-dust core should have a single compact
morphology, and the fragmentation is not evident on our observational scale
(0.05 to 0.5 pc).
More diffused dust component can be revealed by the SCUBA 450 $\micron$
continuum emission. As shown in Figure 4, the 450 $\micron$ emission is more
extended and less elongated than the 1.3 mm emission. We use the average
deconvolved FWHM (full width at half maximum) radius $\langle r\rangle$ to
represent the extent of the continuum and molecular line emissions. Normally
$\langle r\rangle$ can be measured from the 50 % contour level of the emission
region. However, the 50 % contour (for the continuum and molecular lines) is
often close to or even smaller than beam size. We thus suggest measuring the
deconvolved radius from the 10 % contour, and adopt its 1/2 as the value of
$\langle r\rangle$. For the continuum images, the 10 % contour is not
specifically plotted in Figure 4, but close to the 14 $\sigma$ and 4 $\sigma$
contour level for the 1.3 mm and 450 $\micron$ emissions, respectively. We
obtained $\langle r\rangle_{\rm 450\micron}=0.23$ pc and $\langle
r\rangle_{\rm 1.3mm}=0.08$ pc.
We also measured the integrated flux density $F(\lambda)$ of the dust core at
wavelength $\lambda$. In general, we use the 4 $\sigma$ emission level as the
integration area for $F(\lambda)$. As an exception, for the IRAC data, we use
the region of the 4.5 $\micron$ emission (green color in Figure 4) to measure
the integrated flux of all four bands, since the 4.5 $\micron$ emission has a
relatively clear boundary. The 5.8 $\micron$ band shows a similar morphology
with the 4.5 $\micron$, while the emissions at other two bands are much
fainter and cannot be well delineated. The IRAC stellar sources in the
vicinity of the core are carefully excluded from the integration area. The
derived $F(\lambda)$ are shown in Table 4.
### 3.2 Molecular lines
#### 3.2.1 The Submillimeter Array
Using the SMA, we have detected a number of molecular transitions of C18O,
C34S, DCN, and CH3OH. Their beam-averaged spectra towards the 1.3 mm continuum
peak and their velocity-integrated intensity images are shown in Figure 5. For
the CH3OH, altogether we have detected 11 rotational transitions. We selected
five of them with largely different $E_{\rm u}$, and presented their images in
Figure 5. The physical parameters of all the molecular transitions are shown
in Table 5.
As shown in Figure 5, the spectra of the molecular tracers of high-density gas
mostly show lines with single peak profiles. As exceptions, there are two
CH3OH lines, $11_{2}-10_{3}$ ($E_{\rm u}=191$ K) and $15_{7,8}-16_{6,11}$
($E_{\rm u}=523$ K) which show double peak profiles. However, since the
remaining lines are all single-peaked, the two lines are more likely to be
blended with other molecular transitions. The possible candidates for the
blenders are NH2CN $14_{2}-13_{2}$ (f=279.35062 GHz, $E_{\rm u}=228$ K) and
HCCNC $29-28$ (f=288.07346 GHz, $E_{\rm u}=207$ K). We used two gaussian
profiles to fit the blended spectra, as plotted in dotted lines in Figure 5.
For each spectrum, the peak velocity of the second component is well
consistent with the anticipated velocity for the blenders. With the
contamination excluded, these two CH3OH lines should also have single gaussian
profiles.
The C34S emission region shows an elongated morphology from the northeast to
southwest ($PA=45^{\circ}$) as labeled in dashed line. The elongation agrees
with the orientation of the CO outflow and 4.5 $\micron$ shock emission
(Figure 4). An elongated morphology towards northeast is also shown in the low
excited CH3OH lines (i.e. $E_{\rm u}=33$ K and 97 K). Therefore, both the
CH3OH and C34S distributions should be affected by the outflow. The C18O
$(2-1)$ also shows a non-regular morphology. However, it is biased to the
south of the dust core, peaked at
offset$=(0^{\prime\prime},-2^{\prime\prime})$, In addition, the C18O shows a
secondary clump in the southeast, peaked at
offset$=(10^{\prime\prime},-8^{\prime\prime})$. This clump is not detected in
C34S or CH3OH lines, indicating that it may be depleted in these species. The
C18O might trace cooler and less dense gas component, thus have a more
extended feature than other dense-core species. For each molecular transition,
we also measured the average deconvolved radius from the 10 % contour level
(and adopted its 1/2 as the value of $\langle r\rangle$). The results are
shown in Table 5.
#### 3.2.2 The single dishes
Figure 2 and 3 show the molecular lines detected from the single dishes. As
shown in Figure 3, prominent double-peak line profiles are observed in the
12CO $(1-0)$, $(2-1)$, and also HCN $(3-2)$. For both the 12CO $(1-0)$ and
$(2-1)$, the blueshifted emission peak is much stronger than the redshifted
one, and the central absorption dip is well coincident with the C18O line peak
($V_{\rm lsr}=37$ km s-1). Such blue asymmetric 12CO lines suggest the
presence of infall motion towards the core center (Zhou et al., 1993; Mardones
et al., 1997). For the physical explanation, when the infall occurs in the
envelope which is cooler than the inner region, the gas in the front part
would absorb the redshifted side of the line profile, whereas the gas in the
rear part (behind the center) would increase the blueshifted emission because
it is moving towards the observer. Besides the infall signature, the 12CO
lines also exhibit high-velocity emission wings extending to $V_{\rm lsr}=$25
and 50 km s-1 for the blue- and redshifted sides, respectively. This velocity
range is comparable to the outflow velocities revealed by L11 (Figure 5
therein).
The two 13CO lines also have a blueshifted emission peak ($V_{\rm lsr}=35$ km
s-1) with respect to the C18O, suggesting that the 13CO is also probably
tracing the infall motion. However, because the 13CO lines are much less
optically thick than the 12CO, they exhibit no central dip, but instead show
an emission shoulder that continuously declines towards the redshifted side.
As shown in Figure 2a, we can see that the offset positions also exhibit self-
absorbed profiles (except the southeast one). However, compared to the central
spectrum, their blue- and redshifted peaks have more similar intensities. As
an extreme, the southeastern spectrum have a flattened top, with the double-
peak feature almost disappeared. This indicates that the infall motion (along
the line of sight) should have a decline towards those offset points. And
their distance from the center (0.7 pc) can therefore be taken as a lower
limit for the radius of the infalling region.
As shown in Figure 3c, the HCN $(3-2)$ has a double peak profile and high-
velocity emission wings extending to $V_{\rm lsr}=29$ and 53 km s-1 (above the
noise level) for the blue and red wings, respectively. However, its double
peaks have different asymmetry with the 12CO lines, i.e., the red peak is
slightly stronger than the blue one. The offset spectra of the HCN (Figure 2b)
have much lower signal-to-noise ratio (mainly due to their shorter integration
time). However, they still evidently show double peak profiles, and have
similar intensities for the blue and red peaks. Like in the case of the CO
lines, the optically thin isotopic lines, i.e., DCN $(3-2)$ and $(4-3)$ are
both single-peaked, hence the HCN $(3-2)$ profile should originate from a self
absorption effect. Compared to the 12CO profiles, the HCN spectra may reflect
different gas motions, including core expansion or/and rotation (Pavlyuchenkov
et al., 2008). In particular, as the most possible case, a cold and
spherically expanding envelope would cause a prominent blueshifted self-
absorption towards the center, while at the offset positions, the expansion
should be on the plane of the sky, thus show a less blueshift due to the lower
radial velocities. This scenario can reasonably explain the observed line
profiles, but still needs a further examination with an improved angular
resolution and spectral sensitivity. In Figure 2b, we also plotted the
velocity-integrated map of HCN $(3-2)$ (discrete gray scales). We measured the
average deconvolved radius $\langle r\rangle$ of HCN from its 50 % contour. As
a result, it has $\langle r\rangle=12^{\prime\prime}$ (and $\eta_{\rm bf}=1$).
Another JCMT line, H13CO+ $(3-2)$, has a regular gaussian profile, indicating
that it may arise from the dense molecular core and is not evidently affected
by the infall or outflow motion. We only have one-point observation for the
H13CO+ $(3-2)$ at the continuum center, and in calculation for its column
density (Section 4.4), we assume $\eta_{\rm bf}=1$.
## 4 Discussion
### 4.1 The physical properties of the dust core
As shown in Figure 4, the 1.3 mm dust core does not coincide with any infrared
stellar sources besides the extended 4.5 $\micron$ shock emission. This
indicates that the stellar emission from the core center is highly obscured by
the dust. In the vicinity of the 1.3 mm dust core, a few stellar objects are
shown in the IRAC RGB image (also labeled with the asterisks). All these
objects are isolated from the 1.3 mm continuum emission, yet the three objects
nearest to the continuum peak are likely to be embedded in the 450 $\micron$
emission region. They might either be more evolved young stars in the same
star forming region or irrelevant foreground stars. Despite this uncertainty,
it is clear that these objects have no significant contribution to the dust
continuum or molecular line emissions. We therefore make no further discussion
for them.
We can fit the Spectral Energy Distribution (SED) of the dust core from its
flux densities at the Spitzer and JCMT/SCUBA wavebands. Assuming a gray body
emission and a uniform dust temperature $T_{\rm d}$, the continuum flux
density would be (Schnee et al., 2007)
$F_{\nu}=\frac{M_{\rm core}\kappa_{\nu}B_{\nu}(T_{\rm d})}{gD^{2}}$ (1)
where $F_{\nu}$ is the flux density at frequency $\nu$. $M_{\rm core}$ is the
total gas-and-dust mass of the core. $g=100$ is commonly adopted gas-and-dust
mass ratio. $B_{\nu}(T_{\rm d})$ is the Planck function at temperature $T_{\rm
d}$. $D=4.5$ kpc is the source distance. The dust opacity $\kappa_{\nu}$ is
assumed to have a power-law shape, i.e. $\kappa_{\nu}=\kappa_{\rm
230GHz}(\nu/{\rm 230GHz})^{\beta}$, with the reference value $\kappa_{\rm
230GHz}=0.9$ cm2 g-1 (Ossenkopf & Henning, 1994). The free parameters in the
fit are $M_{\rm core}$, $T_{\rm d}$, and $\beta$. We found that the emissions
from 8 $\micron$ to 850 $\micron$ can be best fitted with two temperature
components which have $T_{\rm d}=20$ K and 120 K respectively, and
$\beta=2.1$. The best-fit SED is shown in Figure 6.
We did not include the IRAC 3.6, 4.5 or 5.8 $\micron$ emissions in our SED
model. The 4.5 and 5.8 $\micron$ emissions may largely come from the shocked
emission thus are much stronger than the emissions at other two IRAC bands
(Table 4). As for the 3.6 $\micron$ emission, if being thermally excited, it
may arise from some even hotter component which is much fainter and poorly
constrained by our current data. Therefore we also neglected the 3.6 $\micron$
band. With the derived SED, the bolometric luminosity can be estimated using
$L_{\rm bol}=4\pi D^{2}\int F_{\nu}{\rm d}\nu$. As a result we have $L_{\rm
bol}=2.3\times 10^{4}$ and $8\times 10^{2}$ $L_{\odot}$ for the cold (20 K)
and warm (120 K) components, respectively. Using Equation (1), we can also
estimate the total mass of the two temperature components, which turn out to
be $1.3\times 10^{3}$ and $1.0\times 10^{-3}~{}M_{\odot}$ for the 20 K and 120
K components, respectively. One can see that both the core mass and luminosity
are dominated by the gas-and-dust component which is characterized by $T_{\rm
d}=20$ K.
In Equation (1), by replacing the integrated flux density $F_{\nu}$ with the
flux density at the continuum peak (0.32 Jy beam-1), and then dividing the
obtained mass with the beam area and the average molecular mass (1.4 times the
molecular mass of H2), we can derive the H2 column density $N({\rm H_{2}})$
towards the continuum peak. And then, assuming that the core is approximately
spherical, we can derive the volume number density using $n({\rm
H_{2}})=N({\rm H_{2}})/2\langle r\rangle$. The physical parameters of the dust
core are presented in Table 4.
By extrapolating the best-fit SED curve, we can get a flux density of 2.9 Jy
at $\lambda=1.3$ mm. Compared to this value, the SMA observation has recovered
35% of the 1.3 mm continuum emission. Adopting $T_{\rm d}=20$ K, we also
estimated the physical parameters from the SMA 1.3 mm continuum, which are
presented in Table 4. Based on the continuum observations, we suggests that
the gas-and-dust core in G8.68 should consist of a dense inner region
(characterized by the 1.3 mm emission), and a more extended envelope (traced
by the 450 $\micron$ emission).
### 4.2 The CH3OH rotational temperature
The molecular gas temperature can also be estimated from the CH3OH lines using
the rotation diagram. The methanol lines all have linewidths of several km
s-1, with none of them showing abnormally high intensities, therefore the
CH3OH lines are unlikely to have maser excitations.
Assuming optically thin, the column density of the upper-level $N_{\rm u}$ can
be derived from the integrated line intensity using (Tielens, 2005)
$N_{\rm u}=\frac{8\pi k\nu_{\rm ul}^{2}}{hc^{3}A_{\rm ul}}\int T_{\rm b}{\rm
d}V/\eta_{\rm bf}$ (2)
where $T_{\rm b}$ is the observed brightness temperature. $A_{\rm ul}$ is the
Einstein coefficient in s-1. $\eta_{\rm bf}$ is the beam filling factor. All
other constants take their usual values in SI units. Although for an emission
line, $\eta_{\rm bf}$ may vary at different velocities, we approximate it to
be a single value as the ratio between the integrated emission region and the
beam area, i.e., $\eta_{\rm bf}\simeq\pi\langle r\rangle^{2}/A_{\rm beam}$
(used when the emission region is smaller than the beam size, otherwise
$\eta_{\rm bf}=1$). $\eta_{\rm bf}$ is estimated for each transition and the
derived values are presented in Table 5 (Column 11).
Assuming a Local Thermal Equilibrium (LTE, i.e. energy levels are populated
according to a Boltzmann distribution characterized by a single temperature),
the relation between the total column density $N_{\rm T}$ and $N_{\rm u}$ is
$\frac{N_{\rm u}}{g_{\rm u}}=\frac{N_{\rm T}}{Q(T_{\rm
rot})}\exp({-\frac{E_{\rm u}}{kT_{\rm rot}}})$ (3)
and its logarithmic form is
$\ln(\frac{N_{\rm u}}{g_{\rm u}})=\ln(\frac{N_{\rm T}}{Q(T_{\rm
rot})})-\frac{E_{\rm u}}{kT_{\rm rot}}$ (4)
where $g_{\rm u}$ and $E_{\rm u}$ are the degeneracy and the excitation energy
of the upper level, respectively, and $Q(T_{\rm rot})$ is the partition
function. For CH3OH, a good approximation is $Q(T_{\rm rot})\simeq
1.2327\times T_{\rm rot}^{1.5}$ (Townes & Schawlow, 1955).
The rotation diagram for the CH3OH lines is shown in Figure 7. A linear least-
square fit to the data points results in $T_{\rm rot}=130\pm 10$ K and $N_{\rm
T}=(5.3\pm 0.6)\times 10^{15}$ cm-2.
In the calculation, in order to correct for the optical depth effect, one
should multiply $N_{\rm u}/g_{\rm u}$ with a correction factor
$C_{\tau}=\tau/(1-{\rm e}^{-\tau})$ and fit the rotational temperature
iteratively. The optical depth is estimated using (Remijan et al., 2004,
Equation (3) therein, slightly reformed)
$\tau=\frac{c^{3}\sqrt{4\ln 2}}{8\pi\nu^{3}\sqrt{\pi}\Delta V}N_{\rm u}A_{\rm
ul}[\exp(\frac{h\nu}{kT_{\rm rot}})-1]$ (5)
Among all the CH3OH lines, the $(8_{-1}-7_{0})$ transition has the highest
optical depth ($\tau=0.059$). The other lines are even more optically thin. To
take into account the temperature uncertainty, we also estimated the optical
depth assuming $T_{\rm rot}=20$ K which is a lower limit as suggested by the
SED fitting. At 20 K, the optical depths become $\sim 8$ times larger than the
values at $T_{\rm rot}=130$ K. The derived optical depths are listed in Table
5, and the column densities and abundances are listed in Table 6.
The rotational temperature of $T_{\rm rot}=130$ K is close to the SED
temperature of the warm dust component ($T_{\rm d}=120$ K). Therefore it is
possible that the CH3OH emissions are mainly from the region associated with
the warm dust. Moreover, since the cold component ($T_{\rm d}=20$ K) is more
massive than the warm one for orders of magnitude, the CH3OH may have a severe
depletion in the region for the cold dust component. However, it is also
possible that the dust and gas are thermally decoupled, thus exhibit different
temperatures. The collisional excitations of the molecular gas can be
particularly enhanced by the shocks (especially along the outflow direction),
thereby showing a high value of $T_{\rm rot}$. The dust temperature $T_{\rm
d}$, in comparison, may still be largely dominated by the stellar heating thus
has a much lower value.
### 4.3 The CO emission and the infall motion
As shown in Section 3.2.2, both the infall and outflow signatures are detected
in the 12CO $(2-1)$ and $(1-0)$ lines. In this paper we mainly discuss the
infall properties based on the 12CO $(2-1)$. We first make attempt to separate
the different components from the observed spectrum, then estimate the infall
rate.
Following the procedure of Purcell et al. (2006), we used a broad gaussian
profile to fit the outflow wings (velocity range of $V<32$ and $V>42$ km s-1),
and then subtracted it from the spectrum. The residual line profile (green
line in Figure 8) should mainly represent the emission from the dense
molecular core. One can then mask the velocity range possibly affected by the
infall motion (34 to 43 km s-1), and make a gaussian fit to the spectrum
outside this velocity range. The fitted spectrum is speculated to roughly
represent the molecular core emission unaffected by the infall signature.
However, for the 12CO lines, due to its large optical depth, we cannot
directly apply a Gaussian fit to the spectrum. Instead, one should model the
spectrum using the radiation transfer function. In this case, the line profile
can be expressed as
$T_{\rm mb}(V)=[T_{\rm mb,0}-J(T_{\rm CMB})][1-{\rm e}^{-\tau(V)}]$ (6)
$J(T)=T_{0}/[\exp(T_{0}/T)-1]$ is the Planck-corrected brightness temperature,
and $T_{0}=h\nu/k$. $T_{\rm CMB}=2.7$ K is the temperature of the cosmic
background. At the frequency of CO $(2-1)$ (230 GHz), we have $J(T_{\rm
CMB})=0.2$ K. Compared to the intensity of the CO emission, the contribution
from the cosmic background can be almost neglected.
We also assume the dense molecular core to have a uniform gas distribution
along the line of sight, with central velocity $V_{0}$, velocity dispersion
$\sigma$ and peak optical depth $\tau_{0}$. Then the optical depth is
$\tau(V)=\tau_{0}\exp[-\frac{(V-V_{0})^{2}}{2\sigma^{2}}]$ (7)
where $\sigma$ is related to the (intrinsic) line width $\Delta V$ by
$\sigma=\Delta V/\sqrt{8\ln 2}$. In an optically thick case, the line emission
could be largely saturated. $\tau_{0}$ is thus poorly constrained by the
observed spectrum. However, it can be estimated from comparison to the CO
isotopologues following Garden et al. (1991, Equation (4) therein). Since the
13CO is also affected by the infall motion, we used the C18O $(2-1)$ instead.
Assuming an abundance ratio of $[{\rm{}^{12}CO}/{\rm C^{18}O}]=490$(Garden et
al., 1991), the equation will be
$\frac{T_{\rm mb,0}({\rm{}^{12}CO})}{T_{\rm mb,0}({\rm
C^{18}O})}=\frac{1-\exp[-\tau_{0}({\rm{}^{12}CO})]}{1-\exp[-\tau_{0}({\rm
C^{18}O})]}=\frac{1-\exp[-\tau_{0}({\rm{}^{12}CO})]}{1-\exp[-\tau_{0}({\rm{}^{12}CO})/490]}$
(8)
To fit the line profile, we first take an arbitrary, but reasonable value of
$\tau_{0}$, and then fit the line profile by adjusting the values of $T_{\rm
mb,0}$, $V_{\rm 0}$ and $\Delta V$ in Equation (6) and (7). The best-fit
$T_{\rm mb,0}$ is then used to estimate $\tau_{0}$ again using Equation (8).
The final best fit can be reached after two or three iterations. Eventually,
we have $T_{\rm mb,0}=32$ K, $\Delta V=5.5$ km s-1, $V_{0}=37$ km s-1, and
$\tau_{0}({\rm{}^{12}CO})=68$. In Figure 8, the best fit spectrum is shown in
dashed line. And a sum of dense-core and outflow components is shown in red
dot-dashed line. The output spectrum has an apparent line width of 8.5 km s-1
which is indeed much broader than the intrinsic $\Delta V$. We fit the 12CO
$(1-0)$ using the same method. All their line parameters are listed in Table 5
(Column 5 to 8).
The infall rate is estimated using (Klaassen & Wilson, 2007)
$\dot{M}_{\rm inf}=\frac{4}{3}\pi n({\rm H_{2}})\mu m_{\rm H}r_{\rm
gm}^{2}V_{\rm in}$ (9)
wherein $r_{\rm gm}$ is geometric mean radius of the core, $n({\rm H_{2}})$ is
the ambient source density, and $V_{\rm in}$ is the typical infall velocity.
In calculation we estimated $V_{\rm in}$ from the outflow-subtracted line
profile using Equation (9) in Myers et al. (1996). As a result we have $V_{\rm
in}=0.45$ km s-1. In addition, we assume that the more diffused gas traced by
the 450 $\micron$ emission which has $n({\rm H_{2}})=0.8\times 10^{6}$ cm-3,
is collapsing towards the dense inner region characterized by the 1.3 mm
continuum (Figure 4), thus we have $r_{\rm gm}=\langle r\rangle_{\rm
1.3mm}=0.08$ pc. With these assumptions, we derived an infall rate of
$\dot{M}_{\rm inf}=7.0\times 10^{-4}~{}M_{\odot}$ yr-1.
As seen in Equation (9), the derived infall rate is sensitive to the adoption
of $r_{\rm gm}$, and our currently adopted $r_{\rm gm}$ is relatively
conservative. Adopting $r_{\rm gm}=\langle r\rangle_{450\micron}=0.23$ pc, we
would have $\dot{M}_{\rm inf}=5\times 10^{-3}~{}M_{\odot}$ yr-1. However, we
note that such a large-scale estimate may deviate from the small-scale infall
rate. To resemble the mass infall onto the central stars, it may be more
reasonable to adopt the first value ($r_{\rm gm}=0.08$ pc). With the obtained
infall rate, we then estimate the accretion luminosity, using $L_{\rm
acc}=GM_{*}\dot{M}_{\rm inf}/R_{*}$, and assuming a mass-radius relation of
$R_{*}/R_{\odot}=(M_{*}/M_{\odot})^{0.8}$. As a result, we have $L_{\rm
acc}=(4\pm 2)\times 10^{4}~{}L_{\odot}$. The uncertainty in $L_{\rm acc}$
corresponds to a stellar mass varying between 10 an 100 $M_{\odot}$. It is
likely that the bolometric luminosity of the dust core ($2.3\times
10^{4}~{}L_{\odot}$, see Section 4.1) should have a major energy supply from
the accretion process.
Considering the existence of the outflow, there should be a strong interaction
between the infall and the outflow. And the interaction may be responsible for
the 4.5 $\micron$ shock emission. Chen et al. (2010) have performed an HCO+
(1-0) survey towards the Extended Green Objects (EGOs), i.e., the massive YSO
candidates with the 4.5 $\micron$ shock emissions. They found that nearly one
third of the sample (29 out of 69 sources) exhibit a significant blue
asymmetry. While in an HCO+ $(1-0)$ survey towards 82 massive YSOs selected
from the methanol masers, only 12 sources have infall signatures (Purcell et
al., 2006). Comparing these results, it is likely that the shocks are prone to
take place in the YSOs with infall motions. This is theoretically expected,
since compared to an interaction between the outflow and quiescent gas, an
outflow-infall interaction would more efficiently convert the kinetic energy
into heat and radiation.
### 4.4 The molecular abundances and DCN overabundance
The total column density $N_{\rm T}({\rm X})$ and abundance $f({\rm X})=N_{\rm
T}({\rm X})/N({\rm H_{2}})$ of the C18O, HCN, DCN, H13CO+, and C34S are
calculated from their emission lines at the continuum emission peak using
equation (2) and (3). And a correction for the optical depth is done using
Equation (4). To derive the HCN and H13CO+ abundances, we used the $N({\rm
H_{2}})$ value for the SCUBA 450 $\micron$ continuum (Table 3). While for the
SMA lines, we adopted $N({\rm H_{2}})$ from the 1.3 mm continuum ($0.95\times
10^{24}$ cm-2). We also note that $N_{\rm T}({\rm H^{13}CO^{+}})$ may be
underestimated due to the beam dilution thus should be regarded as a lower
limit. To take into account the temperature uncertainty, we also estimated
$N_{\rm T}({\rm X})$ at the lower limit of $T_{\rm rot}=20$ K (suggested by
the SED fitting). The $N_{\rm T}({\rm X})$ and $f({\rm X})$ values are shown
in Table 6.
In calculation of the HCN, its line profile should be corrected for the self
absorption. We modeled its original line profile using the same method for the
12CO $(2-1)$ (Section 4.2). However, since the abundance ratio between DCN and
HCN is much more uncertain than [C18O/12CO], we cannot use DCN to reliably
determine the optical depth of HCN $(3-2)$. Nevertheless, we expect the HCN
$(3-2)$ to have a low optical depth due to two reasons. Firstly, since the HCN
$(3-2)$ likely traces denser and hotter gas than the 12CO $(2-1)$, if the HCN
$(3-2)$ has a very large optical depth, it should have a comparable intensity
with the 12CO $(2-1)$. Nevertheless, even after the self-absorption
correction, the HCN $(3-2)$ is still much weaker than the 12CO $(2-1)$.
Second, with an apparent line width ($\Delta V=6.2$ km s-2) being close to
$\Delta V$ of the C18O and CH3OH lines (as shown in Table 5), the optical-
depth broadening should be insignificant for the HCN $(3-2)$. We therefore
directly calculate $N_{\rm T}({\rm HCN})$, and then estimate the optical depth
using Equation (5). As a result, we found $\tau=0.78$ at $T_{\rm rot}=20$ K
and 0.10 at 130 K. This result is consistent with our expectation. However, to
more accurately determine the HCN optical depth, one should consider to
observe some other isotopologues such as HC15N (Hatchell et al., 1998).
From the derived abundances, we can get a relative abundance ratio between DCN
and HCN which is [DCN/HCN]$\simeq 0.07$. The values derived at the two
temperature limits are similar to each other (Table 6). Compared to the cosmic
[D/H] ratio ($10^{-5}$, Linsky, 1998), the [DCN/HCN] in G8.68 implies a
deuteration for orders of magnitudes. The [DCN/HCN] in G8.68 is also much
higher that the values detected in hot molecular cores ($10^{-4}$ to
$10^{-3}$, Hatchell et al., 1998). However, it is much more comparable to the
abundance ratio of [N2D+/N2H+] detected in high-mass YSOs in the infrared dark
clouds (Chen et al., 2011). Overabundant DCN was previously detected in a
number low-mass YSOs (Roberts et al., 2002), while in high-mass star-forming
regions, the DCN is not frequently detected.
The overabundant DCN in G8.68 may originate from a high level of deuterium
fractionation in the previous cold pre-stellar phase. In highly deuterated gas
(abundant in H2D+, CH2D+, C2HD+ etc.), DCN can be produced via D-H
substitution of the HCN, or through more complex reaction pathways (Albertsson
et al., 2011, Reaction (17) to (21) therein). Finally the DCN molecules would
mostly reside on the grain mantles (Hatchell et al., 1998; Roberts et al.,
2002). During the star formation, the DCN can be released into the gas-phase
again. However, once the temperature slightly increases, the gas-phase DCN can
be easily destroyed via reactions such as
${\rm H+DCN\rightarrow HCN+D}$ (Charnley et al., 1992; Roberts et al., 2002,
Figure 5 therein). In this sense, to maintain the DCN overabundance in the
gas, two conditions may have to be satisfied. First, there should be a high-
level deuterium fractionation in the previous dark-cloud phase. Second, in
order to compensate the chemical destruction due to the stellar heating, a
rapid sublimation for the dust grains should be necessary. Again, the outflow
and shocks may have a potential contribution to this process. However, unlike
the C34S and CH3OH, the DCN emission has a compact and spherical morphology
which is not evidently coherent with the outflow. Therefore it is possible
that the DCN enhancement is less affected by the shocks and/or more sensitive
to the stellar heating. A higher sensitivity and spatial resolution may help
better reveal the DCN morphology and determine whether it is associated with
the outflow.
As another possibility, the DCN can also be synthesized in the recent gas
phase chemistry. Parise et al. (2009) show that the gas-phase reactions may
sufficiently account for the enhanced D-H ratio in the molecular gas in the
Orion Bar PDR, which has a [DCN/HCN] ratio of $10^{-2}$, comparable with that
in G8.68. However, the gas-phase enhancement may have to proceed in an
environment with stable lukewarm heating. This condition may hardly be
satisfied in regions with rapid evolution of the high-mass stars. Therefore
the grain mantle sublimation may still be the major process for the DCN
enhancement in G8.68. In the future study, one can compare other chemical
products from the grain sublimation and gas-phase chemistry in order to
determine the relative importance of these two processes.
The C34S abundance in G8.68 is much lower than the average C34S abundance in
the UC Hii regions (Olmi & Cesaroni, 1999). $f({\rm C^{18}O})$ is also much
smaller than the typical ISM value ($1.7\times 10^{-7}$, Frereking et al.,
1982). Compared to the ISM abundance, it has a depletion factor of $f_{\rm
D}({\rm C^{18}O})=5\pm 2$.
## 5 Summary
We have investigated the dust continuum and molecular line emissions towards
the high mass YSO G8.68-0.37. We have revealed a dense compact gas-and-dust
core in the SMA 1.3 mm continuum emission, and its more extended envelope in
the SCUBA 450 $\micron$ emission. At our angular resolution (spatial scale
$>0.05$ pc), there is no evident fragmentation structures. We find that an SED
with at least two temperature components is necessary to account for the dust
continuum emissions from mid-IR to submillimeter wavelengths. The best-fit
temperatures for the two components are $T_{\rm d}=20$ K and 120 K. The core
mass and luminosity are mainly contributed by the cold component ($T_{\rm
d}=20$ K).
Prominent infall signatures and outflow wings are detected in both 12CO
$(1-0)$ and $(2-1)$ lines. We separated the outflow and dense-core components
and measured their line parameters. The 12CO $(2-1)$ yields an infall velocity
of 0.45 km s-1. Assuming that the extended envelope is collapsing towards the
inner dense region, we can derive an infall rate of $7\times 10^{-4}$
$M_{\odot}$ year-1. It is possible that the 4.5 $\micron$ shock emission is
largely enhanced by a strong interaction between the infall and outflow
motions. In addition, we also suggest that the infall motion may be important
for suppressing the stellar emissions thereby protecting the DCN and other
fragile species.
We estimated a rotational temperature of 130 K from the CH3OH lines. We
derived the abundances of the molecular species from their spectra, and in
particular, we found a high abundance ratio of [DCN/HCN]$=0.07$. The over
abundant DCN may originate from a high-level of deuterium fractionation in the
previous pre-protostellar phase, as well as the recent grain mantle
sublimation and/or gas-phase chemistry. More details in this chemical process
are still to be further investigated.
## Acknowledgment
We are grateful to the SMA observers and the SMA data archive. We would thank
the anonymous reviewer for the detailed, thoughtful comments and suggestions
helping us to largely improve the presentation and interpretation. This work
is supported by the NSFC grants of No.10733033, 10873019, 10973003, and the
NKBRP grandts of 2009CB24901 and 2012CB821800.
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Table 1: General information of the single dish observations. Instrument | PMO | CSO | JCMT
---|---|---|---
Obs. date | Dec 2009 | May 2011 | Aug 2005
Beam size | $56^{\prime\prime}$ | $30^{\prime\prime}$ | $21^{\prime\prime}$
$\eta_{\rm mb}$ | 0.62 | 0.698 | 0.63
Pointing center | R.A.$=~{}18:06:22.87$ | R.A.$=~{}18:06:23.5$ | R.A.$=~{}18:06:23.46$
| Dec.$=-21:37:20.7$ | Dec.$=-21:37:10.7$ | Dec.$=-21:37:09.64$
Table 2: observational parameters for the molecular lines from the single dishes. Transition | Instrument | Atmosphere | Band width | $\Delta V_{\rm res}$ | rms noise
---|---|---|---|---|---
| | opacity | (MHz) | (km s-1) | per channel (K)
12CO $(2-1)$ | CSO | 0.167 | 500 | 0.079 | 0.2
13CO $(2-1)$ | CSO | 0.152 | 500 | 0.083 | 0.2
C18O $(2-1)$ | CSO | 0.149 | 500 | 0.083 | 0.2
12CO $(1-0)$ | PMO | 0.015 | 145 | 0.370 | 0.1
13CO $(1-0)$ | PMO | 0.015 | 43 | 0.115 | 0.1
C18O $(1-0)$ | PMO | 0.015 | 43 | 0.115 | 0.2
HCN $(3-2)$ | JCMT | 0.111 | 160 | 0.088 | 0.5
H13CO+ $(3-2)$ | JCMT | 0.067 | 160 | 0.090 | 0.3
Table 3: Observational parameters of the SMA.
Epoche | Frequency bands (GHz) | Bandpass | Flux | Phase & synthesized | beam size | rms noise
---|---|---|---|---|---|---
| LSB, USB | Calibrator | Calibrator | Calibrator | (arcsec) | per channel (K)a
2007 | (279.4,281.4), (289.4,291.4) | 3c273 | Neptune | 1733-130,1911-201 | $7.0\times 5.8$ | 0.110
2008 | (219.5,221.5), (229.5,231.5) | 3c454.3 | Neptune | 1733,1911 | $4.8\times 3.6$ | 0.017
2009 | (217.5,219.5), (227.5,229.5) | 3c273 | Uranus | 1733,1911 | $6.8\times 3.6$ | 0.150
$a.$ For the unit conversion, 1 K=0.367, 1.43 and 1.04 Jy beam-1 for the data
in 2007, 2008, and 2009 respectively.
Table 4: Physical parameters of the dust core.
Parameter | Value | Unit
---|---|---
$F(3.6\micron)$ | $13\pm 0.1$ | mJy
$F(4.5\micron)$ | $78\pm 2$ | …
$F(5.8\micron)$ | $68\pm 2$ | …
$F(8.0\micron)$ | $10\pm 1$ | …
$F(24\micron)$ | $657\pm 40$ | …
$F(70\micron)$ | $196\pm 15$ | Jy
$F(450\micron)$ | $144\pm 15$ | …
$F(850\micron)$ | $15\pm 4$ | …
$F(1.3{\rm mm})^{a}$ | $0.65\pm 0.03$ | …
$F(1.3{\rm mm})^{b}$ | $2.9$ | …
| $(450~{}\micron$ / 1.3 mm)c |
$\langle r\rangle$ | $0.23\pm 0.05$ / $0.08\pm 0.02$ | pc
$M_{\rm core}$ | $1.5\pm 0.2$ / $0.30\pm 0.01$ | $10^{3}~{}M_{\odot}$
$N({\rm H_{2}})$ | $1.2\pm 0.2$ / $0.95\pm 0.03$ | $10^{24}$ cm-2
$n({\rm H_{2}})$ | $0.8\pm 0.1$ / $~{}3.8\pm 0.3~{}$ | $10^{6}$ cm-3
Note. To measure the integrated flux density of the dust core, we use the 4
$\sigma$ emission level as the integration area (aperture for the photometry).
As an exception, we use the 4 $\sigma$ level of the 4.5 $\micron$ emission as
the area for all four IRAC bands, which roughly equals to the emission region
with green color in Figure 4. The nearby point sources carefully excluded from
this aperture.
$a.$ The flux density of the SMA continuum observation.
$b.$ The flux density extrapolated from the SED fitting (Figure 6).
$c.$ For the last 4 parameters, the first and second values are derived from
the 450 $\micron$ and 1.3 mm continuum data, respectively.
Table 5: Observed parameters of the molecular lines.
Molecule | Transition | Rest frequency | $E_{\rm u}$ | $V_{\rm LSR}$ | $T_{\rm b,peak}$ | $\Delta V_{\rm FWHM}$ | $\int T_{\rm b}{\rm d}V$ | $\tau^{a}$ | $\langle r\rangle^{b}$ | $\eta_{\rm bf}^{c}$
---|---|---|---|---|---|---|---|---|---|---
| | (GHz) | (K) | (km s-1) | (Kelvin) | (km s-1) | (K km s-1) | | (arcsec) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (9) | (8) | (10) | (11)
12CO(core) | $1-0$ | 115.27120 | 5.5 | 37.0 | 26.0(1.5) | 5.0(0.5) | 134(15) | 75 | – | –
12CO(outflow) | $1-0$ | … | … | 37.3 | 4.0(1.5) | 9.2(1.0) | 37(5) | 0.15 | – | –
12CO(core) | $2-1$ | 230.53800 | 17 | 37.0 | 32.0(0.5) | 5.5(0.5) | 240(30) | 68 | – | –
12CO(outflow) | $2-1$ | … | … | 37.0 | 5.0(0.5) | 9.0(0.5) | 45(6) | 0.15 | – | –
HCN(core) | $3-2$ | 265.88643 | 26 | 38.5 | 12.0(0.2) | 6.2(0.5) | 73(9) | 0.78/0.10 | 12 | 1
HCN(outflow) | $3-2$ | … | … | 39.5 | 0.8(0.2) | 15.0(1.0) | 12(5) | 0.053/0.007 | – | –
13CO | $1-0$ | 110.20135 | 5.3 | 35.0 | 12.4(0.5) | 6.8(0.1) | 85(10) | 0.85 | – | –
13CO | $2-1$ | 220.39968 | 16 | 35.0 | 8.5(0.5) | 7.5(0.5) | 60(15) | 0.70 | – | –
C18O | $1-0$ | 109.78217 | 5.3 | 37.0 | 4.5(0.5) | 5.6(0.4) | 23(4) | 0.17 | – | –
C18O(CSO) | $2-1$ | 219.56036 | 16 | 38.0 | 6.0(0.3) | 5.5(0.4) | 34(3) | 0.140 | – | –
C18O(SMA) | $2-1$ | 219.56036 | 16 | 38.0 | 5.2(0.02) | 4.8(0.3) | 30(3) | 0.423/0.065 | 3.4 | 1
CH3OH | $8_{0}-7_{1}$ | 220.07849 | 97 | 37.5 | 1.6(0.22) | 5.0(1.3) | 8(2) | 0.209/0.028 | 3.1 | 1
CH3OH | $8_{-1}-7_{0}$ | 229.75880 | 89 | 37.0 | 3.42(0.15) | 6.0(0.3) | 18.0(1.5) | 0.453/0.059 | 2.8 | 1
CH3OH | $3_{-2}-4_{-1}$ | 230.02706 | 39 | 38.5 | 0.9(0.03) | 6.0(0.4) | 5.8(0.5) | 0.036/0.004 | 3.2 | 1
CH3OH | $10_{2}-9_{3}$ | 231.28110 | 165 | 39.0 | 0.53(0.06) | 6.0(0.9) | 3.4(0.5) | 0.087/0.011 | 2.2 | 0.95
CH3OH | $9_{-1}-8_{0}$ | 278.30451 | 110 | 38.0 | 1.39(0.06) | 5.4(0.4) | 7.2(0.8) | 0.245/0.031 | 2.9 | 0.89
CH3OH | $2_{-2}-3_{-1}$ | 278.34226 | 33 | 39.0 | 0.24(0.02) | 5.5(0.4) | 1.5(0.2) | 0.045/0.006 | 3.2 | 1
CH3OH | $21_{-2}-20_{-3}$ | 278.48023 | 563 | 40.0 | 0.09(0.02) | 4.0(0.5) | 0.4(0.05) | 0.024/0.002 | 2.2 | 0.48
CH3OH | $15_{7,8}-16_{6,11}$ | 288.07677 | 523 | 37.5 | 0.24(0.02) | 5.5(0.3) | 1.2(0.2) | 0.102/0.012 | 2.3 | 0.52
CH3OH | $14_{4}-15_{3}$ | 278.59906 | 340 | 39.5 | 0.29(0.02) | 6.3(1.3) | 1.8(0.2) | 0.073/0.009 | 2.5 | 0.62
CH3OH | $11_{2}-10_{3}$ | 279.35191 | 191 | 39.5 | 0.28(0.02) | 6.5(1.9) | 2.3(0.3) | 0.081/0.009 | 2.8 | 0.78
CH3OH | $4_{3}-5_{2}$ | 288.70557 | 71 | 39.5 | 0.28(0.02) | 6.5(1.5) | 1.8(0.3) | 0.068/0.008 | 2.8 | 0.77
DCN | $3-2$ | 217.23854 | 81 | 39.0 | 2.1(0.4) | 3.5(0.2) | 7.5(1) | 0.132/0.018 | 3.1 | 0.92
DCN | $4-3$ | 289.64492 | 35 | 39.0 | 1.0(0.04) | 3.5(0.2) | 5.0(0.4) | 0.072/0.009 | 3.3 | 1
C34S | $6-5$ | 289.20907 | 38 | 39.0 | 1.4(0.02) | 4.5(0.5) | 6.5(0.7) | 0.123/0.016 | 3.3 | 1
H13CO+ | $3-2$ | 260.25534 | 25 | 38.0 | 4.7(0.3) | 4.4(0.3) | 20(2) | 0.260/0.033 | – | –
Note. The 12CO, 13CO lines, and C18O lines are from the PMO and CSO
observations (see Table 2). The C18O $(2-1)$ line from the SMA observation is
also presented. The HCN and H13CO+ lines are observed with the JCMT. For the
double-peaked lines, including HCN $(3-2)$, 12CO $(2-1)$ and $(1-0)$, and two
CH3OH lines ($E_{\rm u}=191$ K and $E_{\rm u}=523$ K), the parameters are
measured from the fitted spectra, while for the single-peak lines, we directly
measured the observed spectra.
$a.$ The optical depth at the line peak. For the transitions with two values,
the first and second one are the results for $T_{\rm rot}=20$ K and 114 K,
respectively (see Section 4.4). While for the CO $(1-0)$ and $(2-1)$, the
optical depths are calculated from comparing their isotopic lines (Section
4.3).
$b.$ The effective radius of the emission region, measured from the
deconvolved average radius of the 10% contour region (1/2 times the value). An
exception is the HCN $(3-2)$, for which we directly measured the 50 % contour.
The average uncertainty level is $\sim 2$ arcsec. For G8.68 at $D=4.5$ kpc, 1
arcsec$=0.02$ pc.
$c.$ The beam filling factor, calculated from the ratio between the area of
the deconvolved emission region ($\pi\langle r\rangle^{2}$) and the beam size.
Table 6: Collum density and abundance of the molecular species.
| $T_{\rm rot}=20$ Ka | $T_{\rm rot}=130$ K
---|---|---
Molecule | —————————————————– | —————————————————–
(X) | $N_{\rm T}({\rm X})$ (cm-2) | $f({\rm X})$ | $N_{\rm T}({\rm X})$ (cm-2) | $f({\rm X})$
C18O | $(2.2\pm 0.3)\times 10^{16}$ | $(2.3\pm 0.3)\times 10^{-8}$ | $(5.4\pm 0.6)\times 10^{16}$ | $(5.6\pm 0.7)\times 10^{-8}$
CH3OHb | $(1.8\pm 0.2)\times 10^{15}$ | $(2.0\pm 0.2)\times 10^{-9}$ | $(5.3\pm 0.6)\times 10^{15}$ | $(5.8\pm 0.6)\times 10^{-9}$
C34S | $(2.3\pm 0.3)\times 10^{13}$ | $(2.5\pm 0.4)\times 10^{-11}$ | $(2.6\pm 0.3)\times 10^{13}$ | $(2.9\pm 0.4)\times 10^{-11}$
H13CO+ | $(9.5\pm 1.0)\times 10^{12}$ | $(7.9\pm 0.7)\times 10^{-12}$ | $(1.7\pm 0.2)\times 10^{13}$ | $(1.4\pm 0.1)\times 10^{-11}$
HCN | $(5.6\pm 0.5)\times 10^{14}$ | $(4.6\pm 0.3)\times 10^{-10}$ | $(1.1\pm 0.1)\times 10^{15}$ | $(8.9\pm 0.8)\times 10^{-10}$
DCN | $(3.1\pm 0.1)\times 10^{13}$ | $(3.4\pm 0.2)\times 10^{-11}$ | $(5.6\pm 0.3)\times 10^{13}$ | $(6.2\pm 0.5)\times 10^{-11}$
${\rm[DCN/HCN]}^{c}$ | – | $0.07\pm 0.01$ | – | $0.07\pm 0.01$
$a.$ A lower limit as suggested by the dust continuum SED.
$b.$ At $T_{\rm rot}=20$ K, $N_{\rm T}({\rm CH_{3}OH})$ is derived from
$3_{-2}-4_{-1}$ line.
$c.$ Abundance ratio between DCN and HCN.
Figure 1: The beam size and pointing center of each instrument. The cross
symbol marks the center of the 1.3 mm dust core (Figure 4) which is coincident
with the CSO pointing center. The white ellipse is the synthesized beam of the
SMA in the 2008 observation, and the gray filled circle is the primary beam.
The JCMT beam size is for the frequency of 289 GHz.
Figure 2: (a) Grid spectra of 12CO $(2-1)$ observed from the CSO. The red
cross labels the pointing center of each spectrum. The gray contours are the
SCUBA 450 $\micron$ emission (specified in Figure 4). The green dashed line
represents the beam size. (b) Grid spectra of HCN $(3-2)$ observed from the
JCMT. The red cross labels the pointing center of the each spectrum. The
intensity map (gray scales) is made from interpolating the line intensity at
each point. The integration for the spectra is from 25 to 50 km s-1. The gray-
scale levels are from 30 % to 90 % of the maximum intensity (46.8 K km s-1).
The thick contour is the 50 % level. The blue dashed circle is the beam size.
Figure 3: Molecular lines observed from the PMO, CSO, and JCMT, which are
shown in left, middle, and right panels, respectively. The observing centers
and the beam size of each telescope are shown in Figure 1.
Figure 4: Continuum emissions detected towards G8.68 from infrared to
millimeter wavelengths. The image is centered at the emission peak of the 1.3
mm continuum, the coordinates of which are RA.(J2000)=18h06m23.5s and
Decl.(J2000)=$-21^{\circ}37^{\prime}10.7^{\prime\prime}$. The white contours
are the 1.3 mm emission observed from the SMA. The contour levels are -4, 4,
14, 24… 104 $\sigma$ (0.003 Jy beam-1). The -4 $\sigma$ contour is due to the
insufficient (u,v) coverage and is plotted in dotted line. The square denotes
the strongest CH3OH maser (Walsh et al., 1998). The dashed contours are the
JCMT/SCUBA 450 $\micron$ emission. The levels are 4, 8, 12… 36 $\sigma$ (1.2
Jy beam-1). The IRAC 3.6 (blue), 4.5 (green) and 8.0 (red) $\micron$ images
are shown together in RGB colors (also seen in Figure 1 and 2 in L11). The red
cross labels the SMA phase center. The synthesized beam of the SMA (white
ellipse) and SCUBA (blue circle) beam are plotted in the right corner.
Figure 5: Molecular lines and integrated images observed from the SMA. For
each transition, the contours are 10, 20… 90 percent of the peak intensity.
The integration range is $(37,~{}43)$ km s-1 for all the transitions except
the two blended CH3OH lines. For these two lines the integration range is
$(37,40)$ km s-1 as to eliminate the contamination. The dashed line in the
C34S labels the orientation of the outflow in L11. The DCN $(3-2)$ line is
shifted above the zero level for 2 K. The negative contours due to the missing
flux are omitted to more clearly show the emission features. The gray-scale
image in each panel is the SMA 1.3 mm continuum.
Figure 6: The spectra energy distribution of the dust core. The black squares
with error bars are the data points (the SMA 1.3 mm flux density is marked
with the diamond). The black line is the fitted SED curve. The red dashed
lines are the SEDs of the two temperature components, with $T_{\rm d}=20$ K
and 120 K, respectively.
Figure 7: The rotation diagram of the CH3OH lines. The black squares with
error bars are the data points. The red line is the least-square fit.
Figure 8: The two-component fit to the 12CO $(2-1)$ spectrum towards the
center. The meaning of each line type is labeled in the legend.
|
arxiv-papers
| 2012-03-13T13:42:15 |
2024-09-04T02:49:28.589833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhiyuan Ren, Yuefang Wu, Ming Zhu, Tie Liu, Ruisheng Peng, Shengli,\n Qin, and Lixin Li",
"submitter": "Zhiyuan Ren",
"url": "https://arxiv.org/abs/1203.2805"
}
|
1203.2853
|
# Deep phase modulation interferometry
Gerhard Heinzel gerhard.heinzel@aei.mpg.de Felipe Guzmán Cervantes
felipe.guzman@aei.mpg.de felipe.guzman@nasa.gov Antonio F. García Marín
Joachim Kullmann Karsten Danzmann Albert-Einstein-Institut Hannover (Max-
Planck-Institut für Gravitationsphysik, and Leibniz Universität Hannover),
Callinstraße 38, 30167 Hannover, Germany †NASA Goddard Space Flight Center,
8800 Greenbelt Road, Greenbelt, MD 20771, USA Wang Feng Purple Mountain
Observatory, CAS, 2 West Beijing Road, Nanjing 210008, China
###### Abstract
We have developed a method to equip homodyne interferometers with the
capability to operate with constant high sensitivity over many fringes for
continuous real-time tracking. The method can be considered as an extension of
the “$J_{1}\dots J_{4}$” methods, and its enhancement to deliver very
sensitive angular measurements through Differential Wavefront Sensing is
straightforward. Beam generation requires a sinusoidal phase modulation of
several radians in one interferometer arm. On a stable optical bench, we have
demonstrated a long-term sensitivity over thousands of seconds of 0.1
mrad$/\sqrt{\rm Hz}$ that correspond to 20 pm$/\sqrt{\rm Hz}$ in length, and
10 nrad$/\sqrt{\rm Hz}$ in angle at millihertz frequencies.
###### pacs:
(120.0120) Instrumentation, measurement, and metrology, (120.3180)
Interferometry, (120.4640) Optical instruments, (120.5050) Phase measurement,
(120.5060) Phase modulation
## I Introduction
Optical interferometers with sub-wavelength resolution are useful in many
optical metrology applications, such as, for example, length measurements,
gravitational wave detection, wavefront sensing, and surface profiling, among
others. Our technique was developed in the context of continuously measuring
the position and orientation of a free-floating test mass for space-based
gravitational wave detection anza , although the method is useful for other
applications as well. Other techniques for the optical readout of free-
floating test masses at millihertz frequencies are currently under
investigation, such as a polarizing heterodyne interferometer reaching a
sensitivity of about $300\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ oro_berlin , a
compact homodyne interferometer with a sensitivity of
$100\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ oro_bmh , and a robust implementation of
an optical lever with a readout noise level of
$100\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ oro_napoli . Another method to do this
is heterodyne interferometry as developed for LISA Pathfinder ltp with a
sensitivity of better than $5\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ ltpsubtraction
. The method we present here achieves an optical pathlength measurement
sensitivity of the order of $20\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$, and with an
angular resolution better than $10\,\mathrm{nrad}/\sqrt{\mathrm{Hz}}$, both
above 3 mHz. The conversion from real test mass motion to optical pathlength
is given by the interferometer topology, and is in our case about a factor of
2, which yields a test mass motion resolution of approximately
$10\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$. Those interferometers with the highest
accuracy, namely Fabry-Perot interferometers on resonance or recycled
Michelson interferometers on a dark fringe geo , have a dynamic range of a
small fraction of one fringe only. High resolution and wide dynamic range can
be simultaneously achieved by, e.g., active feedback or heterodyning, each of
which has disadvantages. Active feedback transfers the inherent non-linearity
of the feedback actuator to the output signal or requires another stabilized
laser and a measurement of the high-frequency beat note. Heterodyning, on the
other hand, requires a complex setup to generate the two coherent beams with a
constant frequency difference, typically involving two acousto-optic
modulators (AOMs) with associated frequency generation and RF power
amplification. Other methods to overcome these limitations involve variations
of sinusoidal phase shifting interferometry sasaki_1986 ; sasaki_1987 ;
deGroot_2008 ; deGroot_2009 ; Falaggis_2009 , reporting accuracies of the
order of 1 nm. These methods are typically used in “single-shot” mode for
static applications such as surface profiling, whereas our method is designed
for continuous real-time, long-term tracking of a moving target with low noise
at millihertz frequencies. In particular, the so-called “$J_{1}\dots J_{4}$”
method su89 ; jin91 ; su93 , involves a sinusoidal phase modulation at a fixed
frequency $f_{\rm mod}$ with modulation depths $m\approx 1\dots 5$ in one arm
of the interferometer. The spectrum of the resulting photocurrent has
components at integer multiples of $f_{\rm mod}$, with amplitudes that can be
written in terms of the Bessel functions $J_{n}(m)$ (hence the name) and the
phase difference $\varphi$ due to the optical pathlength difference. The
methods then proceed to use analytical formulae to solve for the unknowns $m$
and $\varphi$, after obtaining the harmonic amplitudes from a spectrum
analyzer or a Fast Fourier Transform (FFT) of the digitized time series. The
accuracies reported are of order 10…100 mrad (1.7 …17 nm) for a laser
wavelength of 1064 nm. We generalize this approach by using a higher
modulation index $m$ (up to 10 or 20) and making use of all harmonics up to an
order $N\approx m$. These are more observations than the four unknowns ($m$,
$\varphi$, modulation phase and a common factor), making an analytical
solution impossible. Instead we use a numerical least-squares solution which
allows consistency checks and improves the signal-to-noise ratio. For our
typical applications we keep $m$ near constant at an optimal value and take
$\varphi$ as useful output, achieving an accuracy of better than 0.12
mrad$/\sqrt{\rm Hz}$ that correspond to 20 pm$/\sqrt{\rm Hz}$ in length, and
10 nrad$/\sqrt{\rm Hz}$ in angle at millihertz frequencies. As compared to
heterodyne interferometers, more complex data processing is necessary to
recover the optical pathlength from the measured photocurrent. However, with
the availability of inexpensive processing power, this computational
complexity is often preferable to additional optics and electronics hardware
needed for the optical heterodyning.
## II Theory
The signal $V_{\mathrm{PD}}(t)$ of a photodetector at the output of a phase-
modulated homodyne interferometer can be expressed as
$V_{\mathrm{PD}}(t)=A\,\left[\,1+C\,\mathrm{cos}\left(\varphi+m\,\mathrm{cos}\left(\omega_{\mathrm{m}}\,t+\psi\right)\,\right)\,\right],$
(1)
where $\varphi$ is the interferometer phase, $m$ is the modulation depth,
$\omega_{\mathrm{m}}=2\pi\,f_{\mathrm{m}}$ is the modulation frequency, $\psi$
is the modulation phase, $C\leq 1$ is the contrast, and $A$ combines nominally
constant factors such as light powers and photodiode efficiencies. The
interferometer output is periodic with $f_{\mathrm{m}}$ and its signal
waveform characteristically depends on the interferometer phase $\varphi$.
Figure 1 illustrates typical waveforms obtained for various states of
$\varphi$.
Figure 1: Waveform of the obtained interferogram for different operating
points of the interferometer phase $\varphi$ with a modulation depth
$m=6\,\mathrm{rad}$.
The expression of Equation 1 can be expanded into its harmonic components as:
$V_{\rm PD}(t)=V_{\rm
DC}(\varphi)+\sum\limits_{{n}=1}^{\infty}a_{n}(m,\varphi)\cos({n}(\omega_{\rm
m}t+\psi))$ (2)
with
$\displaystyle a_{n}(m,\varphi)$ $\displaystyle=$ $\displaystyle
k\,J_{n}(m)\,\cos\left(\varphi+n\frac{\pi}{2}\right),\,\,\mathrm{and}$ (3)
$\displaystyle V_{\rm DC}(\varphi)$ $\displaystyle=$ $\displaystyle
A\left(1+C\,J_{0}(m)\cos\varphi\right),$ (4)
where $k=2CA$, and $J_{\mathrm{n}}(m)$ are the Bessel functions. Figure 2
shows the dependence of the harmonic amplitudes $a_{n}(m,\varphi)$ in terms of
$\varphi$. Our technique is centered around these harmonic amplitudes
$a_{n}(m,\varphi)$ which on the one hand can be directly measured by numerical
Fourier analysis of the photocurrent, and on the other hand have the above
analytical relationships to the unknowns $\varphi$, $m$, $\psi$, $k$.
Figure 2: Dependence of the harmonics amplitudes $a_{n}(m,\varphi)$ with
respect to the interferometer phase $\varphi$ with a modulation depth
$m=6\,\mathrm{rad}$.
The technique we present here uses higher modulation depths $m\geq 6$ to set
up an overdimensioned system of equations that can be numerically solved for
the four sought parameters $\varphi$, $m$, $\psi$, and the common factor $k$
by a least-squares fit algorithm. The information of the harmonic amplitude
$a_{0}(m,\varphi)$, corresponding to the DC component $V_{\rm DC}(\varphi)$ is
not used by the fit algorithm, since it usually contains a higher noise level
due to large variations in environmental and equipment conditions such as room
illumination and electronic noise, among others. However, it is useful for
computation of the interferometer visibility and alignment signals.
## III Data processing
The signal $V_{\rm PD}(t)$ measured at the photodetector is digitized after
appropriate analog processing and anti-alias filtering. The sampling rate
$f_{\rm samp}$ is arranged to be coherent to the modulation frequency $f_{\rm
mod}$. The time series is split in segments of length $N_{\mathrm{FFT}}$
samples that are processed by a Discrete Fourier Transform in order to compute
$N=N_{\mathrm{FFT}}/2$ complex amplitudes $\tilde{\alpha}_{n}(m,\varphi)$. A
non-linear fit algorithm is applied to match the measured
$\tilde{\alpha}_{n}(m,\varphi)$ to the complex amplitudes $c_{n}$ computed
from the model
$\alpha_{n}(m,\varphi)=a_{n}(m,\varphi)\,e^{{\rm i}n\psi}.$ (5)
There is a total of $2N$ equations that can be set up in two uncorrelated
system of equations:
$\displaystyle n\psi$ $\displaystyle=$
$\displaystyle\arctan\left(\frac{\Im\\{\alpha_{n}(m,\varphi)\\}}{\Re\\{\alpha_{n}(m,\varphi)\\}}\right),n=1,2,3\dots
N,\,\,\,\,\,\,\,\,$ (6) $\displaystyle a_{n}(m,\varphi)$ $\displaystyle=$
$\displaystyle\alpha_{n}(m,\varphi)\,e^{{\rm-i}n\psi},\,\,n=1,\,2,\,3\dots N,$
(7)
where $\alpha_{n}(m,\varphi)\,e^{{\rm-i}n\psi}$ is a real number. For the
measured $\tilde{\alpha}_{n}(m,\varphi)\,e^{{\rm-i}n\psi}$, this is not
exactly the case due to noise and phase distortions introduced by the analog
electronics. In order to solve the system of equations, a Levenberg-Marquardt
fit algorithm marquardt ; numrec is applied to minimize the least-squares
expression
$\chi^{2}=\sum\limits_{n=1}^{N}{\left(\alpha_{n}(m,\varphi)-\tilde{\alpha}_{n}(m,\varphi)\right)}^{2},$
(8)
where $\chi^{2}$ is a four dimensional function of $m$, $\varphi$, $\psi$, and
$k$. In practice, these parameters barely vary between consecutive segments of
length $N_{\mathrm{FFT}}$, giving good starting values for a rapid convergence
of the fit. Only in the case this is not accomplished such as upon
initialization or after large disturbances, a modified version of the more
robust Nelder-Mead Simplex algorithm nelder is applied as initial step. In
order to find best values of the modulation index $m$ and the number of bins
$N$ for optimum performance, we conducted a numerical analysis of the $4\times
4$ Hessian matrix of $\chi^{2}$ that is given by
$H=(H_{ij})=\left(\frac{\partial^{2}\chi^{2}}{\partial\Omega_{i}\partial\Omega_{j}}\right),$
(9)
where $\Omega=\\{m,\varphi,\psi,k\\}$ are the four parameters. The inverse of
the Hessian matrix $H^{-1}=(\eta_{ij}$) yields information about the parameter
estimates on the variances $\sigma^{2}$ and correlation coefficients
$\rho_{ij}$:
$\displaystyle\sigma^{2}_{\Omega_{i}}$ $\displaystyle\propto$
$\displaystyle\eta_{ii},$ (10) $\displaystyle\rho_{ij}$ $\displaystyle=$
$\displaystyle\frac{\eta_{ij}}{\sqrt{\eta_{ii}}\sqrt{\eta_{jj}}}.$ (11)
An excursion of $\varphi$ over the range $[0,2\pi]$ –which corresponds to one
interferometer fringe– was conducted in 64 steps by fixed $N$ and $m$ in order
to compute the best, worst and average values of the standard deviation
$\sigma_{\Omega_{i}}(N,m,\varphi)$, which are shown in Figure 3. Assuming
worst case values of the variances
$\widehat{\sigma}^{2}_{\Omega_{i}}(N,m)=\max_{\varphi\in[0,2\pi]}\sigma^{2}_{\Omega_{i}}(N,m,\varphi),$
(12)
we run a similar analysis varying $N$ and $m$ to evaluate for best resolution
of any value of $\varphi$, which is our main measurement and often not
entirely under control. The results are shown in Figure 4.
Figure 3: Ideal resolution in $\varphi$ as function of the modulation index
$m$ for $N=10$, for the best and worst $\varphi$ as well as the average for
all $\varphi\in[0,2\pi]$. Figure 4: Ideal resolution in $\varphi$ as function
of the modulation index $m$ for different orders $N$, for the worst value of
$\varphi$ at each point of each curve.
This analysis revealed useful parameter estimates for $3\leq m\leq N$, and
possible best values of $m$ for minimum $\sigma_{\varphi}$ in the cases of
$m=6,N\geq 8$ and $m=9,N\geq 10$, suggesting best resolutions of $\varphi$.
These results are only rough guidelines, since real instrument noise has not
been yet considered. The dominant noise sources have, however, been
investigated experimentally, as discussed in Section V below. In addition,
software simulations of the fit routine were run with synthetic data as input.
Hardware characteristics of the data acquisition system (DAQ) such as
digitization effects and frequency response of the anti-aliasing filter were
considered in the generation of mock-data, using Equation 1 as nominal noise-
free model. We introduced two independent mock-data sets into two virtual DAQ
channels, by linearly increasing $m$ with $N=10$ bins, and recorded the fit
output (phase $\varphi$). We then computed their phase difference and
extracted the nominal offset in oder to obtain the dependence of the phase
fluctuations (noise) with $m$, which is shown in Figure 5.
Figure 5: Dependence of the measured phase noise with $m$.
A minimum can be observed around $m=9.5-10$, which is consistent with the
analysis presented in Figure 4. Hence, a DAQ test system for real optical
length measurements was set up with $N=10$ bins, and a modulation index
$m\approx 9.7$.
## IV Experimental setup
We have applied this technique to a very stable interferometer, namely the
engineering model of the LISA Pathfinder (LPF) optical bench ltp , which
consists of a 20 cm$\times$20 cm Zerodur${}^{\scriptsize{\ooalign{\hfil\raise
0.0pt\hbox{\tiny R}\hfil\crcr\text{$\mathchar 525\relax$}}}}$ baseplate with
optical components fixed by hydroxide-catalysis bonding bonding . This optical
bench has been extensively characterized as part of ground testing campaigns
for the optical metrology of the LISA Pathfinder mission ltptests , and its
optical pathlength stability has been measured to be better than 5
pm$/\sqrt{\rm Hz}$ above 1 mHz. A non-planar ring oscillator (NPRO) Nd:YAG
laser producing 300 mW at 1064 nm was used as light source. For the
experimental test, we chose a two-beam Mach-Zehnder interferometer, using
self-assembled fiber-coupled phase modulators consisting of single-mode fiber
optics coiled around ring piezo-electric transducers (RPZT) in order to reach
high modulation depths (up to 10 or 20). Figure 6 shows a schematic overview
of the setup.
Figure 6: Schematic overview of the experiment.
The laser beam is split into two equal parts at the first beamsplitter BS1. A
RPZT driven by a sinusoidal voltage of approximately 4.5
$\mathrm{V}_{\mathrm{pp}}$ at $f_{\rm mod}=280\,$Hz, produces a phase
modulation of modulation depth $m\approx 9.7$ in one of the two beams. This
portion of the optical setup denoted as modulation bench, contains the first
beamsplitter BS1, phase modulator, and corresponding fiber coupling devices
which are all mounted on a standard metal optical breadboard. A single-mode
fiber feed-through is used to bring the main laser beam into a vacuum chamber
where both, the modulation bench and the optical bench reside. The LISA
Pathfinder optical bench is a set of four non-polarizing Mach-Zehnder
interferometers, three of which have been used in these experiments. The first
one –denoted M– measures distance fluctuations between two mirrors mounted on
3-axes piezo-electric actuators ltpsubtraction . A second one –denoted R–
serves as phase reference to cancel common-mode pathlength fluctuations that
arise at the modulation bench, such as in metal mounts, phase modulator, and
fiber optics. The third interferometer –denoted F– has an intentionally large
optical pathlength difference of approximately 38 cm, and is used to measure
laser frequency fluctuations. If we denote by $s_{M}$ and $s_{R}$ the optical
pathlengths of the measurement and reference interferometer, respectively, the
phases emerging from the fit algorithm are given by
$\displaystyle\varphi_{M}$
$\displaystyle=\frac{2\,\pi}{\lambda}\left\\{(s_{1}+s_{M})-(s_{2}+s_{3})\right\\}=\frac{2\,\pi}{\lambda}\left\\{(s_{M}-s_{3})+\Delta\right\\},\,\,\,\,\,\,\,\,$
(13) $\displaystyle\varphi_{R}$
$\displaystyle=\frac{2\,\pi}{\lambda}\left\\{(s_{1}+s_{R})-(s_{2}+s_{4})\right\\}=\frac{2\,\pi}{\lambda}\left\\{(s_{R}-s_{4})+\Delta\right\\},\,\,\,\,\,\,\,\,$
(14)
where $\lambda=1064$ nm is the laser wavelength, $s_{\mathrm{x}}$ are the
optical paths outlined in Figure 6, and $\Delta=s_{1}-s_{2}$ represents the
common-mode pathlength difference between the two beams that includes
everything starting from the first beamsplitter BS1, the modulator, fiber
optics up to the beamsplitters on the stable optical bench. Typically, the
fluctuations of $\Delta$ are several $\mu$m on 10…1000 second time scales and
thus much larger than what we want to measure. However, the optical
pathlengths $s_{R}$, $s_{3}$ and $s_{4}$ are confined to the stable optical
bench and have only negligible fluctuations. By measuring both $\varphi_{M}$
and $\varphi_{R}$ and computing their difference
$\varphi=\varphi_{M}-\varphi_{R}=\frac{2\,\pi}{\lambda}\left\\{s_{M}-(s_{R}+s_{3}-s_{4})\right\\}$
(15)
it is possible to cancel the common-mode fluctuations $\Delta$ and to obtain a
measurement that is dominated by the fluctuations of $s_{M}$ as desired. All
photodetectors are indium gallium arsenide (InGaAs) quadrant diodes with 5 mm
diameter. The photocurrent of each quadrant is converted to a voltage with a
low-noise transimpedance amplifier, filtered with a 9-pole 8 kHz Tschebyscheff
anti-aliasing filter and digitized at a rate $f_{\mathrm{samp}}=20\,$kHz by a
commercial 16-channel, 16-bit analog-to-digital converter (ADC) card installed
in a standard PC running Linux. The time series are split in segments of
$N_{\mathrm{FFT}}=1000$ samples and transformed by a FFT algorithm fftw . The
$N=10$ complex amplitudes of bins 1…10 of $f_{\mathrm{mod}}$ at frequencies
$280\dots 2800$ Hz are then fitted. This configuration allows us to reach a
real-time phase measurement rate
$f_{\varphi}=f_{\mathrm{samp}}/N_{\mathrm{FFT}}=20\,\mathrm{Hz}$.
## V Noise investigations
During test and debugging experiments, two main noise sources were identified
to limit the interferometer sensitivity with this technique, which are laser
frequency noise, and the frequency response (transfer function) of the DAQ
analog electronics, including photodiode transimpedance amplifiers and anti-
aliasing filters. In the following we explain the coupling mechanism of these
noise sources, and the mitigation strategies we implemented to counteract
them.
### V.1 Laser frequency noise
Laser frequency noise translates into phase readout noise in any
interferometer, whose pathlength difference $\Delta s$ between the two
interfering beams is not exactly zero. In the case of the LPF optical bench,
this pathlength mismatch has been determined to be approximately
$10\,\mathrm{mm}$ ltp . The free-running frequency noise $\delta\nu$ of an
unstabilized Nd:YAG NPRO laser at $10\,\mathrm{mHz}$ has been measured to be
of the order of $2\times 10^{6}\,\mathrm{Hz}/\sqrt{\mathrm{Hz}}$ ghh:2004 .
The conversion factor from laser frequency fluctuations $\delta\nu$ into phase
fluctuations $\delta\varphi$ is given by the difference in time of travel
between the two beams $\Delta s/c$, such that an estimate of the noise level
can be calculated as
$\delta\varphi=2\pi\frac{\Delta s}{c}\delta\nu\approx
2\pi\frac{\,10^{-2}\mathrm{m}}{3\times 10^{8}\,\mathrm{m/s}}2\times
10^{6}\,\mathrm{Hz}=0.4\,\mathrm{mrad}/\sqrt{\mathrm{Hz}},$ (16)
which limits the interferometer optical pathlength resolution $\delta s$ to
$\delta
s=\frac{\lambda}{2\pi}\,\delta\varphi=\frac{1064\,\mathrm{nm}}{2\pi}\,0.4\,\mathrm{mrad}/\sqrt{\mathrm{Hz}}=68\,\mathrm{pm}/\sqrt{\mathrm{Hz}}.$
(17)
We implemented two mitigations strategies to correct for this effect and
improve the length resolution. Both methods worked similarly well and allowed
suppression of this error below the other noise terms. The first one is based
on the active laser frequency stabilization, for which we have used a
commercial iodine-stabilized Nd:YAG laser. The second method uses the third
interferometer F (mentioned above) to independently measure a phase
proportional to the amplified laser frequency fluctuations, and applies a
noise subtraction technique ltpsubtraction ; felipe_phd ; noisesub that
properly estimates the coupling factor and removes the contribution from the
final data stream. The phase of this interferometer was read out with the same
deep modulation method as the main channels, and is dominated by laser
frequency fluctuations due to its large pathlength difference ($\approx
38\,$cm). A third method can also be easily implemented as an active
stabilization loop by feeding back to the laser over a digital-to-analog
converter the output of a digital controller that uses the difference phase
extracted from interferometers F and R as error signal.
### V.2 Frequency response of data acquisition system
The dominant error was identified to be the frequency response of the analog
electronics of the data acquisition system, in particular the contribution of
the photodiode transimpedance amplifiers and the anti-aliasing filters. The
transfer function (TF) of this analog portion of the DAQ shows small ripples
in its magnitude of the order of 0.9 dB. The desired parameter $\varphi$ is
essentially determined by running a fit onto the relative amplitudes of the 10
measured harmonic components $\tilde{\alpha}_{n}(m,\varphi)$. The ripples are,
however, large enough to alter the ratio between the harmonic amplitudes, such
that the fit algorithm is disturbed, resulting in a high noise level. We
removed this error by separately measuring the transfer function of each
channel of the analog front end, fitting it to a model, and correcting
accordingly the measured complex amplitudes $\tilde{\alpha}_{n}(m,\varphi)$
before entering the fit routine. Thus, we obtained the corresponding TF
complex values
$\beta_{\mathrm{n}}=b_{\mathrm{n}}\,\mathrm{e}^{i\theta_{\mathrm{n}}}$ for the
10 frequency bins of interest $280-2800$ Hz. Hence, the measured complex
amplitudes $\tilde{\alpha}_{n}(m,\varphi)$ were corrected as
$\tilde{\alpha}^{\prime}_{n}(m,\varphi)=\frac{\tilde{\alpha}_{n}(m,\varphi)}{\beta_{n}},$
(18)
By using complex numbers, this correction also accounts for the TF phase
shift, and improves the estimation capability of the modulation phase $\psi$.
## VI Optical length and attitude measurements
The experimental setup of Figure 6 was used to conduct long-term
interferometric length measurements on the LPF optical bench. Figure 7 shows
the results obtained in form of linear spectral densities.
Figure 7: Sensitivity of real optical pathlength measurements. Dashed curve
with crosses: initial sensitivity prior to noise correction techniques. Dashed
curve: sensitivity upon correction of DAQ frequency response. Solid curve:
sensitivity reach after application of noise mitigation strategies -laser
frequency noise and DAQ frequency response-.
The dashed curve with crosses is the sensitivity obtained initially with this
method, without applying any of the noise mitigation strategies explained in
Section V. The dashed curve is the sensitivity achieved after applying the
complex value correction of the DAQ frequency response to the measured
harmonic amplitudes $\tilde{\alpha}_{n}(m,\varphi)$ (as given by Equation 18),
resulting in a sensitivity improvement of about one order of magnitude. The
solid curve is the measurement length sensitivity reached upon subtraction of
laser frequency noise, which increases the length resolution in an additional
factor of approximately 3.5 at 10 mHz. The measured optical pathlength
sensitivity of this technique is of the order of
$20\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ above 3 mHz and approximately a factor of
2 above the performance required to the LPF interferometry, which has been
plotted for comparison purposes. As mentioned above, all photodetectors at the
interferometer outputs on the LPF optical bench are quadrant cell diodes. The
phases extracted from each individual quadrant cell are processed by a
differential wavefront sensing (DWS) algorithm morrison-1 ; morrison-2 , in
order to measure the interferometer alignment with high angular resolution.
The results of this measurement are shown on Figure 8 as a linear spectral
density.
Figure 8: Angular resolution obtained by applying a DWS algorithm to the
phases extracted from individual cells of a quadrant photodetector.
As it can be read from the plot, this technique reaches an angular sensitivity
better than $10\,\mathrm{nrad}/\sqrt{\mathrm{Hz}}$ above 3 mHz, meeting with
sufficient margin the requirements set to the LPF interferometry that have
been also included in the graph as a comparison.
## VII Comparison with other techniques
The only method known to the authors that allows length and angular
measurements at arbitrary operating points with low noise at millihertz
frequencies is heterodyne interferometry as described in Ref. ltp . The deep
phase modulation method presented here needs, in comparison, much simpler beam
generation hardware, namely one low-frequency phase modulator like a piezo-
electric transducer, as opposed to two AOMs with RF driving electronics. On
the other hand, the data processing for phase extraction is more complicated,
which, however, becomes a smaller disadvantage with cheap processing power.
The heterodyne method typically requires additional stabilization loops wand ;
ghh-ltpnoise to reach noise levels at $\mathrm{pm}/\sqrt{\mathrm{Hz}}$, e.g.
for the laser power and certain common-mode pathlengths (see Ref. ltp ). The
experiments described above in Section IV show that these stabilizations are
not required for the deep phase modulation technique.
## VIII Conclusions
We have presented an interferometry technique for high sensitivity length and
angular optical measurements. This technique is based on the deep phase
modulation (over several radians) of one interferometer arm and can be
considered as an extension of the well-known “$J_{1}\dots J_{4}$” method su89
; jin91 ; su93 . The harmonic amplitudes are used to numerically solve an
overdimensioned system of equations to extract the interferometer phase and
other useful interferometer variables. This technique has been applied to
experiments conducted on a very stable interferometer (the engineering model
of the LISA Pathfinder optical bench), achieving an optical pathlength readout
sensitivity of the order of $20\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ (which
translates to $10\,\mathrm{pm}/\sqrt{\mathrm{Hz}}$ for free-floating test mass
displacement), and alignment measurements with an angular resolution better
than $10\,\mathrm{nrad}/\sqrt{\mathrm{Hz}}$ in the millihertz frequency band.
This performance is comparable to the best heterodyne interferometers, and,
e.g., only a factor of 2 above the LISA Pathfinder pathlength measurement
requirements. Two main noise sources were identified, namely laser frequency
fluctuations and the frequency response of the analog portion of the data
acquisition system, which both were completely mitigated by appropriate data
processing methods, hence improving the performance of this technique by over
a factor 35. Unlike other interferometry techniques, no additional control
loops, for instance, to actively stabilize the optical pathlength difference
or laser power fluctuations, have been implemented or are required to reach
the current sensitivity. Nonetheless, this could easily be done, in order to
further improve the performance of this method.
## Acknowledgments
We gratefully acknowledge support by the Deutsches Zentrum für Luft- und
Raumfahrt (DLR) (references 50 OQ 0501 and 50 OQ 0601).
## References
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* (24) F. Guzmán Cervantes, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA, is preparing a manuscript on “Estimation and subtraction of noise contributions in the LISA Pathfinder optical metrology system.”
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* (27) V. Wand, J. Bogenstahl, C. Braxmaier, K. Danzmann, A. García, F. Guzmán, G. Heinzel, J. Hough, O. Jennrich, C. Killow, D. Robertson, Z. Sodnik, F. Steier, and H. Ward, “Noise sources in the LTP heterodyne interferometer,” Class. Quantum Grav. 23, S159–S167 (2006).
* (28) G. Heinzel, V. Wand, A. García, F. Guzmán, F. Steier, C. Killow, D. Robertson, and H. Ward, “Investigation of noise sources in the LTP interferometer,” Technical note (2008), http://edoc.mpg.de/display.epl?mode=doc&id=395069&col=6&grp=1154.
|
arxiv-papers
| 2012-03-13T16:32:43 |
2024-09-04T02:49:28.600384
|
{
"license": "Public Domain",
"authors": "Gerhard Heinzel, Felipe Guzm\\'an Cervantes, Antonio F. Garc\\'ia\n Mar\\'in, Joachim Kullmann, Wang Feng, Karsten Danzmann",
"submitter": "Felipe Guzman Cervantes",
"url": "https://arxiv.org/abs/1203.2853"
}
|
1203.2862
|
# Characterization of photoreceivers for LISA
Felipe Guzmán Cervantes felipe.guzman@nasa.gov felipe.guzman@aei.mpg.de
Jeffrey Livas Robert Silverberg Ernest Buchanan Robin Stebbins NASA
Goddard Space Flight Center, Code 663, 8800 Greenbelt Road, Greenbelt, MD
20771, USA
###### Abstract
LISA will use quadrant photoreceivers as front-end devices for the phasemeter
measuring the motion of drag-free test masses in both angular orientation and
separation. We have set up a laboratory testbed for the characterization of
photoreceivers. Some of the limiting noise sources have been identified and
their contribution has been either measured or derived from the measured data.
We have built a photoreceiver with a 0.5 mm diameter quadrant photodiode with
an equivalent input current noise of better than
$1.8\,\mathrm{pA}/\sqrt{\mathrm{Hz}}$ below 20 MHz and a 3 dB bandwidth of 34
MHz.
###### pacs:
04.80.Nn, 07.60.-j, 07.87.+v, 85.60.-q, 85.60.Gz, 85.60.Dw, 85.60.Bt, 95.55.-n
## I Introduction
The Laser Interferometer Space Antenna (LISA) is a planned gravitational wave
observatory in the frequency range of 0.1 mHz–100 mHz that consists of three
spacecraft separated by 5 million km in a nearly equilateral triangle whose
center follows the Earth in a heliocentric orbit with an orbital phase offset
of 20 degrees. Gravitational waves will be detected as distance fluctuations
between test masses moving along geodetic trajectories that are located in
different spacecraft. LISA will require low power ultra-low noise
photoreceivers for precision inter-spacecraft heterodyne laser interferometry.
Quadrant photoreceivers will be used to measure the test mass motion with a
sensitivity of 8 $\mathrm{nrad}/\sqrt{\mathrm{Hz}}$ in angular orientation and
10 $\mathrm{pm}/\sqrt{\mathrm{Hz}}$ in displacement over the frequency range
of 0.1 mHz–100 mHz jennrich-2009 . The laser beam at the transmitting
spacecraft will have a diameter of approximately 40 cm and an output laser
power at the telescope of the order of 1 W. Given the laser beam propagation
over $5\times 10^{9}$ m and accounting for losses on the beam path, from the
remote optical signal approximately 50 pW will be detected on the entire
quadrant photoreceiver. LISA will use heterodyne laser interferometry (see
Figure 1) for the inter-spacecraft displacement measurement. The incoming weak
signal will optically interfere with a stronger local oscillator $P_{LO}$. The
combined signal $P(t)$ measured at the photoreceiver can be expressed as
$P(t)=\underbrace{P_{LO}+P_{sig}}_{P_{DC}:\mathrm{\,\,DC\,\,power}\,\sim\,P_{LO}}+\underbrace{2\sqrt{P_{LO}\,P_{sig}}\,\cos\left(\Delta\omega\,t+\varphi\right)}_{P_{AC}:\mathrm{\,\,heterodyne\,\,signal}},$
(1)
where $\Delta\omega$ is the frequency difference between the interfering laser
beams (heterodyne frequency), and $\varphi$ is the interferometer phase
containing the gravitational wave information. Both ports of the beamsplitter
will be measured by quadrant detectors. Their combined information can also be
used for common-mode rejection of laser amplitude noise. The main task of the
photoreceiver development is to maintain nearly shot-noise limited performance
over a measurement bandwidth from 2–20 MHz shaddock-2006 ; bykov-2009 . This
frequency range is driven by the Doppler induced frequency variations of the
optical beat note signal due to the LISA constellation armlength changes,
given by the orbits of each spacecraft.
Figure 1: Descriptive diagram of laser heterodyne interferometric detection.
The local oscillator power $P_{LO}$ can be adjusted, according to the required
photoreceiver performance. To reduce power consumption, temperature gradients
at the optical bench due to hot spots at the photoreceivers, and to provide
additional design margin, we aim for a low-noise wide-bandwidth photoreceiver
development operating at low $P_{LO}$ levels. Using 0.5 mW local oscillator
optical power on the entire quadrant photoreceiver (of the order of 100 $\mu$W
per quadrant), and assuming a responsivity $\rho$ of 0.7 A/W for InGaAs
photodiodes at a laser wavelength of 1064 nm, the shot-noise $i_{SN}$ can be
computed as
$i_{SN}=\sqrt{2e\rho P_{DC}}\approx 10\,\mathrm{pA}/\sqrt{\mathrm{Hz}}.$ (2)
Allocating $30\%$ of the shot-noise level –
$3\,\mathrm{pA}/\sqrt{\mathrm{Hz}}$ – to the input current noise contribution
of a quadrant photoreceiver, and considering this is the quadrature sum of the
current noise of the individual quadrants, we set an input current noise goal
of $1.5\,\mathrm{pA}/\sqrt{\mathrm{Hz}}$ for the single-quadrant photoreceiver
transimpedance amplifier (TIA).
## II Photodetector transimpedance amplifier
We have chosen a conventional DC-coupled TIA topology with a single ultra-low
noise / wide-bandwidth operational amplifier (op-amp), as shown in Figure 2.
Figure 2: Topology and noise model of photoreceiver TIA.
### II.1 Noise model
For the TIA topology shown in Figure 2, two main input current noise sources
have been identified in the electronics:
* •
Johnson noise ($i_{J}$) from feedback resistor $R_{f}$:
$i_{J}=\sqrt{\frac{4\,k\,T}{R_{f}}},$ (3)
where $k$ is the Boltzmann’s constant, and $T$ is the temperature (in Kelvin).
* •
Op-amp noise properties: The current $i_{n}$ and voltage $e_{n}$ noise
properties of the op-amp contribute to the total TIA input current noise.
* –
The op-amp current noise $i_{n}$ sums directly to the TIA input.
* –
The op-amp voltage noise $e_{n}$ translates to current noise at the TIA input
$i_{TIA}(f)$ over the input and feedback impedances as
$i_{TIA}(f)=e_{n}\frac{\sqrt{1+(2\pi\,f\,R_{f}\,C_{T})^{2}}}{R_{f}},$ (4)
where $f$ is the frequency, and $C_{T}$ is the total circuit capacitance
$C_{T}=C_{d}+C_{f}+C_{op}+C_{s},$ (5)
including the photodiode capacitance $C_{d}$, feedback impedance $C_{f}$, op-
amp common-mode input capacitance $C_{op}$, and stray capacitances $C_{s}$
from the board, components and packaging.
The bandwidth $BW$ of the photoreceiver can be estimated as
$BW=\sqrt{\frac{GBWP}{2\pi R_{f}\,C_{T}}},$ (6)
where $GBWP$ is the gain-bandwidth product of the operational amplifier. The
total TIA input current noise $I_{\mathrm{noise}}(f)$ model can be expressed
as
$I_{noise}(f)=\sqrt{i_{T}^{2}+i_{TIA}^{2}(f)}\,\cdot\|\overline{TF}(f)\|,$ (7)
where $\|\overline{TF}(f)\|$ is the normalized TIA transfer function,
$i_{TIA}(f)$ is a frequency dependent component of the input current noise
(see Equation 4), and $i_{T}$ is the quadrature sum of various contributors
that can be approximated by neglecting their frequency dependency for modeling
purposes. For example, the expected current noise $i_{T}$ in the photoreceiver
shown in Figure 2, can be computed as
$i_{T}=\sqrt{i_{n}^{2}+i_{J}^{2}+i_{d}^{2}},$ (8)
where $i_{n}$ is the op-amp current noise, $i_{J}$ is the Johnson noise of the
feedback resistor $R_{f}$ (see Equation 3), and $i_{d}$ is the shot-noise from
the photodiode dark current. It can be seen from Equation 5 that the
photodiode capacitance and the op-amp common-mode input capacitance are
crucial factors in the total noise budget. The challenge for the photodiode
manufacture lays in achieving a minimum capacitance per unit area while
maintaining high responsivity and low leakage if reverse-biased. For the TIA
electronics, it is necessary to identify an op-amp with minimal common-mode
input capacitance, current and voltage noise, and a gain-bandwidth product
large enough to maintain the required sensitivity over the required
measurement bandwidth of 2–20 MHz.
## III Prototype photoreceivers
### III.1 Collaboration with industry
Under a Small Business Innovation Research (SBIR) grant, the company Discovery
Semiconductors has developed a large-area quadrant photodiode (QPD) of 1 mm
diameter and a quadrant capacitance of 2.5 pF when reverse-biased at 5 V. A
first fully integrated quadrant photoreceiver (Figure 3: QPD + TIA
electronics) performs with an equivalent input current noise of less than 3.2
pA/$\sqrt{\mathrm{Hz}}$ below 20 MHz joshi-2009 . The characteristics of this
prototype quadrant photoreceiver are:
* •
Diameter of 1 mm with a 20 $\mu$m inter-quadrant gap.
* •
Individual quadrant capacitance $C_{d}$ = 2.5 pF when reverse-biased at 5 V.
* •
Dark current: 140 nA when reverse-biased at 5 V.
* •
Responsivity at 1064 nm: $\sim 0.7$ A/W (quantum efficiency of 0.8).
* •
TIA characteristics:
* –
feedback impedance: $R_{f}=51\,\mathrm{k\Omega}$, $C_{f}=0.1\,\mathrm{pF}$.
* –
op-amp ADA4817: $e_{n}=4\,\mathrm{nV}/\sqrt{\mathrm{Hz}}$,
$i_{n}=2.5\,\mathrm{fA}/\sqrt{\mathrm{Hz}}$, $C_{op}$ = 1.4 pF,
$GBWP=~{}410\,\mathrm{MHz}$.
Discovery Semiconductors has been awarded a second stage grant to further
develop quadrant photoreceivers. Given the successful development of a large-
area low-capacitance QPD in the first step, the next stage will be focused on
the noise reduction of the electronics, e.g. by integrating op-amps with
better noise properties and studying alternative TIA topologies. We expect to
receive additional devices with lower noise electronics at a later date.
Figure 3: Photograph of a prototype quadrant photoreceiver manufactured by
Discovery Semiconductors.
### III.2 Laboratory prototypes
Working in parallel with Discovery Semiconductors to try to understand the
noise and bandwidth trade-offs in more detail, we have identified the ultra-
low noise / high-bandwidth op-amp EL5135 from Intersil intersil_el5135 with
the following nominal noise properties:
$e_{n}=1.5\,\mathrm{nV}/\sqrt{\mathrm{Hz}}$,
$i_{n}=0.9\,\mathrm{pA}/\sqrt{\mathrm{Hz}}$, $C_{op}$ = 1 pF,
$GBWP=1500\,\mathrm{MHz}$.
A significant reduction in the op-amp voltage noise together with the
bandwidth enhancement (given by the higher $GBWP$) were the main factors
considered for selecting the EL5135 for laboratory prototype photoreceivers.
Despite the higher op-amp current noise $i_{n}$ of the EL5135 compared to the
ADA4817 (mentioned above), the lower voltage noise $e_{n}$ of the EL5135
enables us to achieve significantly better performance over the entire
bandwidth from 2–20 MHz. The penalty is somewhat higher noise at low
frequencies. According to Equation 7, the term $i_{T}$ (corresponding to the
quadrature sum of various current noise contributions, including the op-amp
current noise $i_{n}$) dominates at lower frequencies, while the frequency
dependent term $i_{TIA}$ (consisting of the op-amp voltage noise $e_{n}$ swing
across the total equivalent TIA impedance) becomes the dominant noise
contribution at higher frequencies.
We have designed a TIA with a feedback impedance $R_{f}=40\,\mathrm{k\Omega}$,
$C_{f}=0.1\,\mathrm{pF}$, and an expected bandwidth of 40 MHz (according to
Equation 6). We have built two different prototype boards to test the noise
properties of a TIA with the EL5135 op-amp:
* •
GAP500Q photoreceiver board: we have chosen the commercially available QPD
GAP500Q from GPD Optoelectronics gpdopto_gap500q with a diameter of 0.5 mm
and a responsivity of approximately 0.7 A/W (quantum efficiency of 0.8) at
1064 nm. When reverse-biased at 5 V, this device has a nominal quadrant
capacitance $C_{d}$ = 2.0 pF and a dark current of 2.0 nA, according to the
manufacturer. The purpose of this investigation is to operate the TIA
electronics with a photodiode that approximates the per-quadrant capacitance
and package parasitic capacitance of the larger area Discovery Semiconductors
detector.
* •
Mock-up TIA board: we have built a board for controlled noise investigations
of the TIA performance, shown in Figure 4.
Figure 4: Schematics of TIA mock-up board for noise measurements and frequency
response measurements.
This board has two inputs:
1. 1.
input 1: an input capacitor (2.4 pF) of similar quadrant capacitance is used
to replace the photodiode. For noise measurements, this input can be grounded
while maintaining input 2 open.
2. 2.
input 2: this is used to measure the expected photoreceiver transfer function
(TF) by injecting a signal (maintaining input 1 open), and scaling it
accordingly by the feedback gain ($40\mathrm{k}$). The equivalent input
current noise can be obtained, by dividing the output voltage noise by the
scaled transfer function.
The GAP500Q photodiode was reverse-biased with a battery power supply at 5 V,
and the op-amps in both circuits were driven with a power supply at $\pm$5 V.
## IV Performance measurements of prototype photoreceivers
We operated the photoreceiver using only one quadrant of the GAP500Q QPD with
the EL5135 op-amp for the TIA electronics. The photoreceiver output voltage
noise $V_{n}(f)$ is given by
$V_{n}(f)=TF(f)\cdot\sqrt{i_{SN}^{2}+i_{EN}^{2}(f)},$ (9)
where $i_{SN}$ is the photocurrent shot-noise of the incident light, $TF(f)$
is the photoreceiver transfer function, and $i_{EN}(f)$ is the equivalent
input current noise of the TIA electronics. Operating under dark conditions,
the photoreceiver output voltage noise $V_{EN}(f)$ is given by
$V_{EN}(f)=TF(f)\cdot i_{EN}(f),$ (10)
By dividing Equations 9 and 10 diekmann-2008 , we obtain that the input
current noise of the TIA electronics can be computed as
$i_{EN}(f)=\sqrt{\frac{i_{SN}^{2}}{\left(\frac{V_{n}(f)}{V_{EN}(f)}\right)^{2}-1}}.$
(11)
For equivalent input current noise measurements, we used a light-emitting
diode (LED) at a center wavelength of 1050 nm ($\pm$50 nm) thorlabs_led1050e
as shot-noise-limited light source. We measured the GAP500Q photoreceiver
output voltage noise with ($V_{n}(f)$) and without ($V_{EN}(f)$) LED light,
operating at two different optical power levels of 90 $\mu$W and 60 $\mu$W
that are representative for the expected nominal 100 $\mu$W per quadrant.
These measurements showed equivalent input current noise levels $i_{EN}(f)$
that did not scale with the DC optical power level (90 $\mu$W and 60 $\mu$W).
This also shows that the measured current noise $i_{EN}(f)$ upon subtraction
of the shot-noise contribution, is not dependent on the optical power, which
is consistent with a shot-noise behavior of the light source.
We also measured the output voltage noise and the transfer function ($TF(f)$)
of our mock-up TIA board (see Figure 4). Analogous to Equation 10, the input
current noise can be computed by referring the output voltage noise (upon
subtraction in quadrature of the RF spectrum analyzer voltage noise floor) to
the input dividing by the transfer function. Figure 5 shows the noise
measurements.
Figure 5: Input current noise measurements of photoreceiver prototypes. The
dashed trace is the photoreceiver with one quadrant of the GAP500Q QPD and the
EL5135 op-amp TIA. The solid trace is the mock-up test board (see Figure 4).
The dashed-dotted trace is the photoreceiver noise model for a TIA design with
an EL5135 op-amp and a quadrant capacitance of 2.5 pF (Equations 7). The
traces with crosses and circles show the corresponding photoreceiver noise
models using parameter values obtained from a fit to the data. The dotted
trace is the equivalent current noise floor of the RF spectrum analyzer used
as measurement instrument, referred to the input by the TIA transfer function.
The thick solid traced is our TIA input current noise goal of
$1.5\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ in the measurement band 2-20 MHz.
The GAP500Q photoreceiver reaches a level of about
$1.5\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ up to $\sim$10 MHz, increasing at higher
frequencies. It exceeds the noise goal by approximately 20%
($1.8\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$) at 20 MHz. The mock-up TIA circuit
meets our noise goal over the entire bandwidth (2-20 MHz), however, it shows
noise in excess of the model. At lower frequencies the equivalent input noise
is determined by excess current noise ($i_{T}$: assumed to have no frequency
dependence, Equation 8), while at higher frequencies, the increasing slope is
dominated by the op-amp voltage noise swing across the total circuit
capacitance ($i_{TIA}$: dependent on frequency, Equation 4). We run a set of
measurements in order to determine some of the involved unknowns:
1. 1.
op-amp voltage noise ($e_{n}$): we measured the op-amp voltage noise on a
separate sample of the EL5135, by driving the op-amp as a voltage follower
with grounded input. A low-noise 10x amplifier in series with the op-amp
output was necessary for a voltage noise measurement above the spectrum
analyzer noise floor. The voltage noise level measured was
$e_{n}=2.1\,\mathrm{nV}/\mathrm{\sqrt{Hz}}$ at 10 MHz, which is significantly
higher than the specified $1.5\,\mathrm{nV}/\mathrm{\sqrt{Hz}}$.
2. 2.
photodiode and feedback capacitances ($C_{d}$, $C_{f}$): using a LCR-bridge
impedance measurement instrument, we measured the photodiode quadrant
capacitance $C_{d}$ on a separate sample of the GAP500Q to be 3.2 pF with a 5
V reversed bias, which is higher than the nominal 2 pF. We also measured the
capacitor $C_{d}$ of the mock-up board and the feedback capacitor $C_{f}$ to
be 2.2 pF (nominally 2.4 pF) and 0.1 pF, respectively.
These noise properties are higher than specified and may account for part of
the excess noise. A direct measurement of the op-amp current noise $i_{n}$,
the op-amp input capacitance $C_{op}$, and the stray capacitances $C_{s}$ of
the circuit involve the development of dedicated electronic boards (currently
on-going) for well-controlled measurements. These measurements will be
conducted at a later stage. However, it is possible to obtain an estimate of
these values by fitting them as parameters of the noise model (Equations 7 and
8) to the two measured data sets. The noise level difference (offset) in the
data of the GAP500Q photoreceiver and the mock-up TIA is an indicator of a
significant excess current noise contribution, $i_{X}$, present in the photo-
measurement and not in the measurement of the mock-up TIA. This can also be
included into the fit by considering the following non-frequency dependent
contributions $i_{T}$ (Equation 8) for each case:
$\displaystyle\mathrm{GAP500Q\,\,photoreceiver:}$ $\displaystyle i_{T_{PD}}$
$\displaystyle=\sqrt{i_{n}^{2}+i_{J}^{2}+i_{d}^{2}+i_{X}^{2}},$ (12)
$\displaystyle\mathrm{Mock-up\,\,TIA:}$ $\displaystyle i_{T_{MU}}$
$\displaystyle=\sqrt{i_{n}^{2}+i_{J}^{2}}.$ (13)
The capacitance values $C_{d}$ and $C_{f}$ are assumed to be known from the
LCR-bridge measurements. We also assume similar op-amp and board noise
properties ($e_{n}$, $i_{n}$, $C_{op}+C_{s}$) for the two circuits. From the
fit, we obtained an op-amp current noise of $i_{n}\approx
1.1\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ (about 20% higher than nominal
$0.9\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$) and a combined stray plus op-amp input
capacitance $C_{op}+C_{s}\approx 1.3\,$pF (1 pF nominal $C_{op}$). We also fit
a common op-amp voltage noise for the two data sets to be $e_{n}\approx
1.9\,\mathrm{nV}/\mathrm{\sqrt{Hz}}$, which is comparable (within $<10\%$) to
the independent measurement ($2.1\,\mathrm{nV}/\mathrm{\sqrt{Hz}}$), but about
25% higher than nominal ($1.5\,\mathrm{nV}/\mathrm{\sqrt{Hz}}$). Table 1
summarizes the current best estimates (CBE) of the photoreceiver noise
properties.
parameter | nominal | CBE | method
---|---|---|---
$C_{d}\,\left[\mathrm{pF}\right]$ | 2.0 | 3.2 | measured
$e_{n}\,\left[\mathrm{nV}/\mathrm{\sqrt{Hz}}\right]$ | 1.5 | 1.9 | fit
$i_{n}\,\left[\mathrm{pA}/\mathrm{\sqrt{Hz}}\right]$ | 0.9 | 1.1 | fit
$C_{op}+C_{s}\,\left[\mathrm{pF}\right]$ | 1.0 | 1.3 | fit
$i_{X}\,\left[\mathrm{pA}/\mathrm{\sqrt{Hz}}\right]$ | - | 0.7 | fit
Table 1: Noise parameters: comparison between nominal values and current best
estimates (CBE).
The excess current noise contribution, $i_{X}$, present in the GAP500Q
photoreceiver data was determined to be of the order of $i_{X}\approx
0.7\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$. We have reversed-biased the photodiode
with a battery power supply, therefore, noise on the bias voltage translating
to excess current noise is not the cause. Additional tests are required to
determine the origin of this contribution.
## V Conclusions and Outlook
We have presented the results of noise measurements conducted on different
photoreceiver prototypes. The measurements showed approximately 20% noise in
excess of our goal between 10–20 MHz. Direct measurements of the op-amp
voltage noise and the reverse-biased QPD quadrant capacitance evidenced noise
levels higher than nominal, accounting for part of the excess noise at higher
frequencies. By fitting the parameters of the noise model to the data, we
obtained estimates for the combined stray plus op-amp input capacitance and
the op-amp current noise $i_{n}$, which was determined to be approximately 20%
higher than nominal. Significant excess current noise (50% of total) $i_{X}$
was determined between photoconductive (GAP500Q photoreceiver) and electronic
(mock-up TIA) noise measurements. Additional testing is required to determine
its origin. The measured photoreceiver performance is of the order of
$1.5\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ below 10 MHz, increasing up to
$1.8\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ at 20 MHz with a 3 dB bandwidth of 34
MHz. However, the mock-up TIA performs at a level of
$1.35\,\mathrm{pA}/\mathrm{\sqrt{Hz}}$ below 20 MHz (10% higher than expected
from the nominal model) with a measured 3 dB bandwidth of 38 MHz. This
suggests a significantly better performance of a real photoreceiver with the
current TIA design, depending upon clarification and, if viable, mitigation of
the excess current noise $i_{X}$. In addition, as following steps, we plan to
conduct spatial scanning of the photodiode surfaces, measurement of inter-
quadrant cross-talk, and differential wavefront sensing angle measurements.
## VI Acknowledgements
This research was supported in part by NASA contract ATFP07-0127. F. Guzmán
Cervantes is supported by an appointment to the NASA Postdoctoral Program at
the Goddard Space Flight Center, administered by Oak Ridge Associated
Universities (ORAU) through a contract with NASA. We thank A. Joshi, S. Datta
and J. Rue for stimulating discussions.
## References
* (1) Jennrich O, LISA technology and instrumentation, Class. Quantum Grav. 26 (2009).
* (2) Shaddock D, Ware B, Halverson P, Spero R, Klipstein B, Overview of the LISA phasemeter, AIP Conf Proc. 873 654-60 (2006).
* (3) Bykov I, Esteban Delgado J, García Marín A, Heinzel G, Danzmann K, LISA phasemeter development: Advanced prototyping, J. Phys.: Conf. Ser. 154 (2009).
* (4) Joshi A, Rue J, and Datta S, Low-Noise Large-Area Quad Photoreceivers Based on Low-Capacitance Quad InGaAs Photodiodes, IEEE Photonics Technology Letters, Vol.21, No.21 (2009).
* (5) Datasheet of Intersil device EL5135: http://www.intersil.com/data/fn/fn7383.pdf
* (6) Datasheet of GPD Optoelectonics device GAP500Q: http://www.gpd-ir.com/
* (7) Diekmann C, Phasenstabilisierung und -auslesung für LISA, Diploma Thesis, Leibniz Universität Hannover, Germany (2008).
* (8) Datasheet of THORLABS device LED1050E: http://www.thorlabs.com/Thorcat/16300/16388-S01.pdf
|
arxiv-papers
| 2012-03-13T17:02:33 |
2024-09-04T02:49:28.608186
|
{
"license": "Public Domain",
"authors": "Felipe Guzm\\'an Cervantes, Jeffrey Livas, Robert Silverberg, Ernest\n Buchanan, Robin Stebbins",
"submitter": "Felipe Guzman Cervantes",
"url": "https://arxiv.org/abs/1203.2862"
}
|
1203.2880
|
# Limitations of X-ray reflectometry in the presence of surface contamination
D.L. Gil111Present address: Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ and D. Windover National
Institute of Standards and Technology, 100 Bureau Dr., Stop 8520,
Gaithersburg, MD 20899-8520 windover@nist.gov
###### Abstract
Intentionally deposited thin films exposed to atmosphere often develop
unintentionally deposited few monolayer films of surface contamination. This
contamination arises from the diverse population of volatile organics and
inorganics in the atmosphere. Such surface contamination can affect the
uncertainties in determination of thickness, roughness and density of thin
film structures by X-Ray Reflectometry (XRR). Here we study the effect of a
$0.5\text{\,}\mathrm{nm}$ carbon surface contamination layer on thickness
determination for a $20\text{\,}\mathrm{nm}$ titanium nitride thin film on
silicon. Uncertainties calculated using Markov-Chain Monte Carlo Bayesian
statistical methods from simulated data of clean and contaminated TiN thin
films are compared at varying degrees of data quality to study (1) whether
synchrotron sources cope better with contamination than laboratory sources and
(2) whether cleaning off the surface of thin films prior to XRR measurement is
necessary. We show that, surprisingly, contributions to uncertainty from
surface contamination can dominate uncertainty estimates, leading to minimal
advantages in using synchrotron- over laboratory-intensity data. Further, even
prior knowledge of the exact nature of the surface contamination does not
significantly reduce the contamination’s contribution to the uncertainty in
the TiN layer thickness. We conclude, then, that effective and standardized
cleaning protocols are necessary to achieve high levels of accuracy in XRR
measurement.
###### pacs:
06.20.Dk, 61.05.cm, 68.55.jd, 68.35.Ct
††: JPD
## 1 Introduction
X-ray reflectometry is widely used for characterizing the thickness,
roughness, and density of nanometer scale thin films. Because it uses
wavelengths of a similar or smaller scale relative to the thicknesses of the
layers being studied, the resulting data has a relatively direct connection to
the structure and therefore has a rather straightforward traceability to the
International System of Units (SI) [1, 2, 3]. This is a significant advantage
over other techniques like spectroscopic ellipsometry (SE), whose results are
somewhat more difficult to interpret. But x-ray reflectometry is still
somewhat sensitive to surface contamination. A comprehensive study by Seah et
al. on ultra-thin SiO2 layers on Si has shown offsets between x-ray
reflectometry (XRR), neutron reflectometry (NR), spectroscopic ellipsometry,
and x-ray photoelectron spectroscopy (XPS) [4]. These offsets are believed
likely due to the different effects contamination has for each technique. To
use x-ray reflectometry for high-accuracy measurements of film thickness and
roughness – as the National Institute of Standards and Technology (NIST) plans
to do for thin-film standards – these effects must be quantified.
The primary difficulty in quantifying the effects of contamination is that
thin surface contamination layers are hard to measure. The surface
contamination with which we are concerned typically takes the form of rough,
near-monolayer-thicknes, carbonaceous compounds. This is challenging to
measure with x-ray or neutron reflectivity-based methods: carbon’s low
scattering factor for both techniques and the poor quality of the carbon
layers produce low-contrast fringes.
X-ray photoelectron spectroscopy (XPS) and other inelastic scattering
techniques can be used to measure the quantity per surface area – and thus
relative thicknesses – of contamination layers, but generally require
calibration in order to measure absolute thicknesses [4]. This calibration is
often, for denser materials, conducted by reflectometry measurements. In
addition, these experiments are typically conducted in ultra-high-vacuum,
which may change the properties, or even the very presence, of the adsorbed
volatile contamination layers. This makes using these experiments somewhat
challenging for achieving high accuracy in absolute thickness determination.
But a crucial question is whether these experiments are needed at all: How
sensitive are other parameters determined by modeling of x-ray reflectometry
data to the presence of an unknown, hard-to-measure layer of surface
contamination? (E.g., how much do you have to know about contamination to
achieve a given level of uncertainty in thickness for other layers within a
structure?) And what sort of data quality (i.e., dynamic and $q$-space/angular
range) is needed to achieve desired levels of uncertainties? (E.g., do you
need synchrotron data?)
We study these questions by a simulation-based study of the effects of a
carbon contamination layer on a TiN film on Si substrate structure being
considered by NIST for an X-ray reflectometry standard. Using a Bayesian
statistical approach to XRR data analysis, we estimate uncertainties for
structural parameters of the high-Z TiN layer under various contamination and
data quality conditions.
## 2 Background
### 2.1 X-ray reflectometry
X-ray reflectometry can be used to measure the density, thickness, and
roughness of thin films which are laterally homogeneous at the scale of the
beam [5, 6]. We limit ourselves here to the case of layered structures with
fairly sharp interfaces and no density grading other than that provided by
roughness. Density is measured by the critical angle for total external
reflection and through the careful analysis of oscillation amplitudes;
thickness is measured by the period of intensity oscillations appearing after
the critical angle; and roughness is measured by the rapidity of overall
intensity and oscillation intensity fall-off. The analysis here uses the
Parratt recursion [7] with the perturbation for Gaussian roughness described
by [8].
Analyzing XRR data is an inverse problem: because only the intensity – rather
than the complex amplitude – of the reflected beam can be measured, in general
there is no unique structure determined by an XRR pattern. The typical
approach is to fit the parameters of a multilayer structure using an
optimization approach, the most common being Genetic Algorithms (GAs) [9, 10,
11]. Though optimization finds a best-fit solution – and thus a best estimate
of the parameter values – it does not provide estimates of uncertainties on
the parameters. To calculate uncertainties, a more sophisticated – and
computationally expensive – approach is necessary. NIST has developed
statistical Markov Chain Monte Carlo (MCMC) methods in order to obtain
parameter estimations and uncertainties within a Bayesian formalism [12].
### 2.2 Data analysis
To make inferences about physical structure from XRR data by Genetic Algorithm
(GA) or Markov Chain Monte Carlo (MCMC) methods requires a physical model that
relates structural parameters to idealized non-noisy XRR data. This is
provided by the Parratt recursion with Nevot-Croce roughness described above.
A physical model is not sufficient, however, because the data collected are
(at best) noisy and (at worst) corrupted by systematic instrument effects.
There must be, in addition, a statistical model of the data. For the GA
method, this is a $\chi^{2}$ cost function; for the MCMC method, a probability
density function. The cost function and probability density function employed
here make very similar assumptions about the statistical characteristics of
the data, see [13]. Nonetheless, the GA and MCMC methods each recover
different types of information from the data.
X-ray reflectometry data consists of pairs of angles and measured intensities.
For this study, we assume data free of angular errors with counting-error-
limited measured intensities. The error in measured intensities is modeled
using a log-normal likelihood with standard deviation of the square root of
the calculated intensities; this approximates well a Poisson (i.e., counting-
statistic) likelihood [13].
We use a tiered data analysis architecture. XRR model parameters are first
determined for a given structural model using a GA optimization approach [11].
In this analysis, we used a 1000 genome population evolved over 1000
generations to obtain best-fit structural parameters. This structural
information was then used to initialize the starting parameters for a tuned
Markov Chain Monte Carlo (MCMC) sampler to provide us with probability
distributions for each parameter within a given structural model. Each MCMC
was allowed a 50 000 steps conditioning run to tune the MCMC target
dimensions. The MCMCs were then run for 250 000 steps to obtain adequate
statistics for inter-comparison. The tiered analysis was performed several
times on each data set to validate the refinement stability.
### 2.3 Simulated data
Two structures were simulated for this study: case 1 – a single layer of TiN
on an Si substrate (for film parameters, see Table 1) – and case 2 – a
contaminated single layer of TiN on an Si substrate (see Table 2). Simulations
were performed under two different data quality conditions (see Table 3)
selected to compare parameter refinement results between an advanced
laboratory instrument and a synchrotron measurements from, e.g., a third-
generation bending magnet beamline. By way of illustration, we present XRR
data simulations from our case 2 structure for laboratory (see Figure 1 and
synchrotron (see Figure 2) data quality conditions. By considering these two
cases, we can answer an oft-asked question for XRR data-collection: How many
orders of magnitude data quality are needed to determine a given XRR model
parameter? The Bayesian statistical approach used here can directly answer
this question by determining the ‘best-case’ theoretically possible from XRR
measurements for any refined parameter.
Figure 1: Simulated (Cu radiation) X-ray reflectometry (XRR) data for case 2
structural modal and laboratory quality data (see Tables 2 and 3). XRR
simulated data (plus signs) has been fit using a genetic algorithm refinement
(solid line) and yielded nearly identical parameters to those used in the
simulation (Table 2). Note refinement quality (magnified regions).
Figure 2: Simulated (Cu radiation) X-ray reflectometry (XRR) data for case 2 structural modal and synchrotron quality data (see Tables 2 and 3). XRR simulated data (plus signs) has been fit using a genetic algorithm refinement (solid line) and yielded nearly identical parameters to those used in the simulation (Table 2). Note refinement quality (magnified regions). | $t/\textrm{nm}$ | $\sigma/\textrm{nm}$ | $\rho/\textrm{g cm}^{-2}$
---|---|---|---
TiN | 20.0 | 0.5 | 4.90
Si | – | 0.4 | 2.49
Table 1: Case 1: clean TiN/Si structure. Composition, thickness ($t$), roughness ($\sigma$), and density ($\rho$). | $t/\textrm{nm}$ | $\sigma/\textrm{nm}$ | $\rho/\textrm{g cm}^{-2}$
---|---|---|---
C | 0.5 | 0.1 | 2.25
TiN | 20.0 | 0.5 | 4.90
Si | – | 0.4 | 2.49
Table 2: Case 2: carbon-contaminated TiN/Si structure. | $2\theta$ | Step | Maximum | Background
---|---|---|---|---
| range | size | intensity |
| (degrees) | (degrees) | (counts) | (counts)
Laboratory case | 4.0 | 0.005 | $10^{6}$ | 1
Synchrotron case | 7.0 | 0.005 | $10^{8}$ | 1
Table 3: Parameters of XRR data quality cases used in simulations.
### 2.4 MCMC analysis
The MCMC analysis method has initial optimal parameters input using the
results of a GA. Details of MCMC methods are beyond the scope of this paper.
The most important point is that all MCMC implementations, if properly tuned
and allowed sufficient time, should produce the same result. Most research
into, and the complications in, MCMC methods relate to improving sampler
efficiency and thus the number of samples required. Thus the details of the
particular sampling scheme relate mainly to efficiency; the resulting samples
are from the same Bayesian posterior probability distribution.
It is important for interpreting the probability distributions sampled by MCMC
to know three modeling assumptions: (1) the allowed prior ranges for each
parameter within a model, (2) the assumed prior distributions for each
parameter, and (3) the type of noise assumed within the data. In this work, we
use ranges for a uniform prior which assume the same physical structure (same
number of layers) as the simulated data; the ranges of the priors are generous
to allow the MCMC to sample a wide parameter space. In Table 4 we give the
allowed parameter ranges for case 1. In Table 5, we provide the ranges used in
case 2. In Table 6, we use a highly constrained model for case 2 with all the
values of the carbon contamination layer fixed at their simulated values;
i.e., we assume we know the nature of the contamination exactly. In each
model, we assume a uniform prior distribution for thickness, roughness, and
density. In all cases we assume the noise in actual measured data – and thus
the likelihood of the data for a given set of parameter values – is Poisson.
| $t/\textrm{nm}$ | $\sigma/\textrm{nm}$ | $\rho/\textrm{g cm}^{-2}$
---|---|---|---
TiN | 15.0 to 25.0 | 0.01 to 2.5 | 4.0 to 6.0
Si | – | 0.01 to 2.5 | 2.0 to 3.0
Table 4: Model 1: Allowed MCMC (uniform prior) ranges for TiN/Si structure with no surface contamination layer. | $t/\textrm{nm}$ | $\sigma/\textrm{nm}$ | $\rho/\textrm{g cm}^{-2}$
---|---|---|---
C | 0.0 to 2.0 | 0.01 to 2.5 | 2.0 to 3.0
TiN | 15.0 to 25.0 | 0.01 to 2.5 | 4.0 to 6.0
Si | – | 0.01 to 2.5 | 2.0 to 3.0
Table 5: Model 2: Allowed MCMC (uniform prior) ranges for TiN/Si multilayer with surface contamination layer. | $t/\textrm{nm}$ | $\sigma/\textrm{nm}$ | $\rho/\textrm{g cm}^{-2}$
---|---|---|---
C | 0.5 | 0.1 | 2.25
TiN | 15.0 to 25.0 | 0.01 to 2.5 | 4.0 to 6.0
Si | – | 0.01 to 2.5 | 2.0 to 3.0
Table 6: Model 2a: Allowed MCMC range for TiN/Si multilayer with a known
surface contamination layer.
## 3 Results
Statistically-determined uncertainties for laboratory and synchrotron levels
of data quality have been calculated using the MCMC method for a simulated
clean TiN sample (case 1) with corresponding modeling ranges (model 1) and a
simulated carbon contaminated TiN sample (case 2) with its corresponding
modeling ranges (model 2). Absolute and relative uncertainties for each
structural parameter and each data quality are presented. We also present a
modified analysis for case 2, in which we provide the exact parameters for the
contamination layer as prior information for the MCMC method (model 2a) and
discuss the resulting uncertainties.
### 3.1 Clean sample – case 1
The power of the Bayesian analysis via MCMC is through its generation of
posterior probability distributions for each parameter within a physical
model, clearly showing the uncertainty ranges (for example, see Figure 3). The
expanded uncertainties can be directly calculated by finding the parameter
bounds for the probability distribution plot area representing the 95 %
highest probability. For case 1, these expanded uncertainty ranges are
tabulated in Table 7.
For a clean, single-layer structure, there is a clear (factor of two)
advantage to synchrotron measurements with regards to determination of
accurate thickness and roughness information. This statistical determination
method for uncertainty estimation is absent from optimization refinement
methods such as GAs. Studying 2-dimensional (2 simultaneous parameters)
posterior probability distributions allows us to qualitatively and
quantitatively explore parameter correlations. The clear improvements in TiN
thickness and roughness seen in Table 7 are due to the orthogonal (no
correlation) nature of thickness and roughness (see Figure 4). As a general
rule, when no correlation exists between parameters within a refinement, then
better data quality will directly correspond to reduced uncertainties for the
parameters in question. However, if correlations do exist between two or more
parameters, as for example between film roughness and film density (see Figure
5) in case 1, then correlations will introduce intrinsic parameter
uncertainties which cannot be further reduced with higher data quality.
As seen by studying the ratios of uncertainty estimates (last column in Table
7), when determining the density of either the TiN or the Si substrate, there
is no clear advantage between the laboratory and the synchrotron levels of
data quality. The relative quality of density determination is nearly
identical in both cases (uncertainty ratios equal to 1). This constant nature
of density uncertainty over both levels of data quality is likely caused by
two factors: First, that both datasets have the same spacing between
collection points, so that the critical angle is not much more precisely
determined by synchrotron data than the laboratory data. Second, density
correlates with other modeling parameters, for example, interface roughness
(see Figure 5).
Parameter | U | U | [U(lab) /
---|---|---|---
| (synchrotron) | (laboratory) | U(sync)]
$t_{\textrm{TiN}}$ | $0.048\text{\,}\mathrm{nm}$ | $0.11\text{\,}\mathrm{nm}$ | 2.3
$\sigma_{\textrm{TiN}}$ | $0.033\text{\,}\mathrm{nm}$ | $0.060\text{\,}\mathrm{nm}$ | 1.8
$\sigma_{\textrm{Si}}$ | $0.050\text{\,}\mathrm{nm}$ | $0.139\text{\,}\mathrm{nm}$ | 2.8
$\rho_{\textrm{TiN}}$ | $1.08\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $1.05\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1
$\rho_{\textrm{Si}}$ | $0.90\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.90\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1
Table 7: MCMC-determined expanded uncertainties (95% probability intervals)
for the model parameters of case 1, model 1, and the ratio of these
uncertainties, $[$U(lab)/U(sync)$]$
Figure 3: TiN thickness ($t_{1}$) posterior probability density for case 1
using synchrotron quality data.
Figure 4: 2-dimensional histogram showing no correlation between TiN thickness
($t_{1}$) and TiN interface roughness ($\sigma_{1}$) for case 1. Intensity
scale of histogram shows the relative frequency with which the Monte Carlo
Markov Chain explores a given parameter space. (High frequencies directly
correspond to high probabilities for a well-tuned MCMC.)
Figure 5: 2-dimensional histogram showing correlation between TiN density
($\rho_{1}$) and TiN interface roughness ($\sigma_{1}$) for case 1. Intensity
scale of histogram shows the relative frequency with which the Monte Carlo
Markov Chain explores a given parameter space.
### 3.2 Carbon contaminated film - case 2
For case 2, we present the expanded uncertainty ranges in Table 8 and see
several surprising results. When one introduces a carbon contamination layer,
the advantages in reduced uncertainties from using the higher data quality of
a synchrotron vanishes for all but roughness determination. For ratios of
unity or near unity, (e.g., 0.93, 0.75) there is no clear advantage to
synchrotron data. [Apparent disadvantages to synchrotron data (i.e., ratios
less than one) are artifacts to the coarseness of our sampling analysis.] This
is partially a consequence of high inverse correlation between contamination
layer thickness and TiN thickness (see Figure 6). Correlations also exist
between the contamination density and surface roughness (see Figure 7),
further expanding uncertainties throughout the model parameters.
When one examines only the highest quality data (synchrotron), the effect of
clean vs. contaminated surfaces can be directly compared. In Table 9, we see
that only the TiN thickness and roughness show pronounced reductions in
uncertainty ranges from a contaminated vs. clean structure. An astute observer
may wonder why the uncertainty for Si and TiN density are not improved through
the removal of the carbon contamination layer. This is because XRR, in some
cases, is simply not sensitive to a given model parameter. This sensitivity
issue can be distinguished from a correlation phenomena, again by using the
MCMC posterior probability densities or 2-dimensional histograms, and looking
for parameters which produce uniform posteriors out the analysis. In Figure 8,
we see that the Si density probability density is nearly uniform over the
allowed range of the parameter. This lack of a pronounced peak demonstrates
very little sensitivity to Si density in our data.
Figure 6: 2-dimensional histogram showing inverse correlation between
contamination layer thickness ($t_{1}$) and TiN thickness ($t_{2}$) for for
case 2. Intensity scale of histogram shows the relative frequency with which
the Monte Carlo Markov Chain explores a given parameter space.
Figure 7: 2-dimensional histogram showing correlation between contamination
layer density ($\rho_{1}$) and surface roughness ($\sigma_{0}$) for case 2.
Intensity scale of histogram shows the relative frequency with which the Monte
Carlo Markov Chain explores a given parameter space.
Figure 8: Posterior probability density for Si substrate density ($\rho_{3}$)
showing lack of sensitivity for this parameter within the XRR model for case
2.
### 3.3 Contamination of known thickness, roughness, and density - case 2a
In Table 10, we introduce the results from our known parameter carbon
contamination case. We compare the uncertainties between clean, unknown
contamination, and exactly known contamination cases The most interesting
feature is the TiN thickness. Even for the case where the contamination
thickness, roughness, and density are known _a priori_ , the model still has 5
times higher TiN thickness uncertainty over the clean surface case. There is a
reduction of a factor of 2 over the unknown contamination case; this reduction
indicates that not knowing the exact properties of the contamination layer
does have an effect on uncertainties. (Or, identically, that refining an
unknown contamination layer increases the uncertainties.) But this effect is
much smaller than the effect of the mere presence of the contamination layer.
This is because the presence of the contamination layer causes a decrease in
the contrast of the TiN layer thickness fringes, decreasing the ability of
higher quality data to provide more information. This manifests itself in
correlations in the model which can substantially increase the overall
uncertainties for the TiN layer thickness.
The very presence of contamination on the structure – rather than the need to
fit the contamination – has the largest effect on the uncertainty.
## 4 Conclusions
There are some caveats to this conclusion: Only simulated data has been
considered. All systematic instrumental errors have been neglected. No
instrument response functions have been modeled. The comparison between
laboratory and synchrotron data is made only for the case of a Cu K$\alpha$
laboratory source radiation and an synchrotron beamline set to the same energy
– so any advantage to tuning the energy of the beam for specific materials and
structures has been neglected. (For an example of XRR fit improvement through
judicious source energy selection using a synchrotron, see [14]).
But accepting these limitations, save perhaps source energy tuning, as not
being likely to _improve_ data quality, the MCMC XRR analysis technique
provides a powerful tool for studying the theoretical limitations of XRR
measurements of a structure before taking measurements.
In this case of a carbon contamination layer, we see that the theoretical
uncertainty estimates for parameters are dominated by correlations between the
surface contamination thickness and the TiN thickness increasing the
uncertainty estimates for our thin film of interest whenever the carbon
contamination layer is present. Even when armed with prior knowledge of all
parameters for the contamination layer, we see a five-fold increase in the TiN
thickness uncertainty caused by the introduction of the contamination layer.
Higher data quality will provide significant reductions in parameter
uncertainties for simple XRR models, such as the clean TiN thin film. However,
in the presence of contamination, we see minimal gain through enhanced data
quality for the determination of thin film thickness.
This MCMC simulated data study has shown that removing the contamination is
essential to significantly reducing the uncertainties in the high-Z layer
thickness measurement.
Parameter | U | U | [U(lab) /
---|---|---|---
| (synchrotron) | (laboratory) | U(sync)]
$t_{\textrm{C}}$ | $0.64\text{\,}\mathrm{nm}$ | $0.66\text{\,}\mathrm{nm}$ | 1.0
$t_{\textrm{TiN}}$ | $0.67\text{\,}\mathrm{nm}$ | $0.62\text{\,}\mathrm{nm}$ | 0.93
$t_{\textrm{C}}+t_{\textrm{TiN}}$ | $0.32\text{\,}\mathrm{nm}$ | $0.24\text{\,}\mathrm{nm}$ | 0.75
$\sigma_{\textrm{C}}$ | $0.033\text{\,}\mathrm{nm}$ | $0.16\text{\,}\mathrm{nm}$ | 4.8
$\sigma_{\textrm{TiN}}$ | $0.50\text{\,}\mathrm{nm}$ | $0.48\text{\,}\mathrm{nm}$ | 0.96
$\sigma_{\textrm{Si}}$ | $0.027\text{\,}\mathrm{nm}$ | $0.20\text{\,}\mathrm{nm}$ | 7.4
$\rho_{\textrm{C}}$ | $0.61\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.92\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1.5
$\rho_{\textrm{TiN}}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $1.28\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 0.86
$\rho_{\textrm{Si}}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1
Table 8: MCMC-determined expanded uncertainties (95% probability intervals) for the model parameters of case 2, model 2, and the ratio of these uncertainties, $[$U(lab)/U(sync)$]$. Parameter | U(clean) | U(with carbon) | [U(carbon) /
---|---|---|---
| (synchrotron) | (synchrotron) | U(clean)]
$t_{\textrm{TiN}}$ | $0.048\text{\,}\mathrm{nm}$ | $0.67\text{\,}\mathrm{nm}$ | 14
$\sigma_{\textrm{surface}}$ | $0.033\text{\,}\mathrm{nm}$ | $0.033\text{\,}\mathrm{nm}$ | 1
$\sigma_{\textrm{TiN}}$ | $0.033\text{\,}\mathrm{nm}$ | $0.5\text{\,}\mathrm{nm}$ | 15
$\sigma_{\textrm{Si}}$ | $0.050\text{\,}\mathrm{nm}$ | $0.027\text{\,}\mathrm{nm}$ | 0.54
$\rho_{\textrm{TiN}}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1
$\rho_{\textrm{Si}}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | 1
Table 9: Comparison of MCMC-determined expanded uncertainties (95% probability intervals) between clean vs. contaminated cases for synchrotron quality data Parameter | U(clean) | U(carbon) | U(known
---|---|---|---
| | | carbon)
$t_{\textrm{TiN}}$ | $0.048\text{\,}\mathrm{nm}$ | $0.67\text{\,}\mathrm{nm}$ | $0.31\text{\,}\mathrm{nm}$
$\sigma_{\textrm{surface}}$ | $0.033\text{\,}\mathrm{nm}$ | $0.033\text{\,}\mathrm{nm}$ | fixed
$\sigma_{\textrm{TiN}}$ | same as $\sigma_{\textrm{surface}}$ | $0.5\text{\,}\mathrm{nm}$ | $0.36\text{\,}\mathrm{nm}$
$\sigma_{\textrm{Si}}$ | $0.050\text{\,}\mathrm{nm}$ | $0.027\text{\,}\mathrm{nm}$ | $0.027\text{\,}\mathrm{nm}$
$\rho_{\textrm{TiN}}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $1.49\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$
$\rho_{\textrm{Si}}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.94\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$ | $0.93\text{\,}\mathrm{g}\text{\,}{\mathrm{c}}^{-1}\text{\,}{\mathrm{m}}^{2}$
Table 10: Comparison of MCMC-determined expanded uncertainties (95%
probability intervals) between clean, unknown, and known contamination layer
cases for synchrotron quality data
We would like to thank Victor Vartanian of the International SEMATECH
Manufacturing Initiative (ISMI) (Albany, NY) for providing pre-standard test
structures and XRR measurements for XRR SRM development at NIST. We would also
like to thank P.Y. Hung of Sematech (Albany, NY) for extensive and very
helpful discussions on thin film characterization and for providing numerous
interesting sets of XRR data for the development of analysis techniques.
## References
## References
* [1] D. K. Bowen, K. M. Matney, and M. Wormington. X-ray metrology for ulsi structures. AIP Conference Proceedings, 449:928–932, 1998.
* [2] D. K. Bowen and R. D. Deslattes. X-ray metrology by diffraction and reflectivity. AIP Conference Proceedings, 550(550):570–579, 2001.
* [3] K. Hasche, P. Thomsen-Schmidt, M. Krumrey, G. Ade, G. Ulm, J. Stuempel, S. Schaedlich, W. Frank, M. Procop, and U. Beck. Metrological characterization of nanometer film thickness standards for xrr and ellipsometry applications. Proceedings of the SPIE - The International Society for Optical Engineering, 5190(1):165–172, 2003.
* [4] M. P. Seah and S. J. Spencer. Ultrathin SiO2 on si. i. quantifying and removing carbonaceous contamination. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, 21(2):345–352, March 2003.
* [5] E. Chason and T. M. Mayer. Thin film and surface characterization by specular x-ray reflectivity. Critical Reviews in Solid State and Materials Sciences, 22(1):1–67, 1997.
* [6] Ullrich Pietsch, Holy Vaclav, and Tilo Baumbach. High-resolution X-ray scattering : from thin films to lateral nanostructures. Springer, New York, 2004.
* [7] L.G. Parratt. Surface studies of solids by total reflection of x-rays. Phys. Rev., 95:359, 1954.
* [8] L. Nevot and P. Croce. Characterization of surfaces by grazing x-ray reflection - application to study of polishing of some silicate-glasses. Revue De Physique Appliquee, 15(3):761–779, 1980.
* [9] A. D. Dane, A. Veldhuis, D. K. G. de Boer, A. J. G. Leenaers, and L. M. C. Buydens. Application of genetic algorithms for characterization of thin layered materials by glancing incidence x-ray reflectometry. Physica B, 253(3-4):254–268, 1998.
* [10] A. Ulyanenkov and S. Sobolewski. Extended genetic algorithm: application to x-ray analysis. Journal of Physics D-Applied Physics, 38(10A):A235–A238, 2005.
* [11] M. Wormington, C. Panaccione, K. M. Matney, and D. K. Bowen. Characterization of structures from x-ray scattering data using genetic algorithms. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 357(1761):2827–2848, 1999\.
* [12] D. Windover, D. L. Gil, J. P. Cline, A. Henins, N. Armstrong, P. Y. Hung, S. C. Song, R. Jammy, and A. Diebold. NIST method for determining model-independent structural information by x-ray reflectometry. AIP Conference Proceedings, 931(1):287–291, 2007.
* [13] D. S. Sivia. Data Analysis A Bayesian Tutorial. Oxford University Press, Oxford, 1996.
* [14] M. Krumrey, G. Gleber, F. Scholze, and J. Wernecke. Synchrotron radiation-based x-ray reflection and scattering techniques for dimensional nanometrology. Measurement Science and Technology, 22:094032, 2011.
|
arxiv-papers
| 2012-03-13T18:02:07 |
2024-09-04T02:49:28.614479
|
{
"license": "Public Domain",
"authors": "David L. Gil and Donald Windover",
"submitter": "Donald Windover",
"url": "https://arxiv.org/abs/1203.2880"
}
|
1203.2934
|
# About the probability distribution of a quantity with given mean and
variance
Stefano Olivares 111stefano.olivares@ts.infn.it Dipartimento di Fisica,
Università degli Studi di Trieste, I-34151 Trieste, Italy
Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano,
Italy
CNISM, UdR Milano Statale, I-20133 Milano, Italy Matteo G. A. Paris
222matteo.paris@fisica.unimi.it Dipartimento di Fisica, Università degli Studi
di Milano, I-20133 Milano, Italy
CNISM, UdR Milano Statale, I-20133 Milano, Italy
INRIM, Strada delle Cacce 91, 10135 Torino, Italy.
###### Abstract
Supplement 1 to GUM (GUM-S1) recommends the use of maximum entropy principle
(MaxEnt) in determining the probability distribution of a quantity having
specified properties, e.g., specified central moments. When we only know the
mean value and the variance of a variable, GUM-S1 prescribes a Gaussian
probability distribution for that variable. When further information is
available, in the form of a finite interval in which the variable is known to
lie, we indicate how the distribution for the variable in this case can be
obtained. A Gaussian distribution should only be used in this case when the
standard deviation is small compared to the range of variation (the length of
the interval). In general, when the interval is finite, the parameters of the
distribution should be evaluated numerically, as suggested by I. Lira
[Metrologia, 2009, 46, L27]. Here we note that the knowledge of the range of
variation is equivalent to a bias of the distribution toward a flat
distribution in that range, and the principle of minimum Kullback entropy
(mKE) should be used in the derivation of the probability distribution rather
than the MaxEnt, thus leading to an exponential distribution with non Gaussian
features. Furthermore, up to evaluating the distribution negentropy, we
quantify the deviation of mKE distributions from MaxEnt ones and, thus, we
rigorously justify the use of GUM-S1 recommendation also if we have further
information on the range of variation of a quantity, namely, provided that its
standard uncertainty is sufficiently small compared to the range.
Supplement 1 to GUM (GUM-S1) [1] provides assignments of probability density
functions for some common circumstances. In particular, it is stated that if
we know only the mean value $\bar{x}$ and the variance
$\sigma^{2}_{\scriptstyle X}$ of a certain quantity $X$, we should assign a
Gaussian probability distribution to that quantity, according to the principle
of maximum entropy (MaxEnt) [2, 3]. The derivation is quite simple, as one has
to look for the distribution $p(x)$ maximizing the Shannon entropy:
$\displaystyle S[p]=-\int_{\mathbbm{R}}\\!\\!dx\,p(x)\log p(x)\,,$ (1)
which is given by:
$\displaystyle p(x)=\exp\\{-\lambda_{0}-\lambda_{1}x-\lambda_{2}x^{2}\\}\,,$
(2)
where the values of the coefficients $\lambda_{k}$ should be determined to
satisfy the constraints:
$\displaystyle\int_{\mathbbm{R}}\\!\\!dx\,p(x)\,x^{k}=M_{k}\,,$ (3)
with:
$\displaystyle M_{0}=1,\quad M_{1}=\bar{x},\quad M_{2}=\sigma_{\scriptstyle
X}^{2}+\bar{x}^{2}\,.$ (4)
However, sometimes we also know the range of the possible values of the
quantity $X$. Two relevant examples are given by the phase-shift in
interferometry, which is topologically confined in a $2\pi$-window, and by the
displacement amplitude of a harmonic oscillator, whose range of variation is
dictated by energy constraints. In this case, it has been noticed by I. Lira
in [4] that a Gaussian probability distribution with support on the real axis
can be rigorously justified only if the standard uncertainty is sufficiently
small with respect to the range of variation of the quantity. More in details,
if we have any information about the range of variation, then this information
should be employed in deriving the distribution maximizing the entropy as well
as in evaluating the values of the coefficients
$\\{\lambda_{0},\lambda_{1},\lambda_{2}\\}$ of the distribution.
Let us denote ${\mathbbm{B}}\subset{\mathbbm{R}}$ the range of the quantity
$X$, i.e., the subset of the real line where the values of $X$ have nonzero
probability to occur. The functional form of the distribution is still given
by the exponential function in Eq. (2), however with nonzero support only in
${\mathbbm{B}}$, whereas the coefficients are to be determined by formulas
like those in Eq. (3), again with $\mathbbm{R}$ replaced by $\mathbbm{B}$. It
then follows, e.g., that for a variable which is known a priori to lie in a
given interval, the maximum entropy distribution is not Gaussian, and the
Gaussian approximation may be employed only if the standard deviation is small
compared to range of the possible values of the quantity.
Here we point out that having information about the range of variation may be
expressed as a bias of the distribution toward a flat distribution in that
range and the reasoning presented in [4] may be subsumed by the minimum
Kullback entropy principle (mKE) [5, 6, 7]. The Kullback entropy, or relative
entropy, or Kullback-Leibler divergence, of two distributions $p(x)$ and
$q(x)$ reads:
$K[p|q]=\int_{\mathbbm{R}}\\!\\!dx\,p(x)\log\left[p(x)/q(x)\right].$ (5)
According to the mKE, in order to find the distribution $p(x)$ given a bias
toward $q(x)$, we should minimize the function:
${\cal
K}[p]=K[p|q]+\sum_{k=0}^{2}\lambda_{k}\left[\int_{\mathbbm{R}}\\!\\!dx\,p(x)\,x^{k}-M_{k}\right],$
(6)
with respect to the function $p(x)$, obtaining:
$p(x)=q(x)\exp\\{-\lambda_{0}-\lambda_{1}x-\lambda_{2}x^{2}\\},$ (7)
where the parameters $\lambda_{k}$ can be still (numerically) computed by
using Eq. (3). Eq. (7) represents the probability distribution satisfying the
given constraints, but with a bias toward the distribution $q(x)$, which, for
instance, may contains the information about the range of the variable $x$.
This information, which in the case of the MaxEnt is not explicitly taken into
account, now it is naturally considered from the beginning. Remarkably, this
is a different scenario from that covered in GUM-S1, i.e., when further
information on the quantity is available, namely, the interval of values
within which the quantity is known to lie is finite.
Indeed, as mentioned above, if the standard uncertainty is sufficiently small
with respect to the range of variation of the quantity, we can adopt a
Gaussian probability distribution over the whole real axis and, thus, use the
GUM-S1 recommendation. In order to rigorously justify this statement, which
has been qualitatively addressed in [4], we assess quantitatively how the
knowledge of the range of variation influences the assignment of a probability
distribution by considering the deviation of the mKE distribution from a
Gaussian distribution, which would represents the MaxEnt solution in the
absence of any information about the range of variation. The deviation from
normality of the mKE distribution (7) may be quantified by its negentropy [8]:
$N[p]=\mbox{$\frac{1}{2}$}\left[1+\log\left(2\pi\sigma^{2}_{\scriptstyle
X}\right)\right]-S[p]\,,$ (8)
where $S[p]$ is the Shannon entropy (1) of the distribution (7). As for
example, for a variable known to lie in a given interval
$[a,b]\subset{\mathbbm{R}}$, $a<b$, that corresponds to a bias of $p(x)$
toward the flat distribution:
$q(x)=\left\\{\begin{array}[]{ll}(b-a)^{-1}&\mbox{if $x\in[a,b]$}\\\
0&\mbox{otherwise}\end{array}\right.\,,$ (9)
the negentropy (8) reads:
$N[p]=\mbox{$\frac{1}{2}$}\left[1+\log\left(2\pi\sigma^{2}_{\scriptstyle
X}\right)\right]-\log\left(b-a\right)-\lambda_{0}-\lambda_{1}\bar{x}-\lambda_{2}(\sigma_{\scriptstyle
X}^{2}+\bar{x}^{2})\,.$ (10)
In the simplest case, namely when $\bar{x}=0$ and $x\in[-a,a]$, the dependence
of the coefficients $\lambda_{0}$ and $\lambda_{2}$ is such that we have a
scaling law for negentropy, which depends only on the ratio
$a/\sigma_{\scriptstyle X}$. This is illustrated in Fig. 1, where we report
the negentropy as a function of $a/\sigma_{\scriptstyle X}$ for different
values of $\sigma_{\scriptstyle X}$.
Figure 1: Scaling of the negentropy of mKE distribution for zero mean value
and variable known to lie in a symmetric interval $[-a,a]$. We report the
negentropy of the distribution as a function of the ratio
$a/\sigma_{\scriptstyle X}$ for different values of the variance:
$\sigma_{\scriptstyle X}^{2}=0.5$ (green squares), $1$ (red circles), $2$
(blue triangles).
In conclusion, we have shown that the determination of the probability
distribution of a variable for which we know the first two moments and its
range of variation may be effectively pursued by using the mKE. Furthermore,
the negentropy of the distribution may be used to quantify how much the mKE
solution differs from the MaxEnt one, i.e. to assess how the knowledge of the
range of variation influences the assignment of a probability distribution.
Our analysis quantitatively supports the conclusions of Ref. [4] and
rigorously justifies the use of GUM-S1 recommendation also in the presence of
further information on the range of variation of a quantity, namely, provided
that its standard uncertainty is sufficiently small compared to the range.
## Acknowledgments
The authors thank M. Genovese and I. P. Degiovanni for useful discussions.
This work has been supported by MIUR (FIRB “LiCHIS” - RBFR10YQ3H), MAE
(INQUEST), and the University of Trieste (FRA 2009).
## References
## References
* [1] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML 2008 Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of distributions using a Monte Carlo method Joint Committee for Guides in Metrology, JCGM 101 http://www.bipm.org/utils/common/documents/jcgm/JCGM_101_2008_E.pdf
* [2] Jaynes E T 1957 Phys. Rev. 106 620; Jaynes E T 1957 Phys. Rev. 108 171
* [3] Wöger W 1987 IEEE Trans. Instr. Measurement IM-36 655658
* [4] Lira I 2009 Metrologia 46 L27
* [5] Kullback S Information theory and statistics (Wiley, New York, 1959)
* [6] Jaynes E T 1968 IEEE Trans. Systems Science and Cybernetics SSC-4 227
* [7] Olivares S and Paris M G A 2007 Phys. Rev. A 76 042120
* [8] Hyvarinen A 1998 Adv. Neural Inf. Proc. Syst. 10 273; Hyvarinen A and Oja E 2000 Neural Networks 13 411
|
arxiv-papers
| 2012-03-13T20:05:18 |
2024-09-04T02:49:28.622064
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefano Olivares and Matteo G. A. Paris",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1203.2934"
}
|
1203.2956
|
# Optical interferometry in the presence of large phase diffusion
Marco G. Genoni QOLS, Blackett Laboratory, Imperial College London, London
SW7 2BW, UK Stefano Olivares Dipartimento di Fisica, Università degli Studi
di Milano, I-20133 Milano, Italy CNISM, UdR Milano Statale, I-20133 Milano,
Italy. Dipartimento di Fisica, Università degli Studi di Trieste, I-34151
Trieste, Italy Davide Brivio Dipartimento di Fisica, Università degli Studi
di Milano, I-20133 Milano, Italy Simone Cialdi Dipartimento di Fisica,
Università degli Studi di Milano, I-20133 Milano, Italy INFN, Sezione di
Milano, I-20133 Milano, Italia Daniele Cipriani Dipartimento di Fisica,
Università degli Studi di Milano, I-20133 Milano, Italy Alberto Santamato
Dipartimento di Fisica, Università degli Studi di Milano, I-20133 Milano,
Italy Stefano Vezzoli Dipartimento di Fisica, Università degli Studi di
Milano, I-20133 Milano, Italy Matteo G. A. Paris
matteo.paris@fisica.unimi.it Dipartimento di Fisica, Università degli Studi di
Milano, I-20133 Milano, Italy CNISM, UdR Milano Statale, I-20133 Milano,
Italy.
###### Abstract
Phase diffusion represents a crucial obstacle towards the implementation of
high precision interferometric measurements and phase shift based
communication channels. Here we present a nearly optimal interferometric
scheme based on homodyne detection and coherent signals for the detection of a
phase shift in the presence of large phase diffusion. In our scheme the
ultimate bound to interferometric sensitivity is achieved already for a small
number of measurements, of the order of hundreds, without using nonclassical
light.
###### pacs:
07.60.Ly, 42.87.Bg
## I Introduction
Optical interferometry represents a high accurate measurement scheme with wide
applications in many fields of science and technology Cav81 ; r1 ; r2 ; r3 ;
r4 . Besides, the precise estimation of an optical phase shift is relevant for
optical communication schemes where information is encoded in the phase of
travelling pulses. Several experimental protocols have been proposed and
demonstrated to estimate the value of the optical phase Armen02 ; Mitch04 ;
Nag07 ; Res07 ; Hig07 ; Hig09 and showing the possibility to attain the so-
called Heisenberg limit Z92 ; Sam92 ; Lane93 ; San95 ; Eck06 ; Giov046 ; Guo08
; Sme08 ; Gro11 ; Hay11 ; Bra11 . Recent developments also revealed the
potential advantages of nonlinear interactions Boi08 . However, in realistic
conditions, one has to retrieve phase information that has been unavoidably
degraded by different sources of noise, which have to be taken into account in
order to evaluate the interferometric precision Gio11 . The effects of
imperfect photodetection in the measurement stage, or the presence of
amplitude noise in the interferometric arms have been extensively studied
Par95 ; Cam03 ; Huv08 ; Coo10 ; BanPhNat ; BanPhPRL ; Cab10 ; Joo11 ; DurkinPh
; Bah11 ; Dat11 . Only recently, the role of phase-diffusive noise in
interferometry have been theoretically investigated for optical polarization
qubit Bri10 ; Ber10 ; tes:11 , condensate systems BEC1 ; BEC2 , Bose-Josephson
junctions Fer10 , and Gaussian states of light Gen11 . As a matter of fact,
phase-diffusive noise is the most detrimental for interferometry and any
signal that is unaffected by phase-diffusion, is also invariant under a phase
shift, and thus totally useless for phase estimation.
In this paper, we present an experimental interferometric scheme where phase
diffusion may be inserted in a controlled way, and demonstrate that homodyne
detection and coherent signals are nearly optimal for the detection of a phase
shift in the presence of large phase diffusion. Indeed, while in ideal
conditions squeezed vacuum is the most sensitive Gaussian probe state for a
given average photon number Mon06 , for large phase-diffusive noise, coherent
states become the optimal choice, outperforming squeezed states Gen11 . In our
scheme the ultimate bound to interferometric sensitivity, as dictated by the
Cramér-Rao (CR) theorem, is achieved already for a small number of repeated
measurements, of the order of hundreds, using Bayesian inference on homodyne
data and without the need of nonclassical light.
The paper is structured as follows: In Section II we describe the evolution of
a light beam in a phase diffusing environment as well as the bound to
interferometric precision in the presence of phase noise. In Section III we
describe our experimental apparatus, whereas the experimental results are
reported and discussed in Section IV. Section V closes the paper with some
concluding remarks.
## II Interferometry in the presence of phase diffusion
The evolution of a light beam in a phase diffusing environment is described by
the master equation
$\dot{\varrho}=\Gamma\mathcal{L}[a^{{\dagger}}a]\varrho\,,$
where $\mathcal{L}[O]\varrho=2O\varrho O^{\dagger}-O^{\dagger}O\varrho-\varrho
O^{\dagger}O$ and $\Gamma$ is the phase damping rate. An initial state
$\varrho_{0}$ evolves as
$\varrho_{t}=\mathcal{N}_{\Delta}(\varrho_{0})=\sum_{n,m}e^{-\Delta^{2}(n-m)^{2}}\varrho_{n,m}|n\rangle\langle
m|\,,$
where $\Delta\equiv\Gamma t$, and $\varrho_{n,m}=\langle
n|\varrho_{0}|m\rangle$. The diagonal elements are left unchanged, in fact
energy is conserved, whereas the off-diagonal ones are progressively
destroyed, together with the phase information carried by the state. Phase
diffusion corresponds to the application of a random, zero-mean Gaussian-
distributed phase shift, i.e.,
$\displaystyle\varrho_{t}=\int_{\mathbbm{R}}\\!d\beta\,g(\beta|\Delta)U_{\beta}\varrho_{0}U_{\beta}^{{\dagger}}\qquad
g(\beta|\Delta)=\frac{e^{-\beta^{2}/(4\Delta^{2})}}{\sqrt{4\pi\Delta^{2}}}$
(1)
where $U_{\beta}=\exp\\{-i\beta(a^{\dagger}a)\\}$ is the phase shift operator.
We assume that the phase noise occurs between the application of the phase
shift and the detection of the signal, and consider the estimation of a phase
shift applied to a single-mode coherent state. Homodyne detection is then
performed on the output state
$\varrho_{\Delta,\alpha}(\phi)=\mathcal{N}_{\Delta}(U_{\phi}|\alpha\rangle\langle\alpha|U_{\phi}^{\dagger})\,,$
and the value of the unknown phase shift $\phi$ is inferred using Bayesian
estimation applied to homodyne data. Notice, however, that since the phase
noise map and the phase shift operation commute, our results are valid also
when the phase shift is applied to an already phase-diffused coherent state.
The precision of the above procedure is then compared with the benchmarks
given by i) the quantum CR bound for coherent states and any quantum limited
kind of measurement, ii) the ultimate precision achievable with optimized
Gaussian states, i.e., the quantum CR bound for general Gaussian signals,
where, e.g., we allow for squeezing.
### II.1 Interferometric precision in the presence of phase noise
The quantum CR bound Mal9X ; BC9X ; Bro9X ; LQE ; Dav11 is obtained starting
from the Born rule $p(x|\phi)=\hbox{Tr}[\Pi_{x}\varrho_{\phi}]$ where
$\\{\Pi_{x}\\}$ is the operator-valued measure describing the measurement and
$\varrho_{\phi}$ the density operator of the family of phase-shifted states
under investigation. Upon introducing the (symmetric) logarithmic derivative
$L_{\phi}$ as the operator satisfying
$2\partial_{\phi}\varrho_{\phi}=L_{\phi}\varrho_{\phi}+\varrho_{\phi}L_{\phi}\,,$
one proves that the ultimate limit to precision (independently on the
measurement used) is given by the quantum CR bound
$\textrm{Var}(\phi)\geq[MH(\phi)]^{-1}\,,$
where $H(\phi)=\hbox{Tr}[\varrho_{\phi}\,L_{\phi}^{2}]$ is the quantum Fisher
information (QFI). The ultimate sensitivity of an interferometer thus depends
on the family of signals used to probe the phase shift and thus, as said
above, we are going to compare the precision of our interferometer with the
maximum achievable with coherent states, and with the ultimate precision
achievable with optimized Gaussian states (for more details about the
derivation of the corresponding quantum CR bounds see Gen11 ).
Homodyne detection measures the field quadrature
$x_{\theta}=\frac{1}{2}(ae^{-i\theta}+a^{\dagger}e^{i\theta})\,,$
where $\theta=\arg\alpha+\pi/2$ is set to the optimal value to detect the
imposed phase shift. The likelihood of a set of homodyne data
$X=\\{x_{1},x_{2},\ldots,x_{M}\\}\,,$
is the overall probability of the sample given the unknown phase $\phi$, i.e.,
$L(X|\phi)=\prod_{k=1}^{M}p(x_{k}|\phi)\,,$
where
$\displaystyle
p(x|\phi)=\frac{e^{-2x^{2}}}{\pi\Delta}\int_{\mathbbm{R}}\\!\\!d\beta\>e^{-\frac{\beta^{2}}{2\Delta^{2}}+4\alpha
x\cos(\beta+\phi)-2\alpha^{2}\cos^{2}(\beta+\phi)}\,.$
Assuming that no a priori information is available on the value of the phase
shift (i.e., uniform prior), and using the Bayes theorem, one can write the a
posteriori probability
$\displaystyle P(\phi|X)=\frac{1}{\cal N}\,L(X|\phi)\qquad{\cal
N}=\int_{\Phi}d\phi\,L(\phi|X)\,,$ (2)
$\Phi=[0,\pi]$ being the parameter space. The probability $P(\phi|X)$ is the
expected distribution of $\phi$ given the data sample $X$. The Bayesian
estimator $\phi_{\rm B}$ is the mean of the a posteriori distribution, whereas
the sensitivity of the overall procedure corresponds to its variance
${\rm Var}[\phi_{\rm B}]=\int_{\Phi}d\phi\,(\phi-\phi_{\rm
B})^{2}\,P(\phi|X)\,.$
Bayesian estimators are known to be asymptotically unbiased and optimal,
namely, they allow one to achieve the CR bound as the size of the data sample
increases H9X ; O09 . On the other hand, the number of data needed to achieve
the asymptotic region may depend on the specific implementation Bar00 . In the
following we will experimentally show that our setup achieves optimal
estimation already after collecting few hundreds of measurements.
## III Experimental apparatus
A schematic diagram of the interferometer is reported in Fig. 1. The principal
radiation source is provided by a He:Ne laser (12 mW, 633 nm) shot-noise
limited above 2 MHz. The laser emits a linearly polarized beam in a TEM00
mode. The beam is splitted into two parts of variable relative intensity by a
combination of a halfwave plate (HWP) and a polarizing beam splitter (PBS).
The strongest part is sent directly to the homodyne detector where it acts as
the local oscillator, whereas the ramaining part is used to encode the signal
and will undergo the homodyne detection. The optical paths travelled by the
local oscillator and the signal beams are carefully adjusted to obtain a
visibility typically above 90% measured at one of the homodyne output ports.
The signal is amplitude modulated at 4 MHz with a defined modulation depth to
control the average number of photons in the generate state.
The amplitude modulation system consist of a KDP non-linear crystal with the
$xy$ axes at 45∘, and a PBS. The modulation is applied at the KDP crystal by
means a waveform generator Rohde & Schwarz and a power amplifier Mini-Circuits
ZHL-32A. The modulation depth is imposed at the proper level by a computer
that sends a costant voltage to a mixer (M1) located between the waveform
generator and the power amplifier. One of the mirrors in the signal path is
piezo mounted to obtain a variable phase difference between the two beams. The
piezo is preloaded and its resonance frequency is 13.5 kHz.
The phase difference is controlled by the computer after a calibration stage.
The computer sends a voltage signal between 0 and 10 V that corresponds at the
phase diffusion with a frequency of 5 kHz to a power amplifier based on LM675
integrated circuit that is able to drive the piezo at this frequency. With
this system it is possibile to generate any kind of phase modulation.
Figure 1: (color online). Schematic diagram of the experimental setup. A He:Ne
laser is divided into two beams, one acts as the local oscillator and the
other represents the signal beam. The signal is modulated at $4$MHz with a
defined modulation depth to control the average number of photons in the
generate state. One of the mirrors in the signal path is piezo mounted to
obtain a variable phase difference between the two beams. The data are
recorded by a homodyne detector whose difference photocorrent is demodulated
and then acquired by a computer after a low pass filter. We also show the
typical homodyne samples obtained for coherent signals of different amplitudes
by varying the phase of the local oscillator (these are used to check the
calibration of the piezo, which is performed using signals with a larger
number of photons).
The detector is composed by a 50:50 beams splitter (BS) and a balanced
amplifier detector with a bandwidth of 50 MHz. The difference photocurrent is
filtered with high pass filters, amplified and demodulated at 4 MHz by means
of an electrical mixer (M2). In this way the detection occurs outside any
technical noise and, more importantly, in a spectral region where the laser
does not carry excess noise. The signal is filterd by a low pass filter with a
bandwidth of 300 kHz and sent to the computer through the National Instrument
multichannel data acquisition 6251 with 16 bit of resolution and 1.25 MS/s
sampling rate. The same device is used to send diffusion parameters to the
phase modulator and signal parameters to the amplitude modulator.
## IV Experimental results
In this Section, we present our experimental results, obtained with signals of
different energies and different levels of noise. At first we show homodyne
samples with the corresponding a posteriori distributions and then compare the
precision obtained in our scheme with the ultimate bound imposed by the
(quantum) Cramér-Rao theorem. Finally, we analyze the dependence of precision
on the signal energy and the noise in order to illustrate how in the limit of
large phase diffusion coherent states becomes the optimal Gaussian probe
states. In fact, they outperform squeezed vacuum states, whose non-classical
features are degraded by phase diffusion process, to an extent that make them
useless for quantum metrology.
In Fig. 2 we report typical examples of homodyne samples, referred to a
coherent signal with $N=|\alpha|^{2}$ mean photon number measured at fixed
optimal $\theta$, together with the corresponding Bayesian a posteriori
distribution for the phase shift. The yellow area denotes the portion of data
used to infer the phase shift. We choose this range in order to emphasize that
the optimality region in achieved already in that region. In fact, upon
considering larger samples, precision would be improved, due to the
statistical scaling of the variance ${\rm Var}[\phi]=C/M$, $C$ being a
proportionality constant. On the other hand, optimality, i.e., the fact that
$C\simeq 1/H_{\alpha}\,,$
where $H_{\alpha}$ is the QFI for phase-diffused coherent signals, is achieved
for $M\sim 100$ measurements. In the noiseless case the QFI is given by
$H_{\alpha}=4N$, whereas it decreases monotonically by increasing the value of
the noise parameter $\Delta$. Notice that using optimized Gaussian signals,
i.e. the squeezed vacuum state, one has a QFI given by $H_{g}=8N^{2}+8N$ in
the noiseless case. However, in the presence of large phase diffusion, i.e.
for large values of $\Delta$, $H_{\alpha}$ is larger than the QFI obtained for
phase-diffused squeezed vacuum states. In other words, coherent states turns
out to be the optimal Gaussian probe states Gen11 .
Figure 2: (color online) Typical examples of homodyne samples measured at
fixed optimal $\theta$, together with the corresponding Bayesian a posteriori
distribution for the phase shift. The phase diffusion is $\Delta=\pi/6$ rad
and the yellow area denotes the portion of data used to infer the phase shift.
In Fig. 3 we plot the quantity
$K_{M}=M\,{\rm Var}[\phi_{\rm B}]H_{\alpha}\,,$
i.e., the variance of the Bayesian estimator from homodyne data multiplied by
the number of data (measurements) and by the coherent states quantum Fisher
information, as a function of $M$. $K_{M}$ is by definition larger than one
and expresses the ratio between the actual precision of the interferometric
setup and the CR bound. As it is apparent from the plot $K_{M}$ rapidly
decreases with the number of measurements, almost independently on the value
of the number of photons $N$ and of the noise parameter $\Delta$. The
optimality region, i.e., $K_{M}\simeq 1$ is achieved already for $M\simeq 100$
measurements, and the asymptotic value of $K_{M}$ is closer to 1 for
increasing $N$ and $\Delta$. Furthermore, the number of measurements needed to
achieve the optimal region may be (slightly) reduced by using the Jeffreys
prior Jpp
$p(\phi)\propto\sqrt{F(\phi)}\,$
instead of the uniform one, where
$F(\phi)=\int\\!dx\,p(x|\phi)[\partial_{\phi}\log p(x|\phi)]^{2}$
is the Fisher information of the homodyne distribution.
Figure 3: (color online) The noise ratio $K_{M}=({\rm Var}[\phi_{\rm
B}]MH_{\alpha})$ as a function of the number of data $M$ and for different
values of the number of photons $N$ and the noise parameter $\Delta$. Blue
circles: $N=0.90$, $\Delta=\pi/18$ rad; red squares: $N=0.90$, $\Delta=\pi/9$
rad; yellow diamonds: $N=4.12$, $\Delta=\pi/18$ rad; green triangles:
$N=4.12$, $\Delta=\pi/9$ rad.
In Fig. 4 we show the variance of the Bayesian estimator from homodyne data
$V_{M}=M{\rm Var}[\phi_{\rm B}]$
obtained after $M$ measurements, together with the CR bound $1/H_{\alpha}$ for
coherent states, and for the (phase-diffused) optimized Gaussian states, i.e.,
$1/H_{g}$. In particular, the top panel shows the behaviour as a function of
$\Delta$ for different values of the number of photons $N$, while in the
bottom panel we plot the same quantities as a function of the number of
photons $N$ and for different values of the noise $\Delta$. As it is apparent
from the plots, nearly optimal inferferometric precision is achieved for
increasing energy or phase diffusion, i.e., for larger values of $N$ or
$\Delta$.
## V Conclusions
In conclusion, we have demonstrated a nearly optimal interferometric scheme
based on homodyne detection and coherent signals for the detection of a phase
shift in the presence of large phase diffusion. Our scheme does not require
nonclassical light and achieve the ultimate bound to interferometric
sensitivity using Bayesian analysis on small samples of homodyne data, where
the number of measurements is of the order of few hundreds.
It is worth noting that for large phase diffusion coherent states are the
optimal Gaussian probe states. Indeed they outperform squeezed vacuum states,
whose non-classical features are degraded by phase diffusion process, such
that they become completely useless for quantum metrology.
Optical interferometry represents a high accurate measurement scheme with wide
applications in many fields of science and technology, including high
precision measurements and communication channels. On the other hand, phase
diffusion represents a crucial obstacle towards the implementation of high
precision interferometric measurements and phase shift based communication
channels. Our results allow to design feasible, high-performance,
communication channels also in the presence of phase noise, which cannot be
effectively controlled in realistic conditions. Therefore, besides fundamental
interest, our results also represent a benchmark for realistic phase based
communication or measurement protocols.
Figure 4: (color online) Variance $V_{M}=M{\rm Var}[\phi_{\rm B}]$ of the
Bayesian estimator from homodyne data after $M=100$ measurements (points),
together with the CR bound for coherent states (solid lines) and for optimized
Gaussian states (dashed lines). The top panel of shows the behaviour of
$V_{100}$ as a function of $\Delta$ for different values of the number of
photons (top red lines/squares: $N=0.90$; bottom blue lines/circles:
$N=14.11$). The bottom panel shows $V_{100}$ as a function of the number of
photons $N$ and for different values of the noise (top blue lines/squares:
$\Delta=\pi/9$ rad; bottom green lines/circles: $\Delta=\pi/18$ rad).
## Acknowledgements
This work has been supported by MIUR (FIRB “LiCHIS” - RBFR10YQ3H), the UK
EPSRC (EP/I026436/1), MAE (INQUEST), UniMi (PUR2009 SIN.PHO.NANO), UIF/UFI
(Vinci Program), and the University of Trieste (FRA2009).
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|
arxiv-papers
| 2012-03-13T21:06:12 |
2024-09-04T02:49:28.627711
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco G. Genoni, Stefano Olivares, Davide Brivio, Simone Cialdi,\n Daniele Cipriani, Alberto Santamato, Stefano Vezzoli, and Matteo G. A. Paris",
"submitter": "Matteo G. A. Paris",
"url": "https://arxiv.org/abs/1203.2956"
}
|
1203.3083
|
# Clustering in networks with the collapsed Stochastic Block Model
Aaron F. McDaid111Correspondence to: CASL UCD, 8 Belfield Office Park,
Clonskeagh, Dublin 4, Ireland. _Email:_ aaronmcdaid@gmail.com. _Tel:_
+35385775686 , Thomas Brendan Murphy, Nial Friel and Neil J. Hurley
Clique Research Cluster
University College Dublin
(September 22 2012)
###### Abstract
An efficient MCMC algorithm is presented to cluster the nodes of a network
such that nodes with similar role in the network are clustered together. This
is known as _block-modelling_ or _block-clustering_. The model is the
stochastic blockmodel (SBM) with block parameters integrated out. The
resulting marginal distribution defines a posterior over the number of
clusters and cluster memberships. Sampling from this posterior is simpler than
from the original SBM as transdimensional MCMC can be avoided. The algorithm
is based on the _allocation sampler_. It requires a prior to be placed on the
number of clusters, thereby allowing the number of clusters to be directly
estimated by the algorithm, rather than being given as an input parameter.
Synthetic and real data are used to test the speed and accuracy of the model
and algorithm, including the ability to estimate the number of clusters. The
algorithm can scale to networks with up to ten thousand nodes and tens of
millions of edges.
###### keywords:
Clustering , Social networks , Blockmodelling , Computational Statistics ,
MCMC.
††journal: Computational Statistics and Data Analysis
## 1 Introduction
This paper is concerned with _block-modelling_ – an approach to clustering the
nodes in a network, based on the pattern of inter-connections between them.
The starting point for the method presented here is the _stochastic block
model_ (SBM) Nowicki and Snijders (2001). The goal is to improve the speed and
scalability, without compromising on accuracy. We use conjugate priors and
integration in order to focus on the marginal distribution of interest, this
marginalization is also referred to as the ‘collapsing’ of the nuisance
parameters (Liu, 1994; Wyse and Friel, 2012). This allows us to implement an
efficient algorithm based on the _allocation sampler_ of Nobile and Fearnside
(2007). We incorporate existing extensions, such as the weighted-edge model of
Mariadassou et al. (2010), and show how this extension can be incorporated
within our collapsing and within our algorithm. As required by the _allocation
sampler_ , we place a prior on the number of clusters, allowing the number of
clusters to be directly estimated. Together, these techniques allow us to
avoid the more complex forms of transdimensional MCMC and they also allow us
to avoid the need for post-hoc model selection via criteria such as the ICL.
We show that our method can accurately and efficiently estimate the number of
clusters – an improvement over many existing methods. Our algorithm, and the
data we have used in 6.4 and our survey data used in section 7, are available
at http://sites.google.com/site/aaronmcdaid/sbm.
The concept of clustering is broad and originated outside of network analysis,
where the input data is in the form of real-valued vectors describing the
location of the data points in a Euclidean space. Network clustering takes a
set of connected nodes as input and finds a partition of the nodes based on
the network structure. This finds application in many different contexts. For
instance, in bio-informatics, networks of protein-protein interactions are
analysed and clustering is applied to find functional groups of proteins.
Interest in social network analysis has grown greatly in recent years, with
the availability of many networks, such as Facebook datasets, of human
interactions. Clustering of such social networks has been applied in order to
find social communities. In the following, we will distinguish the community-
finding problem from the more general setting of block-modelling.
In network analysis, the input data may be described mathematically as a
graph, which is a set of nodes (where each node represents an entity, say, a
person) and a set of edges linking pairs of nodes together. An edge might
represent a friendship on Facebook or a phone call on a mobile phone network.
In section 7, we apply our method to the network of interactions between
participants at a summer school.
Given a network, the goal in block-modelling is to cluster the nodes such that
pairs of nodes are clustered together if their connectivity pattern to the
clusters in the rest of the network is similar. A cluster might, for example,
consist of a set of nodes which do not tend to have connections among
themselves at all. Given two nodes in this cluster (node $i$ and node $j$),
the neighbours of $i$ tend to be in the same clusters as are the neighbours of
$j$. Community-finding has focussed on finding clusters of high internal edge
density, where an edge between two nodes will tend to pull the two nodes into
the same cluster, and a non-edge will tend to push them into separate
clusters. This contrasts with block-modelling, which allows clusters to have
_low_ internal edge density. Block-modelling is able to find such community
structure, but it is a more general method that is also able to find other
types of structure.
A variety of other, non-probabilistic, approaches have been used to tackle the
broad problem of block-modelling (Everett, 1996; Chan et al., 2011). Outside
of block-modelling, there are other solutions for community-finding in
networks (Newman and Girvan, 2004; Girvan and Newman, 2002; Newman, 2004).
Many probabilistic clustering models have also been applied (Handcock et al.,
2007; Hoff et al., 2002; Airoldi et al., 2008).
There is a huge variety of methods, and we will not attempt to summarize them
further; for the rest of this paper, we will focus on the SBM and on
algorithms for the SBM. For more details, in particular about community
finding, see the excellent review article of Fortunato (2010).
The remainder of the paper is structured as follows. In section 2, we define
the SBM and define the notation used in the paper. We then define, in section
3, the conjugate priors and integration that we use in order to access the
relevant marginal distribution. Section 4 discusses other closely-related
models and algorithms and in particular gives consideration to the issue of
how to select the number of clusters (model selection), comparing the approach
we have used to other approaches and noting connections among the methods.
Section 5 describes the algorithm we use; without collapsing, it would have
been necessary to use full Reversible Jump MCMC (Green (1995)) to search a
sample space of varying dimension and this could be much slower.
In section 6, we evaluate our method on synthetic networks, showing how the
number of clusters can be estimated accurately and the nodes assigned to their
correct cluster with high probability. We also test the scalability and
efficiency of the algorithm by considering synthetic datasets with ten
thousand nodes and ten million edges.
In section 7, we evaluate our method on a dataset of interactions, gathered by
a survey, of participants at a doctoral summer school attended by one of the
authors of this paper. The method is able to detect interesting structures,
demonstrating the differences between _block-modelling_ and _community-
finding_. Section 8 draws some conclusions.
## 2 Stochastic Block Model(SBM)
As formulated in Nowicki and Snijders (2001), a network describes a relational
structure on a set of nodes. Each edge in the network describes a relationship
between the two nodes it links. A general case of a finite alphabet of states
relating a pair of nodes is considered but in the simplest case, discussed by
the same authors in Snijders and Nowicki (1997), relationships are binary – an
edge joining a pair of nodes either exists or not. The network can be
undirected, corresponding to symmetric relationships between the nodes, or may
be _directed_ , where a relationship from node $i$ to node $j$ does not
necessarily imply the same relationship exists from node $j$ to node $i$.
Finally, a _self-loop_ – a relationship from node to itself – may or may not
be allowed.
Throughout the paper, we use $\mathrm{P}(\cdot)$ to refer to probability mass
(i.e. of discrete quantities) and $\mathrm{p}(\cdot)$ to refer to probability
density (i.e. of continuous quantities). $N$ is the number of nodes in the
network and $K$ is the number of clusters. In the algorithm proposed in
Nowicki and Snijders (2001), these are given input values, although in our
approach, we treat $K$ as a random variable with a given prior distribution.
Given $N$ and $K$, the SBM describes a random process for assigning the nodes
to clusters and then generating a network. Specifically, the cluster
memberships are represented by a random vector $z$ of length $N$ such that
$z_{i}\in\\{1,\dots,K\\}$ records the cluster containing node $i$. $z_{i}$
follows a multinomial distribution,
$z_{i}\overset{iid}{\sim}\text{Multinomial}(1;\theta_{1},\dots,\theta_{K})\,,$
such that $\theta_{i}$ is the probability of a node being assigned to cluster
$i$ ($1=\sum_{k=1}^{K}\theta_{k}$). The vector $\theta$ is itself a random
variable drawn from a Dirichlet prior with dimension $K$. The parameter to the
Dirichlet is a vector $(\alpha_{1},\dots,\alpha_{K})$ of length $K$. We follow
Nowicki and Snijders (2001) by fixing the components of this vector to a
single value $\alpha$, and by default $\alpha=1$,
$\theta\sim\text{Dirichlet}(\alpha_{1}=\alpha,\alpha_{2}=\alpha,\dots,\alpha_{K}=\alpha)\,.$
This describes fully how the $N$ nodes are assigned to the $K$ clusters. Next
we describe how, given this clustering $z$, the edges are added between the
nodes.
A network can be represented as an $N\times N$ adjacency matrix, $x$, such
that $x_{ij}$ represents the relation between node $i$ and node $j$ (taking
values 1 or 0 in the binary case). Denote by $x_{(kl)}$ the submatrix
corresponding to the _block_ of connections between nodes in cluster $k$ and
nodes in cluster $l$. If the network is undirected, there are
$\frac{1}{2}K(K+1)$ blocks, corresponding to each pair of clusters; and if the
network is directed, there are $K^{2}$ clusters, corresponding to each
_ordered_ pair of clusters.
It is generally simpler to discuss the directed model; unless otherwise
stated, the formulae presented here apply only to the directed case. The
definitions and derivations can easily be applied to the undirected case,
provided that care is taken only to consider each pair of nodes exactly once.
If self-loops are not allowed, then the diagonal entries of $x$, $x_{ii}$, are
excluded from the model. It is assumed that, given $K$ and $z$, connections
are formed independently within a block so that
$P(x|z,K,\pi)=\prod_{k.l}P(x_{(kl)}|z,K,\pi_{kl})\,,$
where
$P(x_{(kl)}|z,K,\pi_{kl})=\prod_{\\{i|z_{i}=k\\}}\prod_{\\{j|z_{j}=l\\}}P(x_{ij}|z,K,\pi_{kl})\,,$
and the matrix $\pi=\\{\pi_{kl}\\}$ describes the cluster-cluster
interactions. $\pi$ is a $K\times K$ matrix, but for undirected networks only
the diagonal and upper triangle are relevant. Specifically, for binary
networks, $\pi_{kl}$ represents the edge density within the block, and edges
follow the Bernoulli distribution,
$x_{ij}|z,K,\pi\sim\mathrm{Bernoulli}(\pi_{z_{i}z_{j}})\,.$
Each of the $\pi_{kl}$ is drawn from the conjugate
$\text{Beta}(\beta_{1},\beta_{2})$ prior,
$\pi_{kl}\overset{iid}{\sim}\text{Beta}(\beta_{1},\beta_{2})\,.$
Again we follow Nowicki and Snijders (2001) and choose
$\beta_{1}=\beta_{2}=1$, giving a Uniform prior.
This completes the description of the Bayesian presentation of the SBM. A
different approach is taken in other work, such as that of Daudin et al.
(2008), where, using essentially the same model, the goal is to take a point
estimate of the parameters, $(\pi,\theta)$, for a given number of clusters
$K$. Specifically, the aim is to find the MLE; the value of
$(\hat{\pi},\hat{\theta})$ which maximizes $\mathrm{P}(x|\pi,\theta,K)$. This
is described as the frequentist approach, in contrast to the fully Bayesian
approach where a distribution of parameter values is allowed instead of a
point estimate. We will return to this issue in a little more detail in
section 4 in order to discuss the practical differences from an algorithmic
point of view.
### 2.1 Data model variations
The model is naturally extended in Nowicki and Snijders (2001) to allow for a
finite alphabet of two or more relational states, where instead of using a
Bernoulli with a Beta prior for $x$ and $\pi$, we can use a Multinomial and a
Dirichlet to model this alphabet. The Bernoulli-and-Beta-prior model is just a
special case of the Multinomial-and-Dirichlet-prior model. Alternatively, we
can allow an infinite support and extend the model to allow for non-negative
integer weights on the edges, by placing a Poisson distribution on
$P(x|\pi,z)$, as seen in Mariadassou et al. (2010). Now $\pi_{kl}$ represents
the edge rate and is drawn from a Gamma prior,
$\begin{split}x_{ij}|z,K,\pi&\sim\text{Poisson}(\pi_{z_{i}z_{j}})\\\
\pi_{kl}&\sim\mbox{Gamma}(s,\phi)\,.\end{split}$
We do not suggest any default for the hyperparameters $s$ and $\phi$. A
further extension to real-valued weights is also possible, by using a Gaussian
for $p(x|\pi,z)$ and suitable prior on $\pi$, following Wyse and Friel (2012).
These variations, and others, are described in Mariadassou et al. (2010), but
they do not discuss conjugate priors.
The integration approach and algorithm described later in this paper can be
applied to many variants of edge model, however we focus in the remainder of
the paper on the Bernoulli and Poisson models that are supported in our
software.
In summary, given $N$ and $K$ the random process generates $\theta$, $z$,
$\pi$ and ultimately the network $x$. The two main variables of interest are
the clustering $z$ and the network $x$. In a typical application, we have
observed a network $x$ and perhaps we have an estimate of $K$, and our goal is
to estimate $z$.
## 3 Collapsing the SBM
In this section, we show how _collapsing_ can be used to give a more
convenient and efficient expression for the model. This refers to the
integration of nuisance parameters from the model, see Wyse and Friel (2012)
for an application to a different, but related, bipartite model. The SBM has
been partially collapsed by Kemp et al. (2004), but we will consider the full
collapsing of both $\pi$ and $\theta$. As our primary interest is in the
clustering $z$ and the number of clusters $K$, we integrate out $\pi$ and
$\theta$, yielding an explicit expression for the marginal
$\mathrm{P}(x,z,K)$. We emphasize that integration does not change the model,
it merely yields a more convenient representation of the relevant parts of the
posterior. This integration is made possible by the choice of conjugate priors
for $\pi$ and $\theta$. We treat $K$ as a random variable and place a Poisson
prior on $K$ with rate $\lambda=1$, conditioning on $K>0$,
$K\sim\mbox{Poisson}(1)\,|\,K>0\,,$ (1)
which gives us
$\mathrm{P}(K)=\frac{\frac{\lambda^{K}}{K!}e^{-\lambda}}{1-\mathrm{P}(K=0)}=\frac{1}{K!(e-1)}\,.$
We are only interested in these expressions as functions of $K$ and $z$ up to
proportionality, as this will be sufficient for our Markov Chain over
$(K,z|x)$, and hence we can simply use $\mathrm{P}(K)\propto\frac{1}{K!}$.
The Poisson prior is used in the _allocation sampler_ , the algorithm upon
which our method is based (Nobile and Fearnside, 2007). This allows the
estimation of the number of clusters as an output of the model rather than
requiring a user to specify $K$ as an input or to to use a more complex form
of model selection. Thus, we have a fully Bayesian approach where, other than
$N$, which is taken as given, every other quantity is a random variable with
specified priors where necessary,
$\begin{split}\mathrm{p}(x,\pi,z,\theta,K)=\mathrm{P}(K)&\times\mathrm{p}(z,\theta|K)\\\
&\times\mathrm{p}(x,\pi|z)\,.\end{split}$ (2)
With eq. 2 we could create an algorithm which, given a network $x$, would
allow us to sample the posterior $\pi,z,\theta,K|x$. However, we are only
interested in estimates of $z,K|x$. We now show how to collapse $\pi$ and
$\theta$.
Define $\mathbb{R}_{+}$ to be the set of non-negative real numbers, and write
the set of real numbers between 0 and 1 as $[0,1]$. Define $\Theta$ the _unit
simplex_ i.e. the subset of $\mathbb{R}_{+}^{K}$ where
$1=\sum_{k=1}^{K}\theta_{k}$. Define $\Pi$ to be the domain of $\pi$. For the
Poisson model this is $\mathbb{R}_{+}^{B}$ while for the Bernoulli model this
is $[0,1]^{B}$, where $B$ is the number of blocks.
We can access the same posterior for $z$ and $K$ by _collapsing_ two of the
factors in eq. 2,
$\begin{split}\mathrm{P}(x,z,K)=\mathrm{P}(K)\times\int_{\Theta}\mathrm{p}(z,\theta|K)\;\mathrm{d}\theta\\\
\times\int_{\Pi}\mathrm{p}(x,\pi|z)\;\mathrm{d}\pi\,,\end{split}$ (3)
or, equivalently, using the block-by-block independence $x_{(kl)}|z,K$,
$\begin{split}\mathrm{P}(x,z,K)=\mathrm{P}(K)\times\int_{\Theta}\mathrm{p}(z,\theta|K)\;\mathrm{d}\theta\\\
\times\prod_{k,l}\int_{\Pi_{kl}}\mathrm{p}(x_{(kl)},\pi_{kl}|z)\;\mathrm{d}\pi_{kl}\,.\end{split}$
(4)
This allows the creation of an algorithm which searches only over $K$ and $z$.
The algorithm never needs to concern itself with $\theta$ or $\pi$.
Collapsing greatly simplifies the sample space over which the MCMC algorithm
has to search. Without collapsing, the dimensionality of the sample space
would change if our estimate of $K$ changed; this would require a Reversible-
Jump Markov Chain Monte Carlo (RJMCMC) algorithm (see Green (1995)). Finally,
if estimates for the full posterior, including $\pi$ and $\theta$, are
required, it should be noted that it is very easy to sample
$\pi,\theta|x,z,K$, meaning that nothing is lost by the use of collapsing.
Many of the other models described in section 4 are collapsible, and this may
be an avenue for future research.
The integration of eq. 4 allows an expression for the full posterior
distribution to be obtained. Details of the derivation of this expression are
given in Appendix A. Let $n_{k}$ be the number of nodes in cluster $k$.
$n_{k}$ is a function of $z$. For the Bernoulli model, let $y_{kl}$ be the
number of edges that exist in block $kl$, i.e. the block between clusters $k$
and $l$. For the Poisson model, $y_{kl}$ is the total edge weight. $y$ is a
function of $x$ and $z$. Let $p_{kl}$ be the maximum number of edges that can
be formed between clusters $k$ and $l$. For off-diagonal blocks,
$p_{kl}=n_{k}n_{l}$. For diagonal blocks, $p_{kk}$ depends on the form of the
network as follows,
$p_{kk}=\left\\{\begin{array}[]{ll}\frac{1}{2}n_{k}(n_{k}-1)&\mbox{undirected,
no self-loops}\\\ \frac{1}{2}n_{k}(n_{k}+1)&\mbox{undirected, self-loops}\\\
n_{k}(n_{k}-1)&\mbox{directed, no self-loops}\\\ n_{k}^{2}&\mbox{directed,
self-loops}\end{array}\right.\,.$ (5)
The full posterior may be written as
$\begin{split}\mathrm{P}(x,z,K)\propto{}&\frac{1}{K!}\\\
&\times\frac{\Gamma(\alpha
K)\prod_{k=1}^{K}\Gamma(n_{k}+\alpha)}{\Gamma(\alpha)^{K}\Gamma(N+\alpha
K)}\\\ &\times\prod f(x_{(kl)}|z)\,,\end{split}$ (6)
where the final product is understood to take place over all blocks. The form
of the function $f(x_{(kl)}|z)$ depends on the edge model. If Bernoulli, then
$f(x_{(kl)}|z)=\frac{\text{B}(\beta_{1}+y_{kl},p_{kl}-y_{kl}+\beta_{2})}{\text{B}(\beta_{1},\beta_{2})}\,,$
(7)
where $B(.,.)$ is the Beta function. If Poisson, then
$f(x_{(kl)}|z)=\frac{\Gamma(s+y_{kl})\left(\frac{1}{p_{kl}+\frac{1}{\phi}}\right)^{s+y_{kl}}}{\Gamma(s)\phi^{s}}\,.$
(8)
## 4 Related estimation procedures for the SBM
Before defining our algorithm, we look at related work, particularly other
methods that are based on the SBM. We will focus on models which are
identical, or very similar to, the SBM. Therefore, we will not discuss other
models which are loosely related, such as that of Newman and Leicht (2007), or
the “degree-corrected” SBM of Karrer and Newman (2011).
All methods discussed here are aimed at estimating $z$, but they differ in the
approach they take to the parameters $\pi$ and $\theta$ and in whether they
allow the number of clusters, $K$, to be estimated. We also discuss the issue
of model selection, i.e. how the various methods estimate the number of
clusters. This question was avoided in the original paper of Nowicki and
Snijders (2001), where the number of clusters is fixed to $K=2$ in the
evaluation.
The method of Daudin et al. (2008) takes a network, $x$, and number of
clusters $K$, and applies a variational algorithm. Point estimates are used
for $\pi$ and $\theta$, but the clustering $z$ is represented as a
distribution of possible cluster assignments for each node. This makes the
method analogous to the EM algorithm for the MLE – finding the pair
$(\pi,\theta)$ which maximizes $\mathrm{P}(x|\pi,\theta,K)$.
The model used by Zanghi et al. (2008) is a subset of the model of Daudin et
al. (2008). The cluster-cluster density matrix, $\pi$, is simplified such that
it is represented by two parameters $\lambda$ and $\epsilon$, such that the
on-diagonal blocks $\pi_{kk}=\lambda$ and the off-diagonal blocks
$\pi_{kl}=\epsilon$ (for $k\neq l$). A Classification EM (CEM) algorithm to
maximize
$\underset{z,\pi,\theta}{\operatorname{argmax}}\;\mathrm{P}(x,z|\pi,\theta,K)$
is briefly described in Zanghi et al. (2008) but not implemented. Instead,
they implement an _online_ algorithm. One node of the network is considered at
a time and is assigned to the cluster which maximizes
$\mathrm{P}(x,z|\pi,\theta,K)$, updating estimates of $\pi$ and $\theta$ with
each addition. Implicitly, their goal is to use point estimates both for the
parameters _and_ for the clustering, to find
$(\hat{z},\hat{\pi},\hat{\theta})$ that would maximize
$\mathrm{P}(x,z|\pi,\theta,K)$; as such, it is loosely related to the profile
likelihood (Bickel and Chen, 2009).
The methods just discussed are based, directly or indirectly, on the
frequentist approach of finding the maximum likelihood estimate of the
parameters, $(\pi,\theta)$, i.e. the values $\hat{\pi},\hat{\theta}$ that
would maximize the likelihood of the observed network,
$\mathrm{P}(x|\pi,\theta,K)=\sum_{z}\mathrm{P}(x,z|\pi,\theta,K)\,.$
The estimate of $z$ that is used in this frequentist approach is the
conditional distribution of $z$ based on this point estimate of the parameters
and on the observed network, $z|x,\hat{\pi},\hat{\theta},K$. In practice
though, it is not tractable to calculate or maximize this likelihood exactly,
and hence a variety of different approximations and heuristics have been used.
In a Bayesian method, such as ours, a distribution of estimates for
$(\pi,\theta)$ is used instead of a point estimate. The goal is to directly
sample from $z|x,K$. Another example of this Bayesian approach is the
variational algorithm used in Hofman and Wiggins (2008), which is based on the
simpler $\lambda$ and $\epsilon$ parameterization of the $\pi$ matrix used in
Zanghi et al. (2008).
The modelling choices of Latouche et al. (2012), where a new model selection
criterion called $ILvB$ is introduced, are essentially identical to the
standard SBM; each element of $\pi_{kl}$ is independent, and conjugate priors
are specified. A variety of other variational approximations are considered by
Gazal et al. (2011), where there is more focus on parameter estimation and
less focus on model selection.
A further specialization of this model is possible, by employing the
$\lambda,\epsilon$ parameterization, but where $\lambda>\epsilon$, which
explicity constrains the expected edge density within clusters to be larger
than the expected edge density between clusters. This can be considered to be
_community-finding_ as opposed to _block-modelling_. The authors of this paper
considered this in McDaid et al. (2012).
### 4.1 Model selection
Later, in our experiments in section 6, we will demonstrate the ability of the
allocation sampler to accurately estimate the number of clusters. In this
subsection, we will briefly discuss some of the theoretical issues around the
estimation of the number of clusters.
The methods that involve the MLE for the parameters involve the risk of
overfitting; for larger values of $K$, the parameter space of $\pi$ and
$\theta$ becomes much larger and therefore the estimates of
$\mathrm{P}_{\theta=\theta_{mle}}(x|K)$ will become over-optimistic, and will
tend to overestimate $K$ (Schwarz, 1978). Therefore, an alternative
formulation such as the ICL is needed; see Zanghi et al. (2008) and Daudin et
al. (2008) for derivations of the ICL in the context of models based on the
SBM. Instead of using the MLE directly, those measures apply priors to the
parameters and integrate over the priors, as described in Biernacki et al.
(2000), such that the average likelihood is used instead of the maximum
likelihood.
Typically, such integrations cannot be performed exactly and the ICL criterion
consists of approximations that are based on first finding an estimate to the
MLE, and then adding correction terms to this MLE. For the rest of this
subsection, we will not consider those approximate methods and will instead
consider the exact solutions to the integrations.
The _integrated classification likelihood_ , which the ICL intends to
approximate,
$\mathrm{P}(x,z|K)=\int\int\mathrm{P}(x,z,\pi,\theta|K)\,\mathrm{d}\pi\,\mathrm{d}\pi\,,$
can be solved exactly in some models. The SBM is one of those models, and the
posterior mass that our algorithm samples from is exactly equal to the
integrated classification likelihood (if a uniform prior is used for $K$
instead of the default Poisson). While it is easy to exactly calculate the
integrated classification likelihood for a given $(z,K)$, it would not be
tractable to search across all possible $(z,K)$ to find the state that
maximizes the integrated classification likelihood, except for the smallest of
networks.
The BIC is an attempt to approximate the _integrated likelihood_
$\mathrm{P}(x|K)=\sum_{z}\int\int\mathrm{P}(x,z,\pi,\theta|K)\,\mathrm{d}\pi\,\mathrm{d}\theta.$
An exact solution to the BIC is not tractable for the SBM; the likelihood
would require a summation over all possible clusterings $z$.
If we were to use a uniform prior for $K$, then
$\mathrm{P}(x|K)=\mathrm{P}(K|x)$ and an irreducible ergodic Markov chain
algorithm such as ours would visit each value of $K$ in proportion to the
integrated likelihood for that value of $K$. Of course, our algorithm only
gives a _sample_ from the true posterior, and there cannot be any guarantee
that the distribution of the sample is representative of the true
distribution.
The purpose of these last few paragraphs is to demonstrate that there are
other (approximate) ways to calculate the _integrated likelihood_ and the
_integrated classification likelihood_. The Bayesian methods provide
approximations that may, in practice, be at least as good as the
approximations that would be provided by methods such as the ICL.
The model-selection criterion $ILvb$ Latouche et al. (2012) is based on a
variational approximation to a fully Bayesian model. As a result of its
Bayesian model, it is an approximation of the integrated likelihood and no
further adjustment is required for model selection. As with any variational
Bayes method, we assume that the independence assumptions within the
variational approximation are a good approximation of the true posterior. A
second assumption made by those authors is that the Kullback–Leibler
divergence, the difference between the true posterior and the variational
approximation, is independent of $K$. If these two assumptions hold, then the
measure they use, which they call the $ILvB$, is equivalent to
$\mathrm{P}(x|K)$, the _integrated likelihood_. To select the number of
clusters, they use that value of $K$ which maximizes the $ILvB$.
## 5 Estimation
In this section, we describe our MCMC algorithm which samples, given a network
$x$, from the posterior $K,z|x$. The moves are Metropolis-Hastings moves
(Hastings, 1970). We define the moves and calculate the proposal probabilities
and close the section with a discussion of the label-switching phenomenon,
where we use the method proposed in Nobile and Fearnside (2007) to summarize
the clusterings found by the sampler.
Our algorithm is closely based on the _allocation sampler_ Nobile and
Fearnside (2007), which was originally presented in the context of a mixture-
of-Gaussians model. In fact, it can be applied to any model that can be
collapsed to the form $\mathrm{P}(x,z,K)$ where $x$ is some fixed (observed)
data and the goal is to sample the clustering and the number of clusters
$(z,K)$.
In the Gibbs sampler used in Nowicki and Snijders (2001), the parameters are
not collapsed, and sampling is from
$z,\pi,\theta|x,K.$
In their experiment on the Hansell dataset, $K$ was fixed to 2. As a result of
this value for $K$, $\theta$ reduced to a single real number specifying the
relative expected size of the two clusters. Expressions were presented for
$p(\theta|z,\pi,x,K)$, $P(z|\theta,\pi,x,K)$ and $p(\pi|z,\theta,x,K)$ such
that the various elements $z_{i}$ (or $\pi_{kl}$) are conditionally
independent of each other, given $(\pi,x,K)$ (or $(z,x,K)$), allowing for a
straightforward Gibbs sampler.
In contrast, we develop an algorithm that searches across the full sample
space of all possible clusterings, $z$, for all $K$, drawing from the
posterior,
$z,K|x,$
using eq. 6 as the desired stationary distribution of the Markov Chain.
We use four moves:
* 1.
_MK_ : Metropolis move to increase or decrease $K$, adding or removing an
empty cluster.
* 2.
_GS_ : Gibbs sampling on a randomly-selected node. Fixing all but one node in
$z$, select a new cluster assignment for that node.
* 3.
_M3_ : Metropolis-Hastings on the labels in two clusters. This is the M3 move
proposed in Nobile and Fearnside (2007). Two clusters are selected at random
and the nodes are reassigned to the two clusters using a novel scheme fully
described in that paper. $K$ is not affected by this move.
* 4.
_AE_ : The _absorb-eject_ move is a Metropolis-Hasting merge/split cluster
move, as described in Nobile and Fearnside (2007). This move does affect $K$
along with $z$.
At each iteration, one of these four moves is selected at random and
attempted. All the moves are essentially Metropolis-Hastings moves; a move to
modify $z$ and/or $K$ is generated randomly, proposing a new state
$(z^{\prime},K^{\prime})$, and the ratio of the new density to the old density
$\frac{\mathrm{P}(z^{\prime},K|x)}{\mathrm{P}(z,K|x)}=\frac{\mathrm{P}(x,z^{\prime},K)}{\mathrm{P}(x,z,K)}$
is calculated. This is often quite easy to calculate quickly as, for certain
moves, only a small number of factors in eq. 6 are affected by the proposed
move. We must also calculate the probability of this particular move being
proposed, and of the reverse move being proposed. The _proposal probability
ratio_ is combined with the _posterior mass ratio_ to give us the move
_acceptance probability_ ,
$\operatorname{min}\left(1,\frac{\mathrm{P}(x,z^{\prime},K^{\prime})}{\mathrm{P}(x,z,K)}\times\frac{\mathrm{P}_{\text{prop}}((K^{\prime},z^{\prime})\rightarrow(K,z))}{\mathrm{P}_{\text{prop}}((K,z)\rightarrow(K^{\prime},z^{\prime}))}\right)\,,$
(9)
where $\mathrm{P}_{\text{prop}}((K,z)\rightarrow(K^{\prime},z^{\prime}))$ is
the probability that the algorithm, given current state $(K,z)$, will propose
a move to $(K^{\prime},z^{\prime})$.
In the remainder of this section, we discuss the four moves in detail, derive
the proposal probabilities and describe the computational complexity of the
moves.
### 5.1 MK
The _MK_ move increases or decreases the number of clusters by adding or
removing an empty cluster. If _MK_ is selected, then the algorithm selects
with 50% probability whether to attempt to add an empty cluster, or to delete
one. If it chooses to attempt a delete, then one cluster is selected at
random; if that cluster is not empty, then the attempt is abandoned. If it
chooses to attempt an insert, it selects a new cluster identifier randomly
from $\\{1,\dots,K+1\\}$ for the new cluster and inserts a new empty cluster
with that identifier, renaming any existing clusters as necessary.
The proposal probabilities are
$\begin{split}\mathrm{P}_{\text{prop}}((K,z)\rightarrow(K+1,z^{\prime}))&=\frac{0.5}{K+1}\\\
\mathrm{P}_{\text{prop}}((K^{\prime},z^{\prime})\rightarrow(K^{\prime}-1,z))&=\left\\{\begin{array}[]{rl}\frac{0.5}{K^{\prime}}&\text{if
}K^{\prime}>1\\\ 0&\text{otherwise}\end{array}\right.\,.\end{split}$
By adding an empty cluster, $K$ increases to $K^{\prime}=K+1$ and the
posterior mass change is:
$\begin{split}\frac{\mathrm{P}(x,z,K^{\prime})}{\mathrm{P}(x,z,K)}&=\frac{K!}{(K+1)!}\frac{\left(\frac{\Gamma(\alpha(K+1))\prod_{k=1}^{K+1}\Gamma(n_{k}+\alpha)}{\Gamma(\alpha)^{K+1}\Gamma(N+\alpha(K+1))}\right)}{\left(\frac{\Gamma(\alpha
K)\prod_{k=1}^{K}\Gamma(n_{k}+\alpha)}{\Gamma(\alpha)^{K}\Gamma(N+\alpha
K)}\right)}\\\ &=\frac{\Gamma(\alpha(K+1))\Gamma(N+\alpha
K)}{(K+1)\Gamma(\alpha K)\Gamma(N+\alpha(K+1))}\,.\end{split}$
The computational complexity of this move is constant.
### 5.2 GS
The Gibbs update move, _GS_ , selects a node $i$ at random to be assigned to a
new cluster. All other nodes are kept fixed in their current cluster
assignment i.e. a single element of the vector $z$ is updated. Denote by
$z^{\prime}=z_{\\{z_{i}\rightarrow k\\}}$ the modified clustering resulting
from a move of node $i$ to cluster $k$. For each possible value of
$z_{i}\in\\{1,\dots,K\\}$, $z_{i}$ is chosen with probability proportional to
$\mathrm{P}(x,z_{\\{z_{i}\rightarrow k\\}},K)$. The proposal is then accepted.
Bear in mind that this move often simply reassigns the node to the same
cluster it was in before the _GS_ move was attempted.
The calculations involved in _GS_ are quite complex as many of the factors in
eq. 6 are affected. The sizes of the clusters are changed as the node is
considered for inclusion in each cluster, and the number of edges and pairs of
nodes are changed in many of the blocks. The computational complexity is
$\mathcal{O}(K^{2})+\mathcal{O}(N)$ as every block needs to be considered for
each of the $K$ possible moves and every node may be checked to see if it is
connected or not to the current node.
The $\mathcal{O}(N)$ term is just a theoretical worst case over all possible
networks. Our algorithm iterates over the neighbours of the current node and
this is sufficient to perform all the necessary calculations. There is no need
to iterate over the non-neighbours and therefore the average complexity is
equal to the average degree, which will be much less than $N$ in real-world
sparse networks. For small $K$k, and assuming a given average degree, the
complexity of the _GS_ move is independent of $N$.
### 5.3 M3
_M3_ is a more complex move and was introduced in Nobile and Fearnside (2007).
Two distinct clusters are selected at random, $j$ and $k$. All the nodes in
these two clusters are removed from their current clusters and placed in a
list which is then randomly reordered – call this ordered list
$A=\\{a_{1},\dots,a_{n_{j}+n_{k}}\\}$, of size equal to the total number of
nodes in the two clusters. The software creates a temporary cluster to store
these nodes until they are reassigned to the original two clusters. One node
at a time is selected from $A$ and is assigned to one of the two clusters
according to some assignment probability. As the nodes are assigned (or
reassigned) the new cluster assignments are stored in a list
$B_{h}=\\{b_{1},\dots,b_{h-1}\\}$, where $b_{i}$ is the new cluster assignment
of node $a_{i}$ and $B_{h}$ represents the assignments before the $h^{\rm th}$
node in A is processed.
Iterating through the list $A$, $a_{h}$ is assigned to either cluster $j$ or
cluster $k$ with probability satisfying
$p^{a_{h}\rightarrow j}_{B_{h}}+p^{a_{h}\rightarrow k}_{B_{h}}=1\,,$
conditional on the nodes $B_{h}$ that have already been (re-)assigned.
Conceptually, any arbitrary assignment distribution can be chosen, as long as
the probabilities for each choice are non-zero and sum to one. Once all nodes
in the list have been assigned to the two clusters, the proposal probability
is given by
$\mathrm{P}_{\text{prop}}(z\rightarrow
z^{\prime})=\prod_{h=1}^{n_{j}+n_{k}}p^{a_{h}\rightarrow b_{h}}_{B_{h}}\,.$
We remark that while the order in which the nodes are reinserted is random, it
can be shown that this random ordering does not affect the acceptance
probability.
In Nobile and Fearnside (2007), it is proposed to choose the ratio of the
assignment probabilities as the ratio of the two posterior probabilities
resulting from the assignments of the first $h$ nodes. Specifically, denote by
$z_{\\{a_{h}\rightarrow l,B_{h}\\}}$, the clustering that assigns the first
$h-1$ nodes of A according to $B_{h}$ and assigns $a_{h}$ to cluster $l$. Let
$P(x^{\prime},z_{\\{a_{h}\rightarrow l,B_{h}\\}},K)$ be the posterior
probability of this clustering on the network $x^{\prime}$ _where all
unassigned nodes and edges involving these nodes are ignored_. Then
$\begin{split}\frac{p^{a_{h}\rightarrow j}_{B_{h}}}{p^{a_{h}\rightarrow
k}_{B_{h}}}=\frac{\mathrm{P}(x^{\prime},z_{\\{a_{h}\rightarrow
j,B_{h}\\}},K)}{\mathrm{P}(x^{\prime},z_{\\{a_{h}\rightarrow
j,B_{h}\\}},K)}\,.\end{split}$
This heuristic should guide the selection towards ‘good’ choices. To calculate
the proposal probability of the reverse proposal, the list A is again
traversed to calculate
$\mathrm{P}_{\text{prop}}(z^{\prime}\rightarrow
z)=\prod_{h=1}^{n_{j}+n_{k}}p^{a_{h}\rightarrow
z_{a_{h}}}_{B^{\prime}_{h}}\,,$
where $B^{\prime}_{h}=\\{z_{a_{1}},\dots,z_{a_{h-1}}\\}$.
Our algorithm has been optimized for sparser networks. The complexity of _M3_
is made up of three terms. First, it is possible that many or all nodes will
be reassigned, causing a complexity of $\mathcal{O}(N)$ while updating the
data structure that records the size of each cluster. Second, we keep a record
of the number of edges within each block; the M3 move will consider each edge
in the network at most once, as it moves the edge from one block to another,
leading to a complexity of $\mathcal{O}(M)$, where $M$ is the number of edges
in the network. Finally, once the data structures have been updated, a new
posterior mass must be calculated by iterating over each cluster and over each
block, querying the summary data structures, to sum the new terms in eq. 6;
this has a complexity of $\mathcal{O}(K^{2})$.
Together, this gives a complexity of
$\mathcal{O}(N)+\mathcal{O}(M)+\mathcal{O}(K^{2})$. The first term may be
ignored, since for most networks that are considered here and in the
literature, $M>N$. As long as the number of clusters is small, $K^{2}\ll M$,
the $\mathcal{O}(M)$ term dominates. While in the worst case $M=N^{2}$, in
practice, for the sparse networks we consider, $M\ll N^{2}$.
### 5.4 AE
In the _absorb-eject_ _AE_ move, a cluster is selected at random and split
into two clusters, or else the reverse move can merge two clusters. This move
therefore can both change the number of clusters $K$ and change the clustering
$z$. The move will first choose, with 50% probability, whether to attempt a
merge or split.
In the case of the split move, one of the $K$ clusters is selected at random.
Also, the cluster identifier of the proposed new cluster is selected at random
from $\\{1,\dots,K+1\\}$. Finally, the nodes in the original cluster are
assigned between the two clusters. This is similar to the _M3_ move and a
heuristic to guide the assignment, as in _M3_ , could be considered. Instead,
as in Nobile and Fearnside (2007), we use a _probability of ejection_ ,
$p_{E}$, selected randomly from a $\text{Uniform}(0,1)$ distribution, such
that each node is assigned to the new cluster with probability, $p_{E}$. In
such as move, the proposal probability is dependent on $p_{E}$. Rather than
specify an ejection probability, we integrate over the choice of $p_{E}$ in
much the same manner as collapsing.
Given $(z,K)$ and a proposal to split into $(z^{\prime},K^{\prime}=K+1)$,
where a cluster of size $n_{k}$ is split into clusters of size $n_{j_{1}}$ and
$n_{j_{2}}$, the resulting proposal probability for an eject move is
$\mathrm{P}_{\text{prop}}((z,K)\rightarrow(z^{\prime},K^{\prime}))=\frac{\Gamma(n_{j_{1}}+1)\Gamma(n_{j_{2}}+1)}{K(K+1)\Gamma(n_{k}+2)}\,.$
For a merge, the proposal probability is simply obtained as the probability of
selecting the two clusters for merger from the $K^{\prime}=K+1$ possible
clusters. One cluster is selected which will retain its current nodes and
which will expand to contain the nodes in another, randomly selected, cluster,
$\mathrm{P}_{\text{prop}}((z^{\prime},K^{\prime})\rightarrow(z,K))=\frac{1}{K}\frac{1}{K+1}\,.$
The complexity is similar to that of the M3 move.
### 5.5 Applying the moves
In all simulations, discussed in section 6, the algorithm is seeded by
initializing $K=2$ and assigning the nodes randomly to one of the two initial
clusters. The first two moves, _MK_ and _GS_ , are sufficient to sample the
space but have slow mixing. The _AE_ move is sufficient on its own as it can
add or remove clusters as well as move the nodes to reach any $(z,K)$ state.
In practice, we’ll see in section 6 that the combination of _AE_ and _M3_ is
good in the initialization stages to burn-in to a good estimate of both $z$
and $K$ and lessen the dependence on the initialization. It is possible to
envisage many possible extensions to these moves. For example, a form of _M3_
could be made which selects three clusters to rearrange. The _AE_ move could
be extended to include the assignment heuristic of the _M3_ move.
### 5.6 Label Switching
For any given $z$, with $K$ clusters (assuming they are all non-empty), there
are $K!$ ways to relabel the clusters, resulting in $K!$ effectively
equivalent clusterings. The posterior has this symmetry and as the MCMC
algorithm proceeds it often swaps the labels on two clusters, in particular
during the _M3_ move. This is known as the _label switching phenomena_. The
posterior distribution for any $z_{i}|x,K$ assigns node $i$ to each of the $K$
clusters with probability $\frac{1}{K}$, so in the long run every node is
assigned with equal probability to every cluster. While each $z_{i}$ is
uniformly distributed between 1 and K, the components of $z$ are dependent on
each other and pairs of nodes that tend to share a cluster will tend to have
the same values at their corresponding component of $z$. Depending on the
context, this may not be an issue of concern. For example, if the aim is to
estimate $K$ or to estimate the probability of two nodes sharing the same
cluster, see 3(b), or to estimate the size of the largest cluster, then label
switching is not a problem.
However, it sometimes is desirable to undo this label switching by relabelling
the clusters, such that nodes are typically assigned to a single cluster
identifier along with those other nodes that they typically share a cluster
with. Such a relabelling can, for example, make it easier to identify the
nodes which are not strongly tied to any one cluster.
We use the algorithm in Nobile and Fearnside (2007) to undo the label
switching by attempting to maximize the similarity between pairs of
clusterings, after the burn-in clusterings have been discarded. Given two $z$
vectors, at two different points in the Markov Chain, $t$ and $u$, define the
distance between them to be
$D(z^{(t)},z^{(u)})=\sum_{i=1}^{N}I(z^{(t)}_{i},z^{(u)}_{i})\,,$
where $I$ is an indicator function that returns 0 if node $i$ is assigned to
the same cluster at point $t$ and point $u$; and returns 1 otherwise.
For each $z^{(t)}$, consider $z^{(*t)}$, one of the $K!$ possible relabelled
versions of $z^{(t)}$. The Markov Chain is run for $a$ iterations, discarding
the first $b$ iterations as burn-in.
Ideally, the goal is to find the relabelling that minimizes the sum over all
pairs of $u$ and $t$,
$\sum_{t=b}^{a}\sum_{u=t+1}^{a}D(z^{(*t)},z^{(*u)})\,,$
but it is not computationally feasible to search across the full space of all
relabellings. Each state can be relabelled in approximately $K!$ different
ways, the precise number depends on the number of non-empty clusters. There
are $a$ states altogether, therefore the space of all possible relabellings of
all states will have $(K!)^{a}$ elements; this will be untractable for non-
negligible $a$. In our experiments, $a$ tends to be of the order of one
million.
Instead, we use the _online_ algorithm proposed in Nobile and Fearnside
(2007). It first orders the states from the Markov chain by the number of non-
empty clusters. Then, it iterates through the states, comparing each state to
all the preceding relabelled states and relabelling the current state such
that the total distance to all the preceding relabelled states is minimized.
We will see in section 7 how this algorithm helps to summarize the output of
the Markov Chain. This algorithm is fast. On a 2.4 GHz Intel Zeon in a server
with 128GB RAM, it takes 43 seconds to process the output of 1 million
iterations of that data. In comparison, it takes 610 seconds to run the SBM
MCMC algorithm in order to get the states to feed into the label-unswitching
algorithm. Note also that the algorithm doesn’t take up much memory — even
with a network with 10 million edges, the memory usage doesn’t exceed 2GB.
Once the label-switched set of states is obtained, a posterior distribution of
the clustering for each node, $z_{i}|x$, can be calculated. There is a
similarity here with variational methods Daudin et al. (2008); Latouche et al.
(2012) as they model the posterior in this manner, where each node’s
variational posterior is independent of the other nodes’ variational
posterior. It may be interesting to compare these approximate posteriors to
the approximate posterior found by our method.
In the experiments we perform later in sections 6 and 7, the vast majority of
nodes are strongly assigned by this label-switching algorithm to one of the
clusters with at least 99% probability in the posterior. Therefore, the
distance $D(.,.)$ between each state and this ‘summary’ state is usually quite
small. We take this as an indication that the online heuristic has done a
reasonable job of minimizing the distance between the states, at least for
those networks.
## 6 Evaluation
In this section we first look at experiments based on synthetic data and
follow in the next section with an application of the collapsed SBM to a
survey network gathered by one of the authors at a recent summer school. The
synthetic analysis proceeds by generating networks of various sizes from the
model and examining whether the algorithm can correctly estimate the number of
clusters and the cluster assignments.
As mentioned in the previous section, all our experiments are done on a 2.4
GHz Intel Zeon in a server with 128GB RAM, and the memory usage never exceeded
2GB.
### 6.1 Estimating z
A 40-node directed, unweighted network is generated from the model, containing
4 clusters of 10 nodes each. The block densities $\pi_{kl}$ are generated by
drawing from a $\text{Uniform}(0,1)\equiv\text{Beta}(1,1)$ for each of the
$4\times 4=16$ blocks.
Figure 1: The adjacency matrix (with $\delta=0$) for the four-cluster
synthetic network used in 6.1. Each of the four clusters has 10 nodes.
To challenge the algorithm further we add noise to the synthetic data, similar
to simulation experiment described in section 4 of Wyse and Friel (2012). The
values in the matrix $\pi$ are scaled linearly. For a given $\delta$, define
$\pi^{(\delta)}_{kl}=\delta+\pi_{kl}(1-2\delta)$. While the values in the
original $\pi$ are drawn from the full range, $[0,1]$, the elements in the
matrix $\pi^{(\delta)}$ are in the range $(\delta,1-\delta)$. Various networks
for values of $\delta$ between 0 and 0.5 are generated. The original network
model corresponds to $\delta=0$. The network with $\delta=0.5$ corresponds to
an Erdos-Renyi model with $p=0.5$ — this is a random graph model with no block
structure.
$\delta$ | $\mathrm{P}(K=4|x)$ | $\hat{K}_{\text{mode}}$ | $\mathrm{P}(K=\hat{K}_{\text{mode}}|x)$ | $\mathrm{P}(\hat{z}\equiv z|x)$ | $\tau$
---|---|---|---|---|---
0.0 | 0.8982 | 4 | 0.8982 | 0.974 | 50.12
0.1 | 0.8799 | 4 | 0.8799 | 0.952 | 63.99
0.2 | 0.8769 | 4 | 0.8769 | 0.124 | 80.18
0.3 | 0.0073 | 2 | 0.7865 | 0.000 | 371.96
0.4 | 0.0075 | 1 | 0.6293 | 0.000 | 1365.58
Table 1: The performance decreases as the noise level, $\delta$, increases.
The fifth column, $\mathrm{P}(\hat{z}\equiv z|x)$, reports how often the
sampler visits the ‘correct’ answer; i.e. where the visited state was
equivalent, subject to relabelling, to the model from which the network was
generated.
The algorithm is run for one million iterations, discarding the first 500,000
of these as burn-in.
Table 1 shows how the performance is affected as $\delta$ increases. The first
column is the posterior probability for the “correct” answer for $K$,
$\mathrm{P}(K=4|x)$. As the value of $\delta$ increases, the network
approaches the Erdos Renyi model and therefore there is no longer any
structure to detect; this explains why the accuracy decreases as $\delta$
increases. Next is the modal value of K which maximizes the posterior
$\mathrm{P}(K|x)$, followed by the posterior probability of the modal value,
$\mathrm{P}(K=\hat{K}_{\text{mode}}|x)$ . The fifth column,
$\mathrm{P}(\hat{z}\equiv z|x)$ is the probability that the (non-empty)
clusters are equivalent (allowing for relabelling) to the clustering used to
generate the data. Note that sometimes there are empty clusters in the
estimate and therefore $\mathrm{P}(\hat{z}\equiv z|x)$ can be bigger than
$\mathrm{P}(K=4|x)$.
The final column reports $\tau$, the Integrated Autocorrelation Time (IAT) for
the estimate of $K$, defined as $\tau=1+2\sum_{t=1}^{\infty}\rho(t)$, where
$\rho_{t}$ is the autocorrelation at lag $t$. As the sampler visits the
states, we consider how correlated the estimate of $K$ is with the estimates
for preceding states. A low autocorrelation, as summarized by the IAT, is an
indicator of good mixing.
### 6.2 Estimating K
We perform three different types of experiments to judge the ability of the
algorithm to correctly estimate the number of clusters with networks of
increasing size.
First, we repeat the experiments of Latouche et al. (2012). The true numbers
of clusters, $K_{true}$ is set to range from 3 to 7. For each $K_{true}$, 100
networks are randomly generated. The number of nodes in each network, $N$, is
set to 50. The nodes are assigned to the clusters randomly, with
$\theta_{1}=\dots=\theta_{K}=\frac{1}{K_{true}}$. Two parameters are used to
control the density of the blocks. The first, $\lambda$, is the density within
clusters i.e. $\pi_{kk}=\lambda$. Also, one of the clusters is selected to be
a special cluster of ‘hubs’, well connected to the other nodes in the network,
by setting $\pi_{1k}=\pi_{k1}=\lambda$. The second parameter, $\epsilon$,
represents the inter-block density of all the other blocks i.e.
$\pi_{kl}=\epsilon$ for $k,l\neq 1$. As in the experiments of Latouche et al.
(2012), the parameter values are $\lambda=0.9$, and $\epsilon=0.1$.
| 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---
3 | 100 | 0 | 0 | 0 | 0
4 | 0 | 99 | 1 | 0 | 0
5 | 0 | 4 | 96 | 0 | 0
6 | 0 | 0 | 24 | 76 | 0
7 | 0 | 5 | 29 | 41 | 25
(a) ILvb
| 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---
3 | 99 | 1 | 0 | 0 | 0
4 | 0 | 99 | 1 | 0 | 0
5 | 0 | 4 | 96 | 0 | 0
6 | 0 | 0 | 25 | 75 | 0
7 | 0 | 5 | 27 | 46 | 22
(b) our algorithm
Table 2: The rows represent $K_{true}$ and the columns are the estimates from
the $ILvb$ of Latouche et al. (2012) and from our algorithm.
Each network is run through the variational method of Latouche et al. (2012).
The estimated value of $K$ which maximizes the $ILvB$ measure is taken as the
estimate of the number of clusters. A contingency table, showing the true
number of clusters against the estimate from $ILvB$, is displayed in table 2 r
low $K_{true}$ the algorithm is very accurate, and for larger values there is
a tendency to underestimate the number of clusters. For example, when
$K_{true}=7$, the estimate was $\hat{K}=6$ for 41 of the networks and
$\hat{K}=7$ for only 25 of the 100 networks.
The results from our algorithm, shown in table 2 similar to those obtained
using the $ILvB$.
### 6.3 Synthetic SBM networks
The experiments of 6.2 involve synthetic data generated according to a model
of _community structure_ , where edges tend to form primarily within clusters.
In order to explicitly test our algorithm in the more general setting of
_block structure_ , we generated another set of networks with data generated
directly from the SBM.
Similarly to the previous experiment, for each of a range of values of
$K_{true}$, 100 networks are generated. $K_{True}$ is now set to range from 10
to 20 and the number of nodes is set to $N=100$, in order that the size of
each cluster not be very small. Each element of $\pi_{kl}$ is chosen randomly
from Uniform(0,1) and for each of the 100 networks, a new $\pi$ is created
randomly. As these are undirected networks, only the upper triangular portion
of $\pi$ is used when generating the network. Again, we compared the estimates
of $K$ found by the $ILvB$ to those found by our algorithm.
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
---|---|---|---|---|---|---|---|---|---|---|---
10 | 72 | 15 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
11 | 15 | 75 | 5 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0
12 | 5 | 20 | 64 | 6 | 5 | 0 | 0 | 0 | 0 | 0 | 0
13 | 2 | 3 | 21 | 66 | 8 | 0 | 0 | 0 | 0 | 0 | 0
14 | 0 | 0 | 4 | 21 | 61 | 10 | 4 | 0 | 0 | 0 | 0
15 | 0 | 0 | 2 | 8 | 28 | 51 | 9 | 0 | 2 | 0 | 0
16 | 0 | 0 | 1 | 4 | 15 | 32 | 33 | 11 | 4 | 0 | 0
17 | 0 | 0 | 0 | 2 | 4 | 11 | 30 | 45 | 8 | 0 | 0
18 | 0 | 0 | 0 | 1 | 3 | 12 | 20 | 30 | 23 | 10 | 1
19 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 24 | 38 | 13 | 10
20 | 0 | 0 | 0 | 0 | 0 | 1 | 7 | 6 | 23 | 29 | 23
Table 3: The true number of clusters (rows) against the number estimated by $ILvb$ (columns). The diagonal entries are underlined to aid readability, as these represent the correct answer. We see here a tendency to underestimate the number of clusters, especially for larger $K_{True}$. | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20
---|---|---|---|---|---|---|---|---|---|---|---
10 | 95 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
11 | 6 | 93 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
12 | 1 | 8 | 90 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
13 | 0 | 2 | 12 | 86 | 0 | 0 | 0 | 0 | 0 | 0 | 0
14 | 0 | 0 | 1 | 9 | 90 | 0 | 0 | 0 | 0 | 0 | 0
15 | 0 | 0 | 0 | 1 | 13 | 84 | 2 | 0 | 0 | 0 | 0
16 | 0 | 0 | 0 | 0 | 1 | 22 | 73 | 4 | 0 | 0 | 0
17 | 0 | 0 | 0 | 0 | 0 | 2 | 29 | 65 | 4 | 0 | 0
18 | 0 | 0 | 0 | 0 | 0 | 1 | 9 | 28 | 62 | 0 | 0
19 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 7 | 38 | 51 | 0
20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 11 | 28 | 57
Table 4: The true number of clusters (rows) against the number estimated by
our collapsed MCMC algorithm (columns). The diagonal entries are underlined to
aid readability, as these represent the correct answer. The accuracy is better
here than in table 3; we can see that the numbers on the diagonal are larger.
The results are shown in tables 3 and 4. Each row of data represents the 100
networks generated for a given $K_{True}$. Each column represents the
estimated $\hat{K}$. Ideally, the algorithm would correctly estimate the
number of clusters in most cases, corresponding to large number on the
diagonal. We have underlined the diagonal entries for clarity. Note that the
sum of the entries in each row does not always sum exactly to 100, since there
are cases where the algorithms underestimate or overestimate the number of
clusters, beyond the shown range. For the $ILvb$ algorithm, it is necessary to
specify a range of $K$ to be tested; we specified the range from 5 to 30. Our
MCMC algorithm requires no such hint.
For networks with a small number of clusters, both algorithms perform well,
with 72% accuracy for $ILvb$ and 95% accuracy for our algorithm. As the true
number of clusters increase, the performance decreases. Our algorithm
maintains at least 50% accuracy in all cases, whereas the accuracy for $ILvb$
falls to 23%. When they are incorrect, both algorithms have a tendency to
underestimate the number of clusters.
In 6.4, a more thorough investigation of the speed and scalability of our
algorithm with respect to larger networks is given but we close our comparison
with $ILvb$ with some remarks on speed. For the first set of small networks
above, both methods are very fast; they complete within seconds. For example,
the $ILvb$ can be calculated for all values of $K$ from 10 to 20 in a total
under five seconds. We have not defined a convergence criterion for our MCMC
algorithm, and therefore we make no attempt to halt the sampling early in
order to define a ‘runtime’ for our algorithm. But in the occasions where both
methods get the correct result, our algorithm typically reaches the correct
result within nine seconds; and the sampler remains at, or very close to, the
correct clustering for the remainder of the run.
Finally, to demonstrate the importance of the _AE_ move, in fig. 2, the time
taken by our algorithm to reach the correct clustering for three synthetic
networks is shown. The numbers of clusters in the networks is 5, 20, and 50
respectively, with $\pi$ drawn from $\text{Uniform}(0,1)$. In each case, there
were exactly 10 nodes in each cluster, giving $N=10\times K$ nodes in each
network. The x-axis displays the number of iterations and the y-axis the
number of clusters at that stage in the run of the sampler. The correct
clustering is reached in approximately 10,000 iterations.
We found that the _AE_ move is quite important, at least in the early stages.
If _AE_ is disabled, see 2(b), then it takes about 320,000 iterations for
K=50, instead of just 20,000 iterations when all moves are in effect. For fast
burn-in, _M3_ and _AE_ are necessary. With similar experiments we noticed
that, once the chain has burned in, the _M3_ move is sufficient for good
performance and the other simple moves, _GS_ and _MK_ , do not make major
contributions.
(a) All moves enabled (b) _AE_ move disabled
Figure 2: The estimates of K in the synthetic networks, with $K=5,20,50$. The
x-axis (logarithmic scale) is the number of iterations; as the algorithm
proceeds, in each case it converges on the correct estimate of $K$. The
networks had $10\times K$ nodes each. In the lower plot, we see the
performance where where the _AE_ move has been disabled; demonstrating how it
is important in burnin.
### 6.4 Larger networks
Next, we investigate larger networks to demonstrate the scalability of the
algorithm.
A number of synthetic networks are generated, each with approximately ten
thousand nodes and ten million edges. The number of clusters ranges from 3 to
50, and the number of nodes in each clusters, $O$, is set such that the total
number of nodes, $N=K\times O$, is close to 10,000. If we use the default SBM
edge model, then the number of edges would be approximately 50 million. As
this would take up a lot of computer memory, instead we modify the prior for
the per-block densities to be Uniform(0,0.2) in order to ensure that the
expected number of edges is 10 million. Large real-world networks are
typically quite sparse, even more sparse than this synthetic network. The
details, including the speed and accuracy, are in table 5.
The SBM algorithm is run for 100,000 iterations on each of these networks and
the time to converge is recorded. In each case, when the algorithm first
visits the ‘correct’ state, it remains in that state for practically all the
remaining iterations. We record the number of iterations taken before the
algorithm reaches the correct state, and the time that has elapsed at that
point. It typically converges within one hour, but it takes nearly four hours
for the 50-cluster network. Methods that scale to thousands of nodes have been
presented in the literature, such as Daudin et al. (2008) and Latouche et al.
(2012). To our knowledge, ours is the only method which has been demonstrated
on networks with 10,000 or more nodes.
We have attempted to load these networks into the $R$ software package in
order to run them through $ILvb$. However, the memory requirements for such
large adjacency matrices become prohibitive. For large networks, it may be
necessary to consider a different implementation language and techniques in
order to fully explore the scalability of a variational method such as $ILvb$.
Instead, we generated five 500-node networks, with 20 clusters each, according
to the SBM model and run $ILvB$ on it, using only one value of $K$, namely
$K=20$. It takes between 38 and 56 seconds, depending on which of the five
networks is used. In comparison, on the same data, our algorithm takes between
17 and 35 seconds, despite the fact that it is given no clue as to the correct
value of $K$. With 1,000-node networks, the runtimes for $ILvb$ are between
636 and 814 seconds, whereas our algorithm takes between 55 and 78 seconds.
This suggests our algorithm scales better than the $ILvb$ – although perhaps
this is an implementation issue rather than a limitation of the variational
model.
In practice, it is necessary to run $ILvb$ for every possible value of $K$,
and this fact should be incorporated into any evaluation of its runtime. For
larger networks, the range of possible values of $K$ increases making this a
significant issue. In contrast, an algorithm based on the allocation sampler,
such as ours, does not suffer this limitation, suggesting that that our
algorithm is well suited to large networks.
$K$ | $O$ | $N$ | $E$ | $i$ | $t$
---|---|---|---|---|---
3 | 3,333 | 9,999 | 9,722,580 | 41 | 3,317
4 | 2,500 | 10,000 | 8,526,987 | 149 | 2,977
5 | 2,000 | 10,000 | 8,627,869 | 190 | 2,460
6 | 1,667 | 10,002 | 9,974,998 | 416 | 3,265
7 | 1,429 | 10,003 | 9,316,651 | 749 | 3,449
8 | 1,250 | 10,000 | 11,059,656 | 962 | 3,710
9 | 1,111 | 9,999 | 9,581,440 | 1,383 | 4,052
10 | 1,000 | 10,000 | 9,989,886 | 1,277 | 3,785
20 | 500 | 10,000 | 9,871,938 | 5,655 | 4,779
30 | 333 | 9,990 | 9,821,594 | 12,497 | 6,999
40 | 250 | 10,000 | 9,862,703 | 37,742 | 12,452
50 | 200 | 10,000 | 10,008,963 | 40,958 | 24,028
Table 5: The time-to-convergence for the larger synthetic networks. The
networks have $N=K\times O$ nodes, made up of $K$ clusters each with $O$
nodes. After $i$ iterations ($t$ seconds), the algorithm reached the correct
result and remained in, or close to, that state for the remainder of the
100,000 iterations. It should be noted that much of the runtime is simply
taken up with loading the network into memory; the time spent in the MCMC
algorithm itself is smaller than the $t$ figure presented here.
### 6.5 Autocorrelation in K
(a) Adjacency matrix
| 97 | 4 | 4 | 75 | 75
---|---|---|---|---|---
97 | | 4 | 4 | 75 | 75
4 | 4 | 99 | 99 | 4 | 4
4 | 4 | 99 | 99 | 4 | 4
75 | 75 | 4 | 4 | | 97
75 | 75 | 4 | 4 | 97 |
(b) Percentage posterior probability of two nodes sharing a cluster.
(c) Autocorrelation on $K$.
Figure 3: Adjacency matrix used in the analysis of varying K in 6.5. 3(b)
estimates, for every pair of nodes, the predicted probability of them sharing
a cluster. 3(c) shows the autocorrelation in the estimate of $K$.
An autocorrelation analysis can reveal the mixing properties of the algorithm.
However, in the above examples, and in the survey data discussed in section 7,
the estimates of $K$ are very much peaked around a single value. Often the
larger values of $K$ are associated with empty clusters and the estimate of
the number of non-empty clusters is even more peaked. This makes it difficult
to use $K$ as an interesting variable on which to perform autocorrelation
analysis. To address this, we examine the 6-node network in 3(a), for which a
greater variance in the values of $K$ is observed. Define $K_{1}$ to be the
number of non-empty clusters, $K_{1}\leq K$. The posterior predictive
probability for $K=2$ is 57.0%, and for $K=3$ it is 31.4%. For the non-empty
clusters, it is 73.4% for $K_{1}=2$ and 24.4% for $K_{1}=3$. The
autocorrelation in the estimates of $K$ is shown in 3(c).
The acceptance rates on this small 6-node network are relatively high: 8.1%
for _MK_ , 4.2% for _GS_ , 20.5% for _AE_ , 46.0% for _M3_ . We’ll see lower
acceptance rates in the next section when the algorithm is applied to the
survey network.
## 7 Survey of interaction data
A survey was performed by a team involving one of the authors of this paper at
a summer school. We asked the 74 participants to fill in a survey and record
which other participants they knew before the summer school and also which
participants they met during the school. 40 of the participants responded and
gave us permission to make their survey response available publicly in
anonymized format. We created a directed, unweighted, network from the data by
linking A to B if A recorded either type of relationship with B, resulting in
1,138 edges. This network data is available at
https://github.com/aaronmcdaid/Summer-school-survey-network.
Figure 4: The interation survey network of section 7. Node-to-cluster
membership matrix. 74 rows, one for each participant. There are 8 columns, one
for each of the main seven clusters, and an extra cluster which, with very
small probability, is occupied by some nodes. Most nodes are strongly assigned
to one cluster, but the grey areas off the diagonal show a small number of
nodes that are partially assigned to multiple clusters.
Using the procedure described in 5.6, we are able to summarize the output of
the Markov chain in fig. 4. This is a matrix which records, for each
(relabelled) cluster and node, the posterior probability of that participant
being a member of that cluster. Each row represents one participant of the
summer school, and the total weight in each row sums to 1.0. We have ordered
the rows in this figure in order to bring similar rows together, helping to
highlight the sets of nodes which tend to be clustered together in the Markov
Chain. As may be observed, most of the participants are strongly assigned to
one cluster. Every node is assigned to one of the clusters with at least 75%
posterior probability, and the majority of nodes have at least 99% posterior
probability.
Figure 5: The interaction survey network of section 7 as a 74$\times$74
adjacency matrix for the 74 participants in the summer school. 7 clusters were
found by our method, and this matrix is ordered by the summary clustering
found by the label-unswitching method of 5.6. In the text in section 7, we
interpret the clusters found and show how many of the clusters correspond to
the different types of people that attended the event. There were 33 people
who did not respond, these can be seen in the last two clusters.
The number of clusters selected is 7, with 90.7 % posterior probability. We
can summarize this into a single clustering by assigning each node to its
‘best’ cluster as found by the label-unswitching procedure. In fig. 5, we see
this clustering. This particular clustering (or label-switched equivalents)
has posterior probability of 20.7%. (The order in which the clusters are
presented is different in fig. 5 than in fig. 4)
We then analyzed the clusters to see if they could be meaningfully
interpreted. The first thing that stands out is that the final two rows of
blocks are empty; these are simply the 33 people who did not respond to the
survey. It is interesting to see that the non-respondents have been split into
two clusters. Looking at the final two columns of blocks, the differences in
how other clusters linked to the non-respondents can be seen.
With the help of one of the organizers, we verified that the second cluster
(counting from the top, or from the left) is made up of the _Organizers_ of
the summer school, with one exception. These were people based in the research
institute who were involved in organizing the summer school. Therefore, it is
no surprise that the corresponding rows and columns of the adjacency matrix in
fig. 5 are quite dense. The _Organizers_ interacted with almost everybody.
The third and fourth clusters are also made up of people who are based in the
research institute where the summer school was hosted but who weren’t on the
programme committee. We call these _Locals_. The first cluster is made up of
_Visitors_. These were people from further afield who attended the school and
spoke at the summer school. Looking at the blocks at the top left of fig. 5
you can see that the _Locals_ know each other and the _Visitors_ interacted
with each other. But the two groups do not tend to interact strongly with each
other. The _Organizers_ are the glue that hold everybody together. The fifth
cluster appears simply to be made up of participants who did not interact very
much with anybody – in fact they did not even interact with each other.
We can now interpret the fact that there are two clusters of non-respondents.
One of those clusters (the sixth cluster) is made of up of local people. Their
names appeared in the surveys of the _Organizers_ and _Locals_. The final
cluster, the other non-respondent cluster, is made up of a broader range of
people. It includes many non-responder _Visitors_ , including many of the
speakers at the summer school.
A community finding algorithm would not have been able to find these results,
as it would expect to find dense clusters and is tied to the assumption that
the probability of pairs of nodes being connected is, all other things being
equal, greater if they share a cluster than if they do not share a cluster.
This would manifest as dense blocks on the diagonal of this adjacency matrix.
Clearly, a community-finding algorithm could not find the non-respondent
clusters. Also, a community finding algorithm might have merged the
_Organizers_ and _Locals_ clusters. This is because those two clusters are
quite dense internally and also have many connections between them. The only
difference between these two clusters is how they interact with the rest of
the network; this demonstrates how the rich block structure of the Stochastic
Block Model, including the various cluster-cluster interactions, can be
helpful in clustering this data.
We ran the algorithm for 1 million iterations on this survey data, discarding
the first 500,000 iterations as burn-in. The acceptance rates were as follows:
2.3% for _AE_ , 64.6% for _M3_ , 1.1% for _MK_. In the case of the Gibbs
sampler, 2.5% of the time it assigned a node to a new cluster, otherwise the
node was reassigned to its old cluster.
The _M3_ and _AE_ are both Metropolis-Hastings; a change to the clustering is
proposed and then the change is accepted or rejected. Sometimes the accepted
move actually places all the nodes back to the same position they were in, or
sometimes it merely swaps the labels between the two clusters. If we consider
these as ‘rejections’, then the rate and which new states are reached is just
1.0% for _M3_. So, _M3_ is accepted a lot, but it usually only moves between
label-switched equivalents; this tells us that the algorithm is able to move
quickly between the various modes of the distribution, and also suggests that
the posterior is quite peaked around the modes.
### 7.1 Estimating the Network Probability, $\mathrm{P}(x)$
In Section 4, we discussed how the fully Bayesian approach of the SBM
presented here allows model selection criteria such as the ICL to be avoided
to select between models with different input numbers of clusters $K$. It is
also worth remarking that in certain circumstances, such as our survey data
presented here, it is possible to compute an estimate of the network
probability, $P(x)$; that is, the probability, given just the total number of
nodes $N$, that the network $x$ is observed from an SBM. This provides an
absolute measure of the fit of the SBM to the observed data and could be used
to test the hypothesis that the data is drawn from an SBM against some
alternative model.
In the survey data there is one clustering where it, along with its label-
switched equivalents, take up 20.7% of the posterior probability. Call this
$\hat{z}$. Thus we have a value $\hat{z}$ which is visited very often by the
sampler and this allows an accurate estimate of $\mathrm{P}(K,\hat{z}|x)$ to
be obtained using
$7!\times\mathrm{P}(K=7,z=\hat{z}|x)=0.207\,.$
Now inserting $x$, $K$ and $\hat{z}$ into the expression for the joint
distribution, an estimate of $P(x)$ can be obtained using
$\mathrm{P}(x)\mathrm{P}(K=7,z=\hat{z}|x)=\mathrm{P}(x,z=\hat{z},K=7)\,.$
In the case of the survey data we obtain $\log_{2}\mathrm{P}(x)\approx-2,482$.
To put some perspective on this value, we can compare with a model that
selects $x$ uniformly at random from all possible directed networks over
$N=74$ nodes. In this case, we obtain $\log_{2}\mathrm{P}(x)=-N(N-1)=-5,402$.
As a second alternative, if $x$ were generated from an Erdos-Renyi model,
averaged over all possible edge probabilities drawn uniformly at random, then
$\log_{2}\mathrm{P}(x)\approx-4,130$.
## 8 Conclusion
The original stochastic blockmodel was tested on a small network with two
clusters. We have shown how Bayesian models, collapsing, and modern MCMC
techniques can combine to create an algorithm which can accurately estimate
the clusters, and the number of clusters, without compromising on speed.
It is sometimes stated that MCMC is necessarily slower than other methods,
“effectively leading to severe size limitations (around 200 nodes)” (Gazal et
al., 2011). The MCMC method we have presented scales to thousands of nodes,
and is more scalable than a recent variational method. We do not claim that
MCMC will always be necessarily faster than the alternatives, but we observe
that comments on the scalability of Metropolis-Hastings MCMC depends on the
particular model and on the particular proposal functions used. It may be an
open question as to which methods will prove to be most scalable in the long
term, as further improvements are made to all methods.
Our application to the survey data demonstrated that _block-modelling_ can
detect structure in networks that might be missed by _community-finding_
algorithms. Sometimes the links between clusters are more interesting than the
links within clusters.
### Acknowledgements
This research was supported by Science Foundation Ireland (SFI) Grant No.
08/SRC/I1407 - Clique Research Cluster
## Appendix A
Here, we describe the integrations which show that eq. 4 is equivalent to eq.
6.
### A.1 Collapsing $\theta$
Here, we show how to calculate
$\mathrm{P}(z|K)=\int_{\Theta}\mathrm{p}(z,\theta|K)\;\mathrm{d}\theta\,.$
(10)
This corresponds to the first integration expression in eq. 3. $z$ is a vector
which records, for each of the $N$ nodes, which cluster it has been assigned
to. The probability for each cluster is in a vector $\theta$, where
$1=\sum_{k=1}^{K}\theta_{k}\,.$
We integrate over the support of the Dirichlet distribution, which we have
denoted with $\Theta$ in eq. 10,
$\theta\sim\mbox{Dirichlet}({\alpha,\alpha,\dots})\,.$
where we made the common simplification in our prior that all members of the
vector $\alpha$ are identical; $\alpha_{k}=\alpha$.
$\theta$ is drawn from Dirichlet prior,
$\mathrm{p}(\theta)=\frac{1}{\mathrm{B}(\alpha)}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}-1}\,,$
where the normalizing constant $\mathrm{B}(\alpha)$ is
$\mathrm{B}(\alpha)=\frac{\prod_{k=1}^{K}\Gamma(\alpha_{k})}{\Gamma\left(\prod_{k=1}^{K}\alpha_{k}\right)}\,.$
To collapse $\theta$, the expression for $\mathrm{P}(z|K)$ becomes the
Multivariate Pólya distribution. In the derivation, we have defined $n_{k}$ to
be the number of nodes in cluster $k$, i.e.
$n_{k}=\sum_{i=1}^{N}\left\\{\begin{array}[]{cc}1&\text{if}\;z_{i}=k\\\
0&\text{if}\;z_{i}\neq k\end{array}\right.\,.$
In the following expression, we will also find it useful to define another
vector of length $K$,
$\alpha^{\prime}=(\alpha_{1}+n_{1},\alpha_{2}+n_{2},\dots,\alpha_{K}+n_{K})\,,$
$\begin{split}\int_{\Theta}\mathrm{p}(z,\theta|K)\;\mathrm{d}\theta=&{}\int_{\Theta}\mathrm{p}(\theta|K)\mathrm{P}(z|\theta,K)\;\mathrm{d}\theta\\\
=&{}\int_{\Theta}\mathrm{p}(\theta|K)\prod_{k=1}^{K}\theta_{k}^{n_{k}}\;\mathrm{d}\theta\\\
=&{}\int_{\Theta}\frac{1}{\mathrm{B}(\alpha)}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}-1}\prod_{k=1}^{K}\theta_{k}^{n_{k}}\;\mathrm{d}\theta\\\
=&{}\int_{\Theta}\frac{1}{\mathrm{B}(\alpha)}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}+n_{k}-1}\;\mathrm{d}\theta\\\
=&{}\frac{\mathrm{B}(\alpha^{\prime})}{\mathrm{B}(\alpha)}\int_{\Theta}\frac{1}{\mathrm{B}(\alpha^{\prime})}\prod_{k=1}^{K}\theta_{k}^{\alpha_{k}+n_{k}-1}\;\mathrm{d}\theta\\\
=&{}\frac{\mathrm{B}(\alpha^{\prime})}{\mathrm{B}(\alpha)}\\\
=&{}\frac{\Gamma(\sum_{k=1}^{K}\alpha_{k})}{\Gamma(N+\sum_{k=1}^{K}\alpha_{k})}\prod_{k=1}^{K}\frac{\Gamma(n_{k}+\alpha_{k})}{\Gamma(\alpha_{k})}\\\
=&{}\frac{\Gamma(K\alpha)}{\Gamma(N+K\alpha)}\prod_{k=1}^{K}\frac{\Gamma(n_{k}+\alpha)}{\Gamma(\alpha)}\,.\end{split}$
### A.2 Collapsing $\pi$
Now we look at the second integration expression in eq. 3. This describes how
to calculate the probability of a network, $x$, given a clustering, $z$, and
the number of clusters, $K$.
$\mathrm{P}(x|z,K)=\int_{\Pi}\mathrm{P}(x,\pi|z,K)\;\mathrm{d}\pi\,.$
This depends on whether we’re using the unweighted (Bernoulli) or integer-
weighted(Poisson) model for edges. It is also possible to allow real-valued
weights with a Normal distribution and suitable priors, an example of such a
model is solved in Appendix B.2 of Wyse and Friel (2012); that paper is
relevant for all the derivations here as the collapsing approach is quite
similar as in this paper.
The number of pairs of nodes in block between clusters $k$ and $l$ will be
denoted $p_{kl}$ \- for blocks on the diagonal $p_{kk}$ will depend on whether
the edges are directed and on whether self loops are allowed; see eq. 5 for
details. The relevant probabilities for a given block will be shown to be a
function only of $p_{kl}$ and of the total weight (or total number of edges)
in that block. We’ll denote this total weight as
$y_{kl}=\sum_{i,j|z_{i}=k,z_{j}=l}x_{ij}\,.$
In an undirected graph, we should consider each pair of nodes only once,
$y_{kl}=\sum_{i,j|i<j,z_{i}=k,z_{j}=l}x_{ij}\,.$
We are to calculate the integral for a single block. $x_{(kl)}$ represents the
submatrix of $x$ corresponding to pairs of nodes in clusters $k$ and $l$. Our
goal is to simplify the expression such that there there is one factor for
each block,
$\begin{split}\mathrm{P}(x|z,K)&=\prod\mathrm{P}(x_{(kl)}|z,K)\\\
&=\prod\int\mathrm{P}(x_{(kl)},\pi_{kl}|z,K)\;\mathrm{d}\pi_{kl}\,.\end{split}$
For directed graphs, the product is $\prod_{k,l}$, giving $K\times K$ blocks.
But for undirected graphs, the product is $\prod_{k,l|k\leq l}$, giving
$\frac{1}{2}K(K+1)$ blocks. The domain of the integration will be either
$\int_{0}^{1}$ or $\int_{0}^{\infty}$, depending on which of the two data
models, unweighted or weighted, is in effect.
We’ll first consider the unweighted (Bernoulli) model. The probability of a
node in cluster $k$ connecting to a node in cluster $l$ is constrained by
$0<\pi_{kl}<1\,,$
and each element of $x_{(kl)}$ is drawn from a Bernoulli distribution with
parameter $\pi_{kl}$,
$\mathrm{P}(x_{(kl)}|\pi_{kl},z,K)=\pi_{kl}^{y_{kl}}(1-\pi_{kl})^{p_{kl}-y_{kl}}\,.$
The prior for $\pi_{kl}$ is a Beta($\beta_{1},\beta_{2})$ distribution.
$\begin{split}\mathrm{P}(x_{(kl)}|z,K)=&{}\int_{0}^{1}\mathrm{p}(x_{(kl)},\pi_{kl}|z,K)\;\mathrm{d}\pi_{kl}\\\
=&{}\int_{0}^{1}\mathrm{p}(\pi_{kl})\;\mathrm{P}(x_{(kl)}|\pi_{kl},z,K)\;\mathrm{d}\pi_{kl}\\\
=&{}\int_{0}^{1}\frac{\pi_{kl}^{\beta_{1}-1}(1-\pi_{kl})^{\beta_{2}-1}}{\text{B}(\beta_{1},\beta_{2})}\;\pi_{kl}^{y_{kl}}(1-\pi_{kl})^{p_{kl}-y_{kl}}\;\mathrm{d}\pi_{kl}\\\
=&{}\int_{0}^{1}\frac{\pi_{kl}^{y_{kl}+\beta_{1}-1}(1-\pi_{kl})^{p_{kl}-y_{kl}+\beta_{2}-1}}{\text{B}(\beta_{1},\beta_{2})}\;\mathrm{d}\pi_{kl}\\\
=&{}\frac{\text{B}(y_{kl}+\beta_{1},p_{kl}-y_{kl}+\beta_{2})}{\text{B}(\beta_{1},\beta_{2})}\\\
&{}\times\int_{0}^{1}\frac{\pi_{kl}^{y_{kl}+\beta_{1}-1}(1-\pi_{kl})^{p_{kl}-y_{kl}+\beta_{2}-1}}{\text{B}(y_{kl}+\beta_{1},p_{kl}-y_{kl}+\beta_{2})}\;\mathrm{d}\pi_{kl}\\\
=&{}\frac{\text{B}(y_{kl}+\beta_{1},p_{kl}-y_{kl}+\beta_{2})}{\text{B}(\beta_{1},\beta_{2})}\,,\end{split}$
where
$\mathrm{B}(\beta_{1},\beta_{2})=\frac{\Gamma(\beta_{1})\Gamma(\beta_{2})}{\Gamma(\beta_{1}+\beta_{2})}$
is the Beta function. This result is closely related to the Beta-binomial
distribution.
Next, we’ll consider the Poisson model for edges in more detail. Again, we
will see that $p_{b}$ and $y_{b}$ are sufficient for
$\mathrm{P}(x_{(kl)}|K,z)$.
In this integer-weighted model, an edge (or non-edges) between a node in
cluster $k$ and a node in cluster $l$ gets its weight from a Poisson
distribution
$x_{i}|\pi_{kl}\sim\mbox{Poisson}(\pi_{kl})\,,$
and $\pi_{kl}>0$.
This gives us, iterating over the pairs of nodes in the block,
$\mathrm{P}(x_{(kl)}|\pi_{kl},z,K)=\prod_{i,j\in
k,l}\frac{\pi_{kl}^{x_{ij}}}{x_{ij}!}\mbox{exp}(-\pi_{kl})\,.$
We can combine this expression for every block,
$\begin{split}\mathrm{P}(x|\pi,z,K)&=\prod_{kl}\mathrm{P}(x_{(kl)}|\pi_{kl},z,K)\\\
&=\prod_{kl}\prod_{i,j\in
k,l}\frac{\pi_{kl}^{x_{ij}}}{x_{ij}!}\mbox{exp}(-\pi_{kl})\\\
&=\prod_{ij}\frac{1}{x_{ij}!}\prod_{kl}\prod_{i,j\in
k,l}{\pi_{kl}^{x_{ij}}}\mbox{exp}(-\pi_{kl})\,.\end{split}$
We can ignore the $\prod_{ij}\frac{1}{x_{ij}!}$, as one of those will be
included for every pair of nodes in the network. That will contribute a
constant factor to eq. 6; this factor will depend only on the network $x$, and
it will not depend on $K$ or $z$ or any other variable of interest, and hence
we can ignore it for the purposes of eq. 6. Therefore, for our purposes we
will be able to use the following approximation in the derivation
$\mathrm{P}(x_{(kl)}|\pi_{kl},z,K)=\prod_{i,j\in
k,l}{\pi_{kl}^{x_{ij}}}\mbox{exp}(-\pi_{kl})\,.$
We’ll place a Gamma prior on the rates,
$\pi_{b}\sim\mbox{Gamma}(s,\phi)\,.$
$\begin{split}\mathrm{P}(x_{(kl)}&|z,K)=\int_{0}^{\infty}\mathrm{p}(x_{(kl)},\pi_{kl}|z,K)\mathrm{d}\pi_{kl}\\\
&={}\int_{0}^{\infty}\pi_{kl}^{s-1}\frac{e^{-\pi_{kl}/\phi}}{\Gamma(s)\phi^{s}}\prod_{i,j\in
k,l}\frac{\pi_{kl}^{x_{ij}}}{x_{ij}!}e^{-\pi_{kl}}\mathrm{d}\pi_{kl}\\\
={}\prod&\frac{1}{x_{ij}!}\int_{0}^{\infty}\pi_{kl}^{s-1}\frac{e^{-\pi_{kl}/\phi}}{\Gamma(s)\phi^{s}}\prod_{i,j\in
k,l}{\pi_{kl}^{x_{ij}}}e^{-\pi_{kl}}\mathrm{d}\pi_{kl}\\\
&\propto{}\int_{0}^{\infty}\pi_{kl}^{s-1}\frac{e^{-\pi_{kl}/\phi}}{\Gamma(s)\phi^{s}}\prod_{i,j\in
k,l}{\pi_{kl}^{x_{ij}}}e^{-\pi_{kl}}\mathrm{d}\pi_{kl}\\\
&={}\int_{0}^{\infty}\pi_{kl}^{s-1+\sum
x_{ij}}\;\frac{\exp(-\pi_{kl}p_{kl}-\frac{\pi_{kl}}{\phi})}{\Gamma(s)\phi^{s}}\mathrm{d}\pi_{b}\,.\end{split}$
We said earlier that we’ll define $y_{kl}=\sum_{i,j\in k,l}x_{ij}$. We’ll now
substitute that in and also use the following definitions:
$\begin{split}s^{\prime}&=s+y_{kl}\\\
\frac{1}{\phi^{\prime}}&=p_{kl}+\frac{1}{\phi}\,.\end{split}$
Where $\text{Gamma}(s,\phi)$ was the prior on $\pi_{b}$,
$\text{Gamma}(s^{\prime},\phi^{\prime})$ is the posterior now that we have
observed edges with total weight $y_{kl}$ between $p_{kl}$ pairs of nodes.
Returning to $f$, and rearranging such that we can cancel out the integral
(because it is the integral of a Gamma distribution and hence it will equal
1),
$\begin{split}f(x_{(kl)}|z,K)&={}\int_{0}^{\infty}\pi_{kl}^{s^{\prime}-1}\;\frac{\exp(-\frac{\pi_{kl}}{\phi^{\prime}})}{\Gamma(s)\phi^{s}}\mathrm{d}\pi_{kl}\\\
&={}\frac{\Gamma(s^{\prime})\phi^{\prime
s^{\prime}}}{\Gamma(s)\phi^{s}}\int_{0}^{\infty}\pi_{kl}^{s^{\prime}-1}\;\frac{\exp(-\frac{\pi_{kl}}{\phi^{\prime}})}{\Gamma(s^{\prime})\phi^{\prime
s^{\prime}}}\mathrm{d}\pi_{kl}\\\ &={}\frac{\Gamma(s^{\prime})\phi^{\prime
s^{\prime}}}{\Gamma(s)\phi^{s}}\\\
&={}\frac{\Gamma(s+y_{kl})\left(\frac{1}{p_{kl}+\frac{1}{\phi}}\right)^{s+y_{kl}}}{\Gamma(s)\phi^{s}}\,.\end{split}$
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|
arxiv-papers
| 2012-03-14T13:44:00 |
2024-09-04T02:49:28.639024
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aaron F. McDaid, Thomas Brendan Murphy, Nial Friel and Neil J Hurley",
"submitter": "Aaron Francis McDaid",
"url": "https://arxiv.org/abs/1203.3083"
}
|
1203.3592
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCB-PAPER-2011-024 CERN-PH-EP-2012-070
Measurements of the branching fractions and $C\\!P$ asymmetries
of $B^{\pm}\rightarrow J\\!/\\!\psi\,\pi^{\pm}$ and
$B^{\pm}\rightarrow\psi(2S)\pi^{\pm}$ decays
The LHCb collaboration 111Authors are listed on the following pages.
A study of $B^{\pm}\rightarrow J\\!/\\!\psi\,\pi^{\pm}$ and
$B^{\pm}\rightarrow\psi(2S)\pi^{\pm}$ decays is performed with data
corresponding to $0.37\,{\rm fb}^{-1}$ of proton-proton collisions at
$\sqrt{s}=7\,\mathrm{Te\kern-1.00006ptV}$. Their branching fractions are found
to be
$\displaystyle\mathcal{B}(B^{\pm}\rightarrow J\\!/\\!\psi\,\pi^{\pm})$
$\displaystyle=$ $\displaystyle(3.88\pm 0.11\pm 0.15)\times 10^{-5}\ {\rm
and}$ $\displaystyle\mathcal{B}(B^{\pm}\rightarrow\psi(2S)\pi^{\pm})$
$\displaystyle=$ $\displaystyle(2.52\pm 0.26\pm 0.15)\times 10^{-5},$
where the first uncertainty is related to the statistical size of the sample
and the second quantifies systematic effects. The measured $C\\!P$ asymmetries
in these modes are
$\displaystyle A_{CP}^{J\\!/\\!\psi\,\pi}$ $\displaystyle=$ $\displaystyle
0.005\pm 0.027\pm 0.011\ {\rm and}$ $\displaystyle A_{CP}^{\psi(2S)\pi}$
$\displaystyle=$ $\displaystyle 0.048\pm 0.090\pm 0.011$
with no evidence of direct $C\\!P$ violation seen.
Submitted to Phys. Rev. X
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
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A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
The Cabibbo-suppressed decay $B^{+}\rightarrow\psi\pi^{+}$, where $\psi$
represents either a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or
$\psi{(2S)}$, proceeds via a $b\rightarrow c\bar{c}d$ quark transition. Its
branching fraction is expected to be about 5% of the favoured $b\rightarrow
c\bar{c}s$ mode, $B^{+}\rightarrow\psi K^{+}$ (charge conjugation is implied
unless otherwise stated). The Standard Model predicts that for $b\rightarrow
c\bar{c}s$ decays the tree and penguin contributions have the same weak phase
and thus no direct $C\\!P$ violation is expected in $B^{+}\rightarrow\psi
K^{+}$. For $B^{+}\rightarrow\psi\pi^{+}$, the tree and penguin contributions
have different phases and $C\\!P$ asymmetries at the per mille level may occur
[1]. An additional asymmetry may be generated, at the percent level, from
long-distance rescattering, particularly from decays that have the same quark
content ($D^{0}D^{-}$, $D^{*-}D^{0}$, …) [2]. Any asymmetry larger than this
would be of significant interest.
In this paper, the $C\\!P$ asymmetries
$A^{\psi\pi}=\frac{\mathcal{B}(B^{-}\rightarrow\psi\pi^{-})-\mathcal{B}(B^{+}\rightarrow\psi\pi^{+})}{\mathcal{B}(B^{-}\rightarrow\psi\pi^{-})+\mathcal{B}(B^{+}\rightarrow\psi\pi^{+})}$
(1)
and charge-averaged ratios of branching fractions
$R^{\psi}=\frac{\mathcal{B}(B^{\pm}\rightarrow\psi\pi^{\pm})}{\mathcal{B}(B^{\pm}\rightarrow\psi
K^{\pm})}$ (2)
are measured with the $\psi$ reconstructed in the $\mu^{+}\mu^{-}$ final
state. From the latter, $\mathcal{B}(B^{\pm}\rightarrow\psi\pi^{\pm})$ may be
deduced using the established $B^{\pm}\rightarrow\psi K^{\pm}$ branching
fractions [3]. The $C\\!P$ asymmetry for $B^{+}\rightarrow\psi{(2S)}K^{+}$ is
also reported. $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ acts as a control mode in the asymmetry analysis because it is
well measured and no $C\\!P$ violation is observed [3]. Previous measurements
of the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
branching fractions and $C\\!P$ asymmetries [4, 5] have an accuracy of about
10%. The $B^{+}\rightarrow\psi{(2S)}h^{+}\ (h=K,\pi)$ system is less precisely
known due to a factor ten lower branching fraction to the $h\mu\mu$ final
state. The world average for $A^{\psi{(2S)}K}$ is $-0.025\pm 0.024$ [3] and
there has been one measurement of $A^{\psi{(2S)}\pi}=0.022\pm 0.086$ [6].
The LHCb experiment [7] takes advantage of the high $b\bar{b}$ and $c\bar{c}$
cross sections at the Large Hadron Collider to record unprecedented samples of
heavy hadron decays. It instruments the pseudorapidity range $2<\eta<5$ of the
proton-proton ($pp$) collisions with a dipole magnet and a tracking system
which achieves a momentum resolution of $0.4-0.6\%$ in the range $5-100$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The dipole magnet can be operated in
either polarity and this feature is used to reduce systematic effects due to
detector asymmetries. In the sample analysed here, 55% of data was taken with
one polarity, 45% with the other.
The $pp$ collisions take place inside a silicon-strip vertex detector which
has active material $8\rm\,mm$ from the beam line. It provides measurements of
track impact parameters (IP) with respect to primary collision vertices (PV)
and precise reconstruction of secondary $B^{+}$ vertices. Downstream muon
stations identify muons by their penetration through layers of iron shielding.
Charged particle identification (PID) is realised using ring-imaging Cherenkov
(RICH) detectors with three radiators: aerogel, ${\rm C}_{4}{\rm F}_{10}$ and
${\rm CF}_{4}$. Events with a high transverse energy cluster in calorimeters
or a high transverse momentum ($p_{\rm T}$) muon activate a hardware trigger.
About $1{\rm\,MHz}$ of such events are passed to a software-implemented high
level trigger, which retains about $3{\rm\,kHz}$.
The analysis is performed using $0.37~{}\mbox{\,fb}^{-1}$ of data recorded by
LHCb in the first half of 2011. The decay chain $B^{+}\rightarrow\psi h^{+},\
\psi\rightarrow\mu^{+}\mu^{-}$ is reconstructed from good quality tracks which
have a track-fit $\chi^{2}$ per degree of freedom $<5$. The muons are required
to have momentum, $p>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
$\mbox{$p_{\rm T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Selected hadrons
have $p>5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\mbox{$p_{\rm
T}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The two muon candidates are used
to form a $\psi$ resonance with vertex-fit $\chi^{2}<10$. The dimuon invariant
mass is required to be within ${}^{+30}_{-40}$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $\psi$ mass [3]; the
asymmetric limits allow for a radiative tail.
The reconstructed $B^{+}$ candidate vertex is required to be of good quality
with a vertex-fit $\chi^{2}<10$. It is ensured to originate from a PV by
requiring $\chi_{\rm IP}^{2}<25$ where the $\chi^{2}$ considers the
uncertainty on track IP and the PV position. In addition, the angle between
the $B^{+}$ momentum vector and its direction of flight from the PV must be
$<32~{}(10)$ $\rm\,mrad$ for $\psi{(2S)}h^{+}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}$). Furthermore, neither
the muons nor the hadron track may point back to any primary vertex with
$\chi_{\rm IP}^{2}$ $<4$. It is required that the hardware trigger accepted a
muon from the $B^{+}$ candidate or by activity in the rest of the event.
Hardware-trigger decisions based on the hadron are neglected to remove
dependence on the correct emulation of the calorimeter’s response to pions and
kaons.
The $B^{+}$ candidates are refitted [8] requiring all three tracks to
originate from the same point in space and the $\psi$ candidates to have their
nominal mass [3]. Candidates for which one muon gives rise to two tracks in
the reconstruction, one of which is then assumed to be the hadron, form an
artificial peaking background in the $\psi{(2S)}h^{+}$ analysis. These
candidates peak in the invariant mass distribution of the same-sign muon-pion
combination at $m_{\mu\pi}\sim 245$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, i.e. the sum of the muon and pion
rest masses. Requiring $m_{\mu\pi}>300$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ removes this background. In 2% of
events two $B^{+}$ candidates are found. If they decay within 2 $\rm\,mm$ of
each other the candidate with the poorest quality vertex is removed; otherwise
both are kept.
When selecting ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}$ candidates,
a requirement is made on the decay angle of the charged hadron as measured in
the rest frame of the $B^{+}$ with respect to the $B^{+}$ trajectory in the
laboratory frame, $\cos(\theta^{*}_{h})<0$. This requires the hadron to have
flown counter to the trajectory of the $B^{+}$ candidate, hence lowering its
average momentum in the laboratory frame. At lower momentum, the pion-kaon
mass difference provides sufficient separation in the $B^{+}$ invariant mass
distribution, as shown in Fig. 1. In the $B^{+}\rightarrow\psi{(2S)}h^{+}$
analysis, the average momentum of the hadrons is lower, so such a cut is
unecessary to separate the two modes.
Figure 1: Distribution of $\cos(\theta^{*}_{h})$ versus the invariant mass of
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
candidates. The curved structure contains misidentified
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
which separate from the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ vertical band for $\cos(\theta^{*}_{h})<0$. The partially
reconstructed background, $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K\pi$ enters top left.
Particle identification information is quantified as differences between the
logarithm of likelihoods, $\ln\mathcal{L}_{h}$, under five mass hypotheses,
$h\in\\{\pi,\ K,\ p,\ e,\ \mu\\}$. Separation of $\psi\pi^{+}$ candidates from
$\psi K^{+}$ is ensured by requiring that the hadron track satisfies
$\ln\mathcal{L}_{K}-\ln\mathcal{L}_{\pi}={\rm DLL}_{K\pi}<6$. This value is
chosen to ensure that most ($\sim 95\%$) $B^{+}\rightarrow\psi\pi^{+}$ decays
are reconstructed as such. These events form the “pion-like” sample, as
opposed to the kaon-like events satisfying ${\rm DLL}_{K\pi}>6$ that are
reconstructed under the $\psi K^{+}$ hypothesis.
The selected data are partitioned by magnet polarity, charge and
$\mathrm{DLL}_{K\pi}$ of the hadron track. By keeping the two magnet polarity
samples separate, residual detection asymmetries between the left and right
sides of the detector can be evaluated and hence factor out. Event yields are
extracted by performing an unbinned, maximum-likelihood fit simultaneously to
the eight distributions of $B$ invariant mass in the range
$5000<m_{B}<5780~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [9]. Figure 2
shows this fit to the data for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}$, summed
over magnet polarity. The $B^{+}\rightarrow\psi{(2S)}h^{+}$ data is shown in
Fig. 3.
Figure 2: Distributions of
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{\pm}$
invariant mass, overlain by the total fitted PDF (thin line). Pion-like
events, with DLL${}_{K\pi}<6$ are reconstructed as
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$ and enter in the top
plots. All other events are reconstructed as
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ and are shown in the
bottom plots on a logarithmic scale. $B^{-}$ decays are shown on the left,
$B^{+}$ on the right. The dark [red] curve shows the
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$
component, the light [green] curve represents
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$. The
partially reconstructed contributions are shaded. In the lower plots these are
visualised with a dark (light) shade for $B^{0}_{s}$ ($B^{+}$ or $B^{0}$)
decays. In the top plots the shaded component are contributions from
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}\pi$ (dark)
and $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}\pi$
(light). Figure 3: Distributions of $B^{\pm}\rightarrow\psi{(2S)}h^{\pm}$
invariant mass. See the caption of Fig. 2 for details. The partially
reconstructed background in the pion-like sample is present but negligible
yields are found.
The probability density function (PDF) used to describe these distributions
has several components. The correctly reconstructed, $B^{+}\rightarrow\psi
h^{+}$ events are modelled by the function,
$f(x)\propto\exp\left(\frac{-(x-\mu)^{2}}{2\sigma^{2}+(x-\mu)^{2}\alpha_{L,R}}\right)$
(3)
which describes an asymmetric peak of mean $\mu$ and width $\sigma$, and where
$\alpha_{L}(x<\mu)$ and $\alpha_{R}(x>\mu)$ parameterise the tails. The mean
is required to be the same for $\psi K^{+}$ and $\psi\pi^{+}$ though it can
vary across the four charge$\times$polarity subsamples to account for
different misalignment effects. Table 1 shows the fitted values of the common
tail parameters and the widths of the $B^{+}\rightarrow\psi h^{+}$ peaks
averaged over the subsamples.
Table 1: Signal shape parameters from the $B^{\pm}\rightarrow\psi h^{\pm}$ fits. | | ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ | $\psi{(2S)}$
---|---|---|---
$\sigma_{\psi K}$ | (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) | $7.84\pm 0.04$ | $6.02\pm 0.08$
$\sigma_{\psi\pi}$ | (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) | $8.58\pm 0.27$ | $6.12\pm 0.75$
$\alpha_{\rm L}$ | | $0.12\pm 0.03$ | $0.14\pm 0.01$
$\alpha_{\rm R}$ | | $0.10\pm 0.03$ | $0.13\pm 0.01$
The misidentified $\psi K^{+}$ events form a displaced peaking structure to
the left of the $\psi\pi^{+}$ signal and tapers to lower mass. This is
modelled by a Crystal Ball function [10] which is found to be a suitable
effective PDF. Its yield is added to that of the correctly identified events
to calculate the total number of $\psi K^{+}$ events.
The PDF modelling the small component of $\psi\pi^{+}$ decays with
DLL${}_{K\pi}>6$ is fixed entirely from simulation. It contributes negligibly
to the total likelihood so the yield must be fixed with respect to that of
correctly identified $\psi\pi^{+}$ events. The efficiency of the PID cut is
estimated using samples of pions and kaons from $D^{0}\rightarrow
K^{+}\pi^{-}$ decays which are selected with high purity without using PID
information. These calibration events are reweighted in bins of momentum to
match the momentum distribution of the large
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and $\psi{(2S)}K^{+}$
samples. By this technique, the following efficiencies are deduced for ${\rm
DLL}_{K\pi}<6$: $\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi}=(95.8\pm 1.0)\%$; $\epsilon_{\psi{(2S)}\pi}=(96.6\pm 1.0)\%$. The
errors, estimated from simulation, account for imperfections in the
reweighting and the difference of the signal $K^{+}$ and $\pi^{+}$ momenta.
Partially reconstructed decays populate the region below the $B^{+}$ mass.
$B^{+/0}\rightarrow\psi K^{+}\pi$ decays, where the pion is missed, are
modelled in the kaon-like sample by a flat PDF with a Gaussian edge. A small
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow\psi
K^{+}\pi^{-}$ component is needed to achieve a stable fit. It is modelled with
the same shape as the partially reconstructed $B^{+/0}$ decays except shifted
in mass by the $B^{0}_{s}$-$B^{0}$ mass difference, $+87$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the pion-like sample,
$\psi\pi^{+}\pi$ backgrounds are assumed to enter with the same PDF, and same
proportion relative to the signal, as the $\psi K^{+}\pi$ background in the
kaon-like sample. A component of misidentified
$B^{+/0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi$ is
also included with a fixed shape estimated from the data. Lastly, a linear
polynomial with a negative gradient is used to approximate the combinatorial
background. The slope of this component of the pion-like and kaon-like
backgrounds can differ.
Table 2: Raw fitted yields. The labels ‘D’ and ‘U’ refer to the two polarities of the LHCb dipole. | | | $B^{-}$ | $B^{+}$
---|---|---|---|---
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ | $\pi$ | D | $\phantom{00\,}528\pm\phantom{0}27$ | $\phantom{00\,}518\pm\phantom{0}27$
U | $\phantom{00\,}421\pm\phantom{0}23$ | $\phantom{00\,}428\pm\phantom{0}23$
$K$ | D | $\phantom{}13\,363\pm\phantom{}180$ | $\phantom{}13\,466\pm\phantom{}181$
U | $\phantom{}10\,666\pm\phantom{}148$ | $\phantom{}11\,120\pm\phantom{}155$
$\psi{(2S)}$ | $\pi$ | D | $\phantom{00\,0}94\pm\phantom{0}16$ | $\phantom{00\,0}93\pm\phantom{0}16$
U | $\phantom{00\,0}82\pm\phantom{0}15$ | $\phantom{00\,0}70\pm\phantom{0}13$
$K$ | D | $\phantom{0}2\,331\pm\phantom{0}88$ | $\phantom{0}2\,463\pm\phantom{0}93$
U | $\phantom{0}2\,026\pm\phantom{0}78$ | $\phantom{0}1\,836\pm\phantom{0}71$
The stability of the fit is tested with a large sample of pseudo-experiments.
Pull distributions from these tests are consistent with being normally
distributed, demonstrating that the fit is stable under statistical
variations. The yields obtained from the signal extraction fit are shown in
Table 2.
The observables, defined in Eqs. 1 and 2 are calculated by the fit, then
modified by a set of corrections taken from simulation. The acceptances of
$\psi\pi^{+}$ and $\psi K^{+}$ events in the detector are computed using
Pythia [11] to generate the primary collision and EvtGen [12] to model the
$B^{+}$ decay. The efficiency of reconstructing and selecting $\psi\pi^{+}$
and $\psi K^{+}$ decays is estimated with a bespoke simulation of LHCb based
on Geant4 [13]. It models the interaction of muons and the two hadron species
with the detector material. The total correction $\epsilon^{\psi
K}\\!/\epsilon^{\psi\pi}$ is $0.985\pm 0.012$ and $1.007\pm 0.021$ for
$R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $R^{\psi{(2S)}}$
respectively.
$C\\!P$ asymmetries are extracted from the observed charge asymmetries
$(A_{\rm Raw})$ by taking account of instrumentation effects. The interaction
asymmetry of kaons, $A_{\rm Det}^{K}$ is expected to be non-zero, especially
for low-momentum particles. This asymmetry, measured at LHCb using a sample of
$D^{*+}\rightarrow D^{0}\pi^{+},\ D^{0}\rightarrow K^{+}\pi^{-}$ decays, is
$-0.010\pm 0.002$ if the pion asymmetry is zero [14]. The null-asymmetry
assumption for pions has been verified at LHCb to an accuracy of $0.25$% [15].
These results are used with enlarged uncertainties ($0.004$, for both kaons
and pions) to account for the different momentum spectra of this sample and
those used in the previous analyses.
In summary, the $C\\!P$ asymmetry is defined as
$A^{\psi h}=A_{\rm Raw}^{\psi h}-A_{\rm Prod}-A_{\rm Det}^{h},$ (4)
where the production asymmetry, $A_{\rm Prod}$, describes the different rates
with which $B^{-}$ and $B^{+}$ hadronise out of the $pp$ collisions. The
observed, raw charge asymmetry in
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ is
$-0.012\pm 0.004$. Using Eq. 4 with the established $C\\!P$ asymmetry,
$A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}=0.001\pm 0.007$ [3],
$A_{\rm Prod}$ is estimated to be $-0.003\pm 0.009$. This is applied as a
correction to the other modes reported here.
Table 3: Summary of systematic uncertainties. The statistical fit errors are included for comparison. | | $R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\times 10^{-2})$ | $A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi}$ | | $R^{\psi{(2S)}}(\times 10^{-2})$ | $A^{\psi{(2S)}\pi}$ | $A^{\psi{(2S)}K}$
---|---|---|---|---|---|---|---
Simulation uncertainty | | $0.045$ | - | | $0.088$ | - | -
PID efficiencies | | $0.043$ | - | | $0.052$ | - | -
$A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}$ (PDG [3]) | | - | $0.0070$ | | - | $0.0070$ | $0.0070$
$A_{\rm Raw}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}$ statistical error | | - | $0.0046$ | | - | $0.0046$ | $0.0046$
Detection asymmetries | | - | $0.0056$ | | - | $0.0056$ | -
Relative trigger efficiency | | $0.020$ | $0.0031$ | | $0.050$ | $0.0036$ | $0.0003$
Fixed fit parameters | | $0.005$ | $0.0006$ | | $0.017$ | $0.0013$ | $0.0001$
Sum in quadrature (syst.) | | $0.065$ | $0.0106$ | | $0.115$ | $0.0108$ | $0.0084$
Fit error (stat.) | | $0.110$ | $0.0268$ | | $0.404$ | $0.0901$ | $0.0136$
The different contributions to the systematic uncertainties are summarised in
Table 3. They are assessed by modifying the final selection, or altering fixed
parameters and rerunning the signal yield fit. The maximum variation of each
observable is taken as their systematic uncertainty.
The largest uncertainty is due to the use of simulation to estimate the
acceptance and selection efficiencies. It accounts for any bias due to
imperfect modelling of the detector and its relative response to pions and
kaons. Another important contribution arises from the loose trigger criteria
that are employed. This uncertainty is estimated from the shift in the central
values after rerunning the fit using only those events where the muons passed
the software trigger. The use of the PID calibration to estimate the
efficiency for pions to the DLL${}_{K\pi}<6$ selection also contributes a
significant systematic uncertainty.
The measurements of $A^{\psi\pi}$ depend on the estimation of $A_{\rm Prod}$
from the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
channel. The uncertainty on $A_{\rm Prod}$ is determined by the statistical
error of $A_{\rm Raw}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}$ in the
fit, the uncertainty on the world average of
$A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}$ and the estimation of
$A_{\rm Det}^{h}$. These effects are kept separate in the table where it is
seen that the uncertainty on the nominal value of
$A^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K}$ dominates. Finally, it is
noted that the detector asymmetries cancel for $A^{\psi{(2S)}K}$ and a lower
systematic uncertainty can be reported.
The measured ratios of branching fractions are
$\displaystyle R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
$\displaystyle=$ $\displaystyle(3.83\pm 0.11\pm 0.07)\times 10^{-2}$
$\displaystyle R^{\psi{(2S)}}$ $\displaystyle=$ $\displaystyle(3.95\pm 0.40\pm
0.12)\times 10^{-2},$
where the first uncertainty is statistical and the second systematic.
$R^{\psi{(2S)}}$ is compatible with the one existing measurement, $(3.99\pm
0.36\pm 0.17)\times 10^{-2}$ [6]. The measurement of
$R^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is $3.2\sigma$ lower than
the current world average, $(5.2\pm 0.4)\times 10^{-2}$ [3]. Using the
established measurements of the Cabibbo-favoured branching fractions [3], we
deduce
$\displaystyle\mathcal{B}(B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm})$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle(3.88\pm 0.11\pm 0.15)\times 10^{-5}$
$\displaystyle\mathcal{B}(B^{\pm}\rightarrow\psi{(2S)}\pi^{\pm})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle(2.52\pm 0.26\pm
0.15)\times 10^{-5},$
where the systematic uncertainties are summed in quadrature. The measured
$C\\!P$ asymmetries,
$\displaystyle A_{CP}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi}$
$\displaystyle=$ $\displaystyle 0.005\pm 0.027\pm 0.011$ $\displaystyle
A_{CP}^{\psi{(2S)}\pi}$ $\displaystyle=$ $\displaystyle 0.048\pm 0.090\pm
0.011$ $\displaystyle A_{CP}^{\psi{(2S)}K}$ $\displaystyle=$ $\displaystyle
0.024\pm 0.014\pm 0.008,$
have comparable or better precision than previous results, and no evidence of
direct $C\\!P$ violation is seen.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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|
arxiv-papers
| 2012-03-16T00:27:55 |
2024-09-04T02:49:28.669405
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer,\n Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, 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, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, K. de Bruyn, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G.A. Cowan, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P.\n De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano,\n D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\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, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A.\n Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni,\n G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis,\n M. Veltri, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams,\n F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Malcolm John",
"url": "https://arxiv.org/abs/1203.3592"
}
|
1203.3603
|
# Schauder Bases and Operator Theory
Yang Cao Yang Cao, Department of Mathematics , Jilin university, 130012,
Changchun, P.R.China Caoyang@jlu.edu.cn , Geng Tian Geng Tian, Department
of Mathematics , Jilin university, 130012, Changchun, P.R.China
tiangeng09@mails.jlu.edu.cn and Bingzhe Hou Bingzhe Hou, Department of
Mathematics , Jilin university, 130012, Changchun, P.R.China houbz@jlu.edu.cn
(Date: Oct. 14, 2010)
###### Abstract.
In this paper, we firstly give a matrix approach to the bases of a separable
Hilbert space and then correct a mistake appearing in both review and the
English translation of the Olevskii’s paper. After this, we show that even a
diagonal compact operator may map an orthonormal basis into a conditional
basis.
###### Key words and phrases:
.
###### 2000 Mathematics Subject Classification:
Primary 47B37, 47B99; Secondary 54H20, 37B99
## 1\. Introduction and preliminaries
In operator theory, an invertible operator on an infinite dimensional complex
Hilbert space $\mathcal{H}$ means the bounded operator which has a bounded
inverse operator, and it is well-known that, for an $n\times n$ matrix $M_{n}$
(seen as an operator on finite dimensional Hilbert space $\mathbb{C}^{n}$),
$M_{n}$ is invertible if and only if its column vectors are linearly
independent in $\mathbb{C}^{n}$. In other words, the column vectors of $M_{n}$
comprise a basis of $\mathbb{C}^{n}$. From this point of view, we could
generalize the "invertibility" of $\omega\times\omega$ matrix $M$ (the
representation of a bounded operator on an orthonormal basis of $\mathcal{H}$)
in the following manner: all column vectors of $M$ form some kind of basis of
$\mathcal{H}$. Actually, the invertible operator do have a natural
understanding in the ‘basis’ language. That is, the column (or row) vectors of
the matrix of an invertible operator always comprise a ‘Riesz basis’ (it is a
direct corollary of theorem 2, paper [1], although the authors do not state it
in this way). From the above observation, it suggests us to consider the
$\omega\times\omega$ matrix whose column vectors form more general kind of
bases.
Naturally we consider the $\omega\times\omega$ matrix whose column vectors
comprise a Schauder basis. We shall call them the Schauder matrix therefrom.
An operator which has a Schauder matrix representation under some orthonormal
basis (ONB) will be called a Schauder operator. An easy fact is that an
operator is a Schauder operator if and only if it maps some ONB into a
Schauder basis. Many scholars have studied some kind of these operators. A. M.
Olevskii gave a surprising result on the bounded operators which map some ONB
into a conditional quasinormal basis ([5], theorem 1, p479); Stephane Jaffard
and Robert M. Young proved that a Schauder basis always can be given by an
one-to-one positive transformation ([1], theorem 1, p554). I. Singer gave lots
of examples of bases of $\mathcal{H}$ which can be rewritten into a matrix
form (see, [6], p429, p497). Besides these results, as for a joint research
both on operator theory and the basis theory but not in this direction, the
paper [25], [26] by Gowers, the paper [13] by Kwapien, S. and Pelczynski, A.
and the elegant book [2] by M. Young are remarkable examples.
Nevertheless, there is still a gap between the researches in the field of
basis theory and operator theory. There are few joint works on both basis of
Hilbert space and the operators on the Hilbert space. The reason reflects on
two aspects. One is the different terminology systems and the other one is
that there are scanty common objects to study with. The main purpose of this
paper is to show that the Schauder matrix is a candidate to fill this gap. As
basic and traditional tools, the matrix representation of operators plays an
important role in the study of the operators on the Hilbert space
$\mathcal{H}$. So the matrix approach to the basis theory is a good beginning
to the joint research on the bases of the Hilbert space $\mathcal{H}$ and the
operators on it.
In this paper, the matrix representation of operators and bases will be the
bridge between basis theory and operator theory. We firstly give a matrix
approach to the bases of a separable Hilbert space and then correct a mistake
appearing in both review and the English translation of the Olevskii’s paper.
After this, we follow the Olevskii’s result to consider the operators which
can map some ONB into a conditional Schauder basis. We shall call them
conditional operators therefrom. In matrix language, it is equivalent to study
the operator $T$ which has a matrix representation $M$ under some ONB such
that the column vector sequence of $M$ comprise a conditional Schauder basis.
## 2\. An Operator Theory Description of Schauder basis
### 2.1.
Suppose that $\\{e_{k}\\}_{k=1}^{\infty}$ is an ONB of $\mathcal{H}$. An
$\omega\times\omega$ matrix $M=(m_{ij})$ automatically represents an operator
under this ONB. In more details, for a vector $x\in\mathcal{H}$ there is an
unique $l^{2}-$sequence $\\{x_{n}\\}_{n=1}^{\infty}$ such that
$x=\sum_{n=1}^{\infty}x_{n}e_{n}$ in which the series converges in the norm of
$\mathcal{H}$. Let
$y_{n}=\sum_{k=1}^{\infty}m_{ik}x_{k},y=\sum_{n=1}^{\infty}y_{n}e_{n}$, then
the operator $T_{M}$ defined by $T_{M}x=y$ is just the corresponding operator
represented by $M$. In general, $T$ is not a bounded operator. We shall
identify the $\omega\times\omega$ matrix $M$ and the operator $T_{M}$, and
denote them by the same notation $M$ if we have fixed an ONB and there is no
confusion.
Recall that a sequence $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is called a Schauder
basis of the Hilbert space $\mathcal{H}$ if and only if for every vector
$x\in\mathcal{H}$ there exists an unique sequence
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ of complex numbers such that the partial sum
sequence $x_{k}=\sum_{n=1}^{k}\alpha_{n}f_{n}$ converges to $x$ in norm.
Denote by $P_{k}$ the the diagonal operator with the first $k-$th entries on
diagonal line equal to 1 and $0$ for others. Then as an operator $P_{k}$
represents the orthogonal projection from $\mathcal{H}$ to the subspace
$\mathcal{H}^{(k)}=span\\{e_{1},e_{2},\cdots,e_{k}\\}$.
###### Lemma 2.1.
Assume that $\\{e_{k}\\}_{k=1}^{\infty}$ is a fixed ONB of $\mathcal{H}$.
Suppose that an $\omega\times\omega$ matrix $F=(f_{ij})$ satisfies the
following properties:
1\. Each column of the matrix $F$ is a $l^{2}-$sequence;
2\. $F$ has an unique left inverse matrix $G^{*}=(g_{kl})$ such that each row
of $G^{*}$ is also a $l^{2}-$sequence;
3\. Operators $Q_{k}$ defined by the matrix $Q_{k}=FP_{k}G^{*}$ are well-
defined projections on $\mathcal{H}$ and converges to the unit operator $I$ in
the strong operator topology.
Then the sequence
$\\{f_{k}\\}_{k=1}^{\infty},f_{k}=\sum_{i=1}^{\infty}f_{ik}e_{i}$ must be a
Schauder basis.
Here we use the term “left reverse” in the classical means, that is, the
series $\sum_{j=1}^{\infty}g_{kj}f_{jn}$ converges absolutely to $\delta_{kn}$
for $k,n=1,2,\cdots$. $G^{*}$ does not mean the adjoint of $G$, it is just a
notation.
###### Proof.
Property 1 just ensure that series
$\\{f_{k}=\sum_{j=1}^{\infty}f_{ij}e_{i}\\}_{k=0}^{\infty}$ converges to a
well-defined vector $f_{k}$ in $\mathcal{H}$ by norm. Property 2 implies that
$span\\{f_{n};n=1,2,\cdots\\}=\mathcal{H}$ by the uniqueness of the left
inverse. Moreover, the $k-$th row of the matrix $G^{*}$ is just the vector
$g_{k}^{*}$ such that $(g_{k}^{*},f_{n})=\delta_{kn}$. Therefore the vector
sequence $\\{f_{n}\\}_{n=1}^{\infty}$ must be minimal by the Hahn-Banach
theorem(cf, [7] corollary6.8, p82) and the Riesz representation theorem(see,
[7], theorem3.4, p12).
Now for each vector $x=(x_{1},x_{2},\cdots)$ denote by
$\alpha_{k}^{x}=(g_{k}^{*},x),$ it is easy to check that $Q_{k}^{2}=Q_{k}$ and
$Q_{k}x=FP_{k}G^{*}x=\sum_{n=1}^{k}\alpha^{x}_{k}f_{k}.$
By property 3, we have $Q_{k}x\rightarrow x$ since $Q_{k}$ converges to $I$ in
strong operator topology(SOT). That is, series
$\sum_{n=1}^{\infty}\alpha^{x}_{n}f_{n}$ converges to the vector $x$ in norm.
So we have proved that each vector $x$ in $\mathcal{H}$ can be represented by
the sequence $\\{f_{n}\\}_{n=1}^{\infty}$ with coefficients
$\\{\alpha_{n}^{x}\\}_{n=1}^{\infty}$.
To show that $\\{f_{n}\\}_{n=1}^{\infty}$ is a Schauder basis, we just need to
show that this representation is unique. Suppose that
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ is a sequence such that the series
$\sum_{n=1}^{\infty}\alpha_{n}f_{n}$ converges to $0$ in the norm of the
Hilbert space $\mathcal{H}$. Assume that the integer $n_{0}$ is the first
number satisfying $\alpha_{n_{0}}\neq 0$. Then we have
$f_{n_{0}}=-\frac{1}{\alpha_{n_{0}}}\cdot\sum_{n=n_{0}+1}^{\infty}\alpha_{n}f_{n}$
in which the series also converges in the norm topology. It counter to the
fact that the sequence $\\{f_{n}\\}_{n=1}^{\infty}$ is a minimal sequence. ∎
Conversely, suppose that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is a basis of
$\mathcal{H}$. For a fixed ONB $\\{e_{n}\\}_{n=1}^{\infty}$, each vector
$f_{n}$ has a representation $f_{n}=\sum_{k=1}^{\infty}f_{kn}e_{k}$. Denote
$F_{\psi}=(f_{kn})$. We shall call $F_{\psi}$ the Schauder matrix
corresponding to the basis $\psi$. The following lemma is the inverse of the
above lemma.
###### Lemma 2.2.
Assume that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is a Schauder basis. Then the
corresponding Schauder matrix $F_{\psi}$ satisfies the following properties:
1\. Each column of the matrix $F_{\psi}$ is a $l^{2}-$sequence;
2\. $F_{\psi}$ has an unique left inverse matrix $G_{\psi}^{*}=(g_{kl})$ such
that each row of $G_{\psi}^{*}$ is also a $l^{2}-$sequence;
3\. Operators $Q_{k}$ defined by the matrix $Q_{k}=F_{\psi}P_{k}G_{\psi}^{*}$
are well-defined projections on $\mathcal{H}$ and converges to the unit
operator $I$ in the strong operator topology.
###### Proof.
Property 1 comes from the fact that $f_{n}$ is a vector in $\mathcal{H}$.
If $\\{f_{k}\\}_{k=1}^{\infty}$ is a Schauder basis, then the subspace
$\widehat{\mathcal{H}}_{k}=span\\{f_{n};n\neq k\\}$ for each $k$ satisfying
$f_{k}\notin\widehat{\mathcal{H}}_{k}$(cf, [6], p50-51). So we must have a
unique linear functional $\varphi_{k}$ such that
$\varphi_{k}(f_{n})=\delta_{kn}$. Then by the Riesz representation theorem,
there is a unique vector $g^{*}_{k}=(g^{*}_{kl})\in\mathcal{H}$ such that
$\sum_{j=1}^{n}g^{*}_{kj},f_{jn}=\delta_{kn}$ in which
$\\{g^{*}_{kl}\\}_{l=1}^{\infty}$ is a $l^{2}-$sequence. The uniqueness holds
because the sequence $\\{f_{k}\\}_{k=1}^{\infty}$ spans the Hilbert space.
Hence a Schauder matrix must have a unique left inverse matrix whose rows are
$l^{2}-$sequence. Then we have proved the property 2.
Property 3 is just a direct corollary of the definition of Schauder basis.
Denote by $G=(G^{*})^{*}=g_{nk}$ the adjoint matrix of $G^{*}$, then we have
$g_{nk}=\overline{g_{kn}}$. Moreover, denote by $g_{n}$ the $n-$th column
vector and for a vector $x=\sum_{n=1}^{\infty}x_{n}e_{n}$ denote by
$y_{n}=\sum_{k=1}^{\infty}g_{nk}^{*}x_{k}$. Then trivially we have
$y_{n}=(x,g_{n})$ and $(f_{k},g_{n})=\delta_{kn}$ Suppose that
$x=\sum_{k=1}^{\infty}\alpha_{k}f_{k}$ is the representation of the vector $x$
under the basis $\psi$. Then we must have $\alpha_{n}=y_{n}$ since
$y_{n}=(x,g_{n})=(\sum_{k=1}^{\infty}\alpha_{k}f_{k},g_{n})=\alpha_{n}.$
Therefore we have $Q_{k}x=\sum_{n=1}^{\infty}\alpha_{n}f_{n}$. Clearly we have
$Q_{k}x\rightarrow x$ in the norm topology. In other words,
$||Q_{k}x-x||\rightarrow 0$ when $k\rightarrow\infty$ which implies
$Q_{k}\rightarrow I$ in SOT(cf, [7], proposition 1.3, p262). ∎
The matrix $G_{\psi}^{*}$ is unique and decided completely by $F_{\psi}$. In
fact the matrix $G^{*}$ is also the “right inverse” of the matrix $F$ in the
classical sense. For more details, let $F=(f_{kn})_{\omega\times\omega}$,
$G^{*}=(g_{mk})_{\omega\times\omega}$, $f_{n}=\\{f_{kn}\\}_{k=1}^{\infty}$ and
$g_{m}^{*}=\\{g_{mk}^{*}\\}_{k=1}^{\infty}$. Moreover, denote their adjoint
matrices by
$F^{*}=(f^{*}_{kn})_{\omega\times\omega}=(\overline{f_{nk}})_{\omega\times\omega}$,
$G=(g^{*}_{mk})_{\omega\times\omega}=(\overline{g_{km})}_{\omega\times\omega}$.
Then both $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ and
$\psi^{*}=\\{g_{m}\\}_{m=1}^{\infty}$ are biorthogonal basis to each other.
That is, $\psi$ and $\psi^{*}$ are bases and we have
$(f_{n},g_{m})=\delta_{nm}$ for all $n,m\in\mathbb{N}$. Now we show that the
series $\sum_{k=1}^{\infty}f_{nk}g^{*}_{km}$ converges to $\delta_{nm}$ as
$k\rightarrow\infty$ for all $n,m\in\mathbb{N}$. Let
$\\{e_{l}\\}_{l=1}^{\infty}$ be the corresponding ONB. We write $e_{n},e_{m}$
into the linearly combinations of basis vector in $\psi$ and
$\psi^{{}^{\prime}}$ as follows:
$e_{n}=\sum_{k=1}^{\infty}\alpha_{nk}f_{k},e_{m}=\sum_{k=1}^{\infty}\beta_{mk}g^{*}_{k}.$
Then we have $\alpha_{nk}=g^{*}_{kn}$ and
$\beta_{mk}=f^{*}_{km}=\overline{f_{mk}}$. Hence for any integer $N$
$\begin{array}[]{rl}\sum_{k=1}^{N}f_{nk}g^{*}_{km}&=(\sum_{k=1}^{N}\alpha_{nk}f_{k},\sum_{k=1}^{N}\beta_{mk}g^{*}_{k})\\\
&=(e_{n}-\sum_{k=N}^{\infty}\alpha_{nk}f_{k},e_{m}-\sum_{k=N}^{\infty}\beta_{mk}g^{*}_{k}).\end{array}$
Now given $\epsilon>0$, we choose an integer $N$ such that inequalities
$||e_{n}-\sum_{k=N}^{\infty}\alpha_{nk}f_{k}||<\frac{\epsilon}{2},||e_{m}-\sum_{k=N}^{\infty}\beta_{mk}g^{*}_{k}||<\frac{\epsilon}{2}$
hold. Then we have
$\begin{array}[]{rl}&|\sum_{k=1}^{N}f_{nk}g^{*}_{km}-(e_{n},e_{m})|\\\
=&|-(\sum_{k=N}^{\infty}\alpha_{nk}f_{k},e_{m}-\sum_{k=N}^{\infty}\beta_{mk}g^{*}_{k})\\\
&~{}~{}~{}~{}-(e_{n}-\sum_{k=N}^{\infty}\alpha_{nk}f_{k},\sum_{k=N}^{\infty}\beta_{mk}g^{*}_{k})+(\sum_{k=N}^{\infty}\alpha_{nk}f_{k},\sum_{k=N}^{\infty}\beta_{mk}g^{*}_{k})|\\\
\leq&\epsilon(|1+\frac{\epsilon}{2}|+\frac{\epsilon}{4}).\end{array}$
For this reason, we have the following definition.
###### Definition 2.3.
For a Schauder matrix $F_{\psi}$, the corresponding matrix $G_{\psi}^{*}$ is
called the inverse matrix of $F_{\psi}$.
If we do not ask that each row of $G^{*}$ is a $l^{2}-$sequence, an
$\omega\times\omega$ matrix may have a “left inverse” in the classical sense.
###### Example 2.4.
Let $F$ be the matrix
$\begin{bmatrix}1&1&0&0&\cdots\\\ 0&-1&1&0&\cdots\\\ 0&0&-1&1&\cdots\\\
0&0&0&-1&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix},$
and $G^{*}$ be the matrix
$\begin{bmatrix}1&1&1&1&\cdots\\\ 0&-1&-1&-1&\cdots\\\ 0&0&-1&-1&\cdots\\\
0&0&0&-1&\cdots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{bmatrix}.$
It is trivial to check $G^{*}F=FG^{*}=I$. Then by above lemma 2.1 we know $F$
is not a Schauder matrix since the rows of its inverse matrix are not
$l^{2}-$sequence. Moreover, if we denote by $g_{n}$ the $n-th$ column vector,
then the sequence $\xi=\\{g_{n}\\}_{n=1}^{\infty}$ is a complete minimal
sequence(see [6], p24 and p50 for definitions). It is easy to check that $\xi$
is complete since the $l^{2}$-sequence
$h_{n}=\\{h_{n}(j)\\}_{j=1}^{\infty},h_{n}(j)=\delta_{nj}$ is in its range; On
the other hand, the row vector sequence $\\{f_{k}\\}_{k=1}^{\infty}$ satisfies
$(g_{n},f_{k})=\delta_{kn}$ which implies $g_{n}\notin\vee_{m\neq n}g_{m}$(or
in notations of singer, we have
$g_{n}\notin[g_{1},\cdots,g_{n-1},g_{n+1},\cdots]$) by the fact $\vee_{m\neq
n}g_{m}=\ker\varphi_{k}$ in which $\varphi_{k}(x)=(x,f_{k})$ is a bounded
functional by Riesz’s theorem. Therefore $\xi$ is an example which is complete
and minimal sequence but not a basis sequence.
By above lemma 2.1 and 2.2, we have
###### Theorem 2.5.
An $\omega\times\omega$ matrix $F$ is a Schauder matrix if and only if it
satisfies property 1, 2 and 3.
For a Schauder matrix $F$, the column vector sequence
$\\{g_{n}\\}_{n=1}^{\infty}$ of $G$ defined in above lemmas is also a Schauder
basis which is called the biorthogonal basis to the basis
$\\{f_{k}\\}_{k=1}^{\infty}$(cf [2], pp23-29, [6] pp23-25).
The projection $FP_{n}G^{*}$ is just the $n-$th “natural projection” so called
in [4](p354). It is also the $n-$th partial sum operator so called in
[6](definition 4.4, p25). Now we can translate theorem 4.1.15 and corollary
4.1.17 in [4] into the following
###### Proposition 2.6.
If $F$ is a Schauder matrix, then $M=\sup_{n}\\{||FP_{n}G^{*}||\\}$ is a
finite const.
The const $M$ is called the basis const for the basis
$\\{f_{n}\\}_{n=1}^{\infty}$.
Assume that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is a basis. For a subset
$\Delta$ of $\mathbb{N}$, denote by $P_{\Delta}$ the diagonal matrix defined
as $P_{\Delta}(nn)=1$ for $n\in\Delta$ and $P_{\Delta}(nn)=0$ for
$n\notin\Delta$. The projection $Q_{\Delta}=F_{\psi}P_{\Delta}G_{\psi}^{*}$
defined in above lemmas is called a natural projection(see, definition 4.2.24,
[4], p378). In fact for a vector $x=\sum_{n=1}^{\infty}x_{n}f_{n}$, it is
trivial to check $Q_{\Delta}x=\sum_{n\in\Delta}x_{n}f_{n}$. Then we have a
same result for the unconditional basis const(cf, definition4.2.28, [4],
p379):
###### Proposition 2.7.
If $F_{\psi}$ is a Schauder matrix, then the unconditional basis const of the
basis $\psi$ is
$M_{ub}=\sup_{\Delta\subseteq\mathbb{N}}\\{||F_{\psi}P_{\Delta}G_{\psi}^{*}||\\}$.
In virtue of the proposition 4.2.29 and theorem 4.2.32 in the book [4], we
have
###### Proposition 2.8.
For a Schauder basis $\psi$, it is an unconditional basis if and only if
$\sup_{\Delta\subseteq\mathbb{N}}\\{||F_{\psi}P_{\Delta}G_{\psi}^{*}||\\}<\infty$.
Following the notations in lemma 2.1, as a direct corollary of lemma 2.1 and
theorem 6 in [2](p28), we have
###### Proposition 2.9.
$F$ is a Schauder matrix if and only if the adjoint matrix (conjugate
transpose) $G$ of its left inverse $G^{*}$ is a Schauder matrix.
As well known that a sequence of operators $T_{n}$ converges to an operator
$T$ in SOT dose not imply $T_{n}^{*}$ converging to $T$ in SOT, so the above
proposition is not trivial.
###### Corollary 2.10.
$M=\sup_{n}\\{||FP_{n}G^{*}||\\}<\infty$ if and only if
$M^{{}^{\prime}}=\sup_{n}\\{||GP_{n}F^{*}||\\}$ $<\infty$.
### 2.2.
From the definition of the Schauder matrix $F_{\psi}$, basic properties of
Schauder matrix have natural relations to the Schauder basis $\psi$. This
understanding lead us to the following definition.
###### Definition 2.11.
A matrix $F$ is called an unconditional, conditional, Riesz, normalized or
quasinormal respectively if and only if the sequence of its column vectors
comprise an unconditional, conditional, Riesz, normalized or quasinormal
basis. Two Schauder matrices $F_{\psi},F_{\varphi}$ are called equivalent if
and only if the corresponding bases $\psi$ and $\varphi$ are equivalent.
Here we use the term quasinormal instead of “bounded” to avoid ambiguity(cf
[5] p476, [6] p21). Arsove use the word “similar” in the same meaning as the
word “equivalent”(cf, [10] p19, [4]p387).
Denote by $\pi_{\infty}$ the set of all permutations of $\mathbb{N}$(see [6],
p361). Denote by $U_{\pi}$ both the unitary operator which maps $e_{\pi(n)}$
to $e_{n}$ and the corresponding matrix under the ONB
$\\{e_{n}\\}_{n=1}^{\infty}$.
###### Theorem 2.12.
Assume that $F$ is a Schauder matrix and $G^{*}$ is its inverse matrix. We
have
1\. For each invertible matrix $X$, $XF$ is also a Schauder matrix. Moreover,
$XF$ is unconditional(conditional) if and only if $F$ is
unconditional(conditional);
2\. For each diagonal matrix $D=diag(\alpha_{1},\alpha_{2},\cdots)$ in which
each diagonal element $\alpha_{k}$ is nonzero, $FD$ is also a Schauder matrix.
Moreover, $FD$ is unconditional(conditional) if and only if $F$ is
unconditional(conditional);
3\. For a unconditional matrix $F$, $FU$ is also a unconditional matrix for
$U\in\pi_{\infty}$;
4\. Two Schauder matrix $F$ and $F^{{}^{\prime}}$ are equivalent if and only
if there is a invertible matrix $X$ such that $XF=F^{{}^{\prime}}$.
###### Proof.
Property 1, 2, 3 and 4 are basic facts about basis just in a matrix language.
Their counterparts are proposition 4.1.8, 4.2.14, 4.1.5, 4.2.12, and corollary
4.2.34 in [4], Theorem 1 in [10]. Some of those facts are easy to check by our
lemma 2.1. As an example, we shall prove property 1. Let $F^{{}^{\prime}}=XF$,
then clearly $G^{*^{\prime}}=GX^{-1}$ is its inverse matrix. Both properties 1
and 2 in lemma 2.1 hold immediately. To verify property 3, we know that
$FP_{n}G^{*}$ converges to $I$ in SOT if and only if $XFP_{n}G^{*}X^{-1}$
converges to $I$ in SOT. Also we have
$||XFPG^{*}X^{-1}||\leq||X||\cdot||X^{-1}||\cdot||FPG^{*}||$
for any natural projection $P$, which implies the last part of property 1(cf,
[4] theorem 4.2.32). ∎
### 2.3.
Now we turn to study the basic properties of Schauder operators. Recall that a
Schauder operator $T$ is an operator mapping some ONB into a Schauder basis.
In his paper [5], Olevskii call an operator to be generating if and only if it
maps some ONB into a quasinormal conditional basis. Hence our definition of
Schauder operator is a generalization of Olevskii’s one.
###### Theorem 2.13.
Following conditions are equivalent:
1\. $T$ is a Schauder operator;
2\. $T$ maps some ONB $\\{e_{n}\\}_{n=1}^{\infty}$ into a basis;
3\. $T$ has a polar decomposition $T=UA$ in which $A$ is a Schauder operator;
4\. Assume that $T$ has a matrix representation $F$ under a fixed ONB
$\\{e_{n}\\}_{n=1}^{\infty}$. There is some unitary matrix $U$ such that $FU$
is a Schauder matrix.
###### Proof.
$2\Rightarrow 1$. The $k-$th column of the matrix of $T$ under the ONB
$\\{e_{n}\\}_{n=1}^{\infty}$ is just the $l^{2}-$coefficients of $Te_{k}$.
$1\Rightarrow 3$. Assume that $\\{f_{n}\\}_{n=1}^{\infty}$ is a basis in which
$f_{n}$ is the $n-$th column of the matrix $F$ of $T$ under some ONB. Then if
we denote the matrix of $U$ and $A$ also by the same notations, we have
$UA=F$. Property 1 of lemma 2.12 tell us $U^{*}F=A$ is also a Schauder matrix.
$3\Rightarrow 4$. Assume that $\\{g_{n}\\}_{n=1}^{\infty}$ is an ONB such that
the matrix of $A$ under it is a Schauder matrix. Then the operator $U$ defined
as $Ue_{n}=g_{n}$ is a unitary operator and the $n-$th column of its matrix
under the ONB $\\{e_{n}\\}_{n=1}^{\infty}$ is just the $l^{2}-$coefficients of
$g_{n}$. Hence we have $AU$ is a Schauder matrix.
$4\Rightarrow 1$. The column vector sequence of the unitary matrix $U$ is an
ONB. The matrix of $T$ under this ONB is just $U^{*}FU$. Property 1 of lemma
2.12 shows that $U^{*}FU$ is a Schauder matrix since $FU$ is a Schauder matrix
itself. ∎
The equivalence $1\Leftrightarrow 3$ had been used in proof of the theorem
$1^{{}^{\prime}}$ of [5], although Olevskii had not given an explanation.
###### Proposition 2.14.
A Schauder operator $T$ must be injective and has a dense range in
$\mathcal{H}$.
###### Proof.
$T$ must be injective since the representation of $0$ is unique. For a basis
$\\{f_{n}\\}_{n=1}^{\infty}$, the finite linear combination of
$\\{f_{n}\\}_{n=1}^{\infty}$ is dense in the Hilbert space $\mathcal{H}$.
Therefore the range of $T$ must be dense in $\mathcal{H}$. ∎
### 2.4.
If $T$ is a Schauder operator, does for each ONB sequence
$\\{e_{n}\\}_{n=1}^{\infty}$ the vector sequence $\\{Te_{n}\\}_{n=1}^{\infty}$
always be a basis? In this subsection, we shall show that the answer is
negative in general and it is true only in the case that $T$ is an invertible
operator.
###### Lemma 2.15.
Assume that $A$ is a positive operator satisfying
$\sigma(A)\subseteq[\lambda_{1},\lambda_{2}]$ and
$\lambda_{1},\lambda_{2}\in\sigma(A)$ for some $\lambda_{1}>0$. Then for any
const $\varepsilon>0$ small enough, there is a rank 1 projection $P$ such that
$\frac{1}{2\sqrt{2}}\frac{\lambda_{2}}{\lambda_{1}}-\varepsilon<||APA^{-1}||$.
###### Proof.
Let $e_{1},e_{2}$ be two normalized vectors in $\mathcal{H}$ such that
$e_{1}\in E_{[\lambda_{1},\lambda_{1}+\delta]},e_{2}\in
E_{[\lambda_{2}-\delta,\lambda_{2}]}.$
in which $E_{[\lambda_{1},\lambda_{1}+\delta]}$ and
$E_{[\lambda_{2}-\delta,\lambda_{2}]}$ is the spectral projection of $A$ on
the interval $[\lambda_{1},\lambda_{1}+\delta]$ and
$[\lambda_{2}-\delta,\lambda_{2}]$ respectively(cf, [7], pp269-272). Then for
$\delta<\frac{\lambda_{2}-\lambda_{1}}{2}$, we have $(e_{1},e_{2})=0$ and
$\lambda_{1}\leq||Ae_{1}||\leq\lambda_{1}+\delta,\lambda_{2}-\delta\leq||Ae_{2}||\leq\lambda_{2}.$
Consider the vector $e=\frac{1}{\sqrt{2}}e_{1}+\frac{1}{\sqrt{2}}e_{2}$ and
the operator $P=e\otimes e$ defined as: $Px=(x,e)e.$ It is trivial to check
that $P$ is a rank 1 orthogonal projection. Now we have
$APA^{-1}(x)=(A^{-1}x,e)Ae,$ hence $||APA^{-1}||=\sup_{||x||=1}||APA^{-1}x||$.
Then we have
$\begin{array}[]{rl}(A^{-1}e,e)=&\frac{1}{\sqrt{2}}(A^{-1}e_{1},e)+\frac{1}{\sqrt{2}}(A^{-1}e_{2},e)\\\
=&\frac{1}{2}\\{(A^{-1}e_{1},e_{1})+(A^{-1}e_{2},e_{2})\\}\\\
\geq&\frac{1}{2}\\{\frac{1}{\lambda_{1}+\delta}+\frac{1}{\lambda_{2}}\\}\end{array}$
and
$||Ae||^{2}\geq\frac{1}{2}\lambda_{1}^{2}+\frac{1}{2}(\lambda_{2}-\delta)^{2}.$
Therefore the following inequality holds:
$\begin{array}[]{rl}||APA^{-1}e||\geq&\frac{1}{2}\\{\frac{1}{\lambda_{1}+\delta}+\frac{1}{\lambda_{2}}\\}\sqrt{\frac{1}{2}\lambda_{1}^{2}+\frac{1}{2}(\lambda_{2}-\delta)^{2}}\\\
\geq&\frac{1}{2\sqrt{2}}\frac{\lambda_{2}-\delta}{\lambda_{1}+\delta}.\end{array}$
Let $\varepsilon$ be a const satisfying $\varepsilon<\frac{1}{2\sqrt{2}}$.
Hence for the positive number
$\delta<\frac{2\sqrt{2}\lambda_{1}^{2}\varepsilon}{(1-2\sqrt{2}\varepsilon)\lambda_{1}+\lambda_{2}}$
the required inequality holds. ∎
###### Theorem 2.16.
If an operator $A$ maps every ONB sequence into a basis, then $A$ must be an
invertible operator.
###### Proof.
A direct result of 2.12 is that if an operator $A$ maps every ONB into a basis
then it maps each ONB into a unconditional basis. By virtue of theorem 2.13,
we can assume that $T$ is a positive operator. We need to show that
$0\notin\sigma(A)$. Firstly, we have $0\notin\sigma_{p}(A)$ by above
proposition 2.14 since $A$ is a Schauder operator. If $0\in\sigma(p)$ then $0$
must be an accumulation point of $\sigma(T)$. Hence we can choose a sequence
$\\{\lambda_{k}\\}_{k=1}^{\infty}$ such that:
1\. $\\{\lambda_{k}\\}_{k=1}^{\infty}\subseteq\sigma(A)$; and
2\. $\lambda_{k+1}<\lambda_{k}$ and
$\frac{\lambda_{2n}}{\lambda_{2n-1}}<\frac{1}{n+1}$.
Denote by $I_{0}=\sigma(A)-\cup_{n=1}^{\infty}[\lambda_{2n},\lambda_{2n-1}]$
and $A_{0}=AE_{I_{0}}$. Let $A_{n}=AE_{[\lambda_{2n},\lambda_{2n-1}]}$, then
we have $A=A_{0}\oplus A_{1}\oplus A_{2}\oplus A_{3}\cdots$. And each operator
$A_{n}$ is an invertible positive operator for $n\geq 1$. Now by above lemma
2.15, we can choose a vector $e_{1}^{(n)}\in
RanE_{[\lambda_{2n},\lambda_{2n-1}]}$ such that the projection
$P^{(n)}_{1}=e_{1}^{(n)}\otimes e_{1}^{(n)}$ satisfying
$AP^{(n)}_{1}A^{-1}=A_{n}P^{(n)}_{1}A_{n}^{-1}>n$
for each n. Here we use the fact
$E_{[\lambda_{2n},\lambda_{2n-1}]}P^{(n)}_{1}=P^{(n)}_{1}E_{[\lambda_{2n},\lambda_{2n-1}]}=P^{(n)}_{1}.$
Now for each subspace $RanE_{[\lambda_{2n},\lambda_{2n-1}]}$ we choose an ONB
$\\{f_{k}^{(n)}\\}_{k=1}^{\alpha_{k}}$ such that $e^{(n)}_{1}=f^{(n)}_{1}$.
Moreover, choose an ONB $\\{e^{(0)}_{k}\\}_{k=1}^{\alpha_{0}}$ of the subspace
$RanE_{I_{0}}$. Here $\alpha_{k}$ is a finite number or the countable cardinal
which is equal to the dimension of the subspace
$RanE_{[\lambda_{2n},\lambda_{2n-1}]}$ and $RanE_{I_{0}}$ respectively.
Clearly the set $\\{f^{(n)}_{k};n=0,1,2,\cdots\hbox{ and
}k=1,2,\cdots,\alpha_{k}\\}$ is an ONB for $\mathcal{H}$ itself. It is a
countable set and each its arrangement $\psi$ give an ONB sequence of
$\mathcal{H}$. In more details, denote by
$\Delta=\\{(n,k);n=0,1,2,\cdots\hbox{ and }k=1,2,\cdots,\alpha_{k}\\}$. For
any bijection $\sigma:\Delta\rightarrow\mathbb{N}$, define
$g_{n}=f_{t}^{(s)},(s,t)=\sigma^{-1}(n)$. Then
$\psi_{\sigma}=\\{g_{n}\\}_{n=1}^{\infty}$ is an ONB sequence.
###### Claim 2.17.
For each ONB sequence $\psi_{\sigma}$, $\\{Ag_{n}\\}_{n=1}^{\infty}$ is not a
basis.
We have shown that if $\\{Ag_{n}\\}_{n=1}^{\infty}$ is a basis it must be a
unconditional one. So it is enough to show that it is not a unconditional
basis, which can be verified by its unconditional const. Assume that the claim
is not true, that is, $\\{Ag_{n}\\}_{n=1}^{\infty}$ is a basis. It is trivial
to check that $A_{n}P^{(n)}_{1}A_{n}^{-1}$ is a natural projection
corresponding to the basis $\\{Ag_{n}\\}$. In fact, we have
$A_{n}P^{(n)}_{1}A_{n}^{-1}=P_{\sigma(n,1)}-P_{\sigma(n,1)-1}.$
Here we denote by $P_{n}$ the $n-th$ partial sum operator so called in the
book [6]. But now we have $||A_{n}P^{(n)}_{1}A_{n}^{-1}||\rightarrow\infty$
which counters to the fact that a unconditional basis must have a finite
unconditional const (cf, [4], corollary4.2.26). ∎
###### Corollary 2.18.
If an operator $T$ is not invertible, then there is some ONB
$\\{e_{n}\\}_{n=1}^{\infty}$ such that the sequence
$\\{Te_{n}\\}_{n=1}^{\infty}$ is not a basis.
By the theorem 1 of [5], a generating operator never be invertible. Hence we
have
###### Corollary 2.19.
For a generating operator $T$, there is some ONB $\\{e_{n}\\}_{n=1}^{\infty}$
such that the sequence $\\{Te_{n}\\}_{n=1}^{\infty}$ is not a basis.
Both the English translation and the review(MR0318848) of the paper [5] by A.
M. Oleskii make a pity clerical mistake:
Review(MR0318848):“The author obtains a spectral characterization for the
linear operators that transform $\mathbf{every}$ complete orthonormal system
into a conditional basis in a Hilbert space.”
The English translation: “Definition. A bounded noninvertible linear operator
$T:\mathcal{H}\rightarrow\mathcal{H}$ is said to be generating if it maps
$\mathbf{every}$ orthonormal basis $\varphi$ into a quasinormed basis $\psi$.”
The word “every” should be “some” in both of them. Note that in the proof of
the theorem 1 ([5]), Olevskii had shown that an operator never can maps every
ONB into a conditional basis. Even the theorem 1 of [5] itself shows it, but
need a little operator theory discussion.
Since in the Hilbert space $\mathcal{H}$ all quasinormal unconditional bases
are equivalent(cf, Theorem 18.1, [6], p529) and in addition with theorem 2.12,
we have
###### Proposition 2.20.
An $\omega\times\omega$ matrix $F$ is a Riesz matrix if and only if it
represents an invertible operator.
Above result also can be obtained directly form theorem 2 of the paper [1].
###### Corollary 2.21.
An operator $T$ is invertible if and only if there is some ONB such that the
matrix $F$ under this ONB of $T$ is a Riesz matrix.
###### Corollary 2.22.
For an invertible operator $T$, its matrix always be a Riesz matrix under any
ONB.
### 2.5.
Conditional and unconditional bases have very different behaviors. On the
other side, properties of operators given by Schauder matrices are strongly
dependent on the related bases. Both the theorem 1 of the paper [5] and the
behaviors of Riesz matrix(cf, proposition 2.20) support this observation. In
this subsection, we give a same classification of operators dependent on their
matrix representation(Or equivalently, on their actions on ONBs). And then we
give some more remarks on Olevskii’s paper.
###### Definition 2.23.
A Schauder operator $T$ will be called a conditional operator if and only if
there is some ONB $\\{e_{n}\\}_{n=1}^{\infty}$ such the column vector sequence
of its matrix representation $F$ of $T$ under the ONB comprise a conditional
basis. Otherwise, $T$ will be called a unconditional operator.
By the theorem 2.13, we have
###### Corollary 2.24.
A Schauder operator $T$ is conditional if and only if it maps some ONB
$\\{e_{n}\\}_{n=1}^{\infty}$ into a conditional basis
$\\{Te_{n}\\}_{n=1}^{\infty}$.
For convenience, we correct the error appearing in the translation and rewrite
Olevskii’s definition as follows:
###### Definition 2.25.
A bounded operator $T\in\mathcal{L}(\mathcal{H})$ is said to be generating if
and only if it maps some ONB into a quasinormal conditional basis.
Above definition modifies slightly from the original form on the Olevskii’s
paper. We write down the original one to compare them in details:
###### Definition 2.26.
([5], p476) A bounded non-invertible operator
$T:\mathcal{H}\rightarrow\mathcal{H}$ is said to be generating if and only if
it maps some ONB into a quasinormal basis.
###### Proposition 2.27.
Above two definitions are equivalent.
###### Proof.
If a bounded operator $T\in\mathcal{L}(\mathcal{H})$ maps some ONB into a
quasinormal conditional basis, then it must be non-invertible since an
invertible operator maps each ONB into a Riesz basis(hence a unconditional
basis) by proposition 2.20; On the other side, If a bounded non-invertible
operator $T:\mathcal{H}\rightarrow\mathcal{H}$ maps some ONB into a
quasinormal basis. Then the quasinormal basis must be a conditional one
otherwise $T$ must be invertible again by proposition 2.20. ∎
###### Corollary 2.28.
A generating operator is a conditional operator; An invertible operator is a
unconditional operator.
## 3\. A Criterion for Operators to be Conditional
### 3.1.
Question: Is $K=diag\\{1,\frac{1}{2},\frac{1}{3},\cdots\\}$ a conditional
operator?
From Olevskii’s result, we can not obtain the confirm answer. In this section,
we will improve the Olevskii’s technology and gain a confirm answer.
First, let us recall some notations in the line of Olevskii.
Let $A_{k}=\left(\begin{array}[]{c}a_{ij}\\\ \end{array}\right)\in
M_{2^{k}}(\mathbb{C})$ (where $1\leq i,j\leq 2^{k}$) be defined as follows:
$a_{i1}=2^{-\frac{k}{2}},1\leq i\leq 2^{k}$; and if $j=2^{s}+v(1\leq v\leq
2^{s})$, then
$\displaystyle a_{ij}$
$\displaystyle=\left\\{\begin{array}[]{ll}2^{\frac{s-k}{2}},\hskip
28.45274pt(v-1)2^{k-s}<i\leq(2v-1)2^{k-s-1},\\\\[5.69054pt]
-2^{\frac{s-k}{2}},\hskip 28.45274pt(2v-1)2^{k-s-1}<i\leq v2^{k-s}.\\\
\end{array}\right.$
For $\alpha,\frac{1}{\sqrt{2}}<\alpha<1$, let $T_{(k,\alpha)}\in
M_{2^{k}}(\mathbb{C})$ be defined as follows:
$T_{(k,\alpha)}=\begin{bmatrix}\begin{bmatrix}\alpha^{k}&\\\ &\alpha^{k}\\\
\end{bmatrix}\\\ &\begin{bmatrix}\alpha^{k-1}\\\ &\alpha^{k-1}\\\
\end{bmatrix}&\\\ &&\ddots&\\\ &&&\begin{bmatrix}\alpha&\\\ &\ddots\\\
&&\alpha\end{bmatrix}_{2^{k-1}\times 2^{k-1}}\\\ \end{bmatrix}.$
In this section, we will show that if the positive operator $T$ does not admit
the eigenvalue zero and $\sigma(T)$ has a decreasing sequence
$\\{\lambda_{n},n=1,2,\ldots\\}$ which converges to zero and
$\lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=1,$
then $T$ must be a conditional operator. Thus the compact operator
$K=diag\\{1,\frac{1}{2},\frac{1}{3}$, $\cdots\\}$ is a conditional operator.
### 3.2.
Now, we give a key lemma.
###### Lemma 3.1.
Let $T$ be a diagonal operator with entries
$\\{\lambda_{1},\lambda_{2},\lambda_{3},\ldots\\}$ under the ONB
$\\{e_{k}\\}_{k=1}^{\infty}$, where $\lambda_{n}>0$. Given $\alpha$,
$\frac{1}{\sqrt{2}}<\alpha<1$. If for each $k\geq 1$, there exist positive
numbers $c_{k}\leq d_{k}$, such that
a) $sup_{k}\frac{d_{k}}{c_{k}}<\infty$,
b) there exists subset
$\triangle_{k}=\\{n^{k}_{1},n^{k}_{2},\cdots,n^{k}_{2^{k}}\\}$ of $\mathbb{N}$
such that
$c_{k}\leq\frac{\alpha^{k}}{\lambda_{n^{k}_{2^{k}-1}}},\frac{\alpha^{k}}{\lambda_{n^{k}_{2^{k}}}}\leq
d_{k}$, and $c_{k}\leq\frac{\alpha^{j}}{\lambda_{n^{k}_{i}}}\leq d_{k}$ when
$1\leq j\leq k-1,~{}2^{k}(1-\frac{1}{2^{j-1}})+1\leq i\leq
2^{k}(1-\frac{1}{2^{j}}),$
c) $max{\triangle_{k}}<min{\triangle_{k^{\prime}}}$ when $k<k^{\prime}$,
then $T$ is a conditional operator.
###### Proof.
In this proof, we shall identify the operators and the $\omega\times\omega$
matrix representation of the operators under ONB $\\{e_{k}\\}_{k=1}^{\infty}$.
We rearrange $n^{k}_{1},n^{k}_{2},\cdots,n^{k}_{2^{k}}$ into a increasing
sequence and denote it by $m^{k}_{1},m^{k}_{2}$, $\cdots,m^{k}_{2^{k}}$
($m^{k}_{1}<m^{k}_{2}<\cdots<m^{k}_{2^{k}}$). Let $m^{0}_{1}=1$ and
$\mathcal{H}_{k}=span\\{e_{m^{k}_{1}},e_{m^{k}_{1}+1}$,
$\ldots,e_{m^{k+1}_{1}-1}\\}$ for $k\geq 0$, then since
$max{\triangle_{k}}<min{\triangle_{k^{\prime}}}$ when $k<k^{\prime}$, we know
$\mathcal{H}_{k}\cap\mathcal{H}_{k^{\prime}}=(0)$ when $k\neq k^{\prime}$ and
$\oplus_{k\geq 0}\mathcal{H}_{k}=\mathcal{H}$. Moreover,
$\\{\lambda_{n^{k}_{1}},\lambda_{n^{k}_{2}},\ldots,\lambda_{n^{k}_{2^{k}}}\\}\subseteq\\{\lambda_{m^{k}_{1}},\lambda_{m^{k}_{1}+1},\ldots,$
$\lambda_{m^{k+1}_{1}-1}\\}$ for any $k\geq 1$.
Let $T_{k}\in\mathcal{L}(\mathcal{H}_{k})$ the k-th block of $T$ on
$\mathcal{H}_{k}$, i.e.
$T_{k}=\begin{bmatrix}\lambda_{m^{k}_{1}}&&&\\\ &\lambda_{m^{k}_{1}+1}&&\\\
&&\ddots\\\ &&&\lambda_{m^{k+1}_{1}-1}\\\
\end{bmatrix}\begin{matrix}e_{m^{k}_{1}}\\\ e_{m^{k}_{1}+1}\\\ \vdots\\\
e_{m^{k+1}_{1}-1}\end{matrix},$
then $\oplus_{k\geq 0}T_{k}=T$. Denote $\widetilde{T}_{0}=T_{0}$. For $k\geq
1$, let
$\widetilde{T}_{k}=\begin{bmatrix}\lambda_{n^{k}_{2^{k}}}&&&\\\
&\lambda_{n^{k}_{2^{k}-1}}&&\\\ &&\ddots\\\ &&&\lambda_{n^{k}_{1}}&\\\
&&&&S_{k}\\\ \end{bmatrix}\begin{matrix}e_{m^{k}_{1}}\\\ e_{m^{k}_{1}+1}\\\
\vdots\\\ e_{m^{k}_{1}+2^{k}-1}\\\ \widetilde{\mathcal{H}}_{k}\end{matrix},$
where
$\widetilde{\mathcal{H}}_{k}=\bigvee\\{e_{m^{k}_{1}+2^{k}},\ldots,e_{m^{k+1}_{1}-1}\\}$
and $S_{k}$ is a diagonal operator with entries
$\\{\lambda_{m^{k}_{1}},\lambda_{m^{k}_{1}+1},\ldots,$
$\lambda_{m^{k+1}_{1}-1}\\}\backslash\\{\lambda_{n^{k}_{1}},\lambda_{n^{k}_{2}}$,
$\ldots,\lambda_{n^{k}_{2^{k}}}\\}$. It is easy to see that the entries of
$\widetilde{T}_{k}$ are just a rearrangement of entries of $T_{k}$ for $k\geq
1$.
We will prove $\widetilde{T}\triangleq\oplus_{k\geq 0}\widetilde{T}_{k}$ is a
conditional operator and then show $T$ is a conditional operator.
Let $X_{0}=I\in\mathcal{L}(\mathcal{H}_{0})$. For $k\geq 1$, let
$X_{k}=\begin{bmatrix}c_{k}\cdot\begin{bmatrix}\frac{\lambda_{n^{k}_{2^{k}}}}{\alpha^{k}}&&\\\
&\frac{\lambda_{n^{k}_{2^{k}-1}}}{\alpha^{k}}\\\ &&\ddots&\\\
&&&\frac{\lambda_{n^{k}_{i}}}{\alpha^{j}}\\\ &&&&\ddots\\\
&&&&&\frac{\lambda_{n^{k}_{1}}}{\alpha}\\\ \end{bmatrix}&\\\ &I\\\
\end{bmatrix}\in\mathcal{L}(\mathcal{H}_{k}),$
since
$Sup_{k}max\\{c_{k}\frac{\lambda_{n^{k}_{2^{k}}}}{\alpha^{k}},\ldots,c_{k}\frac{\lambda_{n^{k}_{1}}}{\alpha},c_{k}^{-1}\frac{\alpha^{k}}{\lambda_{n^{k}_{2^{k}}}},\ldots,c_{k}^{-1}\frac{\alpha}{\lambda_{n^{k}_{1}}}\\}\leq
Sup_{k}max\\{1,\frac{d_{k}}{c_{k}}\\}<\infty,$
we have $X\triangleq\oplus_{k\geq 0}X_{k}$ is an invertible operator.
Moreover for $k\geq 1$,
$\widetilde{T}_{k}=X_{k}\cdot\begin{bmatrix}T_{(k,\alpha)}c_{k}^{-1}&\\\
&S_{k}\\\ \end{bmatrix},$
so
$\widetilde{T}=\oplus_{k\geq 0}\widetilde{T}_{k}=X\cdot\oplus_{k\geq
0}\begin{bmatrix}T_{(k,\alpha)}c_{k}^{-1}&\\\ &S_{k}\\\ \end{bmatrix},$
where we denote $\begin{bmatrix}T_{(k,\alpha)}c_{k}^{-1}&\\\ &S_{k}\\\
\end{bmatrix}$ by $\widetilde{T}_{0}$ when $k=0$.
Let
$U=\oplus_{k\geq 0}\begin{bmatrix}A_{k}^{*}&\\\ &I\\\ \end{bmatrix},$
where we denote $\begin{bmatrix}A_{k}^{*}&\\\ &I\\\
\end{bmatrix}=I\in\mathcal{L}(\mathcal{H}_{0})$ when $k=0$, then it is an
unitary operator and
$\displaystyle\widetilde{T}U$ $\displaystyle=$ $\displaystyle
X\cdot\oplus_{k\geq 0}\begin{bmatrix}T_{(k,\alpha)}c_{k}^{-1}&\\\ &S_{k}\\\
\end{bmatrix}\cdot\oplus_{k\geq 0}\begin{bmatrix}A_{k}^{*}&\\\ &I\\\
\end{bmatrix}$ $\displaystyle=$ $\displaystyle X\cdot\oplus_{k\geq
0}\begin{bmatrix}T_{(k,\alpha)}A_{k}^{*}c_{k}^{-1}&\\\ &S_{k}\\\
\end{bmatrix}$ $\displaystyle=$ $\displaystyle X\cdot\oplus_{k\geq
0}\begin{bmatrix}T_{(k,\alpha)}A_{k}^{*}&\\\ &S_{k}\\\
\end{bmatrix}\cdot\oplus_{k\geq 0}\begin{bmatrix}c_{k}^{-1}I&\\\ &I\\\
\end{bmatrix}.$
To show $\widetilde{T}$ is conditional, from theorem 2.12, it suffices to show
that
$F\triangleq\oplus_{k\geq 0}\begin{bmatrix}T_{(k,\alpha)}A_{k}^{*}&\\\
&S_{k}\\\ \end{bmatrix}$
is a conditional matrix.
We will deal with it by theorem 2.5 and proposition 2.8. First, one can easily
see that $F$ has an unique left inverse matrix
$G^{*}=\oplus_{k\geq 0}\begin{bmatrix}A_{k}T_{(k,\alpha)}^{-1}&\\\
&S_{k}^{-1}\\\ \end{bmatrix}$
where each row is a $l^{2}-$ sequence.
Second, $Q_{n}=FP_{n}G^{*}$ are obviously projections. Let
$\displaystyle\Lambda_{1}=\\{m^{k}_{1},m^{k}_{1}+1,\ldots,m^{k}_{1}+2^{k}-1;~{}k\geq
1\\}\subseteq\mathbb{N},$
$\displaystyle\Lambda_{2}=\\{m^{k}_{1}+2^{k},m^{k}_{1}+2^{k}+1,\ldots,m^{k+1}_{1}-1;~{}k\geq
1\\}\subseteq\mathbb{N}.$
For any $x\in\mathcal{H}$, we have
$x=\sum\limits_{j=1}^{\infty}x_{j}e_{j}=\sum\limits_{j\in\Lambda_{1}}x_{j}e_{j}+\sum\limits_{j\in\Lambda_{2}}x_{j}e_{j},$
and
$\displaystyle FP_{n}G^{*}(x)$
$\displaystyle=FP_{n}G^{*}(\sum\limits_{j\in\Lambda_{1}}x_{j}e_{j}+\sum\limits_{j\in\Lambda_{2}}x_{j}e_{j})$
$\displaystyle=(\oplus_{k\geq
0}T_{(k,\alpha)}A_{k}^{*})P^{(1)}_{n}(\oplus_{k\geq
0}A_{k}T_{(k,\alpha)}^{-1})(\sum\limits_{j\in\Lambda_{1}}x_{j}e_{j})+P^{(2)}_{n}(\sum\limits_{j\in\Lambda_{2}}x_{j}e_{j}),$
where $\oplus_{k\geq 0}T_{(k,\alpha)}A_{k}^{*}$ and $P^{(1)}_{n}$ are the
operators on $\mathcal{H}^{(1)}=\bigvee_{j\in\Lambda_{1}}\\{e_{j}\\}$,
$P^{(1)}_{n}$ converges to $I$ in the strong operator topology; $P^{(2)}_{n}$
is the operator on $\mathcal{H}^{(2)}=\bigvee_{j\in\Lambda_{2}}\\{e_{j}\\}$
and also converges to $I$ in the strong operator topology.
It follows from the result of Olevskii that $\oplus_{k\geq
0}T_{(k,\alpha)}A_{k}^{*}$ is quasinormal conditional matrix. Then from
theorem 2.5, we have
$\lim\limits_{n\rightarrow\infty}(\oplus_{k\geq
0}T_{(k,\alpha)}A_{k}^{*})P^{(1)}_{n}(\oplus_{k\geq
0}A_{k}T_{(k,\alpha)}^{-1})(\sum\limits_{j\in\Lambda_{1}}x_{j}e_{j})=\sum\limits_{j\in\Lambda_{1}}x_{j}e_{j}.$
Thus $FP_{n}G^{*}(x)$ converges to $x$ as $n\rightarrow\infty$ and $F$ is a
Schauder matrix.
Moreover, since the unconditional basis const of $\oplus_{k\geq
0}T_{(k,\alpha)}A_{k}$ is smaller than the unconditional basis const of $F$
and the unconditional basis const of $\oplus_{k\geq 0}T_{(k,\alpha)}A_{k}$ is
infinity, we have that the unconditional basis const of $F$ is infinity. Thus
from proposition 2.8, we know that $F$ is a conditional matrix and
$\widetilde{T}U$ is a conditional matrix.
Since the entries of $\widetilde{T}$ is just a rearrangement of $T$, one can
easily find an unitary matrix (operator) $\widetilde{U}$ such that
$\widetilde{U}\widetilde{T}\widetilde{U}^{*}=T$, it follows that
$\widetilde{U}^{*}T\widetilde{U}U$ is a conditional matrix. Again from theorem
2.12, $T\widetilde{U}U$ is a conditional matrix. Thus $T$ is a conditional
operator, since it maps orthonormal basis $\\{(\widetilde{U}U)e_{1}$,$\ldots$,
$(\widetilde{U}U)e_{n}$,$\ldots\\}$ into a conditional basis. ∎
Now, we come to the main results.
###### Theorem 3.2.
Let $T\geq 0$ belong to $\mathcal{L}(\mathcal{H})$ which does not admit the
eigenvalue zero. If there exists a constant $\delta>1$ such that
$\lim\limits_{t\rightarrow
0^{+}}Card\\{[\frac{t}{\delta},t]\cap\sigma(T)\\}=\infty,$
then $T$ is a conditional operator.
###### Proof.
First step, we choose a sequence $\\{\lambda_{n}\\}\subseteq\sigma(T)$
satisfying the conditions of lemma 3.1. We will find it by induction.
For $k=1$, $\Delta_{1}=\\{\lambda_{1},\lambda_{2}\\}\subseteq\sigma(T)$ and
$c_{1},d_{1}$ can be easily chosen such that
$\displaystyle\frac{d_{1}}{c_{1}}\leq\delta~{}{\rm
and}~{}c_{1}\leq\frac{\alpha}{\lambda_{1}},\frac{\alpha}{\lambda_{2}}\leq
d_{1}.$
Suppose we have found
$\Delta_{k-1}=\\{\lambda_{2^{k-1}-1},\lambda_{2^{k-1}},\lambda_{2^{k-1}+1},\cdots,\lambda_{2^{k}-2}\\}\subseteq\sigma(T)$
which satisfies
$\Delta_{k-1}\cap\bigcup_{1\leq j\leq k-2}\Delta_{j}=\emptyset,$
and $c_{k-1},d_{k-1}$ such that the first two conditions of lemma 3.1 are
satisfied. Since
$\lim\limits_{t\rightarrow
0^{+}}Card\\{[\frac{t}{\delta},t]\cap\sigma(T)\\}=\infty,$
we can find $t_{0}<min\\{\lambda;~{}\lambda\in\bigcup_{1\leq j\leq
k-1}\Delta_{j}\\}$ such that $t\leq t_{0}$,
$Card\\{[\frac{t}{\delta},t]\cap\sigma(T)\\}\geq 2^{k}.$
Choose arbitrary two elements
$\\{\lambda_{2^{k+1}-3},\lambda_{2^{k+1}-2}\\}\subseteq\sigma(T)\cap[\frac{t_{0}\alpha^{k}}{\delta},t_{0}\alpha^{k}]$,
then choose one after one as follows,
$\displaystyle\\{\lambda_{2^{k+1}-5},\lambda_{2^{k+1}-4}\\}\subseteq\\{\sigma(T)\cap[\frac{t_{0}\alpha^{k-1}}{\delta},t_{0}\alpha^{k-1}]\\}\backslash\\{\lambda_{2^{k+1}-3},\lambda_{2^{k+1}-2}\\}$
$\displaystyle\hskip 142.26378pt\vdots$
$\displaystyle\\{\lambda_{(2^{j}-1)2^{k-j+1}-1},\lambda_{(2^{j}-1)2^{k-j+1}},\lambda_{(2^{j}-1)2^{k-j+1}+1},\ldots,\lambda_{(2^{j+1}-1)2^{k-j}-2}\\}\subseteq\\{\sigma(T)$
$\displaystyle\cap[\frac{t_{0}\alpha^{j}}{\delta},t_{0}\alpha^{j}]\\}\backslash\\{\lambda_{(2^{j+1}-1)2^{k-j}-1},\lambda_{(2^{j+1}-1)2^{k-j}},\lambda_{(2^{j+1}-1)2^{k-j}+1},\ldots,\lambda_{2^{k+1}-2}\\}$
$\displaystyle\hskip 142.26378pt\vdots$
$\displaystyle\\{\lambda_{2^{k}-1},\lambda_{2^{k}},\ldots,\lambda_{3\cdot
2^{k-1}-2}\\}\subseteq\\{\sigma(T)\cap[\frac{t_{0}\alpha}{\delta},t_{0}\alpha]\\}\backslash\\{\lambda_{3\cdot
2^{k-1}-1},\lambda_{3\cdot 2^{k-1}},\ldots,$
$\displaystyle\lambda_{2^{k+1}-2}\\}.$
Since
$Card\\{[\frac{t_{0}\alpha^{j}}{\delta},t_{0}\alpha^{j}]\cap\sigma(T)\\}$ is
more than $2^{k}$, the above process is reasonable.
Denote $c_{k}=t_{0}^{-1},d_{k}=\delta t_{0}^{-1}$, then obviously
$\displaystyle
c_{k}\leq\frac{\alpha}{\lambda_{2^{k}-1}},\ldots,\frac{\alpha}{\lambda_{3\cdot
2^{k-1}-2}},\frac{\alpha^{2}}{\lambda_{3\cdot
2^{k-1}-1}},\ldots,\frac{\alpha^{2}}{\lambda_{7\cdot 2^{k-2}-2}},$
$\displaystyle\cdots\cdots,\frac{\alpha^{k-1}}{\lambda_{2^{k+1}-5}},\frac{\alpha^{k-1}}{\lambda_{2^{k+1}-4}},\frac{\alpha^{k}}{\lambda_{2^{k+1}-3}},\frac{\alpha^{k}}{\lambda_{2^{k+1}-2}}\leq
d_{k}.$
Thus we have found a sequence $\\{\lambda_{n}\\}\subseteq\sigma(T)$ satisfying
the conditions of lemma 3.1. Obviously, $\lambda_{n}$ converges to zero as
$n\rightarrow\infty$.
Second step, we will complete the proof.
We rearrange the sequence $\\{\lambda_{n}\\}\subseteq\sigma(T)$ into a
decreasing sequence $\\{\mu_{n}\\}$. Fix a constant $M>\frac{||T||}{\mu_{1}}$.
For $n\geq 1$, cut each segment $[\mu_{n+1},\mu_{n}]$ into smaller subsegments
(many enough and we denote them by
$[\nu_{m^{n}_{j+1}},\nu_{m^{n}_{j}}],~{}1\leq j\leq k(n)-1$,
$\nu_{m^{n}_{1}}=\mu_{n}$, $\nu_{m^{n}_{k(n)}}=\mu_{n+1}$) in order that
$\dfrac{\nu_{m^{n}_{j}}}{\nu_{m^{n}_{j+1}}}\leq M,~{}1\leq j\leq
k(n)-1,n=1,2,\ldots.$
From the spectral decompose theorem of self-adjoint operator, we have
$T=\oplus_{n\geq 0}\oplus_{1\leq j\leq k(n)-1}T_{(n,j)},$
where $T_{(n,j)}$ is the operator on the subspace $\mathcal{H}_{(n,j)}$
corresponding to $[\nu_{m^{n}_{j+1}},\nu_{m^{n}_{j}}]\cap\sigma(T)$ for $n\geq
1$ and $T_{(0)}$ is the operator on the subspace $\mathcal{H}_{(0)}$
corresponding to $[\mu_{1},\infty)\cap\sigma(T)$.
Denote
$X=\oplus_{n\geq 0}\oplus_{1\leq j\leq k(n)-1}\xi_{(n,j)}^{-1}T_{(n,j)},$
where $\xi_{(0)}=\mu_{1}$,
$\xi_{(n,j)}\in[\nu_{m^{n}_{j+1}},\nu_{m^{n}_{j}}]\cap\sigma(T)$ and
$\xi_{(n,1)}=\mu_{n}$. Then since
$\displaystyle||\xi_{(n,j)}^{-1}T_{(n,j)}||\leq M~{}{\rm
and}~{}||(\xi_{(n,j)}^{-1}T_{(n,j)})^{-1}||\leq M,~{}1\leq j\leq
k(n)-1,~{}n\geq 0,$
we have $X$ is an invertible operator. Moreover,
$S\triangleq\oplus_{n\geq 0}\oplus_{1\leq j\leq
k(n)-1}\xi_{(n,j)}I_{(n,j)}=X^{-1}T,$
where $I_{(n,j)}$ is the identity operator on $\mathcal{H}_{(n,j)}$.
Obviously, $S$ is a diagonal operator with $\\{\lambda_{n}\\}$ its
subsequence. Thus $S$ satisfies the conditions of lemma 3.1 and hence it is a
conditional operator. From theorem 2.12, we obtain that $T$ is a conditional
operator. ∎
Following is a easier criterion for an operator to be conditional.
###### Theorem 3.3.
Let $T\geq 0$ belong to $\mathcal{L}(\mathcal{H})$ which does not admit the
eigenvalue zero. If $\sigma(T)$ has a decreasing sequence $\\{\lambda_{n}\\}$
which converges to zero such that
$\lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=1,$
then $T$ is a conditional operator.
###### Proof.
It suffices to show that there exists a constant $\delta>1$ such that
$\lim\limits_{t\rightarrow
0^{+}}Card\\{[\frac{t}{\delta},t]\cap\\{\lambda_{n},n\geq 1\\}\\}=\infty.$
If not, then there exists $N>0$, such that for any $t_{0}>0$, there is a
$t\leq t_{0}$,
$Card\\{[\frac{t}{\delta},t]\cap\\{\lambda_{n},n\geq 1\\}\\}<N.$
Thus there exist sequences $a_{k},b_{k}$ converge to zero, such that for all
$k$
$\displaystyle\frac{b_{k}}{a_{k}}=\delta,Card\\{[a_{k},b_{k}]\cap\\{\lambda_{n},n\geq
1\\}\\}<N,$ $\displaystyle
b_{k+1}<a_{k},Card\\{[b_{k+1},a_{k}]\cap\\{\lambda_{n},n\geq 1\\}\\}\geq 1.$
Choose $\lambda_{n_{1}}$ such that
$\lambda_{n_{1}}=min\\{\lambda_{n};~{}\lambda_{n}\geq b_{1}\\}$, choose
$\lambda_{n_{2}}$ such that
$\lambda_{n_{2}}=max\\{\lambda_{n};~{}\lambda_{n}\leq a_{1}\\}$. Generally,
choose $\lambda_{n_{2k-1}}=min\\{\lambda_{n};~{}\lambda_{n}\geq b_{k}\\}$ and
$\lambda_{n_{2k}}=max\\{\lambda_{n};~{}\lambda_{n}\leq a_{k}\\}$. It is easy
to see that $n_{2k}-n_{2k-1}\leq N$.
On the other hand, since
$\lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=1,$
we have
$\lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+j}}=1,$
for any $1\leq j\leq N$ and hence
$\lim\limits_{k\rightarrow\infty}\frac{\lambda_{n_{2k-1}}}{\lambda_{n_{2k}}}=1.$
But
$\frac{\lambda_{n_{2k-1}}}{\lambda_{n_{2k}}}\geq\frac{b_{k}}{a_{k}}=\delta>1$
for any $k$, it is a contradiction.
Thus $T$ is a conditional operator. ∎
###### Remark 3.4.
Actually, suppose the limit of $\frac{\lambda_{n}}{\lambda_{n+1}}$ exists,
then
$\lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=1$
if and only if there exists a constant $\delta>1$ such that
$\lim\limits_{t\rightarrow
0}Card\\{[\frac{t}{\delta},t]\cap\\{\lambda_{n},n\geq 1\\}\\}=\infty.$
One can easily prove it. Thus the condition of theorem 3.3 is a little
stronger than theorem 3.2.
###### Corollary 3.5.
Let $T\in\mathcal{L}(\mathcal{H})$ such that $T$ and $T^{*}$ do not admit the
eigenvalue zero. If $\sigma((T^{*}T)^{\frac{1}{2}})$ has a decreasing sequence
$\lambda_{n}$ which converges to zero such that
$\limsup\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=1,$
then $T$ is a conditional operator.
###### Proof.
From the polar decomposition theorem,
$T=U(T^{*}T)^{\frac{1}{2}},$
where $U$ is a unitary operator. Thus from theorem 3.3 and theorem 2.12, we
obtain the result. ∎
###### Corollary 3.6.
Compact operator $K=diag\\{1,\frac{1}{2},\frac{1}{3},\cdots\\}$ is a
conditional operator.
## References
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* [2] Young, Robert M. An introduction to nonharmonic Fourier series. Pure and Applied Mathematics, 93. Academic Press, Inc. , New York-London, 1980.
* [3] Joachim Weidmann, Linear Operators in Hilbert Space, GTM68, Springer-Verlag, 1980.
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* [6] I. Singer, Bases in Banach Space I, Springer-verlag, 1970.
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* [8] Allen L. Shields, “Weighted shift operators and analytic function theory”, in: Topics in Operator Theory, Math. Surveys No. 13, 49-128, Amer. Math. Soc., Providence (1974).
* [9] Garling, D. J. H., Symmetric bases of locally convex spaces, Studia Math. 30, 1968, 163-181.
* [10] Arsove, Maynard G. Similar bases and isomorphisms in Fr chet spaces. Math. Ann. 135, 1958, 283-293.
* [11] Beurling, Arne On two problems concerning linear transformations in Hilbert space. Acta Math. 81, (1948).
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* [16] Jiang, Chunlan; Ji, Kui, Similarity classification of holomorphic curves. Adv. Math. 215 (2007), no. 2, 446-468.
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|
arxiv-papers
| 2012-03-16T02:27:38 |
2024-09-04T02:49:28.683705
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Cao, Geng Tian, Bingzhe Hou",
"submitter": "Cao Yang",
"url": "https://arxiv.org/abs/1203.3603"
}
|
1203.3628
|
# Second Order Corrections to the Magnetic Moment of Electron at Finite
Temperature
Samina S. Masood∗ and Mahnaz Q. Haseeb∗∗
∗Department of Physics, University of Houston Clear Lake, Houston TX 77058,
masood@uhcl.edu;
∗∗Department of Physics, COMSATS Institute of Information Technology,
Islamabad
mahnazhaseeb@comsats.edu.pk.
###### Abstract
Magnetic moment of electron at finite temperature is directly related to the
modified electron mass in the background heat bath. Magnetic moment of
electron gets modified at finite temperature also, when it couples with the
magnetic field, through its temperature dependent physical mass. We show that
the second order corrections to the magnetic moment of electron is a
complicated function of temperature. We calculate the self-mass induced
thermal contributions to the magnetic moment of electron, up to the two loop
level, for temperatures valid around the era of primordial nucleosynthesis. A
comparison of thermal behavior of the magnetic moment is also quantitatively
studied in detail, around the temperatures below and above the nucleosynthesis
temperature.
## 1 Introduction
Quantum Electrodynamics (QED) is well known as the simplest representative and
the most accurate gauge theory. Thermal medium effects are incorporated in QED
by taking into account the vacuum fluctuations of propagating particles along
with the hot particles in the background heat bath. For simplicity, all the
particles in the background are assumed to be in thermal equilibrium with the
heat bath. These particles are virtually created and annihilated continuously
due to the effects of heat bath at finite temperature. The interactions with
the background electrons, positrons and photons are included through
statistical distribution functions of fermions and bosons, known as Fermi-
Dirac distribution and Bose-Einstein distribution. These distribution
functions represent the possibility of exchange of virtual particles with the
real hot particles from the heat bath. Electromagnetic interactions of
particles get modified at finite temperature because of the many-body aspects
of the statistical background possessed by the hot medium. This replaces the
notion of one-particle systems adopted for particle interactions in the vacuum
with many particle aspects.
The techniques for handling the particle interactions in the background medium
have extensively evolved over the last few decades and are a part of standard
literature (see for example [1], [2] and references therein). It has been
explicitly demonstrated [3-10] that in the hot background, the density-of-
states factors have to be modified to include the real emission and absorption
of particles which are in thermal equilibrium with the heat bath. The validity
of QED renormalization in a background of particles is refined by comparing
the second order (in $\alpha$) corrections with the first order corrections
from the heat bath. It is explicitly seen, as expected, that one-loop
radiative corrections are significantly larger than the two-loop corrections.
Our scheme of calculations is based on the real part of the propagator and the
results are valid, at least below the decoupling temperature [11], i.e.,
around 2 MeV.
Renormalization techniques of vacuum theory are extended to include finite
temperature effects in a standard manner. Regular renormalization procedure
for QED in vacuum is used at finite temperature to study the background
effects on electron mass, charge and wave function renormalization constants.
The modifications in the electromagnetic properties are estimated in terms of
the renormalized values of QED parameters up to the two loop level [3-19].
Feynman rules at finite temperature remain the same as those in vacuum except
that the particle propagators are appropriately modified. We work in Minkowski
space where the Green’s functions depend on real Minkowski momenta $p^{\mu}$.
Therefore, the dynamical processes such as particles propagating in the heat
bath may be more conveniently dealt with. Moreover, in the real-time
formulation, the thermal corrections can be easily kept separate from the
vacuum corrections. Therefore, order by order cancellation of temperature
dependent singularities, and the convergence of perturbative expansion can be
straight away tracked down.
In this paper, thermal contributions to the anomalous magnetic moment of
electron are specifically studied. The magnetic moment of electron modifies
through the radiative corrections to the electron mass both at the one loop
and two loop levels. We analyze net effect from the first order and the second
order radiative contributions, including both the irreducible and disconnected
graphs up to the two loop level. Second order selfmass corrections due to
finite temperature (of the order $\alpha^{2}$) are used here to estimate
finite temperature effects on the magnetic moment of electron.
## 2 Self-Mass of an Electron at Finite Temperature
The renormalized mass of electron is represented by a physical mass given as
$m_{phys}=m+\delta m,$ (1)
where $m$ is the electron mass at zero temperature. Radiatively corrected
physical mass up to order $\alpha^{2}$ is
$m_{phys}\cong m+\delta m^{(1)}+\delta m^{(2)},$
with $\delta m^{(1)}$ and $\delta m^{(2)}$ as the shifts in the electron mass
at one and two loop level respectively. The physical mass is deduced by
locating pole of the propagator $\frac{i(\leavevmode\hbox{\hbox
to0.0pt{\thinspace/\hss}{$p$}}+m)}{p^{2}-m^{2}+i\varepsilon}$. For this
purpose, all finite terms in electron self-energy are combined together. The
physical mass of the electron at one loop was obtained by writing
$\Sigma(p)=A(p)E\gamma_{0}-B(p)\vec{p}.\vec{\gamma}-C(p),$ (2)
where $A(p)$, $B(p)$, and $C(p)$ are the relevant coefficients. Taking the
inverse of the propagator with momentum and mass term separated as
$S^{-1}(p)=(1-A)E\gamma_{0}-(1-B)\vec{p}.\vec{\gamma}-(m-C).$ (3)
The temperature-dependent radiative corrections to the electron mass up to the
first order in $\alpha$, are obtained from the temperature modified
propagator. These corrections are rewritten in the form of boson and fermion
loop integrals at the one loop level as
$\displaystyle E^{2}-|\mathbf{p}|^{2}$ $\displaystyle=$ $\displaystyle
m^{2}+\frac{\alpha}{2\pi^{2}}\left(I.p+J_{B}.p+m^{2}J_{A}\right)$ (4)
$\displaystyle\equiv$ $\displaystyle m_{phys}^{2},$
where
$I.p=\frac{4\pi^{3}T^{2}}{3},$ (5)
and
$J_{B}.p=8\pi\left[\frac{m}{\beta}a(m\beta)-\frac{m^{2}}{2}b(m\beta)-\frac{1}{\beta^{2}}c(m\beta)\right].$
(6)
Thus up to the first order in $\alpha,$ thermal corrections to the mass of
electron were obtained in ref. [3] to be
$m_{phys}^{2}=m^{2}\left[1-\frac{6\alpha}{\pi}b(m\beta)\right]+\frac{4\alpha}{\pi}mT\text{
}a(m\beta)+\frac{2}{3}\alpha\pi
T^{2}\left[1-\frac{6}{\pi^{2}}c(m\beta)\right].$ (7)
The first order correction at finite temperature is calculated as
$\displaystyle\frac{\delta m}{m}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2m^{2}}\left(m_{phys}^{2}-m^{2}\right)$ (8)
$\displaystyle\simeq$ $\displaystyle\frac{\alpha\pi
T^{2}}{3m^{2}}\left[1-\frac{6}{\pi^{2}}c(m\beta)\right]+\frac{2\alpha}{\pi}\frac{T}{m}a(m\beta)-\frac{3\alpha}{\pi}b(m\beta),$
with $\frac{\delta m}{m}$ as the relative shift in electron mass due to finite
temperature which was originally determined in ref. [3] with
$a(m\beta)=\ln(1+e^{-m\beta}),$ (9)
$b(m\beta)=\mathop{\displaystyle\sum}\limits_{n=1}^{\infty}(-1)^{n}\mathop{\mathrm{E}i}(-nm\beta),$
(10)
$c(m\beta)=\mathop{\displaystyle\sum}\limits_{n=1}^{\infty}(-1)^{n}\frac{e^{-nm\beta}}{n^{2}},$
(11)
At low temperature, the functions $a(m\beta)$, $b(m\beta)$, and $c(m\beta)$
fall off in powers of $e^{-m\beta}$ in comparison with
$\left(\frac{T}{m}\right)^{2}$ and can be neglected so that
$\frac{\delta m}{m}\overset{T\ll m}{\longrightarrow}\frac{\alpha\pi
T^{2}}{3m^{2}}.$ (12)
Moreover, in the high-temperature limit, $a(m\beta)$ and $b(m\beta)$ are
vanishingly small whereas $c(m\beta)\longrightarrow-\pi^{2}/12$, yielding
$\frac{\delta m}{m}\overset{T>m}{\longrightarrow}\frac{\alpha\pi
T^{2}}{2m^{2}}.$ (13)
Eq. (8) is valid for large temperatures relevant in QED including $T\sim$ $m.$
This range of temperature is particularly interesting from the point of view
of primordial nucleosynthesis. It has been found that some parameters in the
early universe such as the energy density and the helium abundance parameter
$Y$ become slowly varying functions of temperature [20-22] whereas they remain
constant in both extreme limits given by $T\ll m$ and $T\gg m$.
Using the same procedure as the one used for one loop calculations, the
relative shift in electron mass at the two loop level was obtained in ref.
[17]. This relative shift in electron mass introduces temperature dependence
in the magnetic moment of electron up to two loops. However, the two-loop
order result is very complicated and cannot be easily simplified. Therefore,
we will use the complete expression for the two loop calculations of electron
selfmass, near the nucleosynthesis temperature, given in refs. [12-14]. In the
following section, we compute the magnetic moment of electron from the self-
mass of electron, up to the two loop level, in thermal background.
## 3 Magnetic Moment of Electron in the Heat Bath
The anomalous magnetic moment of an electron is generated due to the coupling
of electron with the magnetic field through the radiative corrections. Some of
these results that are used here were given in ref. [17]. The electromagnetic
coupling is affected by the electron mass and the radiative corrections to the
electron mass. The coupling of electron mass with the external magnetic field
is regulated through mass of the particle itself. It is known from the
calculation of the radiative corrections that the self-mass corrections to the
electron are contributed by the distribution of hot bosons and fermions in the
background medium. This effect, in turn, changes the electromagnetic
properties of the medium itself. Therefore the magnetic moment gets changed
with the finite temperature effects. The magnetic moment of electron is
related to the relative shift in electron mass at finite temperature
$\frac{\delta m}{m}$ as:
$\mu_{a}=\frac{\alpha}{2\pi}-\frac{2}{3}\frac{\delta m}{m}.$ (14)
The leading order contributions to the magnetic moment up to the one loop
level is
$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{2}{3}\alpha\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-b(m\beta)\right]$
(15)
which can be shown to be:
$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{2}{9}\frac{\alpha\pi T^{2}}{m^{2}}$ (16)
for $T<m$ while it becomes:
$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{1}{3}\frac{\alpha\pi T^{2}}{m^{2}}$ (17)
for $T>m.$ First order in $\alpha$ contribution to $\mu_{a}$ around the
temperature range relevant for primordial nucleosynthesis (i.e., $T\sim m$),
soon after the big bang is given by expression in eq. (15). The two loop
contribution to the magnetic moment can simply be added to the magnetic moment
in terms of the relative shift in electron mass $\frac{\delta m^{(2)}}{m}$ as
$\mu_{a}=\frac{\alpha}{2\pi}-\frac{2}{3}\left(\frac{\delta
m^{(1)}}{m}+\frac{\delta m^{(2)}}{m}\right).$ (18)
Now using the expression for $\frac{\delta m}{m}$ at finite temperature in
ref. [14], we get thermal contributions to the magnetic moment up to the two-
loop level as
$\displaystyle\mu_{a}$ $\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}-\
\frac{2}{3}\alpha\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
(19) $\displaystyle-\frac{2\alpha^{2}}{3}\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
$\displaystyle\times\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
$\displaystyle-\frac{4}{3}\alpha^{2}\mathop{\displaystyle\sum}\limits_{r=1}^{\infty}[-\frac{m^{2}\beta^{2}}{\pi^{2}}c(m\beta)+T^{2}\\{\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\text{
}(-1)^{n+r+1}\frac{\pi\text{ }}{6mEv}\frac{e^{-\beta(rE+mn)}}{n}$
$\displaystyle-\frac{3}{8}(-1)^{r}\frac{e^{-r\beta
E}}{E^{2}v^{2}}[\frac{9E^{2}}{2m^{2}}+6\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}+4\mathop{\displaystyle\sum}\limits_{n,s=3}^{r+1}\frac{1}{ns}+(-1)^{s-r}\\{\frac{9E}{m}\left(3+4\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}\right)$
$\displaystyle+2\left(\frac{E^{2}v^{2}}{m^{2}}-3\right)\left(9+18\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}+8\mathop{\displaystyle\sum}\limits_{n,s=3}^{r+1}\frac{1}{ns}\right)\\}]+\frac{4}{E^{2}v^{2}}\\}$
$\displaystyle-\frac{T}{m}\\{\frac{\pi m\text{
}}{6Ev}\mathop{\displaystyle\sum}\limits_{s=2}^{r+1}\mathop{\displaystyle\sum}\limits_{n=1}^{s+1}\frac{e^{-\beta(rE+mn)}}{n}\left[1\
-\left\\{(-1)^{r+n}-(-1)^{s+n}\right\\}\right]$
$\displaystyle+[\left\\{\mathop{\mathrm{E}i}(-m\beta)-\mathop{\mathrm{E}i}(-2m\beta)\right\\}\\{\frac{9E}{4}\left(\frac{E}{E^{2}v^{2}}-\frac{1}{m}\right)$
$\displaystyle+\left(\frac{5E}{m}-21+\frac{E^{2}}{2m^{2}}\right)\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\\}$
$\displaystyle+\left\\{\frac{9}{4v^{2}}-\mathop{\displaystyle\sum}\limits_{n=1}^{s+1}\left[\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}1-E^{2}\left(\frac{1}{2m^{2}}+\frac{3}{E^{2}v^{2}}\right)+\frac{3E}{m}\right]\right\\}(-1)^{s}\mathop{\mathrm{E}i}\text{
}(-sm\beta)]$
$\displaystyle+\frac{e^{-rm\beta}}{m}\\{\left[\frac{9E}{2v^{2}}+2\left(\frac{3E}{v^{2}}+\frac{3E^{2}v^{2}}{m}-5E\right)\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\right]\mathop{\displaystyle\sum}\limits_{s=1}^{\infty}\sinh
sm\beta$
$\displaystyle-\frac{3m^{3}}{E^{2}v^{2}}\left(\frac{3}{4}-\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\right)\mathop{\displaystyle\sum}\limits_{s=1}^{\infty}\cosh
sm\beta\\}]\\}+\frac{1}{m^{2}}\\{\frac{9m}{4E^{2}v^{2}}\left(E^{3}+\frac{m^{3}}{2}\right)$
$\displaystyle+\left[\frac{3m}{E^{2}v^{2}}(E^{3}+m^{3})+5mE-3E^{2}v^{2}\right]\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\\}\left\\{\mathop{\mathrm{E}i}(-m\beta)-2\mathop{\mathrm{E}i}(-2m\beta)\right\\}$
$\displaystyle-\frac{1}{m^{2}}\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\\{\mathop{\displaystyle\sum}\limits_{s=1}^{r+1}\frac{(-1)^{s}}{n}[\frac{m^{2}re^{-sm\beta}}{2}$
$\displaystyle+\left\\{sE\left(2m-\frac{E^{2}}{m}\right)+\frac{m^{2}(s-r)}{2}\right\\}\mathop{\mathrm{E}i}(-sm\beta)]$
$\displaystyle-\frac{\pi m^{3}\text{ }}{3Ev}\left[e^{-\beta rE}\text{
}(-1)^{n+r}(n+1)\
-\mathop{\displaystyle\sum}\limits_{s=2}^{r+1}(-1)^{n+s}\right]\mathop{\mathrm{E}i}(-nm\beta)\\}].$
It can be clearly seen from eq. (19) that the second order corrections are
suppressed by at least two orders of magnitudes as compared to the one loop
contributions. Dependence of the self-mass induced thermal contributions to
the anomalous magnetic moment of electron is very complicated at the two loop
level, as indicated by eq. (19). The exact estimate of this magnetic moment
for application to the primordial nucleosynthesis is very involved and
probably is not really so significant at two loop level. However, low
temperature $(T<m)$ and high temperature $(T>m)$ values of the magnetic moment
can be quantitatively analyzed to prove the validity of the renormalization
scheme. We give the plotting for magnetic moment $\mu_{a}$ vs $\frac{T}{m}$ in
low temperature and the high temperature regions in the next section. This
analysis indicates that the magnetic moment of electron changes its behavior
around nucleosynthesis.
## 4 Results and Discussion
The electron mass acquires a significant contribution from the heat bath even
for temperatures that are smaller than the electron mass. However, this
dependence becomes very complicated as soon as the background temperature
approaches the value of electron mass. One loop corrections to the electron
mass at finite temperature are presented in eq. (8). The low ($T<m$) and high
($T>m$) temperature values of self-mass of electron at the one loop level are
given in eqs. (12) and (13), respectively as limiting cases of eq. (8).
Incorporating the second order relation for the physical mass of electron [17]
into eq. (14) leads to eq. (19) which gives the general form of the anomalous
magnetic moment at finite temperature up to order $\alpha^{2}$. When an
electron couples with the magnetic field at finite temperature, a nonzero
contribution to the magnetic moment is picked up due to the coupling of
electron mass with the thermal background. Eq. (14) presents the acceptable
relation of the magnetic moment with that of the self-mass of electron.
Figure 1: Low temperature behavior of the magnetic moment of electron at the
two loop level.
Figure 2: High temperature behavior of the magnetic moment of electron at the
two loop level.
The behavior of the magnetic moment of electron near the nucleosynthesis
temperatures (eq.(19)) is very complicated and it can be fitted through a
single mathematical function under some special conditions only. However, we
can extract the quantitative behavior of magnetic moment of electron for low
and high temperatures in a comparatively simple form. Using previously studied
second order contributions to the electron mass at low temperature $(T<m)$,
leading order contributions to the magnetic moment of electron can be computed
as
$\mu_{a}\overset{T<m}{\longrightarrow}-\ \frac{2}{9}\frac{\alpha\pi
T^{2}}{m^{2}}-10\alpha^{2}\left(\frac{T^{2}}{m^{2}}\right),$ (20)
whereas, the leading order contributions at high temperature $(T>m)$ comes out
to be
$\mu_{a}\overset{T>m}{\longrightarrow}-\ \frac{1}{3}\frac{\alpha\pi
T^{2}}{m^{2}}-\frac{\alpha^{2}\pi^{2}}{6}\left(\frac{T^{2}}{m^{2}}\right)^{2}+\frac{\alpha^{2}}{6}\frac{m^{2}}{T^{2}}.$
(21)
Eqs. (20) and (21) are used for a quantitative study of magnetic moment of
electron at low temperature and high temperature, respectively. We plot the
temperature dependence of magnetic moment of electron versus $\frac{T}{m}.$ A
plot of eq. (20) is given in fig. 1, whereas, eq. (21) is plotted in fig. 2\.
Both of these graphs give a sort of quadratic behavior in the negative sense.
Eqs. (21) and (22), fig. 1 and fig. 2. indicate the difference between low
temperature and high temperature behavior. Major difference in the behavior
occurs due to the $m^{2}/T^{2}$ term at high temperature. Contribution of this
term reduces with increasing temperatures and becomes totally ignorable at
very high temperatures. This is obvious from fig. 1 and fig. 2 that the
magnetic moment of electron falls off rapidly with temperature after
nucelosynthesis as compared to that before nucleosynthesis. This rapid
decrease in magnetic moment after the nucleosynthesis is not the same as it is
compared in eqs. (16) and (17), implying that the one loop and two loop
behaviors are not exactly similar.
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|
arxiv-papers
| 2012-03-16T08:04:28 |
2024-09-04T02:49:28.693162
|
{
"license": "Public Domain",
"authors": "Samina S. Masood and Mahnaz Q. Haseeb",
"submitter": "Mahnaz Haseeb",
"url": "https://arxiv.org/abs/1203.3628"
}
|
1203.3662
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2012-001 CERN-PH-EP-2012-071
Observation of $C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays
The LHCb collaboration 111Authors are listed on the following pages.
An analysis of $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow
D\pi^{\pm}$ decays is presented where the $D$ meson is reconstructed in the
two-body final states: $K^{\pm}\pi^{\mp}$, $K^{+}K^{-}$, $\pi^{+}\pi^{-}$ and
$\pi^{\pm}K^{\mp}$. Using $1.0{\rm\,fb}^{-1}$ of LHCb data, measurements of
several observables are made including the first observation of the suppressed
mode $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$. $C\\!P$ violation in
$B^{\pm}\rightarrow DK^{\pm}$ decays is observed with $5.8\,\sigma$
significance.
Submitted to Physics Letters B
Keywords: LHC, $C\\!P$ violation, hadronic $B$ decays
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S.
Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11,
S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E.
Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38,
M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M.
Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J.
Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H.
Voss10, R. Waldi55, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42,
A.D. Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
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
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
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
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
A fundamental feature of the Standard Model and its three quark generations is
that all $C\\!P$ violation phenomena are the result of a single phase in the
CKM quark-mixing matrix [1, *Kobayashi:1973fv]. The validity of this model may
be tested in several ways, and one — verifying the unitarity condition
$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0$ — is readily applicable
to $B$ mesons. This condition describes a triangle in the complex plane whose
area is proportional to the amount of $C\\!P$ violation in the model [3].
Following the observation of $C\\!P$ violation in the $B^{0}$ system [4,
*Abe:2001xe], the focus has turned to testing the unitarity of the theory by
over-constraining the sides and angles of this triangle. Most related
measurements involve loop or box diagrams, and for which the CKM model is
typically assumed when interpreting data [6, *Bona:2005vz]. This means the
least-well determined observable, the phase
$\gamma=\arg\left(-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*}\right)$ is of particular
interest as $\gamma\neq 0$ can produce direct $C\\!P$ violation in tree
decays.
One of the most powerful methods for determining $\gamma$ is measurements of
the partial widths of $B^{\pm}\rightarrow DK^{\pm}$ decays where the $D$
signifies a $D^{0}$ or $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
meson. In this case, the amplitude for the $B^{-}\rightarrow D^{0}K^{-}$
contribution is proportional to $V_{cb}$ whilst the $B^{-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}$ amplitude depends on
$V_{ub}$. If the $D$ final state is accessible for both $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons, the interference of these
two processes gives sensitivity to $\gamma$ and may exhibit direct $C\\!P$
violation. This feature of open-charm $B^{-}$ decays was first recognised in
its application to $C\\!P$ eigenstates, such as $D\rightarrow K^{+}K^{-}$,
$\pi^{+}\pi^{-}$ [8, *Gronau:1991dp] but can be extended to other decays, e.g.
$D\rightarrow\pi^{-}K^{+}$. This second category, labelled “ADS” modes in
reference to the authors of [10, *Atwood:2000ck], requires the favoured,
$b\rightarrow c$ decay to be followed by a doubly Cabibbo-suppressed $D$
decay, and the suppressed $b\rightarrow u$ decay to precede a favoured $D$
decay. The amplitudes of such combinations are of similar total magnitude and
hence large interference can occur. For both the $C\\!P$-mode and ADS methods,
the interesting observables are partial widths and $C\\!P$ asymmetries.
In this paper, we present measurements of the $B^{\pm}$ decays in the $C\\!P$
modes, $[K^{+}K^{-}]_{D}h^{\pm}$ and $[\pi^{+}\pi^{-}]_{D}h^{\pm}$, the
suppressed ADS mode $[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ and the favoured
$[K^{\pm}\pi^{\mp}]_{D}h^{\pm}$ combination where $h$ indicates either pion or
kaon. Decays where the bachelor — the charged hadron from the $B^{-}$ decay —
is a kaon carry greater sensitivity to $\gamma$. $B^{-}\rightarrow D\pi^{-}$
decays have some limited sensitivity and provide a high-statistics control
sample from which probability density functions (PDFs) are shaped. In total,
13 observables are measured: three ratios of partial widths
$R_{K/\pi}^{f}=\frac{\Gamma(B^{-}\rightarrow[f]_{D}K^{-})+\Gamma(B^{+}\rightarrow[f]_{D}K^{+})}{\Gamma(B^{-}\rightarrow[f]_{D}\pi^{-})+\Gamma(B^{+}\rightarrow[f]_{D}\pi^{+})},$
(1)
where $f$ represents $KK$, $\pi\pi$ and the favoured $K\pi$ mode, six $C\\!P$
asymmetries
$A_{h}^{f}=\frac{\Gamma(B^{-}\rightarrow[f]_{D}h^{-})-\Gamma(B^{+}\rightarrow[f]_{D}h^{+})}{\Gamma(B^{-}\rightarrow[f]_{D}h^{-})+\Gamma(B^{+}\rightarrow[f]_{D}h^{+})},$
(2)
and four charge-separated partial widths of the ADS mode relative to the
favoured mode
$R_{h}^{\pm}=\frac{\Gamma(B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm})}.$
(3)
Elsewhere, similar analyses have established the $B^{\pm}\rightarrow
D_{C\\!P}h^{\pm}$ modes [12, 13, 14] and found evidence of the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$ decay [15, 16, 17]. Analyses
of $B^{\pm}\rightarrow[K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}]_{D}K^{\pm}$
decays [18, 19] have yielded the most precise measurements of $\gamma$ though
a $5\sigma$ observation of $C\\!P$ violation from a single analysis has not
been achieved. This work represents the first simultaneous analysis of
$B^{\pm}\rightarrow D_{C\\!P}h^{\pm}$ and $B^{\pm}\rightarrow D_{\\!{\rm
ADS}}h^{\pm}$ modes. It is motivated by the future extraction of $\gamma$
which, with this combination, may be determined with minimal ambiguity.
This paper describes an analysis of 1.0 $\mbox{\,fb}^{-1}$ of
$\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$ data collected by LHCb in 2011.
The 2010 sample of 35 $\mbox{\,pb}^{-1}$ is used to define the selection
criteria in an unbiased manner. The LHCb experiment [20] takes advantage of
the high $b\bar{b}$ and $c\bar{c}$ cross sections at the Large Hadron Collider
to record large samples of heavy hadron decays. It instruments the
pseudorapidity range $2<\eta<5$ of the proton-proton ($pp$) collisions with a
dipole magnet and a tracking system which achieves a momentum resolution of
$0.4-0.6\%$ in the range $5-100$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
dipole magnet can be operated in either polarity and this feature is used to
reduce systematic effects due to detector asymmetries. In 2011, 58% of data
were taken with one polarity, 42% with the other. The $pp$ collisions take
place inside a silicon microstrip vertex detector that provides clear
separation of secondary $B$ vertices from the primary collision vertex (PV) as
well as discrimination for tertiary $D$ vertices. Two ring-imaging Cherenkov
(RICH) detectors with three radiators (aerogel, $C_{4}F_{10}$ and $CF_{4}$)
provide dedicated particle identification (PID) which is critical for the
separation of $B^{-}\\!\rightarrow\\!DK^{-}$ and
$B^{-}\\!\rightarrow\\!D\pi^{-}$ decays.
A two-stage trigger is employed. First a hardware-based decision is taken at a
frequency up to 40 MHz. It accepts high transverse energy clusters in either
an electromagnetic calorimeter or hadron calorimeter, or a muon of high
transverse momentum ($p_{\rm T}$). For this analysis, it is required that one
of the three tracks forming the $B^{\pm}$ candidate points at a deposit in the
hadron calorimeter, or that the hardware-trigger decision was taken
independently of these tracks. A second trigger level, implemented entirely in
software, receives 1 MHz of events and retains $\sim 0.3\%$ of them. It
searches for a track with large $p_{\rm T}$ and large impact parameter (IP)
with respect to the PV. This track is then required to be part of a secondary
vertex with a high $p_{\rm T}$ sum, significantly displaced from the PV. In
order to maximise efficiency at an acceptable trigger rate, the displaced
vertex is selected with a decision tree algorithm that uses $p_{\rm T}$,
$\chi^{2}_{\rm IP}$, flight distance and track separation information. Full
event reconstruction occurs offline, and after preselection around $2.5\times
10^{5}$ events are available for final analysis.
Approximately one million simulated events for each
$B^{\pm}\rightarrow[h^{+}h^{-}]_{D}h^{\pm}$ signal mode are used as well as a
large inclusive sample of generic $B\rightarrow DX$ decays. These samples are
generated using a tuned version of Pythia [21] to model the $pp$ collisions,
EvtGen [22] encodes the particle decays and Geant4 [23] to describe
interactions in the detector. Although the shapes of the signal peaks are
determined directly on data, the inclusive sample assists in the understanding
of the background. The signal samples are used to estimate the relative
efficiency in the detection of modes that differ only by the bachelor track
flavour.
## 2 Event selection
During event reconstruction, 16 combinations of $B^{\pm}\rightarrow Dh^{\pm}$,
$D\rightarrow h^{\pm}h^{\mp}$ are formed with the candidate $D$ mass within
$1765-1965$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. $D$ daughter tracks
are required to have $\mbox{$p_{\rm T}$}>250$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ but this requirement is tightened to
$0.5<\mbox{$p_{\rm T}$}<10$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$5<p<100$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for bachelor tracks to ensure
best pion versus kaon discrimination. The decay chain is refitted [24]
constraining the vertices to points in space and the $D$ candidate to its
nominal mass, $m^{D^{0}}_{\rm PDG}$ [25].
Reconstructed candidates are selected using a boosted decision tree (BDT)
discriminator [26]. It is trained using a simulated sample of
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}K^{\pm}$ and background events from
the $D$ sideband ($35<|m(hh)-m^{D^{0}}_{\rm PDG}|<100$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the independent sample
collected in 2010. The BDT uses the following properties of the candidate
$B^{\pm}$ decay:
* •
From the tracks, the $D$ and $B^{\pm}$: $p_{\rm T}$ and $\chi^{2}_{\rm IP}$
with respect to the PV;
* •
From the $B^{\pm}$ and $D$: decay time, flight distance from the PV and vertex
quality;
* •
From the $B^{\pm}$: the angle between the momentum vector and a line
connecting the PV to its decay vertex.
Information from the rest of the event is employed via an isolation variable
that considers the imbalance of $p_{\rm T}$ around the $B^{\pm}$ candidate,
$A_{\mbox{$p_{\rm T}$}}=\frac{\mbox{$p_{\rm T}$}(B)-\sum_{n}\mbox{$p_{\rm
T}$}}{\mbox{$p_{\rm T}$}(B)+\sum_{n}\mbox{$p_{\rm T}$}},$ (4)
where the $\sum_{n}\mbox{$p_{\rm T}$}$ sums over the $n$ tracks within a cone
around the candidate excluding the three signal tracks. The cone is defined by
a circle of radius 1.5 in the plane of pseudorapidity and azimuthal angle
(measured in radians). As no PID information is used as part of the BDT, it
performs equally well for all modes considered here.
The optimal cut value on the BDT response is chosen by considering the
combinatorial background level ($b$) in the invariant mass distribution of
favoured $B^{\pm}\rightarrow[K\pi]_{D}\pi^{\pm}$ candidates. The large signal
peak in this sample is scaled to the anticipated ADS-mode branching fraction
to provide a signal estimate ($s$). The quantity $s/\sqrt{s+b}$ serves as an
optimisation metric. The BDT response peaks towards 0 for background and 1 for
signal. The optimal cut is found to be $>0.92$ for the ADS mode; this is also
applied to the favoured mode. For the cleaner $C\\!P$ modes, a cut of ${\rm
BDT}>0.80$ gives a similar background level but with a 20% higher signal
efficiency.
PID information is quantified as differences between the logarithm of
likelihoods, $\ln\mathcal{L}_{h}$, under five mass hypotheses,
$h\in\\{\pi,K,p,e,\mu\\}$ (DLL). Daughter kaons of the $D$ meson are required
to have ${\rm DLL}_{K\pi}=\ln\mathcal{L}_{K}-\ln\mathcal{L}_{\pi}>2$ and
daughter pion must have ${\rm DLL}_{K\pi}<-2$. Multiple candidates are
arbitrated by choosing the candidate with the best-quality $B^{\pm}$ vertex;
only 26 events in the final sample of $157\,927$ require this consideration.
Candidates from $B$ decays that do not contain a true $D$ meson can be reduced
by requiring the flight distance significance of the $D$ candidate from the
$B^{-}$ vertex to be $>2$. The effectiveness of this cut is monitored in the
$D$ sideband where it is seen to remove significant structures peaking near
the $B^{-}$ mass. A simulation study of the $B^{-}\rightarrow
K^{-}K^{+}K^{-}$, $K^{-}\pi^{+}\pi^{-}$ and $K^{-}K^{+}\pi^{-}$ modes suggests
this cut leaves 2.5, 1.3 and 0.8 events respectively under the
$B^{-}\rightarrow[K^{+}K^{-}]_{D}K^{-}$, $[\pi^{+}\pi^{-}]_{D}K^{-}$ and
$[\pi^{+}K^{-}]_{D}K^{-}$ signals. This cut also removes cross feed (e.g.
$B^{-}\rightarrow[K^{-}\pi^{+}]_{D}\pi^{-}$ as a background of
$[\pi^{+}\pi^{-}]_{D}K^{-}$) which occurs when the bachelor is confused with a
$D$ daughter at low decay time. Finally, the combination of the bachelor and
the opposite-sign $D^{0}$ daughter is made under the hypothesis they are
muons. The parent $B$ candidate is vetoed if the invariant mass of this
combination is within $\pm 22$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of
either the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$ mass
[25].
Due to misalignment, the reconstructed $B^{\pm}$ mass is not identical to the
established value, $m^{B^{\pm}}_{\rm PDG}$ [25]. As simulation is used to
define background shapes, it is useful to apply linear momentum scaling
factors separately to the two polarity datasets so the $B^{\pm}$ mass peak is
closer to $m^{B^{\pm}}_{\rm PDG}$. After this correction, the
$D^{0}\rightarrow K^{-}\pi^{+}$ mass peak is measured at 1864.8
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ with a resolution of 7.4
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Selected $D$ candidates are
required to be within $\pm 25$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of
$m^{D^{0}}_{\rm PDG}$. This cut is tight enough that no cross feed occurs from
the favoured mode into the $C\\!P$ modes. In contrast, the ADS mode suffers a
potentially large cross feed from the favoured mode in the circumstance that
both $D$ daughters are misidentified. The invariant mass spectrum of such
cross feed is broad but peaks around $m^{D^{0}}_{\rm PDG}$. It is reduced by
vetoing any ADS candidate whose $D$ candidate mass under the exchange of its
daughter track mass hypotheses, lies within $\pm 15$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of $m^{D^{0}}_{\rm PDG}$.
Importantly for the measurements of $R_{h}^{\pm}$, this veto is also applied
to the favoured mode. With the $D$ mass selection and the $D$ daughter PID
requirements, this veto reduces the rate of cross feed to an almost negligible
rate of $(6\pm 3)\times 10^{-5}$.
Partially reconstructed events populate the invariant mass region below the
$B^{\pm}$ mass. Such events may enter the signal region, especially where
Cabibbo-favoured $B\rightarrow XD\pi^{\pm}$ modes are misidentified as
$B\rightarrow XDK^{\pm}$. The large simulated sample of inclusive
$B_{q}\rightarrow DX$ decays, $q\in\\{u,d,s\\}$, is used to model this
background. After applying the selection, two non-parametric PDFs [27] are
defined (for the $D\pi^{\pm}$ and $DK^{\pm}$ selections) and used in the
signal extraction fit. These PDFs are applied to all four $D$ modes though two
additional contributions are needed in specific cases. In the $D\rightarrow
K^{+}K^{-}$ mode,
$\Lambda_{b}^{0}\rightarrow[p^{+}K^{-}\pi^{+}]_{\Lambda_{c}}h^{-}$ enters if
the pion is missed and the proton is reconstructed as a kaon. In the
$B^{\pm}\rightarrow D_{\\!{\rm ADS}}K^{\pm}$ mode, partially reconstructed
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
D^{0}K^{+}\pi^{-}$ decays represent an important, Cabibbo-favoured background.
PDFs of both these sources are defined from simulation, smeared by the modest
degradation in resolution observed in data. When discussing these
contributions, inclusion of the charge conjugate process is implied
throughout.
## 3 Signal yield determination
The observables of interest are determined with a binned maximum-likelihood
fit to the invariant mass distributions of selected $B$ candidates [28].
Sensitivity to $C\\!P$ asymmetries is achieved by separating the candidates
into $B^{-}$ and $B^{+}$ samples. $B^{\pm}\rightarrow DK^{\pm}$ events are
distinguished from $B^{\pm}\rightarrow D\pi^{\pm}$ using a PID cut on the
${\rm DLL}_{K\pi}$ of the bachelor track. Events passing this cut are
reconstructed as $DK^{\pm}$, events failing the cut are reconstructed as the
$D\pi^{\pm}$ final state. The fit therefore comprises four subsamples —
$(\mathrm{plus,minus})\\!\times\\!(\mathrm{pass,fail})$ — for each $D$ mode,
fitted simultaneously and displayed in Figs. 1–4. The total PDF is built from
four or five components representing the various sources of events in each
subsample.
1. 1.
$B^{\pm}\rightarrow D\pi^{\pm}$. In the sample failing the bachelor PID cut, a
modified Gaussian function,
$f(x)\propto\exp\left(\frac{-(x-\mu)^{2}}{2\sigma^{2}+(x-\mu)^{2}\alpha_{L,R}}\right)$
(5)
describes the asymmetric peak of mean $\mu$ and width $\sigma$ where
$\alpha_{L}(x<\mu)$ and $\alpha_{R}(x>\mu)$ parameterise the tails.
True $B^{\pm}\rightarrow D\pi^{\pm}$ events that pass the PID cut are
reconstructed as $B^{\pm}\rightarrow DK^{\pm}$. As these events have an
incorrect mass assignment they form a displaced mass peak with a tail that
extends to higher invariant mass. These events are modelled by the sum of two
Gaussian PDFs also altered to include tail components. All parameters are
allowed to vary except the lower-mass tail which is fixed to ensure fit
stability and later considered amongst the systematic uncertainties. These
shapes are considered identical for $B^{-}$ and $B^{+}$ decays and for all
four $D$ modes. This assumption is validated with simulation.
2. 2.
$B^{\pm}\rightarrow DK^{\pm}$: In the sample that passes the ${\rm
DLL}_{K\pi}$ cut on the bachelor, the same modified Gaussian function is used.
The mean and the two tail parameters are identical to those of the larger,
$B^{\pm}\rightarrow D\pi^{\pm}$ peak. The width is $0.95\pm 0.02$ times the
$D\pi^{\pm}$ width, as determined by a standalone study of the favoured mode.
Its applicability to the $C\\!P$ modes is checked with simulation and a 1%
systematic uncertainty assigned. Events failing the PID cut are described by a
fixed shape that is obtained from simulation and later varied to assess the
systematic error.
3. 3.
Partially reconstructed $B\rightarrow DX$: A fixed, non-parametric PDF,
derived from simulation, is used for all subsamples. The yield in each
subsample varies independently, making no assumption of $C\\!P$ symmetry.
4. 4.
Combinatoric background: A linear approximation is adequate to describe the
slope across the invariant mass spectrum considered. A common parameter is
used in all subsamples, though yields vary independently.
5. 5.
Mode-specific backgrounds: In the $D\rightarrow KK$ mode, two extra components
are used to model $\Lambda^{0}_{b}\rightarrow\Lambda_{c}^{+}h^{-}$ decays.
Though the total contribution is allowed to vary, the shape and relative
proportion of $\Lambda_{c}^{+}K^{-}$ and $\Lambda_{c}^{+}\pi^{-}$ are fixed.
This latter quantity is estimated at $0.060\pm 0.015$, similar to the
effective Cabibbo suppression observed in $B$ mesons. For the
$B^{\pm}\rightarrow D_{\\!{\rm ADS}}K^{\pm}$ mode, the shape of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\pi^{-}$
background is taken from simulation. In the fit, this yield is allowed to vary
though the reported yield is consistent with the simulated expectation, as
derived from the branching fraction [29] and the $b\overline{}b$ hadronisation
[30].
The proportion of $B^{\pm}\rightarrow Dh^{\pm}$ passing or failing the PID
requirement is determined from a calibration analysis of a large sample of
$D^{*\pm}$ decays reconstructed as $D^{*\pm}\rightarrow D\pi^{\pm},\
D\rightarrow K^{\mp}\pi^{\pm}$. In this calibration sample, the $K$ and $\pi$
tracks may be identified, with high purity, using only kinematic variables.
This facilitates a measurement of the RICH-based PID efficiency as a function
of track momentum, pseudorapidity and number of tracks in the detector. By
reweighting the calibration spectra in these variables to match the events in
the $B^{\pm}\rightarrow D\pi^{\pm}$ peak, the effective PID efficiency of the
signal is deduced. This data-driven technique finds a retention rate, for a
cut of ${\rm DLL}_{K\pi}>4$ on the bachelor track, of 87.6% and 3.8% for kaons
and pions, respectively. A $1.0\%$ systematic uncertainty on the kaon
efficiency is estimated from simulation. The $B^{\pm}\rightarrow D\pi^{\pm}$
fit to data becomes visibly incorrect with variations to the fixed PID
efficiency $>\pm 0.2\%$ so this value is taken as the systematic uncertainly
for pions.
A small negative asymmetry is expected in the detection of $K^{-}$ and $K^{+}$
mesons due to their different interaction lengths. A fixed value of $(-0.5\pm
0.7)$% is assigned for each occurrence of strangeness in the final state. The
equivalent asymmetry for pions is expected to be much smaller and ($0.0\pm
0.7$)% is assigned. This uncertainty also accounts for the residual physical
asymmetry between the left and right sides of the detector after summing both
magnet-polarity datasets. Simulation of $B$ meson production in $pp$
collisions suggests a small excess of $B^{+}$ over $B^{-}$ mesons. A
production asymmetry of $(-0.8\pm 0.7)$% is assumed in the fit such that the
combination of these estimates aligns with the observed raw asymmetry of
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decays
at LHCb [31]. Ongoing studies of these instrumentation asymmetries will reduce
the associated systematic uncertainty in future analyses.
The final $B^{\pm}\rightarrow Dh^{\pm}$ signal yields, after summing the
events that pass and fail the bachelor PID cut, are shown in Table 1. The
invariant mass spectra of all 16 $B^{\pm}\rightarrow[h^{+}h^{-}]_{D}h^{\pm}$
modes are shown in Figs. 1–4. Regarding the $B^{\pm}\rightarrow D\pi^{\pm}$
mass resolution: respectively, $14.1\pm 0.1$, $14.2\pm 0.1$ and $14.2\pm 0.2$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are found for the $D\rightarrow
KK$, $K\pi$ and $\pi\pi$ modes with common tail parameters
$\alpha_{L}=0.115\pm 0.003$ and $\alpha_{R}=0.083\pm 0.002$. As explained
above, the $B^{\pm}\rightarrow DK^{\pm}$ widths are fixed relative to these
values.
Table 1: Corrected event yields. $B^{\pm}$ mode | $D$ mode | $B^{-}$ | $B^{+}$
---|---|---|---
$DK^{\pm}$ | $K^{\pm}\pi^{\mp}$ | $\phantom{0}3170\pm\phantom{0}83$ | $\phantom{0}3142\pm\phantom{0}83$
$K^{\pm}K^{\mp}$ | $\phantom{00}592\pm\phantom{0}40$ | $\phantom{00}439\pm\phantom{0}30$
$\pi^{\pm}\pi^{\mp}$ | $\phantom{00}180\pm\phantom{0}22$ | $\phantom{00}137\pm\phantom{0}16$
$\pi^{\pm}K^{\mp}$ | $\phantom{000}23\pm\phantom{00}7$ | $\phantom{000}73\pm\phantom{0}11$
$D\pi^{\pm}$ | $K^{\pm}\pi^{\mp}$ | $40767\pm 310$ | $40774\pm 310$
$K^{\pm}K^{\mp}$ | $\phantom{0}6539\pm 129$ | $\phantom{0}6804\pm 135$
$\pi^{\pm}\pi^{\mp}$ | $\phantom{0}1969\pm\phantom{0}69$ | $\phantom{0}1973\pm\phantom{0}69$
$\pi^{\pm}K^{\mp}$ | $\phantom{00}191\pm\phantom{0}16$ | $\phantom{00}143\pm\phantom{0}14$
The ratio of partial widths relates to the ratio of event yields by the
relative efficiency with which $B^{\pm}\rightarrow D^{0}K^{\pm}$ and
$B^{\pm}\rightarrow D^{0}\pi^{\pm}$ decays are reconstructed. This ratio,
estimated from simulation, is 1.012, 1.009 and 1.005 for $D\rightarrow
KK,K\pi,\pi\pi$ respectively. A 1.1% systematic uncertainty, based on the
finite size of the simulated sample, accounts for the imperfect modelling of
the relative pion and kaon absorption in the tracking material.
The fit is constructed such that the observables of interest are parameters of
the fit and all systematic uncertainties discussed above enter the fit as
constant numbers in the model. To evaluate the effect of these systematic
uncertainties, the fit is rerun many times varying each of the systematic
constants by its uncertainty. The resulting spread (RMS) in the value of each
observable is taken as the systematic uncertainty on that quantity and is
summarised in Table 2. Correlations between the uncertainties are considered
negligible so the total systematic uncertainty is just the sum in quadrature.
For the ratios of partial widths in the favoured and $C\\!P$ modes, the
uncertainties on the PID efficiency and the relative width of the $DK^{\pm}$
and $D\pi^{\pm}$ peaks dominate. These sources also contribute in the ADS
modes, though the assumed shape of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\pi^{-}$
background is the largest source of systematic uncertainty in the
$B^{\pm}\rightarrow D_{\\!{\rm ADS}}K^{\pm}$ case. For the $C\\!P$
asymmetries, instrumentation asymmetries at LHCb are the largest source of
uncertainty.
Table 2: Systematic uncertainties on the observables. PID refers to the fixed efficiency of the ${\rm DLL}_{K\pi}$ cut on the bachelor track. PDFs refers to the variations of the fixed shapes in the fit. “Sim” refers to the use of simulation to estimate relative efficiencies of the signal modes which includes the branching fraction estimates of the $\Lambda_{b}^{0}$ background. $A_{\rm instr.}$ quantifies the uncertainty on the production, interaction and detection asymmetries. $\times 10^{-3}$ | PID | PDFs | Sim | $A_{\rm instr.}$ | Total
---|---|---|---|---|---
$R_{K/\pi}^{K\pi}$ | 1.4 | 0.9 | 0.8 | 0 | 1.8
$R_{K/\pi}^{KK}$ | 1.3 | 0.8 | 0.9 | 0 | 1.8
$R_{K/\pi}^{\pi\pi}$ | 1.3 | 0.6 | 0.8 | 0 | 1.7
$A_{\pi}^{K\pi}$ | 0 | 1.0 | 0 | 9.4 | 9.5
$A_{K}^{K\pi}$ | 0.2 | 4.1 | 0 | 16.9 | 17.4
$A_{K}^{KK}$ | 1.6 | 1.3 | 0.5 | 9.5 | 9.7
$A_{K}^{\pi\pi}$ | 1.9 | 2.3 | 0 | 9.0 | 9.5
$A_{\pi}^{KK}$ | 0.1 | 6.6 | 0 | 9.5 | 11.6
$A_{\pi}^{\pi\pi}$ | 0.1 | 0.4 | 0 | 9.9 | 9.9
$R_{K}^{-}$ | 0.2 | 0.4 | 0 | 0.1 | 0.4
$R_{K}^{+}$ | 0.4 | 0.5 | 0 | 0.1 | 0.7
$R_{\pi}^{-}$ | 0.01 | 0.03 | 0 | 0.07 | 0.08
$R_{\pi}^{+}$ | 0.01 | 0.03 | 0 | 0.07 | 0.07
## 4 Results
The results of the fit with their statistical uncertainties and assigned
systematic uncertainties are:
$\displaystyle R_{K/\pi}^{K\pi}$ $\displaystyle=$
$\displaystyle\phantom{-}0.0774\pm 0.0012\pm 0.0018$ $\displaystyle
R_{K/\pi}^{KK}$ $\displaystyle=$ $\displaystyle\phantom{-}0.0773\pm 0.0030\pm
0.0018$ $\displaystyle R_{K/\pi}^{\pi\pi}$ $\displaystyle=$
$\displaystyle\phantom{-}0.0803\pm 0.0056\pm 0.0017$ $\displaystyle
A_{\pi}^{K\pi}$ $\displaystyle=$ $\displaystyle-0.0001\pm 0.0036\pm 0.0095$
$\displaystyle A_{K}^{K\pi}$ $\displaystyle=$
$\displaystyle\phantom{-}0.0044\pm 0.0144\pm 0.0174$ $\displaystyle
A_{K}^{KK}$ $\displaystyle=$ $\displaystyle\phantom{-}0.148\pm 0.037\pm 0.010$
$\displaystyle A_{K}^{\pi\pi}$ $\displaystyle=$
$\displaystyle\phantom{-}0.135\pm 0.066\pm 0.010$ $\displaystyle A_{\pi}^{KK}$
$\displaystyle=$ $\displaystyle-0.020\pm 0.009\pm 0.012$ $\displaystyle
A_{\pi}^{\pi\pi}$ $\displaystyle=$ $\displaystyle-0.001\pm 0.017\pm 0.010$
$\displaystyle R_{K}^{-}$ $\displaystyle=$ $\displaystyle\phantom{-}0.0073\pm
0.0023\pm 0.0004$ $\displaystyle R_{K}^{+}$ $\displaystyle=$
$\displaystyle\phantom{-}0.0232\pm 0.0034\pm 0.0007$ $\displaystyle
R_{\pi}^{-}$ $\displaystyle=$ $\displaystyle\phantom{-}0.00469\pm 0.00038\pm
0.00008$ $\displaystyle R_{\pi}^{+}$ $\displaystyle=$
$\displaystyle\phantom{-}0.00352\pm 0.00033\pm 0.00007.$
From these measurements, the following quantities can be deduced:
$\displaystyle R_{C\\!P+}$ $\displaystyle\approx$
$\displaystyle<R_{K/\pi}^{KK},R_{K/\pi}^{\pi\pi}>/R_{K/\pi}^{K\pi}$
$\displaystyle=$ $\displaystyle\phantom{-}1.007\pm 0.038\pm 0.012$
$\displaystyle A_{C\\!P+}$ $\displaystyle=$
$\displaystyle<A_{K}^{KK},A_{K}^{\pi\pi}>$ $\displaystyle=$
$\displaystyle\phantom{-}0.145\pm 0.032\pm 0.010$ $\displaystyle R_{{\rm
ADS}(K)}$ $\displaystyle=$ $\displaystyle(R_{K}^{-}+R_{K}^{+})/2$
$\displaystyle=$ $\displaystyle\phantom{-}0.0152\pm 0.0020\pm 0.0004$
$\displaystyle A_{{\rm ADS}(K)}$ $\displaystyle=$
$\displaystyle(R_{K}^{-}-R_{K}^{+})/(R_{K}^{-}+R_{K}^{+})$ $\displaystyle=$
$\displaystyle-0.52\pm 0.15\pm 0.02$ $\displaystyle R_{{\rm ADS}(\pi)}$
$\displaystyle=$ $\displaystyle(R_{\pi}^{-}+R_{\pi}^{+})/2$ $\displaystyle=$
$\displaystyle\phantom{-}0.00410\pm 0.00025\pm 0.00005$ $\displaystyle A_{{\rm
ADS}(\pi)}$ $\displaystyle=$
$\displaystyle(R_{\pi}^{-}-R_{\pi}^{+})/(R_{\pi}^{-}+R_{\pi}^{+})$
$\displaystyle=$ $\displaystyle\phantom{-}0.143\pm 0.062\pm 0.011,$
where the correlations between systematic uncertainties are taken into account
in the combination and angled brackets indicate weighted averages. The above
definition of $R_{C\\!P+}$ is only approximate and is used for experimental
convenience. It assumes the absence of $C\\!P$ violation in
$B^{\pm}\rightarrow D\pi^{\pm}$ and the favoured $B^{\pm}\rightarrow DK^{\pm}$
modes. The exact definition of $R_{C\\!P+}$ is
$\frac{\Gamma(B^{-}\rightarrow D_{C\\!P+}K^{-})+\Gamma(B^{+}\rightarrow
D_{C\\!P+}K^{+})}{\Gamma(B^{-}\rightarrow D^{0}K^{-})}$ (6)
so an additional, and dominant, 1% systematic uncertainty accounts for the
approximation. For the same reason, a small addition to the systematic
uncertainty of $R_{K/\pi}^{K\pi}$ is needed to quote this result as the ratio
of $B^{\pm}$ branching fractions,
$\frac{\mathcal{B}(B^{-}\rightarrow D^{0}K^{-})}{\mathcal{B}(B^{-}\rightarrow
D^{0}\pi^{-})}=(7.74\pm 0.12\pm 0.19)\%.\\\ $
To summarise, the $B^{\pm}\rightarrow DK^{\pm}$ ADS mode is observed with
$\sim 10\sigma$ statistical significance when comparing the maximum likelihood
to that of the null hypothesis. This mode displays evidence ($4.0\sigma$) of a
large negative asymmetry, consistent with the asymmetries reported by previous
experiments [15, 16, 17]. The $B^{\pm}\rightarrow D\pi^{\pm}$ ADS mode shows a
hint of a positive asymmetry with $2.4\sigma$ significance. The $KK$ and
$\pi\pi$ modes both show positive asymmetries. The statistical significance of
the combined asymmetry, $A_{C\\!P+},$ is $4.5\sigma$ which is similar to that
reported in [12, 14] albeit with a smaller central value. All these results
contain dependence on the weak phase $\gamma$ and will form an important
contribution to a future measurement of this parameter.
Assuming the $C\\!P$-violating effects in the $C\\!P$ and ADS modes are due to
the same phenomenon (namely the interference of $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ transitions) we compare the maximum likelihood with
that under the null-hypothesis in all three $D$ final states where the
bachelor is a kaon. This log-likelihood difference is diluted by the non-
negligible systematic uncertainties in $A_{C\\!P+}$ and $A_{{\rm ADS}(K)}$
which are dominated by the instrumentation asymmetries and hence are highly
correlated. In conclusion, with a total significance of $5.8\sigma$, direct
$C\\!P$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays is observed.
Figure 1: Invariant mass distributions of selected
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm}$ candidates. The left plots
are $B^{-}$ candidates, $B^{+}$ are on the right. In the top plots, the
bachelor track passes the ${\rm DLL}_{K\pi}>4$ cut and the $B$ candidates are
reconstructed assigning this track the kaon mass. The remaining events are
placed in the sample displayed on the bottom row and are reconstructed with a
pion mass hypothesis. The dark (red) curve represents the $B\rightarrow
DK^{\pm}$ events, the light (green) curve is $B\rightarrow D\pi^{\pm}$. The
shaded contribution are partially reconstructed events and the total PDF
includes the combinatorial component. Figure 2: Invariant mass distributions
of selected $B^{\pm}\rightarrow[K^{+}K^{-}]_{D}h^{\pm}$ candidates. See the
caption of Fig. 1 for a full description. The contribution from
$\Lambda_{b}\rightarrow\Lambda_{c}^{\pm}h^{\mp}$ decays is indicated by the
dashed line. Figure 3: Invariant mass distributions of selected
$B^{\pm}\rightarrow[\pi^{+}\pi^{-}]_{D}h^{\pm}$ candidates. See the caption of
Fig. 1 for a full description. Figure 4: Invariant mass distributions of
selected $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ candidates. See the
caption of Fig. 1 for a full description. The dashed line here represents the
partially reconstructed, but Cabibbo favoured, $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}\pi^{+}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\pi^{-}$
decays where the pions are lost. The pollution from favoured mode cross feed
is drawn, but is too small to be seen.
## 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 (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal 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 Region Auvergne.
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|
arxiv-papers
| 2012-03-16T10:55:53 |
2024-09-04T02:49:28.701078
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer,\n Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, 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, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, K. de Bruyn, A. B\\\"uchler-Germann,\n I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M.\n Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G.A. Cowan, R. Currie, C. D'Ambrosio, P. David, P.N.Y.\n David, I. De Bonis, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, P.\n De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano,\n D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz\n Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, 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, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, 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, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K.\n Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\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, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A.\n Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni,\n G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis,\n M. Veltri, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams,\n F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Paolo Gandini",
"url": "https://arxiv.org/abs/1203.3662"
}
|
1203.3724
|
8 (1:26) 2012 1–63 Sep. 07, 2011 Mar. 23, 2012
# Static Analysis of Run-Time Errors in Embedded Real-Time Parallel C
Programs*
Antoine Miné CNRS & École Normale Supérieure, 45 rue d’Ulm, 75005 Paris,
France mine@di.ens.fr
###### Abstract.
We present a static analysis by Abstract Interpretation to check for run-time
errors in parallel and multi-threaded C programs. Following our work on
Astrée, we focus on embedded critical programs without recursion nor dynamic
memory allocation, but extend the analysis to a static set of threads
communicating implicitly through a shared memory and explicitly using a finite
set of mutual exclusion locks, and scheduled according to a real-time
scheduling policy and fixed priorities. Our method is thread-modular. It is
based on a slightly modified non-parallel analysis that, when analyzing a
thread, applies and enriches an abstract set of thread interferences. An
iterator then re-analyzes each thread in turn until interferences stabilize.
We prove the soundness of our method with respect to the sequential
consistency semantics, but also with respect to a reasonable weakly consistent
memory semantics. We also show how to take into account mutual exclusion and
thread priorities through a partitioning over an abstraction of the scheduler
state. We present preliminary experimental results analyzing an industrial
program with our prototype, Thésée, and demonstrate the scalability of our
approach.
###### Key words and phrases:
Abstract interpretation, parallel programs, run-time errors, static analysis
###### 1991 Mathematics Subject Classification:
D.2.4, F.3.1, F.3.2
This work was partially supported by the INRIA project “Abstraction” common to
CNRS and ENS in France, and by the project ANR-11-INSE-014 from the French
Agence nationale de la recherche.
*This article is an extended version of our article [mine:esop11] published in the Proceedings of the 20th European Symposium on Programming (ESOP’11).
## 1\. Introduction
Ensuring the safety of critical embedded software is important as a single
“bug” can have catastrophic consequences. Previous work on the Astrée analyzer
[blanchet-al-PLDI03] demonstrated that static analysis by Abstract
Interpretation could help, when specializing an analyzer to a class of
properties and programs — namely in that case, the absence of run-time errors
(such as arithmetic and memory errors) on synchronous control / command
embedded avionic C software. In this article, we describe ongoing work to
achieve similar results for multi-threaded and parallel embedded C software.
Such an extension is demanded by the current trend in critical embedded
systems to switch from large numbers of single-program processors
communicating through a common bus to single-processor multi-threaded
applications communicating through a shared memory — for instance, in the
context of Integrated Modular Avionics [ima]. Analyzing each thread
independently with a tool such as Astrée would not be sound and could miss
bugs that only appear when threads interact. In this article, we focus on
detecting the same kinds of run-time errors as Astrée does, while taking
thread communications into account in a sound way, including accesses to the
shared memory and synchronization primitives. In particular, we correctly
handle the effect of concurrent threads accessing a common variable without
enforcing mutual exclusion by synchronization primitives, and we report such
accesses — these will be called data-races in the rest of the article.
However, we ignore other concurrency hazards such as dead-locks, live-locks,
and priority inversions, which are considered to be orthogonal issues.
Our method is based on Abstract Interpretation [cc-POPL77], a general theory
of the approximation of semantics which allows designing static analyzers that
are fully automatic and sound by construction — i.e., consider a superset of
all program behaviors. Such analyzers cannot miss any bug in the class of
errors they analyze. However, they can cause spurious alarms due to over-
approximations, an unfortunate effect we wish to minimize while keeping the
analysis efficient.
To achieve scalability, our method is thread-modular and performs a rely-
guarantee reasoning, where rely and guarantee conditions are inferred
automatically. At its core, it performs a sequential analysis of each thread
considering an abstraction of the effects of the other threads, called
interferences. Each sequential analysis also collects a new set of
interferences generated by the analyzed thread. It then serves as input when
analyzing the other threads. Starting from an empty set of interferences,
threads are re-analyzed in sequence until a fixpoint of interferences is
reached for all threads. Using this scheme, few modifications are required to
a sequential analyzer in order to analyze multi-threaded programs. Practical
experiments suggest that few thread re-analyses are required in practice,
resulting in a scalable analysis. The interferences are considered in a flow-
insensitive and non-relational way: they store, for each variable, an
abstraction of the set of all values it can hold at any program point of a
given thread. Our method is however quite generic in the way individual
threads are analyzed. They can be analyzed in a fully or partially flow-
sensitive, context-sensitive, path-sensitive, and relational way (as is the
case in our prototype).
As we target embedded software, we can safely assume that there is no
recursion, dynamic allocation of memory, nor dynamic creation of threads nor
locks, which makes the analysis easier. In return, we handle two subtle
points. Firstly, we consider a weakly consistent memory model: memory accesses
not protected by mutual exclusion (i.e., data-races) may cause behaviors that
are not the result of any thread interleaving to appear. The reason is that
arbitrary observation by concurrent threads can expose compiler and processor
optimizations (such as instruction reordering) that are designed to be
transparent on non-parallel programs only. We prove that our semantics is
invariant by large classes of widespread program transformations, so that an
analysis of the original program is also sound with respect to reasonably
compiled and optimized versions. Secondly, we show how to take into account
the effect of a real-time scheduler that schedules the threads on a single
processor following strict, fixed priorities. According to this scheduling
algorithm, which is quite common in the realm of embedded real-time software —
e.g., in the real-time thread extension of the POSIX standard [posix-threads],
or in the ARINC 653 avionic operating system standard [ARINC] — only the
unblocked thread of highest priority may run. This ensures some lock-less
mutual exclusion properties that are actually exploited in real-time embedded
programs and relied on for their correctness (this includes the industrial
application our prototype currently targets). We show how our analysis can
take these properties into account, but we also present an analysis that
assumes less properties on the scheduler and is thus sound for true multi-
processors and non-real-time schedulers. We handle synchronization properties
(enforced by either locks or priorities) through a partitioning with respect
to an abstraction of the global scheduling state. The partitioning recovers
some kind of inter-thread flow-sensitivity that would otherwise be completely
abstracted away by the interference abstraction.
The approach presented in this article has been implemented and used at the
core of a prototype analyzer named Thésée. It leverages the static analysis
techniques developed in Astrée [blanchet-al-PLDI03] for single-threaded
programs, and adds the support for multiple threads. We used Thésée to analyze
in 27 h a large (1.7 M lines) multi-threaded industrial embedded C avionic
application, which illustrates the scalability of our approach.
### Organisation
Our article is organized as follows. First, Sec. 2 presents a classic non-
parallel semantics and its static analysis. Then, Sec. LABEL:sec:shared
extends them to several threads in a shared memory and discusses weakly
consistent memory issues. A model of the scheduler and support for locks and
priorities are introduced in Sec. LABEL:sec:sched. Our prototype analyzer,
Thésée, is presented in Sec. LABEL:sec:result, as well as some experimental
results. Finally, Sec. LABEL:sec:relwork discusses related work, and Sec.
LABEL:sec:conclusion concludes and envisions future work.
This article defines many semantics. They are summarized in Fig. 1, using
$\subseteq$ to denote the “is less abstract than” relation. We alternate
between two kinds of concrete semantics: semantics based on control paths
($\mathbin{{\mathbb{P_{\pi}}}{}}$, $\mathbin{{\mathbb{P_{*}}}{}}$,
$\mathbin{{\mathbb{P_{\mathcal{H}}}}{}}$), that can model precisely thread
interleavings and are also useful to characterize weakly consistent memory
models ($\mathbin{{\mathbb{P^{\prime}_{*}}}{}}$,
$\mathbin{{\mathbb{P^{\prime}_{\mathcal{H}}}}{}}$), and semantics by
structural induction on the syntax ($\mathbin{{\mathbb{P}}{}}$,
$\mathbin{{\mathbb{P_{\mathcal{I}}}}{}}$,
$\mathbin{{\mathbb{P_{\mathcal{C}}}}{}}$), that give rise to effective
abstract interpreters ($\mathbin{{\mathbb{P^{\sharp}}}{}}$,
$\mathbin{{\mathbb{P_{\mathcal{I}}^{\sharp}}}{}}$,
$\mathbin{{\mathbb{P_{\mathcal{C}}^{\sharp}}}{}}$). Each semantics is
presented in its subsection and adds some features to the previous ones, so
that the final abstract analysis
$\mathbin{{\mathbb{P_{\mathcal{C}}^{\sharp}}}{}}$ presented in Sec.
LABEL:sec:schedabs should hopefully not appear as too complex nor artificial,
but rather as the logical conclusion of a step-by-step construction.
Our analysis has been mentioned first, briefly and informally, in [bertrane-
al-aiaa10, § VI]. We offer here a formal, rigorous treatment by presenting all
the semantics fully formally, albeit on an idealised language, and by studying
their relationship. The present article is an extended version of
[mine:esop11] and includes a more comprehensive description of the semantics
as well as the proof of all theorems, that were omitted in the conference
proceedings due to lack of space.
non-parallel semantics | parallel semantics |
---|---|---
|
---
$\mathbin{{\mathbb{P}}{}}$ (§2.2) $\scriptstyle{\subseteq}$ $\mathbin{{\mathbb{P^{\sharp}}}{}}$ (§2.3) $\mathbin{{\mathbb{P_{\pi}}}{}}$ (§LABEL:sec:pathsem) $\scriptstyle{=}$( | |
---|---
$\mathbin{{\mathbb{P_{\mathcal{I}}}}{}}$ (§LABEL:sec:interfersem)
$\scriptstyle{\subseteq}$ $\mathbin{{\mathbb{P_{\mathcal{I}}^{\sharp}}}{}}$
(§LABEL:sec:sharedabs) $\mathbin{{\mathbb{P_{*}}}{}}$
(§LABEL:sec:interleavesem) $\scriptstyle{\subseteq}$
$\mathbin{{\mathbb{P^{\prime}_{*}}}{}}$ (§LABEL:sec:weaksem)
$\scriptstyle{\subseteq}$ $\mathbin{{\mathbb{P_{\mathcal{C}}}}{}}$
(§LABEL:sec:schedinterfersem)
$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$
$\mathbin{{\mathbb{P_{\mathcal{C}}^{\sharp}}}{}}$ (§LABEL:sec:schedabs)
$\mathbin{{\mathbb{P_{\mathcal{H}}}}{}}$ (§LABEL:sec:schedinterleavesem)
$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$
$\mathbin{{\mathbb{P^{\prime}_{\mathcal{H}}}}{}}$ (§LABEL:sec:schedweaksem)
$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$( non-scheduled structured
semantics non-scheduled path-based semantics scheduled structured semantics
scheduled path-based semantics
Figure 1. Semantics defined in the article.
### Notations
In this article, we use the theory of complete lattices, denoting their
partial order, join, and least element respectively as $\sqsubseteq$,
$\sqcup$, and $\bot$, possibly with some subscript to indicate which lattice
is considered. All the lattices we use are actually constructed by taking the
Cartesian product of one or several powerset lattices — i.e., $\mathcal{P}(S)$
for some set $S$ — $\sqsubseteq$, $\sqcup$, and $\bot$ are then respectively
the set inclusion $\subseteq$, the set union $\cup$, and the empty set
$\emptyset$, applied independently to each component. Given a monotonic
operator $F$ in a complete lattice, we denote by $\operatorname{{\it lfp}}F$
its least fixpoint — i.e., $F(\operatorname{{\it lfp}}F)=\operatorname{{\it
lfp}}F$ and $\forall X:F(X)=X\Longrightarrow\operatorname{{\it
lfp}}F\sqsubseteq X$ — which exists according to Tarski [tarski-PJM55, cc-
PJM79]. We denote by $A\rightarrow B$ the set of functions from a set $A$ to a
set $B$, and by $A\stackrel{{\scriptstyle\sqcup}}{{\longrightarrow}}B$ the set
of complete $\sqcup-$morphisms from a complete lattice $A$ to a complete
lattice $B$, i.e., such that
$F(\sqcup_{A}\,X)=\bigsqcup_{B}\;\\{\,F(x)\;|\;x\in X\,\\}$ for any finite or
infinite set $X\subseteq A$. Additionally, such a function is monotonic. We
use the theory of Abstract Interpretation by Cousot and Cousot and, more
precisely, its concretization-based ($\gamma$) formalization [cc-JLC92]. We
use widenings ($\mathbin{\triangledown}$) to ensure termination [cc-PLILP92].
The abstract version of a domain, operator, or function is denoted with a
$\sharp$ superscript. We use the lambda notation $\lambda x:f(x)$ to denote
functions. If $f$ is a function, then $f[x\mapsto v]$ is the function with the
same domain as $f$ that maps $x$ to $v$, and all other elements $y\neq x$ to
$f(y)$. Likewise, $f[\forall x\in X:x\mapsto g(x)]$ denotes the function that
maps any $x\in X$ to $g(x)$, and other elements $y\notin X$ to $f(y)$.
Boldface fonts are used for syntactic elements, such as “${\mathbf{while}}$”
in Fig. 2. Pairs and tuples are bracketed by parentheses, as in $X=(A,B,C)$,
and can be deconstructed (matched) with the notation “$\mbox{let
}(A,-,C){\;=\;}X\mbox{ in }\cdots$” where the “$-$” symbol denotes irrelevant
tuple elements. The notation “$\mbox{let }\forall x\in
X:y_{x}{\;=\;}\cdots\mbox{ in }\cdots$” is used to bind a collection of
variables $(y_{x})_{x\in X}$ at once. Semantic functions are denoted with
double brackets, as in $\mathbin{{\mathbb{X}\llbracket}\,y\,{\rrbracket}}$,
where $y$ is an (optional) syntactic object, and $\mathbin{{\mathbb{X}}{}}$
denotes the kind of objects ($\mathbin{{\mathbb{S}}{}}$ for statements,
$\mathbin{{\mathbb{E}}{}}$ for expressions, $\mathbin{{\mathbb{P}}{}}$ for
programs, $\mathbin{{\mathbbx{\Pi}}{}}$ for control paths). The kind of
semantics considered (parallel, non-parallel, abstract, etc.) is denoted by
subscripts and superscripts over $\mathbin{{\mathbb{X}}{}}$, as exemplified in
Fig. 1. Finally, we use finite words over arbitrary sets, using $\epsilon$ and
$\cdot$ to denote, respectively, the empty word and word concatenation. The
concatenation $\cdot$ is naturally extended to sets of words: $A\cdot
B\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny def}}}{{=}}}\;\,\\{\,a\cdot
b\;|\;a\in A,\,b\in B\,\\}$.
## 2\. Non-parallel Programs
This section recalls a classic static analysis by Abstract Interpretation of
the run-time errors of _non-parallel_ programs, as performed for instance by
Astrée [blanchet-al-PLDI03]. The formalization introduced here will be
extended later to parallel programs, and it will be apparent that an analyzer
for parallel programs can be constructed by extending an analyzer for non-
parallel programs with few changes.
### 2.1. Syntax
For the sake of exposition, we reason on a vastly simplified programming
language. However, the results extend naturally to a realistic language, such
as the subset of C excluding recursion and dynamic memory allocation
considered in our practical experiments (Sec. LABEL:sec:result). We assume a
fixed, finite set of variable names $\mathcal{V}$. A program is a single
structured statement, denoted $\operatorname{{\it body}}\in\mathit{stat}$. The
syntax of statements $\mathit{stat}$ and of expressions $\mathit{expr}$ is
depicted in Fig. 2. Constants are actually constant intervals $[c_{1},c_{2}]$,
which return a new arbitrary value between $c_{1}$ and $c_{2}$ every time the
expression is evaluated. This allows modeling non-deterministic expressions,
such as inputs from the environment, or stubs for expressions that need not be
handled precisely, e.g., $\sin(x)$ could be replaced with $[-1,1]$. Each unary
and binary operator $\diamond_{\ell}$ is tagged with a syntactic location
$\ell\in\mathcal{L}$ and we denote by $\mathcal{L}$ the finite set of all
syntactic locations. The output of an analyzer will be the set of locations
$\ell$ with errors — or rather, a superset of them, due to approximations.
$\begin{array}[]{l}\begin{array}[]{lcl@{\qquad}l}\mathit{stat}&::=&X\leftarrow\mathit{expr}&{\text{{\small(assignment
into $X\in\mathcal{V}$)}}}\\\ &|&{\mathbf{if}}\;\mathit{expr}\bowtie
0\;{\mathbf{then}}\;\mathit{stat}&{\text{{\small(conditional)}}}\\\
&|&{\mathbf{while}}\;\mathit{expr}\bowtie
0\;{\mathbf{do}}\;\mathit{stat}&{\text{{\small(loop)}}}\\\
&|&\mathit{stat};\,\mathit{stat}&{\text{{\small(sequence)}}}\\\ \\\
\mathit{expr}&::=&X&{\text{{\small(variable $X\in\mathcal{V}$)}}}\\\
&|&[c_{1},c_{2}]&{\text{{\small(constant interval,
$c_{1},c_{2}\in\mathbb{R}\cup\\{\pm\infty\\}$)}}}\\\
&|&-_{\ell}\,\mathit{expr}&{\text{{\small(unary operation,
$\ell\in\mathcal{L}$)}}}\\\
&|&\mathit{expr}\diamond_{\ell}\mathit{expr}&{\text{{\small(binary operation,
$\ell\in\mathcal{L}$)}}}\\\ \\\ \bowtie&::=&=|\neq|<|>|\leq|\geq\\\
\diamond&::=&+|-|\times|\;/\\\ \end{array}\end{array}$
Figure 2. Syntax of programs.
For the sake of simplicity, we do not handle procedures. These are handled by
inlining in our prototype. We also focus on a single data-type (real numbers
in $\mathbb{R}$) and numeric expressions, which are sufficient to provide
interesting properties to express, e.g., variable bounds, although in the
following we will only discuss proving the absence of division by zero.
Handling of realistic data-types (machine integers, floats arrays, structures,
pointers, etc.) and more complex properties (such as the absence of numeric
and pointer overflow) as done in our prototype is orthogonal, and existing
methods apply directly — for instance [bertrane-al-aiaa10].
### 2.2. Concrete Structured Semantics $\mathbin{{\mathbb{P}}{}}$
As usual in Abstract Interpretation, we start by providing a concrete
semantics, that is, the most precise mathematical expression of program
semantics we consider. It should be able to express the properties of interest
to us, i.e., which run-time errors can occur — only divisions by zero for the
simplified language of Fig. 2. For this, it is sufficient that our concrete
semantics tracks numerical invariants. As this problem is undecidable, it will
be abstracted in the next section to obtain a sound static analysis.
A program environment $\rho\in\mathcal{E}$ maps each variable to a value,
i.e., $\mathcal{E}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mathcal{V}\rightarrow\mathbb{R}$. The semantics
$\mathbin{{\mathbb{E}\llbracket}\,e\,{\rrbracket}}$ of an expression
$e\in\mathit{expr}$ takes as input a single environment $\rho$, and outputs a
set of values, in $\mathcal{P}(\mathbb{R})$, and a set of locations of run-
time errors, in $\mathcal{P}(\mathcal{L})$. It is defined by structural
induction in Fig. 3. Note that an expression can evaluate to one value,
several values (due to non-determinism in $[c_{1},c_{2}]$) or no value at all
(in the case of a division by zero).
$\begin{array}[]{l}\underline{\mathbin{{\mathbb{E}\llbracket}\,e\,{\rrbracket}}:\mathcal{E}\rightarrow(\mathcal{P}(\mathbb{R})\times\mathcal{P}(\mathcal{L}))}\\\\[4.0pt]
\mathbin{{\mathbb{E}\llbracket}\,X\,{\rrbracket}}\rho\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\\{\,\rho(X)\,\\},\,\emptyset)\\\\[3.0pt]
\mathbin{{\mathbb{E}\llbracket}\,[c_{1},c_{2}]\,{\rrbracket}}\rho\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\\{\,c\in\mathbb{R}\;|\;c_{1}\leq c\leq
c_{2}\,\\},\,\emptyset)\\\\[3.0pt]
\mathbin{{\mathbb{E}\llbracket}\,-_{\ell}\,e\,{\rrbracket}}\rho\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mbox{let
}(V,\Omega){\;=\;}\mathbin{{\mathbb{E}\llbracket}\,e\,{\rrbracket}}\rho\mbox{
in }(\\{\,-x\;|\;x\in V\,\\},\,\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{E}\llbracket}\,e_{1}\diamond_{\ell}e_{2}\,{\rrbracket}}\rho\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\\\ \qquad\mbox{let
}(V_{1},\Omega_{1}){\;=\;}\mathbin{{\mathbb{E}\llbracket}\,e_{1}\,{\rrbracket}}\rho\mbox{
in }\\\ \qquad\mbox{let
}(V_{2},\Omega_{2}){\;=\;}\mathbin{{\mathbb{E}\llbracket}\,e_{2}\,{\rrbracket}}\rho\mbox{
in }\\\ \qquad(\\{\,x_{1}\diamond x_{2}\;|\;x_{1}\in V_{1},\,x_{2}\in
V_{2},\,\diamond\neq/\vee x_{2}\neq 0\,\\},\\\
\qquad\;\Omega_{1}\cup\Omega_{2}\cup\\{\,\ell\;|\;\diamond=/\wedge 0\in
V_{2}\,\\})\\\ \text{where }\diamond\in\\{\,+,-,\times,/\,\\}\end{array}$
Figure 3. Concrete semantics of expressions.
To define the semantics of statements, we consider as semantic domain the
complete lattice:
$\mathcal{D}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mathcal{P}(\mathcal{E})\times\mathcal{P}(\mathcal{L})$ (1)
with partial order $\sqsubseteq$ defined as the pairwise set inclusion:
$(A,B)\sqsubseteq(A^{\prime},B^{\prime})\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{\iff}}}\;\,A\subseteq A^{\prime}\wedge B\subseteq B^{\prime}$. We
denote by $\sqcup$ the associated join, i.e., pairwise set union. The
structured semantics $\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}$ of a
statement $s$ is a morphism in $\mathcal{D}$ that, given a set of environments
$R$ and errors $\Omega$ before a statement $s$, returns the reachable
environments after $s$, as well as $\Omega$ enriched with the errors
encountered during the execution of $s$. It is defined by structural induction
in Fig. 4. We introduce the new statements $e\bowtie 0?$ (where
$\bowtie\,\in\\{\,=,\neq,<,>,\leq,\geq\,\\}$ is a comparison operator) which
we call “guards.” These statements do not appear stand-alone in programs, but
are useful to factor the semantic definition of conditionals and loops (they
are similar to the guards used in Dijkstra’s Guarded Commands [dijkstra-
EWD472]). Guards will also prove useful to define control paths in Sec.
LABEL:sec:pathsem. Guards filter their argument and keep only those
environments where the expression $e$ evaluates to a set containing a value
$v$ satisfying $v\bowtie 0$. The symbol $\not\bowtie$ denotes the negation of
$\bowtie$, i.e., the negation of $=$, $\neq$, $<$, $>$, $\leq$, $\geq$ is,
respectively, $\neq$, $=$, $\geq$, $\leq$, $>$, $<$. Finally, the semantics of
loops computes a loop invariant using the least fixpoint operator
$\operatorname{{\it lfp}}$. The fact that such fixpoints exist, and the
related fact that the semantic functions are complete $\sqcup-$morphisms,
i.e., $\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}(\sqcup_{i\in
I}X_{i})=\sqcup_{i\in
I}\;{\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}X_{i}}$, is stated in
the following theorem:
###### Theorem 1.
$\forall s\in\mathit{stat}:\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}$
is well defined and a complete $\sqcup-$morphism.
###### Proof 2.1.
In Appendix LABEL:proof:morphism.∎
$\begin{array}[]{l}\underline{\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}:\mathcal{D}\stackrel{{\scriptstyle\sqcup}}{{\longrightarrow}}\mathcal{D}}\\\\[4.0pt]
\mathbin{{\mathbb{S}\llbracket}\,X\leftarrow
e\,{\rrbracket}}(R,\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\emptyset,\Omega)\;\sqcup\;\bigsqcup_{\rho\in R}\;\mbox{let
}(V,\Omega^{\prime}){\;=\;}\mathbin{{\mathbb{E}\llbracket}\,e\,{\rrbracket}}\rho\mbox{
in }(\\{\,\rho[X\mapsto v]\;|\;v\in V\,\\},\,\Omega^{\prime})\\\\[3.0pt]
\mathbin{{\mathbb{S}\llbracket}\,e\bowtie
0?\,{\rrbracket}}(R,\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\emptyset,\Omega)\;\sqcup\;\bigsqcup_{\rho\in R}\;\mbox{let
}(V,\Omega^{\prime}){\;=\;}\mathbin{{\mathbb{E}\llbracket}\,e\,{\rrbracket}}\rho\mbox{
in }(\\{\,\rho\;|\;\exists v\in V:v\bowtie
0\,\\},\,\Omega^{\prime})\\\\[3.0pt]
\mathbin{{\mathbb{S}\llbracket}\,s_{1};\,s_{2}\,{\rrbracket}}(R,\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\mathbin{{\mathbb{S}\llbracket}\,s_{2}\,{\rrbracket}}\circ\mathbin{{\mathbb{S}\llbracket}\,s_{1}\,{\rrbracket}})(R,\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{S}\llbracket}\,{\mathbf{if}}\;e\bowtie
0\;{\mathbf{then}}\;s\,{\rrbracket}}(R,\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}\circ\mathbin{{\mathbb{S}\llbracket}\,e\bowtie
0?\,{\rrbracket}})(R,\Omega)\sqcup\mathbin{{\mathbb{S}\llbracket}\,e\not\bowtie
0?\,{\rrbracket}}(R,\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{S}\llbracket}\,{\mathbf{while}}\;e\bowtie
0\;{\mathbf{do}}\;s\,{\rrbracket}}(R,\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mathbin{{\mathbb{S}\llbracket}\,e\not\bowtie
0?\,{\rrbracket}}(\operatorname{{\it lfp}}\lambda
X:(R,\Omega)\sqcup(\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}\circ\mathbin{{\mathbb{S}\llbracket}\,e\bowtie
0?\,{\rrbracket}})X)\\\\[5.0pt] \text{where
}\bowtie\,\in\\{=,\neq,<,>,\leq,\geq\\}\\\ \end{array}$
Figure 4. Structured concrete semantics of statements.
We can now define the concrete structured semantics of the program as follows:
$\mathbin{{\mathbb{P}}{}}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\Omega,\text{ where
}(-,\Omega)=\mathbin{{\mathbb{S}\llbracket}\,\operatorname{{\it
body}}\,{\rrbracket}}(\mathcal{E}_{0},\emptyset)$ (2)
where $\mathcal{E}_{0}\subseteq\mathcal{E}$ is a set of initial environments.
We can choose, for instance, $\mathcal{E}_{0}=\mathcal{E}$ or
$\mathcal{E}_{0}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\\{\,\lambda X\in\mathcal{V}:{0}\,\\}$. Note that all run-time
errors are collected while traversing the program structure; they are never
discarded and all of them eventually reach the end of $\operatorname{{\it
body}}$, and so, appear in $\mathbin{{\mathbb{P}}{}}$, even if
$\mathbin{{\mathbb{S}\llbracket}\,\operatorname{{\it
body}}\,{\rrbracket}}(\mathcal{E}_{0},\emptyset)$ outputs an empty set of
environments. Our program semantics thus observes the set of run-time errors
that can appear in any execution starting at the beginning of
$\operatorname{{\it body}}$ in an initial environment. This includes errors
occurring in executions that loop forever (such as infinite reactive loops in
control / command software) or that halt before the end of $\operatorname{{\it
body}}$.
### 2.3. Abstract Structured Semantics $\mathbin{{\mathbb{P^{\sharp}}}{}}$
The semantics $\mathbin{{\mathbb{P}}{}}$ is not computable as it involves
least fixpoints in an infinite-height domain $\mathcal{D}$, and not all
elements in $\mathcal{D}$ are representable in a computer as $\mathcal{D}$ is
uncountable. Even if we restricted variable values to a more realistic, large
but finite, subset — such as machine integers or floats — naive computation in
$\mathcal{D}$ would be unpractical. An effective analysis will instead compute
an abstract semantics over-approximating the concrete one.
The abstract semantics is parametrized by the choice of an abstract domain of
environments obeying the signature presented in Fig. 5. It comprises a set
$\mathcal{E}^{\sharp}$ of computer-representable abstract environments, with a
partial order $\subseteq^{\sharp}_{\mathcal{E}}$ (denoting abstract
entailment) and an abstract environment
$\mathcal{E}^{\sharp}_{0}\in\mathcal{E}^{\sharp}$ representing initial
environments. Each abstract environment represents a set of concrete
environments through a monotonic concretization function
$\gamma_{\mathcal{E}}:\mathcal{E}^{\sharp}\rightarrow\mathcal{P}(\mathcal{E})$.
We also require an effective abstract version $\cup^{\sharp}_{\mathcal{E}}$ of
the set union $\cup$, as well as effective abstract versions
$\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}$ of the semantic
operators $\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}$ for assignment
and guard statements. Only environment sets are abstracted, while error sets
are represented explicitly, so that the actual abstract semantic domain for
$\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}$ is
$\mathcal{D}^{\sharp}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mathcal{E}^{\sharp}\times\mathcal{P}(\mathcal{L})$, with
concretization $\gamma$ defined in Fig. 5. Figure 5 also presents the
soundness conditions that state that an abstract operator outputs a superset
of the environments and error locations returned by its concrete version.
Finally, when $\mathcal{E}^{\sharp}$ has infinite strictly increasing chains,
we require a widening operator $\mathbin{\triangledown}_{\mathcal{E}}$, i.e.,
a sound abstraction of the join $\cup$ with a termination guarantee to ensure
the convergence of abstract fixpoint computations in finite time. There exist
many abstract domains $\mathcal{E}^{\sharp}$, for instance the interval domain
[cc-POPL77], where an abstract environment in $\mathcal{E}^{\sharp}$
associates an interval to each variable, the octagon domain [mine-HOSC06],
where an abstract environment in $\mathcal{E}^{\sharp}$ is a conjunction of
constraints of the form $\pm X\pm Y\leq c$ with $X,Y\in\mathcal{V}$,
$c\in\mathbb{R}$, or the polyhedra domain [ch:popl78], where an abstract
environment in $\mathcal{E}^{\sharp}$ is a convex, closed (possibly unbounded)
polyhedron.
$\begin{array}[]{ll}\mathcal{E}^{\sharp}&{\text{{\small(set of abstract
environments)}}}\\\\[3.0pt]
\gamma_{\mathcal{E}}:\mathcal{E}^{\sharp}\rightarrow\mathcal{P}(\mathcal{E})&{\text{{\small(concretization)}}}\\\\[3.0pt]
\bot^{\sharp}_{\mathcal{E}}\in\mathcal{E}^{\sharp}&{\text{{\small(empty
abstract environment)}}}\\\ \qquad\text{ s.t.
}\gamma_{\mathcal{E}}(\bot^{\sharp}_{\mathcal{E}})=\emptyset\\\\[3.0pt]
\mathcal{E}^{\sharp}_{0}\in\mathcal{E}^{\sharp}&{\text{{\small(initial
abstract environment)}}}\\\ \qquad\text{ s.t.
}\gamma_{\mathcal{E}}(\mathcal{E}^{\sharp}_{0})\supseteq\mathcal{E}_{0}\\\\[3.0pt]
\subseteq^{\sharp}_{\mathcal{E}}:(\mathcal{E}^{\sharp}\times\mathcal{E}^{\sharp})\rightarrow\\{\,\operatorname{{\it
true}},\operatorname{{\it false}}\,\\}&{\text{{\small(abstract
entailment)}}}\\\ \qquad\text{s.t.
}X^{\sharp}\subseteq^{\sharp}_{\mathcal{E}}Y^{\sharp}\Longrightarrow\gamma_{\mathcal{E}}(X^{\sharp})\subseteq\gamma_{\mathcal{E}}(Y^{\sharp})\\\\[3.0pt]
\cup^{\sharp}_{\mathcal{E}}:(\mathcal{E}^{\sharp}\times\mathcal{E}^{\sharp})\rightarrow\mathcal{E}^{\sharp}&{\text{{\small(abstract
join)}}}\\\ \qquad\text{s.t.
}\gamma_{\mathcal{E}}(X^{\sharp}\cup^{\sharp}_{\mathcal{E}}Y^{\sharp})\supseteq\gamma_{\mathcal{E}}(X^{\sharp})\cup\gamma_{\mathcal{E}}(Y^{\sharp})\\\\[3.0pt]
\mathbin{\triangledown}_{\mathcal{E}}:(\mathcal{E}^{\sharp}\times\mathcal{E}^{\sharp})\rightarrow\mathcal{E}^{\sharp}&{\text{{\small(widening)}}}\\\
\qquad\text{s.t.
}\gamma_{\mathcal{E}}(X^{\sharp}\mathbin{\triangledown}_{\mathcal{E}}Y^{\sharp})\supseteq\gamma_{\mathcal{E}}(X^{\sharp})\cup\gamma_{\mathcal{E}}(Y^{\sharp})\\\
\qquad\text{and }\forall(Y^{\sharp}_{i})_{i\in\mathbb{N}}:\text{ the sequence
}X^{\sharp}_{0}=Y^{\sharp}_{0},\,X^{\sharp}_{i+1}=X^{\sharp}_{i}\mathbin{\triangledown}_{\mathcal{E}}Y^{\sharp}_{i+1}\\\
\qquad\text{reaches a fixpoint }X^{\sharp}_{k}=X^{\sharp}_{k+1}\text{ for some
}k\in\mathbb{N}\\\\[5.0pt]
\mathcal{D}^{\sharp}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\mathcal{E}^{\sharp}\times\mathcal{P}(\mathcal{L})&{\text{{\small(abstraction
of $\mathcal{D}$)}}}\\\\[3.0pt]
\gamma:\mathcal{D}^{\sharp}\rightarrow\mathcal{D}&{\text{{\small(concretization
for $\mathcal{D}^{\sharp}$)}}}\\\\[-2.0pt] \qquad\text{s.t.
}\gamma(R^{\sharp},\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\gamma_{\mathcal{E}}(R^{\sharp}),\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}:\mathcal{D}^{\sharp}\rightarrow\mathcal{D}^{\sharp}\\\
\lx@intercol\qquad\text{s.t. }\forall s\in\\{\,X\leftarrow e,\,e\bowtie
0?\,\\}:(\mathbin{{\mathbb{S}\llbracket}\,s\,{\rrbracket}}\circ\gamma)(R^{\sharp},\Omega)\sqsubseteq(\gamma\circ\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}})(R^{\sharp},\Omega)\hfil\lx@intercol\\\
\end{array}$
Figure 5. Abstract domain signature, and soundness and termination conditions.
In the following, we will refer to assignments and guards collectively as
primitive statements. Their abstract semantics
$\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}$ in
$\mathcal{D}^{\sharp}$ depends on the choice of abstract domain; we assume it
is provided as part of the abstract domain definition and do not discuss it.
By contrast, the semantics of non-primitive statements can be derived in a
generic way, as presented in Fig. 6. Note the similarity between these
definitions and the concrete semantics of Fig. 4, except for the semantics of
loops that uses additionally a widening operator $\mathbin{\triangledown}$
derived from $\mathbin{\triangledown}_{\mathcal{E}}$. The termination
guarantee of the widening ensures that, given any (not necessarily monotonic)
function $F^{\sharp}:\mathcal{D}^{\sharp}\rightarrow\mathcal{D}^{\sharp}$, the
sequence $X^{\sharp}_{0}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\bot^{\sharp}_{\mathcal{E}},\emptyset)$,
$X^{\sharp}_{i+1}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,X^{\sharp}_{i}\mathbin{\triangledown}F^{\sharp}(X^{\sharp}_{i})$
reaches a fixpoint $X^{\sharp}_{k}=X^{\sharp}_{k+1}$ in finite time
$k\in\mathbb{N}$. We denote this limit by $\operatorname{{\it lim}}\lambda
X^{\sharp}:X^{\sharp}\mathbin{\triangledown}F^{\sharp}(X^{\sharp})$. Note
that, due to widening, the semantics of a loop is generally not a join
morphism, and even not monotonic [cc-PLILP92], even if the semantics of the
loop body is. Hence, there would be little benefit in imposing that the
semantics of primitive statements provided with $\mathcal{D}^{\sharp}$ is
monotonic, and we do not impose it in Fig. 5. Note also that
$\operatorname{{\it lim}}F^{\sharp}$ may not be the least fixpoint of
$F^{\sharp}$ (in fact, such a least fixpoint may not even exist).
$\begin{array}[]{l}\underline{\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}:\mathcal{D}^{\sharp}\rightarrow\mathcal{D}^{\sharp}}\\\\[4.0pt]
\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s_{1};\,s_{2}\,{\rrbracket}}(R^{\sharp},\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s_{2}\,{\rrbracket}}\circ\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s_{1}\,{\rrbracket}})(R^{\sharp},\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,{\mathbf{if}}\;e\bowtie
0\;{\mathbf{then}}\;s\,{\rrbracket}}(R^{\sharp},\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\\\
\qquad(\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}\circ\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,e\bowtie
0?\,{\rrbracket}})(R^{\sharp},\Omega)\;\cup^{\sharp}\;\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,e\not\bowtie
0?\,{\rrbracket}}(R^{\sharp},\Omega)\\\\[3.0pt]
\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,{\mathbf{while}}\;e\bowtie
0\;{\mathbf{do}}\;s\,{\rrbracket}}(R^{\sharp},\Omega)\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\\\ \qquad\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,e\not\bowtie
0?\,{\rrbracket}}(\operatorname{{\it lim}}\lambda
X^{\sharp}:X^{\sharp}\mathbin{\triangledown}((R^{\sharp},\Omega)\;\cup^{\sharp}\;(\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,s\,{\rrbracket}}\circ\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,e\bowtie
0?\,{\rrbracket}})X^{\sharp}))\\\\[3.0pt] \text{where:}\\\
\quad(R^{\sharp}_{1},\Omega_{1})\,\cup^{\sharp}\,(R^{\sharp}_{2},\Omega_{2})\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(R^{\sharp}_{1}\cup^{\sharp}_{\mathcal{E}}R^{\sharp}_{2},\;\Omega_{1}\cup\Omega_{2})\\\
\quad(R^{\sharp}_{1},\Omega_{1})\mathbin{\triangledown}(R^{\sharp}_{2},\Omega_{2})\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,(R^{\sharp}_{1}\mathbin{\triangledown}_{\mathcal{E}}R^{\sharp}_{2},\;\Omega_{1}\cup\Omega_{2})\end{array}$
Figure 6. Derived abstract functions for non-primitive statements.
The abstract semantics of a program can then be defined, similarly to (2), as:
$\mathbin{{\mathbb{P^{\sharp}}}{}}\;\,{\stackrel{{\scriptstyle\mbox{\rm\tiny
def}}}{{=}}}\;\,\Omega,\text{ where
}(-,\Omega)=\mathbin{{\mathbb{S^{\sharp}}\llbracket}\,\operatorname{{\it
body}}\,{\rrbracket}}(\mathcal{E}^{\sharp}_{0},\emptyset)\enspace.$
The following theorem states the soundness of the abstract semantics:
###### Theorem 2.
$\mathbin{{\mathbb{P}}{}}\subseteq\mathbin{{\mathbb{P^{\sharp}}}{}}.$
|
arxiv-papers
| 2012-03-16T14:55:20 |
2024-09-04T02:49:28.711916
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antoine Min\\'e",
"submitter": "Antoine Mine",
"url": "https://arxiv.org/abs/1203.3724"
}
|
1203.3878
|
Present address: ]Department of Physics, Florida State University,
Tallahassee, Florida 32306, USA
# Nature of yrast excitations near $N=40$: Level structure of 67Ni
S. Zhu R. V. F. Janssens M. P. Carpenter Physics Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA C. J. Chiara Physics Division,
Argonne National Laboratory, Argonne, Illinois 60439, USA Department of
Chemistry and Biochemistry, University of Maryland, College Park, Maryland
20742, USA R. Broda Institute of Nuclear Physics, Polish Academy of
Sciences, PL-31342 Krakow, Poland B. Fornal Institute of Nuclear Physics,
Polish Academy of Sciences, PL-31342 Krakow, Poland N. Hoteling Physics
Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
Department of Chemistry and Biochemistry, University of Maryland, College
Park, Maryland 20742, USA W. Królas Institute of Nuclear Physics, Polish
Academy of Sciences, PL-31342 Krakow, Poland T. Lauritsen Physics Division,
Argonne National Laboratory, Argonne, Illinois 60439, USA T. Pawłat
Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Krakow,
Poland D. Seweryniak Physics Division, Argonne National Laboratory, Argonne,
Illinois 60439, USA I. Stefanescu Physics Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA Department of Chemistry and
Biochemistry, University of Maryland, College Park, Maryland 20742, USA J. R.
Stone Department of Chemistry and Biochemistry, University of Maryland,
College Park, Maryland 20742, USA Department of Physics, University of
Oxford, OX1 3PU Oxford, UK W. B. Walters Department of Chemistry and
Biochemistry, University of Maryland, College Park, Maryland 20742, USA X.
Wang [ Physics Division, Argonne National Laboratory, Argonne, Illinois
60439, USA Physics Department, University of Notre Dame, Notre Dame, Indiana
46556, USA J. Wrzesiński Institute of Nuclear Physics, Polish Academy of
Sciences, PL-31342 Krakow, Poland
###### Abstract
Excited states in 67Ni were populated in deep-inelastic reactions of a 64Ni
beam at 430 MeV on a thick 238U target. A level scheme built on the previously
known 13-$\mu$s isomer has been delineated up to an excitation energy of 5.3
MeV and a tentative spin and parity of (21/2-). Shell model calculations have
been carried out using two effective interactions in the $f_{5/2}pg_{9/2}$
model space with a 56Ni core. Satisfactory agreement between experiment and
theory is achieved for the measured transition energies and branching ratios.
The calculations indicate that the yrast states are associated with rather
complex configurations, herewith demonstrating the relative weakness of the
$N=40$ subshell gap and the importance of multi particle-hole excitations
involving the $g_{9/2}$ neutron orbital.
###### pacs:
23.20.Lv, 21.60.Cs, 27.50.+e, 25.70.Lm
## I INTRODUCTION
The presence of shell gaps with magic numbers of nucleons is a cornerstone of
nuclear structure. Over the past decade it has increasingly become clear that
magic numbers are not immutable, but depend on the ratio of protons and
neutrons Sorlin and Porquet (2008); Janssens (2009). In discussions of magic
numbers, neutron number $N=40$ has historically been a subject of debate,
especially in the case of the Ni isotopes. Proton number $Z=28$ is magic, and,
for neutrons, a sizable energy gap at $N=40$ is thought to separate the $pf$
shell from the intruder $g_{9/2}$ state, potentially making $Z=28$, $N=40$
68Ni a doubly-magic nucleus. Experimentally, the occurrence of shell closures
in 68Ni was first suggested based on the observation of a 1770-keV
0${}_{2}^{+}$ level as the lowest excited state, followed by a 2${}_{1}^{+}$
state of relatively high excitation energy (2034 keV) Bernas et al. (1982);
Broda et al. (1995). The discovery of several isomeric states in 68Ni and in
neighboring nuclei Grzywacz et al. (1998); Ishii et al. (2000) supported the
case for its magic character further, as did the results from Coulomb
excitation measurements indicating a $B(E2,2_{1}^{+}\rightarrow 0^{+})$
reduced transition probability roughly three times smaller than the
corresponding value for ${}^{56}_{28}$Ni28 Sorlin et al. (2002); Bree et al.
(2008). However, based on recent high-precision mass measurements in the
neutron-rich Ni isotopes (up to 73Ni), the $N=40$ shell closure appears to be
more doubtful when inferred from changes in the two-neutron separation
energies Guénaut et al. (2007); Rahaman et al. (2007). It has been argued in
the literature that the apparent contradiction between the $B(E2)$ value and
the separation energy is a consequence of the parity change across the $N=40$
gap, with a sizable fraction of the low-lying $B(E2)$ strength residing in
excited states around 4 MeV, and the 2${}_{1}^{+}$ level being associated
predominantly with a neutron-pair excitation Sorlin et al. (2002); Langanke et
al. (2003). The size of the $N=40$ gap is then of the order of 2 MeV only and
the corresponding discontinuity in the sequence of orbitals corresponds at
most to a subshell closure.
Beta-decay studies Hannawald et al. (1999); Mueller et al. (1999); Sorlin et
al. (2003); Gaudefroy et al. (2005) and in-beam investigations at intermediate
beam energies Aldrich et al. (2008); Gade et al. (2010); Ljungvall et al.
(2010); Rother et al. (2011) provide evidence for the onset of collectivity
and strong polarization of the 68Ni core in neighboring nuclei of the region.
For example, the 2${}_{1}^{+}$ levels in $N=40$ 64Cr Gade et al. (2010) and
66Fe Hannawald et al. (1999) are located at excitation energies as low as 420
and 573 keV, respectively. More generally, the available low-spin level
structures in these nuclei suggest sizable admixture of spherical and deformed
components in the configurations near their ground states. These observations,
combined with the large $B(E2,2_{1}^{+}\rightarrow 0^{+})$ value measured for
70Ni42 Perru et al. (2006), lead to the conclusion that any “island” of nuclei
with indications of a significant $N=40$ gap is rather localized. A main
contributor to this situation is the monopole tensor force Otsuka et al.
(2001, 2005) between protons in the $pf$ shell and $g_{9/2}$ neutrons, where
the occupation of the latter orbital leads to the onset of deformation, as
evidenced, for example, by the presence of rotational bands in neutron-rich
55-57Cr and 59-61Fe nuclei Deacon et al. (2005); Zhu et al. (2006); Deacon et
al. (2011, 2007); Hoteling et al. (2008). On the other hand, couplings of
protons and/or neutrons to the 68Ni core do not always result in a large
polarization of the core. For example, the first excited state above the
(19/2-) isomer in 71Cu is located 2020 keV higher in energy, herewith
mirroring the location of the 2${}_{1}^{+}$ level in 68Ni Stefanescu et al.
(2009).
From the considerations above, it is clear that a satisfactory description of
nuclear structure in this mass region is still lacking. This is also reflected
in on-going theoretical efforts to determine the most appropriate interactions
for use in the calculations. In this context, the present data on the neutron-
hole nucleus 67Ni provide an opportunity to test the most modern interactions
while investigating the nature of yrast excitations up to moderate spin.
At present, only limited information is available on the low-lying structure
of 67Ni. Using deep-inelastic reactions of a 64Ni beam at 350 MeV on a thick
208Pb target, Pawłat et al. Pawłat et al. (1994) identified a 1008-keV
isomeric state with a half-life of T${}_{1/2}>0.3$ $\mu$s decaying through
coincident 314- and 694-keV transitions towards the 67Ni ground state. The
presence of the isomer was later confirmed in a fragmentation measurement
where a 13.3(2)-$\mu$s half-life was determined Grzywacz et al. (1998).
Feeding of the isomer in 67Co $\beta$ decay was subsequently reported Weissman
et al. (1999). These three studies Grzywacz et al. (1998); Pawłat et al.
(1994); Weissman et al. (1999) proposed spin and parity quantum numbers of
9/2+ for the long-lived state and associated this level with the occupation of
the $g_{9/2}$ orbital by a single neutron. With the NMR/ON technique, the
magnetic dipole moment of the 1/2- ground state was measured to be +0.601(5)
$\mu_{N}$, a value differing only slightly from the $\nu p_{1/2}$ single-
particle value; a fact regarded as evidence for the strength of the $N=40$
shell closure Rikovska et al. (2000). For the isomer, a quenched $g$-factor
value of $|g|$=0.125(6) was reported in Ref. Georgiev et al. (2002) and was
interpreted as evidence for a 2% admixture of a
$\pi(f^{-1}_{7/2}f_{5/2})_{1+}\nu g_{9/2}$ configuration involving a proton
excitation across the $Z=28$ gap into the supposedly pure $\nu g_{9/2}$ state.
Prior to the present work, no transitions feeding the isomeric state had been
reported. Here, despite the long half-life, eight new states have been placed
above the isomer from an investigation of prompt-delayed coincidence events in
a deep-inelastic reaction with a pulsed beam.
## II EXPERIMENT
A number of experiments have demonstrated that the yrast states of hard-to-
reach neutron-rich nuclei can be populated in deep-inelastic processes at beam
energies $15\%-25\%$ above the Coulomb barrier Janssens et al. (2002); Fornal
et al. (2005); Broda (2006), allowing experimental access to high-spin
structures in regions inaccessible with conventional heavy-ion induced,
fusion-evaporation reactions.
The experiment was carried out with a 64Ni beam delivered by the ATLAS
superconducting linear accelerator at Argonne National Laboratory. The 430-MeV
beam energy was chosen to correspond roughly to an energy of 20$\%$ above the
Coulomb barrier in the middle of a 55-mg/cm2 thick 238U target. The beam was
pulsed with a 412-ns repetition rate, each beam pulse being $\sim$$0.3$ ns
wide. Gammasphere Lee (1990), with 100 Compton-suppressed HPGe detectors, was
used to collect events with three or more $\gamma$ rays in coincidence. The
data were sorted into two-dimensional ($E_{\gamma}-E_{\gamma}$ matrices) and
three-dimensional ($E_{\gamma}-E_{\gamma}-E_{\gamma}$ cubes) histograms under
various timing conditions. The prompt $\gamma\gamma\gamma$ cube (PPP cube) was
incremented for $\gamma$ rays observed within $\pm$20 ns of the beam burst
while, in the delayed $\gamma\gamma\gamma$ cube (DDD cube), the transitions
were required to occur in an interval of $\sim$40 to $\sim$800 ns after the
prompt time peak (excluding the subsequent beam pulse), but within $\pm$20 ns
of each other. In this way, events associated with isomeric deexcitations
could be isolated and identified. The prompt-delayed-delayed (PDD) and prompt-
prompt-delayed (PPD) cubes were incremented by combining prompt and delayed
events. These proved critical in identifying prompt $\gamma$ rays feeding
isomeric levels as they revealed themselves in double-gated spectra on the
known transitions below the isomer in the PDD cubes. The relations between
these prompt $\gamma$ rays were subsequently established by examining proper
double coincidence gates in the PPD and PPP cubes. Examples and further
details of this technique can be found in Refs. Hoteling et al. (2008);
Stefanescu et al. (2009).
The spins and parities of the levels were deduced from an angular-correlation
analysis. In addition, considerations based on the fact that the reactions
feed yrast states preferentially, and/or on comparisons with shell-model
calculations, were also taken into account. The projectile-like products of
deep-inelastic reactions are usually characterized by no, or very little,
alignment. Therefore, the analysis of $\gamma$$\gamma$ angular correlations
for selected pairs of transitions is required Fornal et al. (2005); Hoteling
et al. (2006). In practice, in order to avoid as much as possible ambiguities
in the spin assignments, at least one known stretched transition was included
in the analysis.
## III RESULTS
In previous studies Pawłat et al. (1994); Grzywacz et al. (1998); Weissman et
al. (1999), the 314- and 694-keV transitions deexciting the 13.3(2)-$\mu$s
isomer in 67Ni were assigned to the
$9/2^{+}$$\rightarrow$$5/2^{-}$$\rightarrow$$1/2^{-}$ cascade and no
transition above this long-lived state was reported. As stated above,
transitions feeding the isomer were initially identified in the present work
by using the PDD coincidence data. A coincidence spectrum from this PDD cube
with double gates placed on the 314- and 694-keV transitions is presented in
Fig. 1. Besides a 1345-keV line belonging to the $2^{+}\rightarrow 0^{+}$
transition in 64Ni, three $\gamma$ rays are clearly visible at 1210, 1655, and
1667 keV. The 1345-keV line originates from Coulomb excitation of the 64Ni
beam, and is attributed to random coincidences. By double gating the PDD cube
with one of the newly discovered prompt $\gamma$ rays and one of the delayed
transitions, their mutual coincidence relationships can be verified further.
The results, displayed in Fig. 2, establish the feeding of the 67Ni isomer by
the 1210- and 1655-keV transitions. Finally, additional evidence was provided
by the analysis of the PPD cube, where a double gate on the prompt 1655- and
1667-keV $\gamma$ rays yields a delayed spectrum in which the 314- and 694-keV
lines appear, consistent with the expected coincidence relationships for
$\gamma$ rays across the isomer. This observation also implies that the 1655-
and 1667-keV transitions are in mutual, prompt coincidence.
Levels above the 67Ni isomer were investigated further in the PPP cube with
the newly-observed 1210-, 1655- and 1667-keV transitions as a starting point.
A double gate on the latter two $\gamma$ rays reveals the presence of three
additional lines at 63, 172, and 708 keV (see Fig. 3). The 1655- and 1210-keV
transitions are not in mutual coincidence, herewith establishing the presence
of parallel decay sequences. Exploiting additional coincidence relationships,
such as those displayed in Fig. 4, it was possible to propose the level scheme
of Fig. 5. Thus, states at 2218, 2663, 3530, 3913, 4330, and 4502 keV were
firmly established through the various competing decay paths. The ordering of
the highest levels at 4565 and 5273 keV is based on the measured $\gamma$-ray
intensities. The presence of a low-energy transition of 63 keV might suggest a
longer half-life for the 4565-keV state. Unfortunately, at this energy, the
timing signal of the large-volume germanium detectors is rather poor. This
fact, combined with the rather small intensity, made it impossible to obtain
firm information on the level lifetime. However, time spectra gated on the
transitions below the 4502-keV state do not provide evidence for a measurable
lifetime and an upper limit of $\sim$15 ns can be given for the 4565-keV
level.
Angular correlations were used to determine the multipolarity of some of the
newly identified transitions. Because the yield of the 314- and 694-keV
isomeric cascade was sufficient, the relevant coincidence intensities were
grouped into 12 different angles $\theta$. The measured angular-correlation
pattern for this pair strongly favors a sequence with two stretched quadrupole
transitions, as can be seen from the comparison with the theoretical
prediction of Fig. 6(a), which agrees with a
9/2+$\rightarrow$5/2-$\rightarrow$1/2- cascade. In view of the smaller
intensities, the correlation between the 1655- and the 1667-keV lines was
grouped into five angles [Fig. 6(b)]. It is consistent with a quadrupole-
dipole sequence. To be consistent with the decay pattern of the 2663-keV
level, the 1655-keV $\gamma$ ray is proposed as a quadrupole transition,
leading to a 13/2+ assignment for this state, and 15/2+ quantum numbers for
the level at 4330 keV.
Due to the lack of statistics, the correlation data for other transitions were
regrouped into the two angles of 33∘ (from 20∘ to 42∘ in Gammasphere) and 77∘
(69∘ to 87∘). Intensity ratios were obtained for the 1655-1250, 1655-172, and
1210-1695 keV pairs of transitions. The ratio of 0.81(9) measured for the
1655-1250 keV cascade points to a dipole character for the 1250-keV $\gamma$
ray, resulting in a 15/2+ assignment to the 3913-keV level. With this 15/2+
assignment and the measured 1.8(3) ratio indicating a quadrupole-dipole
cascade for the 1210-1695 keV pair where the dipole transition has a large
$E2/M1$ mixing ratio, a consistent picture emerges with the proposed 11/2+
spin and parity for the 2218-keV state. Note that the mixed-dipole character
for the 1210-keV transition is also consistent with the expectations of shell-
model calculations, as will be discussed below. A 17/2 spin assignment to the
4502-keV state was derived from the 0.85(4) correlation ratio measured for the
1655-172 keV pair. Even though correlation data could not be extracted for the
1210-1312 keV cascade, the 13/2+ assignment to the 3530-keV level is supported
by the presence of the weak, 2522-keV decay branch towards the 9/2+ isomer.
Finally, the general agreement between these assignments and the results of
shell-model calculations was used to tentatively propose 19/2- and 21/2-
assignments to the two highest states. The experimental information on levels
in 67Ni is summarized in Table 1.
Table 1: List of levels with the spin-parity assignments and $\gamma$ rays identified in 67Ni, including intensities and placements. Ei | J${}_{i}^{\pi}$ | J${}_{f}^{\pi}$ | Eγ | Iγ
---|---|---|---|---
(keV) | | | (keV) |
0 | 1/2- | | |
694.3(2)111Observed only with beam off | 5/2- | 1/2- | 694.3(2) |
1008.1(3)111Observed only with beam off | 9/2+ | 5/2- | 313.8(2) |
2218.0(4) | 11/2+ | 9/2+ | 1210.0(3) | 66(9)
2662.8(4) | 13/2+ | 11/2+ | 444.9(3) | 13(2)
| | 9/2+ | 1654.7(2) | 100(8)
3530.3(4) | 13/2+ | 11/2+ | 1312.3(3) | 16(3)
| | 9/2+ | 2522(1) | 2.0(5)
3913.0(4) | 15/2+ | 13/2+ | 382.7(2) | 16(3)
| | 13/2+ | 1250.0(3) | 29(5)
| | 11/2+ | 1695.1(5) | 7(1)
4330.1(4) | 15/2+ | 13/2+ | 1667.3(2) | 50(7)
4501.9(4) | 17/2(-) | 15/2+ | 171.8(2) | 48(6)
| | 15/2+ | 588.8(2) | 52(6)
4564.7(5) | (19/2-) | 17/2(-) | 62.8(2) | 31(6)
5273.1(7) | (21/2-) | (19/2-) | 708.4(5) | 20(3)
| (21/2-) | 17/2(-) | 771(1) | $<$3
## IV DISCUSSION
At first glance, the level structure on top of the 9/2+ isomer in 67Ni appears
to be of single-particle character. The yrast sequence does not exhibit any
regularity in the increase in excitation energy with angular momentum, as
would be expected in the presence of collectivity, and states of opposite
parity compete for yrast status. Moreover, in the absence of any notable
Doppler shift for any of the observed transitions, the combined feeding and
level lifetimes must be at least of the order of the stopping time of the
reaction products in the thick uranium target; i.e. 1 ps or longer. It should
also be noted that the sequence of levels above the 9/2+ isomer exhibits
similarities with the structure found above the corresponding 9/2+ long-lived
state in 65Ni. The latter structure was interpreted in terms of single-
particle excitations - see Ref. Pawłat et al. (1994) for details. These
observations would argue in favor of a subshell closure at $N=40$.
In order to gain further insight into the nature of the observed 67Ni states,
large-scale calculations were carried out with the shell-model code ANTOINE
Caurier and Nowacki (1999); Caurier (1989-2004) using both the jj44b Brown
and the JUN45 Honma et al. (2009) effective interactions. Both Hamiltonians
were restricted to the $f_{5/2}$, $p_{3/2}$, $p_{1/2}$, and $g_{9/2}$ valence
space and assume a 56Ni core. However, the required two-body matrix elements
and single-particle energies were obtained from fits to different sets of
data. Specifically, the JUN45 interaction was developed by considering data in
nuclei with $Z\sim 32$ and $N\sim 50$, and excludes explicitly the Ni and Cu
isotopes as the 56Ni core is viewed as being rather “soft” Honma et al.
(2009). In contrast, experimental data from $Z=28-30$ isotopes and $N=48-50$
isotones were incorporated in the fits in the case of the jj44b interaction
Brown .
The results of the calculations are compared with the experimental data in
Fig. 5. With both interactions, the energy of the 9/2+ state is predicted
lower than the measured value. This can be viewed as an indication that the
adopted single-particle energy of the $g_{9/2}$ neutron orbital is too low in
the two Hamiltonians. It is worth noting that the jj44b interaction calculates
this state to lie within 192 keV of the data and indicates about a 25%
admixture of the $\nu g_{9/2}^{3}$ configuration into the 9/2+ wave function.
With the JUN45 interaction, the level is predicted to lie 498 keV lower than
in the data with roughly 33% of the wave function involving three neutrons in
the $g_{9/2}$ orbital. This is possibly the result of the location of the
$g_{9/2}$ orbital at a lower energy in the JUN45 Hamiltonian, as compared to
that used in the jj44b case, which leads to larger configuration mixing in the
wave function of the $9/2^{+}$ state.
Overall, the calculated spectrum with both interactions appears somewhat
compressed when compared to the data, as illustrated on the right side of Fig.
5. Note that for reasons of clarity, only the calculated yrast and near-yrast
excitations are shown; i.e., the states with a likely corresponding level in
the data are plotted. The correspondence between data and calculations is
rather satisfactory when the computed excitation energies are expressed
relative to the 9/2+ isomer as is done on the left-hand side of Fig. 5.
Indeed, both interactions predict close-lying 11/2+ and 13/2+ levels,
separated from the next 13/2+ excitation by roughly 1 MeV, in agreement with
the proposed level scheme. A pair of close-lying 15/2+ levels is also computed
to be located directly above the 13/2${}_{2}^{+}$ state, as seen in the data.
Both interactions also predict a first excited 17/2+ state more than 300 keV
above the 15/2${}_{2}^{+}$ excitation with higher-spin, positive-parity states
another 1.3 MeV or more above this. In contrast, negative-parity levels are
present at lower excitation energies with both effective interactions, leading
to the proposed assignments of 17/2(-), (19/2-), and (21/2-) for the 4502-,
4565-, and 5273-keV states in Fig. 5. As indicated in the figure, these
assignments should be viewed as tentative, especially in the case of the 17/2
level, where calculated 17/2 states of both parities are separated only by
$\sim$200 and $\sim$400 keV, depending on the interaction.
It is of interest to identify in the calculations the main components of the
wave functions of the observed states. For the 11/2+ level, and the non-yrast
13/2${}_{2}^{+}$ and 15/2${}_{2}^{+}$ states, both Hamiltonians result in wave
functions in which the $\nu f_{5/2}^{5}p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{1}$
configurations dominate with a contribution of the order of 50%. Perhaps
surprisingly, the 13/2${}_{1}^{+}$ and 15/2${}_{1}^{+}$ states are computed to
be more fragmented, with respective main contributions by the $\nu
f_{5/2}^{5}p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{1}$ and $\nu
f_{5/2}^{4}p_{3/2}^{4}p_{1/2}^{2}g_{9/2}^{1}$ configurations of $\sim$30%
only. In addition, the JUN45 interaction results in a $\sim$10% admixture of
the $\nu g_{9/2}^{3}$ configuration into the wave functions of these two
levels. This contribution is of the order of 5% with the jj44b interaction.
With this Hamiltonian the wave functions of all the negative-parity states are
mixed with only the 17/2- level having a contribution from the $\nu
f_{5/2}^{4}p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{2}$ of the order of 50%. In
contrast, with the JUN45 interaction, where the ordering of states is computed
in better agreement with the data [see the
15/2${}_{2}^{+}$—17/2(-)—(19/2-)—(21/2-) sequence in Fig. 5], the wave
function of every negative-parity state is characterized by a 40-50% component
from the $\nu f_{5/2}^{4}p_{3/2}^{4}p_{1/2}^{1}g_{9/2}^{2}$ configuration.
Table 2: Relative branching ratios depopulating the 13/2${}_{1}^{+}$, 13/2${}_{2}^{+}$, and 15/2${}_{1}^{+}$ levels derived from experimental measurements, and calculated results using the JUN45 and jj44b effective interactions. J${}_{i}^{\pi}$ | J${}_{f}^{\pi}$ | Measurements | JUN45111Branching ratios are obtained with calculated transition energies. | jj44b111Branching ratios are obtained with calculated transition energies. | JUN45222Branching ratios are obtained with measured transition energies. | jj44b222Branching ratios are obtained with measured transition energies.
---|---|---|---|---|---|---
13/2${}^{+}_{1}$ | 11/2+ | 13(2) | 6 | 23 | 36 | 67
| 9/2+ | 100(8) | 100 | 100 | 100 | 100
13/2${}^{+}_{2}$ | 13/2${}^{+}_{1}$ | $<$3 | 0 | 166 | 0 | 96
| 11/2+ | 100(20) | 100 | 100 | 100 | 100
| 9/2+ | 11(4) | 9 | 105 | 7 | 92
15/2${}^{+}_{1}$ | 13/2${}^{+}_{2}$ | 57(9) | 33 | 11 | 95 | 40
| 13/2${}^{+}_{1}$ | 100(15) | 100 | 100 | 100 | 100
| 11/2+ | 23(2) | 15 | 5 | 28 | 7
In the absence of lifetime information on the 67Ni levels above the isomer,
additional tests of the shell-model calculations are possible by considering
the branching ratios for transitions competing in the deexcitation of specific
levels. For the computation of the $B(E2)$ transition probabilities, proton
and neutron effective charges $e_{p}=1.5e$ and $e_{n}=0.5e$ were adopted as is
usual for nuclei in this region. Comparisons between computed branchings for
the two Hamiltonians and the data are presented in Table 2. Only cases for
which the coincidence yields were sufficient to allow gating on the
transitions directly feeding a state of interest were considered for Table 2.
Note that this table provides shell-model results using either the calculated
or the measured transition energies. The latter values effectively remove the
dependence of the ratios on the transition energies. From the table, it is
clear that calculations with the JUN45 Hamiltonian are consistently in better
agreement with the measured branching ratios. It is also worth pointing out
that both Hamiltonians also compute a $11/2^{+}\rightarrow 9/2^{+}$ transition
of strongly mixed $E2/M1$ character ($|\delta|>0.5$), in agreement with the
angular-correlation data for the 1210-1695 keV cascade (see Section III).
From the discussion above, it is concluded that the levels above the 9/2+
isomeric state can be understood as neutron excitations, with contributions of
protons across the $Z=28$ gap playing a minor role at best. Calculations with
both interactions are in fair agreement with the data. They attribute a
significant role to the $g_{9/2}$ neutron orbital for every state observed in
this measurement. In fact, in most cases, significant $\nu g_{9/2}^{2}$ and
$\nu g_{9/2}^{3}$ configurations are part of the wave functions. Similar
observations have been made for other nuclei close to 68Ni; see, for example,
recent comparisons between calculations with the same jj44b and JUN45
interactions and data for 65,67Cu in Ref. Chiara et al. (2008). From these
findings, it is concluded that even in a nucleus only one neutron removed from
$N=40$, the impact of a neutron shell closure is rather modest. As the
$g_{9/2}$ neutron orbital is shape driving, multi particle-hole excitations
involving this state may be expected to be associated with enhanced
collectivity and it would be of interest to investigate the latter in future
measurements.
## V CONCLUSIONS
A level scheme above the known 13-$\mu$s isomer in 67Ni was established for
the first time by exploring prompt and delayed coincidence relationships from
deep-inelastic reaction products. Spin and parity quantum numbers for the
newly observed states were deduced from an angular-correlation analysis
whenever sufficient statistics was available. Shell-model calculations have
been carried out with two modern effective interactions, JUN45 and jj44b, for
the $f_{5/2}pg_{9/2}$ model space with 56Ni as a core. Satisfactory agreement
between experiment and theory was achieved. Even though the level structure of
67Ni appears to exhibit a single-particle character based on comparisons
between the measured level properties, including branching ratios, with the
results of shell-model calculations, it is suggested that the yrast and near-
yrast states are associated with rather complex configurations. In fact,
calculations indicate that the wave functions of the yrast states involve a
large number of configurations without a dominant ($\sim$50%) specific one;
the latter being more prevalent in the near-yrast levels. It is hoped that the
present data will stimulate additional theoretical work such as comparisons
with calculations using other effective interactions or a different model
space. Further experimental work aimed at the evolution of the degree of
collectivity with spin and excitation energy is highly desirable as well.
###### Acknowledgements.
The authors thank the ATLAS operating staff for the efficient running of the
accelerator and J.P. Greene for target preparation. This work was supported by
the U.S. Department of Energy, Office of Nuclear Physics, under Contract No.
DE-AC02-06CH11357 and Grant No. DE-FG02-94ER40834, by Polish Scientific
Committee Grant No. 2PO3B-074-18, and by Polish Ministry of Science Contract
No. NN202103333.
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* Honma et al. (2009) M. Honma, T. Otsuka, T. Mizusaki, and M. Hjorth-Jensen, Phys. Rev. C 80, 064323 (2009).
* Chiara et al. (2008) C. J. Chiara et al., Phys. Rev. C 85, 024309 (2012).
Figure 1: Partial spectrum with double coincidence gates set on the 314- and
694-keV cascade below the 67Ni 9/2+ isomer in the PDD cube. The spectrum shows
three strong transitions feeding the 9/2+ state; the 1345-keV $\gamma$ ray of
64Ni is due to random coincidences (see text for detail).
Figure 2: Partial coincidence spectra with different gates in the PDD cube
establishing the feeding of the 9/2+ isomeric state, see text for details.
Note the change in energy scales between panels (a, b) and (c, d).
Figure 3: Partial coincidence spectrum with double gates set on the 1655- and
1667-keV lines in the PPP cube showing the transitions from the states with
highest excitation energy observed in the present work.
Figure 4: Partial spectra with different gates in the PPP cube demonstrating
various coincidence relationships used to establish the level scheme of Fig.
5.
Figure 5: The proposed level scheme of 67Ni. Results of the shell-model
calculations with the JUN45 and jj44b effective interactions are shown for
comparison. The set of calculations to the left is identical to that on the
right, except that the excitation energies were offset such that the 9/2+
isomeric state matches the data.
Figure 6: Measured angular correlations for $\gamma\gamma$ cascades in 67Ni.
Dashed lines in the figure correspond to expected patterns associated with
pairs of stretched quadrupole-quadrupole transitions (panels a and b), while
the dot-dashed line is associated with a stretched quadrupole-dipole pair of
$\gamma$ rays (panel b).
|
arxiv-papers
| 2012-03-17T17:39:42 |
2024-09-04T02:49:28.722476
|
{
"license": "Public Domain",
"authors": "S. Zhu, R. V. F. Janssens, M. P. Carpenter, C. J. Chiara, R. Broda, B.\n Fornal, N. Hoteling, W. Krolas, T. Lauritsen, T. Pawlat, D. Seweryniak, I.\n Stefanescu, J. R. Stone, W. B. Walters, X. Wang, J. Wrzesinski",
"submitter": "Shaofei Zhu",
"url": "https://arxiv.org/abs/1203.3878"
}
|
1203.3920
|
Stochastic Characteristics and Simulation of the Random Waypoint Mobility
Model
Ahuja, Aditya 1, Venkateswarlu K. 1, and Venkata Krishna, P. 1
1School of Computing Science and Engineering, VIT University, Vellore - 632
014
aditya.ahuja@intel.com, venkateswarlu.vit@gmail.com, parimalavk@gmail.com
## Abstract
Simulation results for Mobile Ad-Hoc Networks (MANETs) are fundamentally
governed by the underlying Mobility Model. Thus it is imperative to find
whether events functionally dependent on the mobility model ‘converge’ to well
defined functions or constants. This shall ensure the long-run consistency
among simulation performed by disparate parties. This paper reviews a work on
the discrete Random Waypoint Mobility Model (RWMM), addressing its long run
stochastic stability. It is proved that each model in the targeted discrete
class of the RWMM satisfies Birkhoff’s pointwise ergodic theorem [13], and
hence time averaged functions on the mobility model surely converge. We also
simulate the most common and general version of the RWMM to give insight into
its working.
Keywords: Random Waypoint Mobility Model, Asymptotic Mean Stationary, Ergodic,
Simulation
## Introduction
Mobility models are used for the generation of node movement in simulations of
MANETs. Protocol development is a consequence of such a simulation. The
probabilistic aspects of the founding mobility model has direct implications
on the simulation results. Many papers [2]-[5] have already concluded that
stochastically unstable mobility models shall result in simulation results
that diverge in time.
The Random Trip Mobility Model, through the presence of a unique stationary
distribution for the location of nodes, has already been proved to be stable
[5]. _The work presented in this paper is purely a review of the stability of
the discrete version of the RWMM proved by Timo, Blackmore and Hanlen_ [1].
Therein the notion of stability is considered to be the satisfaction of
Birkhoff’s Pointwise Ergodic Theorem by the mobility model. If to the contrary
the mobility model is unstable, the simulation results are bound to be
unreliable.
The stimulus for this line of work is that the stability or lack thereof of
the mobility model is possibly passed up the layers of the protocol stack. For
instance the DSR protocol preserves the mobility model’s stability [6]: if the
node location random process is stable, then so is the route selection random
process. The consequence of this is that the strong law of large numbers also
holds for the simulations at the network layer.
A mobility model is quantified using a random process. It is stationary if the
set of probability laws regulating the movement of the nodes are independent
of time. Many works have come up with the transformation of non-stationary
models into (in some places pointwise ergodic theorem satisfying) stationary
models with the motivation that the strong law of large numbers may be
applicable.
The classic RWM model does display starting transients and local
nonstationarity. Thus we analyze its properties by means of imposing a
mathematically weaker ‘asymptotic stationarity’ property. A random process,
the mean of which is governed asymptotically by a process with a stationary
distribution is called Asymptotically Mean Stationary (AMS). It has been
proved [8][Theorem 1] that a random process is AMS if and only if it satisfies
Birkhoff’s pointwise ergodic theorem. By consequence a mobility model is
stable if and only if it is AMS [1].
In the classic RWMM [1], every node, using an independent and identically
uniformly distributed (IID) random process $\\{W_{k}\\}_{k=0}^{\infty}$,
selects a sequence of waypoints $\mathbf{w}=w_{0},w_{1},w_{2},...$. For every
pair $(w_{i},w_{i+1}),i\in\mathcal{Z^{*}}$, the node chooses a speed randomly
and uniformly distributed from the closed interval $[min\\_s,max\\_s]$. At
this chosen speed it then travels in a straight line from $w_{i}$ to
$w_{i+1}$.
In this review, the main result addressed is: a) The general discrete class of
the RWMM is asymptotically mean stationary (by virtue of which it is stable)
and ergodic. b) For stable node movement the following conditions suffice -
(i) Node waypoint selection is an AMS random process, (ii) Speed selection
random process is stationary.
This paper is organized as follows. The next section introduces the
preliminaries. Following that we describe the general RWMM. Next is the
contribution of the base paper in the form of a theorem. Simulation results
for the classic case and conclusion end this paper.
## Preliminaries
We will adopt the dynamical systems [9]-[10] model for a random process. Given
a discrete finite alphabet $\mathcal{X}$, let
$\mathbf{X}=\\{X_{k}\\}_{k=0}^{\infty}$ be the associated discrete time random
process. The distribution of $\\{X_{n}\\}_{n=0}^{\infty}$ is the set
$\\{\mu^{(k)}:k\geq 0\\}$ where $\mu^{(k)}$ is the probability measure on
$\mathcal{X}^{k}$ given by:
$\mu^{(k)}(x_{0}^{k-1})=Pr[X_{0}=x_{0},X_{1}=x_{1},...,X_{k-1}=x_{k-1}]$
In order to simplify our work, we use the Kolmogorov Representation Theorem
(where certain consistency conditions are satisfied) [9][Theorem I.1.2]. This
enables us to replace the distribution with a unique probability measure $\mu$
on the space $\mathcal{X}^{\infty}=\Pi_{i=0}^{\infty}\mathcal{X}$. Throughout
we shall be dealing with cylinder sets as elementary events:
$[x_{m}^{n}]=\\{\mathbf{\bar{x}}:\bar{x_{i}}=x_{i},m\leq i\leq n\\}$. The
$\sigma$-algebra $\mathcal{F_{X^{\infty}}}$ is generated using these cylinder
sets. Time is incorporated using the shift transform
$T_{\mathcal{X}}^{k}=x_{k},x_{k+1},x_{k+2},...,k\in\mathcal{Z^{*}}$.
Eventually we result with the dynamical system
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ which is
related to the original random process by
$\\{X_{k}\\}_{k=0}^{\infty}=\\{\Pi_{0}(T_{\mathcal{X}}^{k}\mathbf{x})\\}_{k=0}^{\infty}$,
$\Pi_{0}\mathbf{x}=x_{0}$.
Suppose we have a mobility model quantified by the random process
$\\{X_{k}\\}_{k=0}^{\infty}$ and capture the location of each node for the
first $k$ time instances of a simulation given by
$x_{0}^{k-1}=x_{0},x_{1},x_{2},...,x_{k-1}$. The dynamical system associated
with this stochastic experiment is
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ and the
trajectory captured is the elementary event
$[x_{0}^{k-1}]\in\mathcal{F_{X^{\infty}}}$. If variable length shift must be
considered
$T_{\mathcal{X^{*}}}\mathbf{x}=T_{\mathcal{X}}^{f\mathbf{(x)}}\mathbf{x}$ as
is necessitated in certain cases by the random processes associated with the
updation of routing tables of network routers, we may study the probabilistic
properties of
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X^{*}}})$.
Now we come up with certain definitions and lemmas lifted from the base work
which serve as the foundation for future proof developments.
_Definition 1 (Stationarity)_[1]: The system
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is
called stationary, and $T_{\mathcal{X}}$ is said to be measure preserving if,
$\forall A\in\mathcal{F_{X^{\infty}}}$, $\mu(A)=\mu(T^{-1}A)$ .
_Definition 2 (Ergodicity)_[1][13]: The stationary system
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is said
to be ergodic if $A=T^{-1}A\Rightarrow\mu(A)=0$ or $1$ . Equivalently,
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is
ergodic iff $\forall f\in L^{1}(\mu)$ the limit
$<f>=<f>(\mathbf{x})=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}f(T^{k}_{\mathcal{X}}\mathbf{x})$
is a constant almost everywhere in $\mu$.
_Definition 3 (Stability)_[1]: A mobility model associated with the random
process $(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$
is said to be stable if for all bounded and measurable $f$, the limit
$<f>(\mathbf{x})=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}f(T^{k}_{\mathcal{X}}\mathbf{x})$
exists almost everywhere in $\mu$.
_Definition 4 (Asymptotic Mean Stationarity)_[1]: The system
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is said
to be asymptotic mean stationary (AMS) if, $\forall
A\in\mathcal{F_{X^{\infty}}}$ the limit
$\overline{\mu}(A)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}\mu(T^{-k}_{\mathcal{X}}A)$
exists.
Here the probability measure $\overline{\mu}$ is defined on the measurable
space $(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}})$. It is called the
stationary mean of $\mu$ and describes the average of the long run behaviour
of the system.
_Lemma 1 (Birkhoff’s Pointwise Ergodic Theorem)_ [1][13]: Let the dynamical
system $(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$
have $T_{\mathcal{X}}$ as a measure preserving map, and let $f$ be measurable
with $E(|f|)<+\infty$. Then
$\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}f(T^{k}_{\mathcal{X}}\mathbf{x})=E(f|\mathcal{C})$.
Here $\mathcal{C}$ is the $\sigma$-algebra of invariant sets of
$T_{\mathcal{X}}$ . If the random process is ergodic, then $\mathcal{C}$ is
the trivial $\sigma$-algebra, and $E(f|\mathcal{C})=E(f)$ which is a constant.
It has been proved [8] that asymptotic mean stationarity is both a necessary
and sufficient condition for the pointwise ergodic theorem.
_Lemma 2 (AMS Pointwise Ergodic Theorem)_ [8][Theorem 1]: A dynamical system
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is AMS
iff for all measurable $f$ with a finite expectation, the limit
$<f>(\mathbf{x})=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}f(T^{k}_{\mathcal{X}}\mathbf{x})$
exists almost everywhere in $\mu$.
Eventually we conclude, using definitions 3,4 and lemma 2:
Stability: A mobility model with
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu)$ as the associated
probability space is stable with respect to $T_{\mathcal{X}}$ iff
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu,T_{\mathcal{X}})$ is AMS.
## Discrete Version of the RWMM
We now initiate the study of a discrete time space version of the Random
Waypoint Mobility Model. Consider a MANET with each mobile node in the set
$\mathcal{V}=\\{v_{1},v_{2},...,v_{\mathcal{|V|}}\\}$ located in a discrete
finite geographical area described by the set $\mathcal{S}$. The following are
the random processes to exposit the discrete RWMM.
### Waypoint Random Process Per Node
From the geographical space $\mathcal{S}$, each mobile node $v\in\mathcal{V}$
selects an infinite tuple of waypoints $\mathbf{w}=w_{0},w_{1},w_{2},...$
randomly. Let us denote the waypoint selection random process as
$\mathbf{W}=\\{W_{k}\\}_{k=0}^{\infty}$ with the corresponding dynamical
system as
$(\mathcal{W^{\infty}},\mathcal{F_{W^{\infty}}},\mu_{\mathbf{w}},T_{\mathcal{W}})$,
and also $\mathcal{W=S}$.
_RWMM Correlation :_ In the classic RWM model, waypoint selection random
process is IID, and in most cases uniformly distributed. So the stochastic
process
$(\mathcal{W^{\infty}},\mathcal{F_{W^{\infty}}},\mu_{\mathbf{w}},T_{\mathcal{W}})$
is a Bernoulli Scheme[14].
### Path Random Process Per Node
Figure 1: Different paths corresponding to discrete time-space equivalent of
different speeds
In the classic RWMM, whenever an arbitrary node selects a sequence of
waypoints $\mathbf{w}$, then for each consecutive pair
$(w_{i},w_{i+1}),i\in\mathcal{Z^{*}}$, it also selects a speed uniformly
distributed from $[min\\_s,max\\_s]$ and traverses the straight line path
between $w_{i}$ and $w_{i+1}$. In discretized time and space, snapshot of the
node’s position per instance of time is taken during its trip between
waypoints. This shall result in a random path with finite possibilities per
waypoint pair $(w_{i},w_{i+1})$ (figure 1). For each combination of waypoints
$(w,w^{\prime})\in\mathcal{W\times W}$ construct the set of all paths
$\mathcal{P}_{w,w^{\prime}}$ and take the union of all such sets so as to
obtain all admissible paths $\mathcal{P}$: $\hskip
15.0pt\mathcal{P}_{w,w^{\prime}}=\\{p_{1},p_{2},p_{3},...,p_{|\mathcal{P}_{w,w^{\prime}}|}\\}$,
$\hskip 15.0pt\mathcal{P}=\bigcup\limits_{(w,\acute{w})\in\mathcal{W\times
W}}\mathcal{P}_{w,\acute{w}}$. In order to describe the stochastic process
$\mathbf{P}=\\{P_{k}\\}_{k=0}^{\infty}$, noting that $\mathbf{P}$ is
conditioned on $\mathbf{W}$, we first define the set of permitted path
sequences $\mathcal{P}_{\mathbf{w}}^{\infty}\subset\mathcal{P}^{\infty}$ given
$\mathbf{w}$ as
$\mathcal{P}_{\mathbf{w}}^{\infty}=\\{\mathbf{p}\in\mathcal{P}^{\infty}:p_{i}\in\mathcal{P}_{w_{i},w_{i+1}},\forall
i\in\mathcal{Z^{*}}\\}$. Here again let $\mathcal{F_{P^{\infty}}}$ be the
$\sigma$-algebra generated from $p_{m}^{n}\in\mathcal{P^{\infty}}$. Defining a
collection of probability measures
$\nu_{\mathbf{wp}}=\\{\nu_{\mathbf{w}}:\mathbf{w}\in\mathcal{W^{\infty}},\nu_{\mathbf{w}}(\mathcal{P}_{\mathbf{w}}^{\infty})=1\\}$
results in the channel [11] $(\mathcal{W},\nu_{\mathbf{wp}},\mathcal{P})$.
_Definition 5 (Stationary Channel)_[1]: If
$\forall\mathbf{w}\in\mathcal{W^{\infty}},\forall
A\in\mathcal{F_{P^{\infty}}}$, $\hskip
6.0pt\nu_{T_{\mathcal{W}}\mathbf{w}}(A)=\nu_{\mathbf{w}}(T_{\mathcal{P}}^{-1}A)$
, the channel $(\mathcal{W},\nu_{\mathbf{wp}},\mathcal{P})$ is said to be
$(T_{\mathcal{W}},T_{\mathcal{P}})$ stationary.
_RWMM Correlation:_ Considering elementary events
$[p_{0}^{n-1}]\in\mathcal{F_{P^{\infty}}}$:
$\nu_{\mathbf{w}}([p_{0}^{n-1}])=\prod_{i=0}^{n-1}\frac{1}{|\mathcal{P}_{w_{i},w_{i+1}}|}\hskip
10.0ptp_{i}\in\mathcal{P}_{w_{i},w_{i+1}},0\leq i\leq n-1$
And $\nu_{\mathbf{w}}$ is zero otherwise. To prove stationarity, see that on
the transformed $T_{\mathcal{W}}\mathbf{w}$ the non-zero probability for
$[p_{0}^{n-1}]$ is given by:
$\nu_{T_{\mathcal{W}}\mathbf{w}}([p_{0}^{n-1}])=\prod_{i=0}^{n-1}\frac{1}{|\mathcal{P}_{w_{i+1},w_{i+2}}|}\hskip
10.0ptp_{i}\in\mathcal{P}_{w_{i+1},w_{i+2}},0\leq i\leq n-1$
And if $T_{\mathcal{P}}^{-1}[p_{0}^{n-1}]=[\bar{p}_{1}^{n}]$ with
$\bar{p}_{m+1}=p_{m},0\leq m<n$ :
$\nu_{\mathbf{w}}(T_{\mathcal{P}}^{-1}[p_{0}^{n-1}])=\nu_{\mathbf{w}}([\bar{p}_{1}^{n}])=\prod_{i=0}^{n-1}\frac{1}{|\mathcal{P}_{w_{i+1},w_{i+2}}|}\hskip
10.0ptp_{i}\in\mathcal{P}_{w_{i+1},w_{i+2}},0\leq i\leq n-1$
The last two equations are equal which proves that the channel is stationary.
Next it is proved that the channel is output mixing and consequently ergodic.
A channel is said to be output mixing[1] if, $\forall
A,B\in\mathcal{F_{P^{\infty}}},\forall\mathbf{w}\in\mathcal{W^{\infty}}$222Incorrect
sigma field in [1]:
$\lim_{n\to\infty}\left|\nu_{\mathbf{w}}\left(T_{\mathcal{P}}^{-n}A\bigcap
B\right)-\nu_{\mathbf{w}}(T_{\mathcal{P}}^{-n}A)\nu_{\mathbf{w}}(B)\right|=0$
The elementary events in case of the general RWMM are decoupled for $\tau\geq
b$ for $[p_{0}^{a-1}],[p_{0}^{b-1}]\in\mathcal{F_{P^{\infty}}}$ in the
following equation:
$\nu_{\mathbf{w}}\left(T_{\mathcal{P}}^{-\tau}[p_{0}^{a-1}]\bigcap[p_{0}^{b-1}]\right)=\nu_{\mathbf{w}}\left([\acute{p}_{\tau}^{\tau+a-1}]\bigcap[p_{0}^{b-1}]\right)=$
$\left(\prod_{i=\tau}^{\tau+a-2}\frac{1}{|\mathcal{P}_{w_{i+1},w_{i+2}}|}\right)\left(\prod_{j=0}^{b-2}\frac{1}{|\mathcal{P}_{w_{j+1},w_{j+2}}|}\right)\begin{array}[]{c}\acute{p}_{i}\in\mathcal{P}_{w_{i+1},w_{i+2}},\tau\leq
i\leq\tau+a-1\\\ p_{j}\in\mathcal{P}_{w_{j+1},w_{j+2}},0\leq j\leq
b-1\end{array}$
$=\nu_{\mathbf{w}}\left(T_{\mathcal{P}}^{-\tau}[p_{0}^{a-1}]\right)\nu_{\mathbf{w}}\left([p_{0}^{b-1}]\right)$
Hence the channel is output mixing and ergodic [11][Lemma 9.4.3].
Finally we define a probability measure $\mu_{\mathbf{p}}$ conditioning it on
the waypoint selection probability measure $\mu_{\mathbf{w}}$:
$\mu_{\mathbf{p}}(A)=\sum\limits_{\mathbf{w^{\prime}}\in\mathcal{W}^{\infty}}\mu_{\mathbf{w}}(\mathbf{w^{\prime}})\nu_{\mathbf{w^{\prime}}}(A),\hskip
5.0pt\forall A\in\mathcal{F_{P^{\infty}}}$ . Thus we result with
$(\mathcal{P^{\infty}},\mathcal{F_{P^{\infty}}},\mu_{\mathbf{p}},T_{\mathcal{P}})$
as the corresponding dynamical system for
$\mathbf{P}=\\{P_{k}\\}_{k=0}^{\infty}$.
### Location Random Process per node
We define the time taken $t(i)$ to reach $w_{i}$ from $w_{0}$ as a function of
the first $i$ paths $p_{0},p_{1},...,p_{i-1}$. We assume that each path length
$l(p)$ is a positive finite quantity. So $t(i)=\sum_{j=0}^{i-1}l(p_{j}),\hskip
5.0pti\geq 1$. Let the $i^{th}$ path $p_{i}$ take the form
$s_{t(i)},s_{t(i)+1},s_{t(i)+2},...,s_{t(i+1)}$, with $s_{j}\in\mathcal{S}$
and $s_{t(k)}=w_{k}$. Correlating with the given paths’ sequence $\mathbf{p}$,
we arrive at node location sequence $\mathbf{s}=s_{0},s_{1},...$. Thus we have
the node location random process $\mathbf{S}=\\{S_{k}\\}_{k=0}^{\infty}$ given
by the dynamical system
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S}})$.
### Location Random Process for all nodes
Consider the $\mathcal{|V|}$ tuple
$X_{n}=(S_{n,1},S_{n,2},...,S_{n,\mathcal{|V|}})$, with $S_{i,j}$ denoting
node $j$’s location at time $i$. This random variable’s alphabet is
$\mathcal{X}=\mathcal{S^{|V|}}$. Define
$(\mathcal{X^{\infty}},\mathcal{F_{X^{\infty}}},\mu_{\mathbf{p}},T_{\mathcal{X}})$
as the dynamical system for the random process $\\{X_{k}\\}_{k=0}^{\infty}$.
## Main Result and its Proof
Theorem [1]: Suppose the nodes
$\mathcal{V}=\\{v_{1},v_{2},...,v_{\mathcal{|V|}}\\}$ move in agreement with
the discrete RWMM already defined. Let $\mathbf{W}_{v}$ denote the waypoint
selection random process for node $v$ and $\mathbf{P}_{v}$ be the
corresponding path random process. Let
$(\mathcal{W}_{v},\nu_{\mathbf{wp}},\mathcal{P}_{v})$ denote the path and
waypoint stochastic processes’ connecting channel and let $\mathbf{X}$ denote
the location random process for all nodes. Then
* •
If $\forall v\in\mathcal{V}$, $\mathbf{W}_{v}$ is AMS and the channel
$(\mathcal{W}_{v},\nu_{\mathbf{wp}},\mathcal{P}_{v})$ is stationary, then
$\mathbf{X}$ is AMS and stable.
* •
If $\forall v\in\mathcal{V}$, $\mathbf{W}_{v}$ is ergodic and the channel
$(\mathcal{W}_{v},\nu_{\mathbf{wp}},\mathcal{P}_{v})$ is ergodic, then
$\mathbf{X}$ is ergodic.
Proof Sketch [1]:
Dropping the redundant subscript $v$ henceforth.
_Lemma A: If $\mathbf{W}$ is AMS and ergodic and
$(\mathcal{W},\nu_{\mathbf{wp}},\mathcal{P})$ is stationary and ergodic, then
$\mathbf{P}$ is AMS and ergodic._
_Proof:_ [11][Lemmas 9.3.1, 9.3.3] prove this lemma directly as the AMS and
ergodic waypoint random process and the path random process are connected by a
stationary, ergodic channel.
_Lemma B: If $\mathbf{P}$ is ergodic then $\mathbf{S}$ is ergodic._
_Proof:_ Given
$(\mathcal{P^{\infty}},\mathcal{F_{P^{\infty}}},\mu_{\mathbf{p}},T_{\mathcal{P}})$
is AMS. For all $p\in\mathcal{P}$ let $l(p)$ denote path length, let
$L=\max\\{l(p):\mathbf{p}\in\mathcal{P}\\}$ and let
$f:\mathcal{P}\to\bigcup_{i=1}^{L}\mathcal{S}^{i}$ be the breakdown of a path
to its corresponding geographic cells -
$f(p)=s_{0},s_{1},s_{2},...,s_{l(p)-1}$. So
$\mathbf{S}=\\{S_{k}\\}_{k=0}^{\infty}=f(P_{0}),f(P_{1}),f(P_{2}),...=S_{0},S_{1},...,S_{l(P_{0})},...,S_{l(P_{0})+l(P_{1})},...$.
For ease of working, define the encoder
$F:\mathcal{P}^{\infty}\to\mathcal{S}^{\infty}$ as
$\mathbf{s}=F(\mathbf{p})=f(p_{0}),f(p_{1}),f(p_{2}),...$. Now $\forall
A\in\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}}(A)=\mu_{p}(F^{-1}A)$. Here the
mapping $F$ is many to one.
Next it is described a pseudo-inverse
$G^{-1}:\hat{\mathcal{S}}^{\infty}\to\hat{\mathcal{P}}^{\infty}$ as
$G^{-1}\mathbf{s}=\mathbf{p_{s}}$ where
$\hat{\mathcal{P}}^{\infty}\subseteq\mathcal{P}^{\infty}$ according to
[8][Theorem 1] having full measure
$\mu_{\mathbf{p}}(\hat{\mathcal{P}}^{\infty})=1$ such that every bounded
measurable function on this set converges; $\hat{\mathcal{S}}^{\infty}$ is the
induced range of $F$ on $\hat{\mathcal{P}}^{\infty}$ and $\mathbf{p_{s}}$ is a
representative from the partition of $\hat{\mathcal{P}}^{\infty}$ induced by
$\mathbf{s}\in\hat{\mathcal{S}}^{\infty}$.
Define the length of the first $n$ paths in
$\mathbf{p}\in\mathcal{P}^{\infty}$ as
$\gamma_{\mathbf{p}}(n)=\sum_{i=0}^{n-1}l(p_{i})$111Wrong limit in [1]. Then
the variable length shift
$T_{\mathcal{S^{*}}}:\mathcal{S}^{\infty}\to\mathcal{S}^{\infty}$ is given by
$T_{\mathcal{S^{*}}}^{n}\mathbf{s}=T_{\mathcal{S}}^{\Gamma_{n}(\mathbf{s})}\mathbf{s}$
where
$\Gamma_{n}(\mathbf{s})=\begin{array}[]{c}\gamma_{G^{-1}(\mathbf{s})}(n),\hskip
10.0pt\mathbf{s}\in\hat{\mathcal{S^{*}}}\\\ 1,\hskip
50.0pt\mathbf{s}\notin\hat{\mathcal{S^{*}}}\end{array}$
Eventually it is proved that
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}h(T_{\mathcal{S^{*}}}\mathbf{s})$
exists $\forall\mathbf{s}\in\hat{\mathcal{S^{*}}}$ and for all bounded
measurable $h$. Thus
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S^{*}}})$
is AMS. Note that one $T_{\mathcal{S^{*}}}$ shift is equivalent to one path
shift.
_Sublemma: If
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S^{*}}})$
is AMS with stationary mean $\bar{\mu}_{\mathbf{s}}^{*}$ then
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S}})$
is AMS where
$T_{\mathcal{S^{*}}}\mathbf{s}=T_{\mathcal{S}}^{\gamma(\mathbf{s})}\mathbf{s}$
and $1\leq\gamma(\mathbf{s})\leq L\hskip 3.0pt$. 222Typographical error in [1]
for $T_{\mathcal{S^{*}}}$ _
Define a new measure (inspired from [8][Ex.6]):
$\bar{\mu}_{\mathbf{s}}(A)=\frac{1}{E_{\bar{\mu}_{\mathbf{s}}}[\gamma(\mathbf{s})]}\sum_{k=1}^{L}\sum_{i=0}^{k-1}\bar{\mu}_{\mathbf{s}}^{*}(T_{\mathcal{S}}^{-i}A\cap\Delta_{k}^{-1})$
Here
$\Delta_{k}^{-1}=\\{\mathbf{s}\in\mathcal{S^{\infty}}:\gamma(\mathbf{s})=k\\}$,
and $\\{\Delta_{k}^{-1}\\}_{k=1}^{L}$ is a partition of $\mathcal{S^{\infty}}$
[8]. Thus we have
$T_{\mathcal{S^{*}}}^{-1}A=\cup_{k=1}^{L}(T_{\mathcal{S}}^{-k}A\cap\Delta_{k}^{-1})$.
We also have
$\bar{\mu}_{\mathbf{s}}^{*}(T_{\mathcal{S^{*}}}^{-1}A)=\bar{\mu}_{\mathbf{s}}^{*}(A)=\sum_{k=1}^{L}\bar{\mu}_{\mathbf{s}}^{*}(A\cap\Delta_{k}^{-1})=\sum_{k=1}^{L}\bar{\mu}_{\mathbf{s}}^{*}(T_{\mathcal{S}}^{-k}A\cap\Delta_{k}^{-1})$
The first two terms are equal by virtue of transformation invariance of
$\bar{\mu}_{\mathbf{s}}^{*}$. The next two terms are equal by virtue of
intersection distribution of $A$ on $\Delta_{k}^{-1}$. The first and the last
term are equal from the immediately preceding correlation between
$T_{\mathcal{S^{*}}}$ and $T_{\mathcal{S}}$. Substituting
$T_{\mathcal{S}}^{-1}A$ for $A$ in the definition of $\bar{\mu}_{\mathbf{s}}$
and using the above equation we arrive at the $T_{\mathcal{S}}$ invariance of
$\bar{\mu}_{\mathbf{s}}$. Further it is shown that $\bar{\mu}_{\mathbf{s}}$
asymptotically dominates $\bar{\mu}_{\mathbf{s}}^{*}$ under $T_{\mathcal{S}}$
which when taken with the $T_{\mathcal{S}}$ invariance of
$\bar{\mu}_{\mathbf{s}}$ and [8][Theorem 2] proves that
$\bar{\mu}_{\mathbf{s}}^{*}$ is AMS w.r.t. $T_{\mathcal{S}}$ .
Next it is proved that if $\bar{\mu}_{\mathbf{s}}(A)=0$ and
$T_{\mathcal{S}}^{-1}A=A$ then $\mu_{\mathbf{s}}(A)=0$. This in conjunction
with [12][Theorem 2.2] proves that $\mu_{\mathbf{s}}$ is AMS w.r.t.
$T_{\mathcal{S}}$.
Thus from the sublemma
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S}})$
is AMS which completes the proof.
_Lemma C: If $\mathbf{P}$ is ergodic, then $\mathbf{S}$ is ergodic _
_Proof:_ Let $A\in\mathcal{F_{S^{\infty}}}$ be $T_{\mathcal{S}}$ invariant and
let $\mathbf{s}=F(\mathbf{p})$ be an arbitrary member of $A$. Now
$F(\mathbf{p})\in A\Leftrightarrow T_{\mathcal{S}}^{n}F(\mathbf{p})\in A\hskip
3.0pt$ $\Rightarrow F(\mathbf{p})\in A\Leftrightarrow
T_{\mathcal{S}}^{l(p_{0})}F(\mathbf{p})\in A\hskip 3.0pt$ $\Rightarrow
F(\mathbf{p})\in A\Leftrightarrow F(T_{\mathcal{P}}\mathbf{p})\in A$
Taking $F^{-1}$ on both sides (as the equation holds for all $\mathbf{s}$ and
all $\mathbf{p}$ associated with each $\mathbf{s}$ )
$\mathbf{p}\in F^{-1}A\Leftrightarrow T_{\mathcal{P}}\mathbf{p}\in F^{-1}A$ .
Hence $F^{-1}A$ is $T_{\mathcal{P}}$ invariant. By the premise of the lemma,
$\mu_{\mathbf{p}}(F^{-1}A)=0$ or $1$. Hence $\mu_{\mathbf{s}}(A)=0$ or $1$.
Thus by definition,
$(\mathcal{S^{\infty}},\mathcal{F_{S^{\infty}}},\mu_{\mathbf{s}},T_{\mathcal{S}})$
is ergodic.
## Simulation
We have simulated a basic packet exchange in a MANET using NS2 with the node
movement generated according to the general continuous RWMM. The traffic
consisted of constant bitrate UDP packets with IEEE 802.11 protocol at the MAC
layer. The exchanges resulted in bursty traffic. The plot for the number of
bytes received as a function of time for a particular node is given in figure
2.
Figure 2: Number of bytes received as a function of time for a particular node
## Conclusion
In this paper we have successfully demonstrated that the discrete general RWMM
is AMS and hence stable. Thus simulations with RWMM as the underlying node
movement generation algorithm tend to be reliable. The stability preserving
protocols allow higher layers of the protocol stack to propagate this
stability hence permitting reliability of simulations at higher levels also.
Moreover we have simulated the continuous version of the RWMM with the intent
of seeing the local non-stationary properties (which is highlighted by the
bursty traffic).
## References
* [1] R. Timo, K. Blackmore, and L. Hanlen, “Strong Stochastic Stability for MANET Mobility Models,” 15th IEEE International Conference on Networks , DOI: 10.1109/ICON.2007.4444054, pp. 13-18, November 2007
* [2] S. Kurkowski, T. Camp, and M. Colagrosso, “MANET Simulation Studies: The Incredibles,” ACM SIGMOBILE Mobile Comp. and Commun. Review $MC^{2}R$ , vol. 9, no. 4, pp. 50-60, October 2005.
* [3] T. Andel and A. Yasinsac, “On the Credibility of MANET Simulations,” _Computer_ , vol. 39, no. 7, pp. 48-54, July 2006.
* [4] J. Yoon, M. Liu, and B. Noble, “Sound Mobility Models,” in _Proc. IEEE Intl. Symp. Mobile Ad Hoc Net. and Comp., MobiHoc_ , September 2003, pp. 205-216.
* [5] J. Boudec and M. Vojnovic,“The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation,” _IEEE/ACM Trans. Networking_ , vol. 14, no. 6, pp. 1153-1166, December 2006.
* [6] R. Timo, K. Blackmore, and J. Papandriopoulos, “Strong Stochastic Stability for Dynamic Source Routing,” Tech. Rep. PA006280, NICTA, August 2007.
* [7] P. Billingsley, _Probability and Measure_ , Wiley series in probability and mathematical statistics. John Wiley and Sons, 3rd edition, 1995.
* [8] R. Gray and J. Kieffer, “Asymptotically Mean Stationary Measures,” _J. Ann. Prob._ , vol. 8, no. 5, pp. 962-973, October 1980.
* [9] P. Shields, _The Ergodic Theory of Discrete Sample Paths_ , vol. 13 of _Graduate Studies in Mathematics_ , American Mathematical Society, 1996.
* [10] R. Gray, _Probability Random Processes, and Ergodic Properties_ , Springer Verlag, 2001 (Revision 1987), http://ee.stanford.edu/$\sim$gray/
* [11] R. Gray, _Entropy and Information Theory_ , Springer Verlag, 2000 (Revision 1990), http://ee.stanford.edu/$\sim$gray/
* [12] Y. Kakihara, “Ergodicity and Extremality of AMS Sources and Channels,” _International Journal of Mathematics and Mathematical Sciences_ , vol. 2003, no. 28, pp. 1755-1770, 2003.
* [13] Wikipedia The Free Encyclopedia - Ergodic Theory, Retrieved April 5, 2011 http://en.wikipedia.org/wiki/Ergodic_theory
* [14] Wikipedia The Free Encyclopedia - Bernoulli Scheme, Retrieved April 5, 2011 http://en.wikipedia.org/wiki/Bernoulli_scheme
|
arxiv-papers
| 2012-03-18T06:41:21 |
2024-09-04T02:49:28.730000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Ahuja, K. Venkateswarlu and P. Venkata Krishna",
"submitter": "Venkata Krishna P",
"url": "https://arxiv.org/abs/1203.3920"
}
|
1203.3973
|
# Local Optical Probe of Motion and Stress in a multilayer graphene NEMS
Antoine Reserbat-Plantey Laëtitia Marty Olivier Arcizet Nedjma Bendiab
Vincent Bouchiat Institut Néel, CNRS et Université Joseph Fourier, BP 166,
F-38042 Grenoble Cedex 9, France
Nanoelectromechanical systems (NEMSs)reviewNEMS are emerging nanoscale
elements at the crossroads between mechanics, optics and electronics, with
significant potential for actuation and sensing applications. The reduction of
dimensions compared to their micronic counterparts brings new effects
including sensitivity to very low massbachtoldbalance ; zettlbalance ,
resonant frequencies in the radiofrequency range zettlNTRF , mechanical non-
linearitiesBachtoldNonlinear and observation of quantum mechanical
effectsquantumNEMS . An important issue of NEMS is the understanding of
fundamental physical properties conditioning dissipation mechanisms, known to
limit mechanical quality factors and to induce aging due to material
degradation. There is a need for detection methods tailored for these systems
which allow probing motion and stress at the nanometer scale. Here, we show a
non-invasive local optical probe for the quantitative measurement of motion
and stress within a multilayer graphene NEMS provided by a combination of
Fizeau interferences, Raman spectroscopy and electrostatically actuated
mirror. Interferometry provides a calibrated measurement of the motion,
resulting from an actuation ranging from a quasi-static load up to the
mechanical resonance while Raman spectroscopy allows a purely spectral
detection of mechanical resonance at the nanoscale. Such spectroscopic
detection reveals the coupling between a strained nano-resonator and the
energy of an inelastically scattered photon, and thus offers a new approach
for optomechanics.
Graphene’sNovoselov2004p2970 outstanding mechanical HoneAFM2008 , electrical
castroneto2009 and optical Nair2008 properties, make it an ideal material for
flexible, conductive and semi-transparent films. Multilayer graphene (MLG),
which has a thickness of several tens of atomic layers, is sufficiently stiff
Booth2008 to produce free-standing cantilevers, with an unprecedented aspect
ratio. Such structures can be used to make suspended mirrors, with a mass
ranging from tens to hundreds of femtograms. When suspended over silica, such
cantilevers form optical cavities which can be electrostatically actuated and,
are thus ideal for the implementation of NEMSBunch . Previous attempts to
probe local motion of graphene resonatorsBachtoldAFM have reached nanometer
scale but cannot measure directly the stress and remained confined to a
limited range of pressure and temperature. Nevertheless, recent studiesBolotin
based on hybrid graphene-metallic cantilevers has brought promising results on
static stress graphene using optical profilometry. In the present work, we use
Raman spectroscopy to probe the local stress within a MLG cantilever. We
explore mechanical regimes from DC up to MHz frequencies by taking advantage
of the large dynamical range of optical detection. The MLG displacement is
considerably greater than previously reported, and optical interferences allow
self-calibration of displacements, while Raman spectroscopy gives quantitative
analysis of the local stress within the structure.
Figure 1: Fizeau fringes in a MLG cantilever overhanging silicon oxide. a:
typical SEM micrograph of a sample, consisting in two reflectors: an oxidized
silicon back-mirror and a MLG cantilever, having a semi-transparent behavior.
b: Schematic view of the device : electrical excitation and optical detection,
either with a photodiode (intensity) or a Raman spectrometer (intensity and
spectral data). c: White light optical image of a device showing iridescence.
d: Reflectance profile is measured along the dashed-line of the inset. The
reduction in signal strength observed at large distances from the hinge is due
to reduced spatial mode matching. The fringe contrast is however preserved.
Inset: Reflectance confocal (X,Y) scan at 633 nm. The scale length is 5
$\mu$m.
Samples were prepared from micron-sized MLG planar flakes clamped on one side
by a gold film and overhanging silicon oxide (see Methods). Typical samples
had a thickness of approximately 100 monolayers (ca. 30 nm), as verified by
atomic force microscopy (cf Supp. Info.). Their thicknesses were adjusted so
as to prevent collapse whilst maintaining semi-transparency with an optical
reflectance (transmission) coefficient $R$ = 0.22 ($T$ = 0.61) for a 30 nm
thick MLGSkulason . Some flakes tend to stick up after the fabrication process
(see Fig. 1a), at a wedge angle $\alpha\rm{\in[5^{\circ};35^{\circ}]}$, and
these leave a wedge gap of length $h(x,y)$ in the range between 0.3 and 3
microns. The resulting structures form an optical cavity, characterized by a
low optical interference order ($2h/\lambda$¡10, where $\lambda$ is the probe
wavelength), with MLG top mirror of extremely low mass (10-100 fg), and of
high mechanical resonance frequencies (1-100 MHz). With approximately 100
measured samples, we have observed a variety of geometries (Fig. 1), allowing
us to explore various mechanical regimes, with a wide range of wedge angles
$\alpha$, sizes, and shapes. Iridescence is observed under white light
illumination (Fig. 1c), and the interference pattern observed under
monochromatic light (Fig. 1d) has contrasted equal-thickness fringes (so-
called Fizeau fringes, see Supp. Info.). Unlike conventional graphene-based
optical cavities with fixed geometriesLing2010 , the optical length of the
cavity increases linearly along the cantilever, which allows the observation
of multiple interference fringes (cf. Fig. 1cd). Interference patterns are
observed both from the reflection of the pump laser and from Raman scattered
light (see Supp. Info.), the latter having the considerable advantage of
carrying local informations related to the material (stress, doping, defects,
temperature).
Furthermore, the optical length of the cavity can be adjusted through
electrostatic actuation of the cantilever, thus producing rigid shift of the
interference fringes pattern (see video in Supp. Info.). This is achieved by
applying a DC or AC voltage $V$ (typically up to $30$ Volts) to the clamp
electrode (Fig. 1b) while the SiO2 capped silicon substrate is grounded Bunch
. This results in an attractive electrostatic force $F$, which produces
reduction of the cavity length with respect to the equilibrium position
$h_{0}$ in absence of driving. We measure the response of a harmonic drive,
which create of force quadratic in voltage $F(2\omega)\propto V(\omega)^{2}$
through the local light intensity variation, $\Delta I(x,y,2\omega,h_{0})$:
$\Delta
I(x,y,2\omega,h_{0})\propto\chi_{mec}(x,y,2\omega)\chi_{opt}(x,y,h_{0})\
V(\omega)^{2},$ (1)
where $\chi_{mec}$ is the mechanical susceptibility (see Supp. Info.) and
$\chi_{opt}$ is the optical susceptibility defined as
$\chi_{opt}(x,y,h_{0})=\partial g_{opt}/\partial h$, where $g_{opt}$ is a
periodic interferometric function of $h(x,y)$ defined as the normalized
reflected light ($I_{r}$) or Raman scattered light ($I_{G}$)
$I_{r,G}/I_{0}=g_{opt}(h)$ (see Fig. 1d).
Figure 2: Quasi static actuation and stress mapping of wedged MLG NEMS. a:
Variations in G peak intensity (black) and position (red) under MLG actuation,
revealing peak softening. The lower dashed line represents the drive voltage.
b: Map of G peak produced by confocal (X,Z) scan mapping of the cantilever
cross section. Inset: G peak position along the cantilever (purple, and along
the same sample following collapse of the cantilever onto the silica substrate
(green). Black marks indicate the hinge position. The scale bars represent a
length of 5 $\rm{\mu}$m.
The quadratic dependence of $\Delta I$ upon voltage is systematically
observed, both for reflected light (cf. Fig. S5) and for the MLG Raman lines
(cf. Fig. 2a). Since $g_{opt}$ is $\lambda/2$ periodic, a precise calibration
of the low frequency motion response under electrostatic actuation can be
obtained and is found to be of the order of 20 nm.V-2. (cf. Supp. Info.).
Interestingly, energy of the stress-sensitive optical phonon (so called Raman
G peak) also follows quadratic behavior. The G peak Raman shift is indeed
synchronized with the interferential response $I_{G}(t)$ (Fig. 2a), and
exhibits softening of approximately 1.9 cm-1 at the maximum cantilever
deflection. This Raman peak softening cannot be interpreted as a doping effect
since the doping level necessary to induce the observed Raman shifts would
correspond to surface charge incompatible with the one induced by the gate
driveKim . Moreover, the doping induced during AC gating would directly follow
gate variation and therefore be $\omega$ periodic which is in disagreement
with the observed $2\omega$ periodic Raman shift (cf. Fig. 2a). This Raman
peak softening is interpreted as a stress/strain effect and, by analogy with
strained graphene measurementsOtakar2010 ; Otakar2011 ; Hone2009 , it is thus
possible to extract a corresponding strain value of $0.06\%$ at maximum
deviation resulting from a quasi static stress of 600 MPa. For such low
strain, the G band splitting is not resolved. Besides, the stress exerted at
the hinge scales like $LF/[2t^{2}\rm{sin}(2\alpha)]$ where $F$, $L$ and $t$
are the electrostatic force, the cantilever length and thickness, respectively
(see Supp. Info.). For our large aspect ratio structures ($L/t\gg 1$), the
local stress can be very intense and reaches hundreds of MPa for electrostatic
forces estimated here, which are about 25 nN.$\mu$m-1. This value is in
agreement with the quasi-static stress of 600 MPa deduced above.
Nevertheless, the MLG Raman signature depends on the position along the flake.
A micro-Raman confocal (X,Z) scan (Fig. 2b) reveals a linear increase in the
position of the G peak along the cantilever axis, from the free-end of the
cantilever to the hinge, which is not observed when the MLG is collapsed
(inset of Fig. 2b). This linear shift could be interpreted as a continuously
increasing electrostatic field effectKim ; Ferrari , due to charge within the
substrate which also influences the position of the Raman G peakBerciaud2009 .
However, in this experiment, the G peak shows local hardening around the hinge
position, in the suspended case, whereas local softening is observed at the
same location, after collapse. Indeed, uniaxial strain in MLG also induces
symmetry breaking of the Raman G peak, leading to a mode splitting and, each
component (G+, G-) softens or hardens under tensile or compressive strain,
respectivelyOtakar2010 ; HoneAFM2008 . That stress induced Raman shift is
characterized by an average mode shift rate aboutOtakar2011 -3.2 cm-1.GPa-1.
This outcome is in agreement with a maximum compressive strain at the hinge in
the suspended case and, a transition toward a tensile strain when it
collapses. By converting these Raman shifts into stress at the hinge, this
gives an equivalent built-in stress of 300 MPa.
Figure 3: Detection of mechanical resonance by Fizeau interferometry. a:
Amplitude at $2\omega$ versus drive frequency for different rf drive voltages
showing the non-linear behavior of the fundamental and the two first harmonic
mechanical modes. Laser probe is focused close to the hinge. Schematics of the
deformed shape are associated to each mode and dark dots represent the laser
probe position. b: Amplitude at $2\omega$ versus drive frequency, for an
increasing range of drive voltages (bottom to top), revealing signal folding
due to optical interferences. The laser spot is located close to the free end
of the cantilever. c: Evolution of MLG cantilever resonance frequency
$\omega_{0}$ (red) and its associated dissipation $Q^{-1}$(blue) as a function
of the temperature of the optical cryostat. Measurements a-c are performed
under vacuum and the laser spot is positioned at the edge of a fringe
($\chi_{opt}$ is optimal).
Interestingly, changing both the laser spot position and the drive amplitude
allows probing in a separated fashion the non-linearities arising from
mechanical (Fig. 3a) and from optical (Fig. 3b) origins. It is worth noting
that optical non-linearities are observed when the probe is far from the hinge
(see Fig. 3b) where, due to the lever-arm effect, oscillation amplitude of
$h(x,y)$ becomes comparable to the probe wavelength. Like in quasi-static
regime, it is possible to calibrate the displacement amplitude with respect to
the driving excitation $\delta V_{AC}$ by using the periodic nature of
$\chi_{opt}$. Peak folding in indeed observed for increasing drive beyond 3V
(cf. Fig. 3b). Assuming mechanical linear response, the drive increase to
produce two successive foldings (corresponding to $\lambda/4$ in amplitude)
provides calibration of the drive efficiency, which equals 150 nm.V-2 in the
case presented in Fig. 3b. Interestingly, the entire signature of the optical
non-linearities is visible for a restricted range of drive voltage which
ensures to neglect mechanical non-linearities.
Close to the hinge, optical non-linearities are extinguished due to smaller
variations of $h$ and thus reveal non-linearities of mechanical origin, which
are observed on each mode for drive voltages higher than 4V (cf. Fig. 3a).
This measurement highlights the wide range of mechanical non-linearities
observed in MLG structuresLifshitz ; BachtoldNonlinear ; Landau , and it is
worth noting that the detection efficiency strongly depends on the mode
profile since it is based on Fizeau fringes pattern modulation.
$\chi_{mac}(x,y,2\omega)$ can exhibit important variations due to the spatial
nature of the probed vibration. In particular, $\chi_{mec}(x,y,2\omega)$ can
be strongly reduced when the laser probe is focused at a node of the
mechanical resonance. As an example, the first harmonic ($\omega_{1}$), found
to be a torsional mode via finite elements analysis, implies cantilever
deformation with singular position where the cavity length does not vary
(typically, a node region ($x_{n},y_{n}$)). Thus, according to Eq. 1,
$\chi_{mec}(x_{n},y_{n},2\omega_{1})\ll\chi_{mec}(x_{n},y_{n},2\omega_{0})$,
whereas focusing the laser at a different position allows to enhance the local
optical response. This particular extinction feature of the detection has a
great potential for further mapping of MLG deformation associated with a
single mechanical mode.
In order to investigate the influence of the laser probe in our all-optical
method, cryogenic measurements has been carried out as shown in Fig. 3c. The
fundamental resonant frequency exhibits a linear upshift upon cooling from
300K to 70K, below which it saturates due to extrinsic heating (see Supp.
Info.). In contrast to doubly clamped graphene-based NEMSBunch ;
BachtoldNonlinear ; Mandar , it is not possible to discuss the frequency
hardening observed in Fig. 3c in terms of cantilever tensioning induced by
differential thermal expansion since we study a simply clamped geometry. An
important feature of any resonator is the measurement of the quality factor,
defined as $Q=\omega/\Delta\omega$, which characterizes the high sensitivity
(high $Q$) of the resonator to its environment. A linear decrease of the
dissipation, $Q^{-1}$, is observed upon cooling to 70K. Both effects,
frequency hardening and decrease of the dissipation, are possibly a
consequence of the stiffening of the clamp electrode. Further measurements
will allow to investigate both extrinsic effects (clamp stiffening losses) and
mechanical intrinsic properties of MLG which should bring new insights to
understand damping mechanisms in NEMS. Effective substrate temperature is
obtained by measuring Stokes and Anti-Stokes Raman intensities ratio (Supp.
Info.) and indicates a temperature threshold of 70K. Interestingly, all the
physical quantities (resonant frequencies, quality factor $Q$, Raman shift)
are sensitive to the environmental temperature until 70K. This demonstrates
experimentally that room temperature experiments discussed in this letter are
not altered by laser heating. Concerning absorption of mechanical energy at
resonance, we have seen no change in the Raman Stokes/Anti-Stokes measurements
when sweeping the excitation frequency through the mechanical resonance,
indicating no increase of phonon bath temperature.
Figure 4: Detection of mechanical resonance and dynamic stress using Raman
spectroscopy. a: Raman spectra of the G peak under MLG actuation at mechanical
resonance frequency (red) and off resonance (detuned by 360 kHz) (blue). For
the resonating case, signal to noise ratio is smaller than off-resonance case
due to larger oscillation amplitude at resonance which takes the resonator out
of focus. b-c: Lock-in amplitude at $2\omega$ (dark line) as a function of
drive frequency, for a 5V rf drive voltage under vacuum (b) and in air (c).
The position of the Raman G peak (green) shows a softening which coincides
with the mechanical resonance. This softening is even more marked under
vacuum. Blue and red circles, shown in caption b, correspond to the Raman
spectra plotted in Fig. 3a. This sample is the same as presented in Fig. 3a.
To demonstrate the spectral detection of mechanical resonance, Raman response
of MLG cantilever under vacuum is plotted (Fig. 4a) at fundamental mechanical
resonance $\omega_{0}$ = 1.2 MHz (red curve) and off resonance (blue curve).
At $\omega_{0}$, G peak softening is -5 cm-1 in position and about +10 cm-1 in
width (peak’s FWHM) (Fig. 4a) which takes into account the averaging induced
broadening (see Supp. Info.). This Raman softening estimated at -1 cm-1.V-2,
is attributed to corresponding variation in stress within the cantilever,
induced at mechanical resonance according to universal stress behavior in sp2
carbon materialsOtakar2011 ; Hone2009 ; Ferrari2009 (shift rate: 0.003
cm-1.MPa-1). This dynamical stress is thus about 1.6 GPa, and therefore
provides a quantitative means of detecting NEMS resonances stress effects. It
is worth noting that the measured stress in MLG cantilever at mechanical
resonance is more than one order of magnitude higher than previously
reportedPomeroy in silicon-based MEMS devices.
In Figs. 4b-c, we have detected the fundamental mechanical resonance of this
MLG cantilever using both reflected and Raman scattered light under different
experimental conditions. As the lifetime of optical phonons is much shorter (1
ps) than $\omega_{0}^{-1}$ ($\rm{\sim\ 100\ ns}$), the Raman scattering
process provides instantaneous information related to stress in the vibrating
cantilever. For each excitation frequency, we record a Raman spectrum (1s
averaging), which reflects stress at the cantilever position. For several
samples, we were able to check that this softening behavior (green curve, Fig.
4b-c), observed under mechanical excitation, coincides with the mechanical
resonance width irrespective of the chamber pressure (cf. Fig. 4b-c).
In contrast to the vacuum case (Fig. 4b), where the quality factor is about
$Q_{vac}$ = 26.1, the same sample in air (Fig. 4c) has a reduced quality
factor ($Q_{air}$ = 2.3) as well as a shifted Raman G peak, which indicates
that the dynamical stress depends on the oscillation amplitude (as also
suggested by the Fig. S9 in Supp. Info.). The value of $Q_{air}$ agrees with
typical viscous damping modelHosaka for that particular geometry, thus
confirming that viscous damping is the predominant mechanism for limiting the
quality factor in air (see Supp. Info.). Nevertheless, this damping mechanism
is no longer the main one under vacuum where the dissipation may be governed
by clamping losses. One can compare the ratio of the drive efficiency at low
frequency (20 nm.V-2) and at resonance (150 nm.V-2) with the ratio of the G
peak shift sensitivity at low frequency (1 cm-1.V-2) and at resonance (5
cm-1.V-2). Both ratios equal to the measured quality factor $Q$ ($\rm{\sim
6}$), as expected for a mechanical resonatorLandau . To demonstrate the
versatility of the Raman-based spectral detection of the mechanical resonance,
we have investigated a similar effect on two other types of NEMS (Si nano
cantilevers and SiC nanowires, see Supp. Info.).
There exists a fundamental coupling between the device position (directly
given by the cavity length $h(x,y)$) and a spectroscopic property (measured by
the Raman peak shift). In our case, the flake displacement generates a
mechanical stress that causes a shift of stress-sensitive Raman peaks. We
therefore extract a coupling constant that is the ratio between the Raman peak
shift (in Hz) and the estimated displacement (in meters). The novel
optomechanical coupling linking the Raman G peak position shift to the
cantilever displacement reaches $\rm{\sim 10^{17}}$ Hz.m-1 in magnitude, which
compares favorably to similar quantities involving other optomechanical
systemsPRL arcizet 2006 . This large optomechanical coupling, in which all the
isotropically scattered Raman photons carry informations on the nano-resonator
dynamics, enables mechanical stress information to be spectrally encoded.
Interestingly, this provides an efficient rejection of background signal even
in a backscattering configuration for on-chip devices. For the detection of
submicron NEMS, this generates in many cases better signal to noise ratio,
compared to diffraction-limited elastic optical detection techniques. Finally,
the resonant nature of Raman scattering in graphene preserves a large
interaction cross-section, allowing the optomechanical coupling to be
maintained even when working with nanosized oscillators, which is not the case
in standard optomechanical approachesPRL arcizet 2006 in which only a small
fraction of the detected photons carries the optomechanical information.
In conclusion, we demonstrate a non invasive and high bandwidth optical probe,
enabling imaging of dynamical stress and motion in a NEMS. This probe,
combining Raman spectroscopy with Fizeau interferometry, is applied to
multilayer graphene NEMS and is found to be compatible with two other types of
NEMSs. Calibrated motion and stress can be measured and mechanical resonances
can be detected through optical mode shifting and mapped as a local stress
along a vibrating cantilever. The reliability of Raman spectroscopy in this
context finds its origins in the large optomechanical coupling between strain
modulation and mechanical displacements. We demonstrate the coupling between
flexural vibrational modes and optical phonons. This localized probe of
material stress is furthermore expected to preserve its large coupling
strength when working with even smaller oscillators. Due to its high
stiffness, semi-transparency behavior and, extremely low mass, MLG emerges as
an ultra-sensitive platform for the simultaneous exploration of the spatial,
temporal and spectral properties of NEMS, this system is thus promising for
the detection of ultralow forces and could be used as carbon-based molecular
sensors. Moreover, this probe allows low temperature measurements, thus paving
the way for stress mapping of other high quality factor resonators and
understanding of dissipation factors in such systems.
###### Acknowledgements.
This work is partially supported by ANR grants (MolNanoSpin, Supergraph,
Allucinan), ERC Advanced Grant No. 226558 and, the Nanosciences Foundation of
Grenoble. Samples were fabricated in the NANOFAB facility of the Néel
Institute. We thank A. Allain, D. Basko, C. Blanc, E. Bonet, O. Bourgeois, E.
Collin, T. Crozes, L. Del-Rey, M. Deshmukh, E. Eyraud, C. Girit, R. Haettel,
C. Hoarau, D. Jeguso, D. Lepoittevin, R. Maurand, J-F. Motte, R. Piquerel, Ph.
Poncharal, V. Reita, A. Siria, C. Thirion, P. Vincent, R. Vincent and W.
Wernsdorfer for help and discussions.
Methods Multilayered Graphene flakes are deposited on 280 nm thick oxidized
silicon wafer by micro-mechanical exfoliationNovoselov2004p2970 of Kish
graphite. Electrical contacts are made using deep UV lithography and e-beam
deposition of 50 nm Au film. Samples are suspended by etching (buffered
hydrogen fluoride at concentration 1:3 HF/NH4F) and drying using CO2 critical
point drying. Experiments have been performed on approximately 100 samples
(see Supp. Info). Micro-Raman spectroscopy was performed with a commercial
Witec Alpha 500 spectrometer setup with a dual axis XY piezo stage in a back-
scattering/reflection configuration. Grating used has 1800 lines/mm which
confer a spectral resolution of 0.01 cm-1 for 10 s integration time. Two laser
excitation wavelengths are used, 633 nm (He-Ne) and 532 nm (Solid state argon
diode). Raman spectra are recorded in air with a Nikon x100 objective
($NA=0.9$) focusing the light on a 320 nm diameter spot (532 nm light) and,
with a Mitutoyo x50 objective (M plan APO NIR) in vacuum. All measurements
made under vacuum (Fig. 3, Fig. 4a-b) are under active pumping at residual
pressure equals to 10-6 bar. For Raman (reflectance) measurements laser power
is kept below 1mW.$\mu$m-2 (1 $\mu$W.$\mu$m-2). For rf measurements in air or
vacuum, optical response is recorded with a silicon fast photodiode and a
lock-in detector synchronized at twice the rf drive frequency. The signal is
maximum when 2$\ \omega_{AC}$ coincides with the fundamental mechanical
resonance frequency $\omega_{0}$ of the cantilever. Cryogenic measurements
involve an optical continuous He flow Janis cryostat with electrical contacts.
## References
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* (28) Hosaka, H. Itao, K. and Kuroda, S. Damping characteristics of beam-shaped micro-oscillators. Sens. Actuators A 49, 87 (1995).
* (29) Pomeroy, J.W. et al. Dynamic Operational Stress Measurement of MEMS Using Time-Resolved Raman Spectroscopy. Jour. of Micro. syst. 17, 6 (2008).
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|
arxiv-papers
| 2012-03-18T17:00:04 |
2024-09-04T02:49:28.737417
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antoine Reserbat-Plantey, Laetitia Marty, Olivier Arcizet, Nedjma\n Bendiab and Vincent Bouchiat",
"submitter": "Antoine Reserbat-Plantey",
"url": "https://arxiv.org/abs/1203.3973"
}
|
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